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Their impressive performance suggests that these models should also constitute powerful priors for inverse imaging problems, but that approach has not yet led to comparable fidelity. There are several key obstacles: (i) the data likelihood term is usually intractable; (ii) learned generative models cannot be directly conditioned on the distorted observations, leading to conflicting objectives between data likelihood and prior; and (iii) the reconstructions can deviate from the observed data. We present FLAIR, a novel, training-free variational framework that leverages flow-based generative models as prior for inverse problems. To that end, we introduce a variational objective for flow matching that is agnostic to the type of degradation, and combine it with deterministic trajectory adjustments to guide the prior towards regions which are more likely under the posterior. To enforce exact consistency with the observed data, we decouple the optimization of the data fidelity and regularization terms. Moreover, we introduce a time-dependent calibration scheme in which the strength of the regularization is modulated according to off-line accuracy estimates. Results on standard imaging benchmarks demonstrate that FLAIR consistently outperforms existing diffusion- and flow-based methods in terms of reconstruction quality and sample diversity. Source code is available at https://inverseflair.github.io/. + +# 1 Introduction + +Flow-based generative models are at the core of modern image generators like Stable Diffusion or FLUX [14]. Beyond image generation based on text prompts, these models have emerged as powerful data-driven priors for a whole range of visual computing tasks. Their comprehensive representation of the visual world, learned from internet-scale training datasets, makes them an attractive alternative to traditional handcrafted image priors. Often, they can be used without any task-specific retraining. + +While it is evident that a model capable of generating photorealistic images should be suitable as prior (a.k.a. regularizer) for inverse imaging problems, a practical implementation faces several challenges. On the one hand, flow-based models normally operate in the lower-dimensional latent space of a variational autoencoder (VAE), which means that the forward operator (the relationship between the observed, degraded image and the desired, clean target image) is no longer linear. On the other hand, the iterative nature of the generative process means that intermediate stages are corrupted with (time-dependent) random noise. Hence, one cannot explicitly evaluate their data likelihood, which renders the data term intractable. Moreover, learned generative models tend to overly favor regions of the training distribution that have a high sample density. For test samples that fall in low-density regions, the prior will have a too strong tendency to pull towards outputs with higher a-priori likelihood, compromising fidelity to the input observations. + +![](images/845cd1572ebf6c0bf66c97f0a7879f043bdef2515d06dabb26c96763bfe68b17.jpg) +Figure 1: Starting from the adjoint based initialization, we alternate between (i) regularizer updates via a flow-matching loss that aligns the velocity $u_{t}$ of the variational distribution with the learned velocity field $v_{\theta}$ , and (ii) hard data consistency steps that project the current estimate onto the measurement manifold. + +Here, we propose flow-based latent adaptive inference with deterministic re-noising (FLAIR), a novel, training-free variational framework explicitly tailored to integrate flow-based latent diffusion models into inverse problem-solving. To the best of our knowledge, FLAIR is the first scheme that combines latent generative modeling, flow matching and variational inference into a unified formulation for inverse problems. Our main contributions are + +- A novel variational objective for inverse problems with flow-matching priors. +- Deterministic trajectory adjustments guide the prior towards regions which are more consistent with the observed data. +- Decoupled optimization of data and regularization terms, enabling hard data consistency. +- A novel, time-dependent weighting scheme, calibrated via offline accuracy estimates, that adapts the regularization along the flow trajectory to match the changing reliability of the model's predictions, ensuring robust inference. + +# 2 Related Work + +Deep learning based priors. Deep learning-based methods typically follow one of two main approaches: they either directly learn an inverse mapping [27, 18, 29, 4, 59], or aim to learn a suitable prior, either through non-generative approaches like unrolled optimization networks [15, 28, 1, 35] or through generative models such as generative adversarial networks [36, 7, 45], or diffusion [17, 48, 51] respectively flow based models [30, 32]. The latter have demonstrated impressive performance in image generation tasks, sparking growing interest in leveraging them as priors for solving inverse problems, particularly through posterior sampling techniques. + +Posterior sampling. Although incorporating the prior learned from a diffusion or flow-based model seems straightforward, problems arise due to the inherent time-dependent structure of diffusion models, which makes the likelihood term intractable [11]. A variety of approaches have been proposed for diffusion-based posterior sampling [57, 19, 34, 20]: enforcing the trajectory to stay on the respective noise manifold [11, 12, 60], applying an SVD to run diffusion in the spectral domain [24], utilizing range-null space decomposition during the reverse diffusion process [24], guidance by the pseudo-inverse of the forward operator [49]. + +Many prior methods perform well in pixel space but are difficult to apply in latent diffusion models due to VAE non-linearity or memory constraints. In order to circumvent this issue, the authors of ReSample [47] rely on enforcing hard data consistency through optimization and resampling during the reverse diffusion process. PSLD [41], introduces additional objectives terms to ensure that all gradient updates point to the same optima in the latent space. FlowChef [38] incorporates guidance + +into the flow trajectory during inference, whereas FlowDPS [25] separates the update step into two components: one for estimating the clean image and another for estimating the noise. + +In contrast FLAIR follows another class of posterior sampling-based methods, which integrate diffusion priors with inverse problems by directly optimizing a variational objective that approximates the data posterior [33]. This framework was recently extended by RSD [61], which incorporates a repulsion mechanism to promote sample diversity and applied to latent diffusion models. A known issue with this type of optimization is mode collapse [39], which leads to blurry results for these methods. Our method targets this problem by introducing a deterministic trajectory adjustment. + +# 3 Background + +# 3.1 Inverse problems + +In many imaging tasks, such as inpainting [5], super-resolution [37] or tomographic reconstruction [46], one aims to recover a target signal $x \in \mathbb{R}^n$ from a distorted observation $y \in \mathbb{R}^m$ . The observation is regarded as the result of applying a forward operator $\mathcal{A} : \mathbb{R}^n \mapsto \mathbb{R}^m$ to the target signal, corrupted by additive Gaussian noise $\nu \in \mathbb{R}^m$ with standard deviation $\sigma_{\nu}$ . + +$$ +y = \mathcal {A} x + \nu . \tag {1} +$$ + +In most practical applications, the forward operator $\mathcal{A}$ is either non-invertible or severely ill-conditioned, making (1) generally ill-posed. + +Variational methods solve ill-posed inverse problems by minimizing an energy functional + +$$ +\mathcal {E} (x, y) = \mathcal {D} (x, y) + \mathcal {R} (x). \tag {2} +$$ + +to recover the solution. + +Interpreted probabilistically via Bayes' theorem, the posterior distribution $p(x|y)$ is proportional to the product $p(y|x)p(x)$ . In the negative log-domain, this yields the data term $\mathcal{D}(x,y) = -\log p(y|x)$ and the regularizer $\mathcal{R}(x) = -\log p(x)$ . Handcrafted priors based on regularity assumptions like sparsity [43, 44, 10, 13, 9] have long been the standard, but have largely been replaced by deep learning-based methods in modern data-driven schemes. + +# 3.2 Flow based priors + +Models based on flow matching [30] learn a time-dependent vector field $v_{\theta}(x_t, t)$ that continuously transforms samples from a simple initial distribution $p_1(x)$ to a complex target data distribution $p_0(x)$ . Formally, this transformation is described by solving the ordinary differential equation (ODE): + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} \psi_ {t} (x) = v _ {\theta , t} \left(\psi_ {t} (x)\right), \quad t \in [ 0, 1 ], \tag {3} +$$ + +where $\psi_t(x)$ represents the trajectory of a sample, evolving smoothly from an initial value drawn at $t = 1$ toward a target value at $t = 0$ . + +Since the integrated ODE path maps the simple distribution $p_1(x)$ to the complex target $p_0(x)$ , the learned flow-based model captures the structure of the data and can therefore serve as a powerful prior for solving inverse problems. To make this approach tractable for high-resolution data, we adopt the latent diffusion model (LDM) framework [40], which shifts the generative process to a lower-dimensional latent space using a pretrained autoencoder with encoder $E: \mathbb{R}^n \mapsto \mathbb{R}^d$ and decoder $D: \mathbb{R}^d \mapsto \mathbb{R}^n$ , where $d \ll n$ . However, applying such priors to inverse problems introduces challenges, as the non-linearity of the VAE disrupts the linear relationship between measurements and the target signal, resulting in a nonlinear forward operator. + +# 3.3 Variational flow sampling + +To solve inverse problems from a Bayesian perspective, we aim to sample from the posterior + +$$ +p \left(x _ {0} \mid y\right) \propto p \left(y \mid x _ {0}\right) p \left(x _ {0}\right), \tag {4} +$$ + +where the likelihood is given by $p(y|x_0) = \mathcal{N}(\mathcal{A}x_0,\sigma^2\mathrm{Id})$ , and $p(x_0)$ represents the prior modeled by the flow-based generative model. + +Inspired by previous work [33, 61] we introduce a variational distribution $q(x_0|y) = \mathcal{N}(\mu_x,\sigma_x^2)$ to approximate the true posterior $p(x_0|y)$ , by minimizing their Kullback-Leibler divergence: + +$$ +q \left(x _ {0} | y\right) \in \arg \min _ {q \left(x _ {0} | y\right)} \mathrm {K L} \left(q \left(x _ {0} | y\right) \| p \left(x _ {0} | y\right)\right). \tag {5} +$$ + +Rewriting the KL divergence by means of the variational lower bound leads to: + +$$ +K L (q \left(x _ {0} | y\right) \| p \left(x _ {0} | y\right)) = - \underbrace {\mathbb {E} _ {q \left(x _ {0} | y\right)} \left[ \log p \left(y \mid x _ {0}\right) \right]} _ {\mathcal {D} (x, y)} + \underbrace {K L \left(q \left(x _ {0} | y\right) \| p \left(x _ {0}\right)\right)} _ {\mathcal {R} (x)} + \underbrace {\log p (y)} _ {\text {c o n s t}}. \tag {6} +$$ + +Since a single Gaussian cannot capture a multi-modal posterior, we simplify to a deterministic approximation, setting $\sigma_x^2 = 0$ . Equivalently, this corresponds to a single-particle approximation in the sense of Stein variational methods [31]. As shown in [50], rewriting Equation 6 under this approximation and extending it to the time-dependent noisy posterior yields: + +$$ +\arg \min _ {q (x _ {0} | y)} \underbrace {\mathbb {E} _ {q (x _ {0} | y)} \left[ \frac {\| y - f (\mu_ {x}) \| ^ {2}}{2 \sigma_ {\nu} ^ {2}} \right]} _ {\mathcal {D} (x, y)} + \underbrace {\int_ {0} ^ {T} \omega (t) \mathbb {E} _ {q (x _ {t} | y)} \left[ \| \nabla_ {x} \log q (x _ {t} | y) - \nabla_ {x} \log p (x _ {t}) \| ^ {2} \right] d t} _ {\mathcal {R} (x)} \tag {7} +$$ + +The first term in Equation 7 describes the data term $\mathcal{D}(x,y)$ and the second the regularizer $\mathcal{R}(x)$ , where the integral ensures optimization over the entire diffusion trajectory. Notably, the latter constitutes a weighted score-matching objective, where $\nabla_{x}\log p(x_{t})$ represents the score function [51], which may be extracted from a pretrained diffusion or flow model. + +The score of the noisy variational distribution depends on the forward diffusion process and can be computed analytically. + +Note that for $\omega (t) = \beta (t) / 2$ the weighted score-matching loss recovers the gradient of the diffusion model's evidence lower bound, so that optimizing it yields the maximum likelihood estimate of the data distribution [50]. However, optimizing Equation 7 is costly, as it requires computing the gradient through the flow model. As shown in [53] this can be circumvented by reformulating the regularizer in terms of the Wasserstein gradient flow: + +$$ +\nabla_ {\mu_ {x}} \mathcal {R} (x) = \mathbb {E} _ {t, q \left(x _ {t} \mid y\right)} \left[ \omega (t) \left(\underbrace {\nabla_ {x} \log q \left(x _ {t} \mid y\right)} _ {\text {s c o r e o f n o i s y v a r i a t i o n a l d i s t r i b u t i o n}} - \underbrace {\nabla_ {x} \log p \left(x _ {t}\right)} _ {\text {s c o r e o f n o i s y p r i o r d i s t r i b u t i o n}}\right) \right] \tag {8} +$$ + +Note that optimizing only the regularization term, without the data term, at test time is equivalent to the objective of Score Distillation Sampling (SDS) [39]. + +# 4 Method + +Flow Formulation. The variational formulation in Equation 7 is formulated for the score, but can be reformulated into a denoising or $\epsilon_{\theta}$ parameterization [51, 33]. However, we are interested in a variational objective that depends on the velocity field $v_{\theta}(x_t,t)$ , which characterizes the probabilistic trajectory that connects the noise and data distributions. + +Proposition 1. We propose to replace the score-based regularizer in the standard variational objective with a flow matching formulation, resulting in the following objective function: + +$$ +\arg \min _ {q (x _ {0} | y)} \underbrace {\mathbb {E} _ {q (x _ {0} | y)} \left[ \frac {\| y - f (\mu_ {x}) \| ^ {2}}{2 \sigma_ {\nu} ^ {2}} \right]} _ {\mathcal {D} (x, y)} + \underbrace {\int_ {0} ^ {T} \lambda_ {\mathcal {R}} (t) \mathbb {E} _ {q (x _ {t} | y)} \left[ \| v _ {\theta} (x _ {t} , t) - u _ {t} (x _ {t} | \epsilon) \| ^ {2} \right] d t} _ {\mathcal {R} (x)} \tag {9} +$$ + +$$ +\nabla_ {\mu_ {x}} \mathcal {R} (x) = \mathbb {E} _ {t, q \left(x _ {t} \mid y\right)} \left[ \lambda_ {\mathcal {R}} (t) v _ {\theta} \left(x _ {t}, t\right) - u _ {t} \left(x _ {t} \mid \epsilon\right) \right] \tag {10} +$$ + +The flow-matching term that defines the regularizer arises by reparameterizing the variational distribution to $q(x_{t}|y) = \mathcal{N}((1 - t)\mu_{x},t^{2}I)$ . This corresponds to sampling via the deterministic map + +$\psi_t(x_0 \mid \epsilon) = (1 - t)x_0 + t\epsilon$ , with $\epsilon \sim \mathcal{N}(0, I)$ . By reformulating the score in terms of the target velocity field $u_t$ , we get: + +$$ +\nabla_ {x} \log q (x _ {t} | y) = - \frac {(1 - t) u _ {t} (x _ {t} | \epsilon) + x _ {t}}{t} \tag {11} +$$ + +For the learned velocity $v_{\theta}(x_t,t)$ a similar approximation holds - for a full derivation, see the supplementary material subsection A.3. + +$$ +v _ {\theta} \left(x _ {t}, t\right) \approx \frac {- t \nabla_ {x} \log p \left(x _ {t}\right) - x _ {t}}{1 - t} \tag {12} +$$ + +We can therefore approximate the score of the noisy prior with our learned velocity field $v_{\theta}$ + +$$ +\nabla_ {x} \log p (x _ {t}) \approx - \frac {(1 - t) v _ {\theta} (x _ {t} , t) + x _ {t}}{t}. \tag {13} +$$ + +**Hard Data Consistency.** Existing variational posterior sampling approaches [33, 61] impose soft constraints on the data fidelity term $\mathcal{D}(x,y)$ . In contrast, recent work [47] has demonstrated that, when sampling from latent diffusion models, enforcing hard data consistency generally leads to better reconstructions with improved visual fidelity. Our method shares this motivation, but differs in that we optimize over a variational distribution, i.e., we compute $\min_{q(x_0|y)}[-\log p(y|x_0)]$ . An additional advantage of this variational setup is that it allows us to initialize the optimization variable with an adjoint based initialization $\mu_x = E(A^\top y)$ , with $E$ being the encoder of the VAE and $A^\top$ the adjoint of the linear forward operator in pixel space. Other initialization strategies are also possible. + +Accuracy Calibration. As our framework evaluates the trajectory at each time step, we aim to weight the regularizer's contribution according to its reliability. The difficulty of the prediction task has been shown to depend on the network parameterization, as well as on the specific time step $t$ [21]. Since the regularization term $\mathcal{R}(x)$ in our approach is equivalent to the training objective of the pre-trained flow model, we can easily weight it by the expected model error, which we calibrate on a small set of images. Specifically, we sample $N$ calibration images and compute the conditional flow matching objective for 100 linearly spaced time steps between 0 and 1, then average the error over all images to obtain the expected model error at each time step. Different functions of the model error can be chosen as weight for the regularizer. We choose: + +$$ +\lambda_ {\mathcal {R}} (t) = \frac {1}{N} \left(\sum_ {i = 1} ^ {N} \left\| v _ {\theta} \left(x _ {t} ^ {(i)}, t\right) - u _ {t} \left(x _ {t} ^ {(i)} \mid \epsilon\right) \right\| ^ {2}\right) ^ {- 1} \tag {14} +$$ + +and set $\lambda_{\mathcal{R}}(t) = 0$ for all $t < 0.2$ , since the accuracy of SD3 is heavily degraded for low noise levels. + +Deterministic Trajectory Adjustment. Score distillation sampling relies on the assumption that $x_{t} = (1 - t)\mu_{x} + t\epsilon$ lies in a region of the learned prior that has reasonably high support/density. In practice, this is not always the case. When not tightly conditioned (usually with extensive text prompts), even the best available diffusion models assign low density to many plausible regions of the latent space, leading to bad gradient steps. Therefore, we increase the probability of $p(x_{t})$ by additionally conditioning $x_{t}$ on the estimated "end-point" $\mu$ . + +Proposition 2. We introduce a reparameterized variational distribution with a mean that linearly interpolates between the posterior mean $\mu_{x}$ and a model-guided $\hat{x}_{1}$ : + +$$ +q \left(x _ {t} \mid y\right) = \mathcal {N} \left(\left(1 - t\right) \mu_ {x} + t \alpha \hat {x} _ {1}, t ^ {2} \left(1 - \alpha^ {2}\right) I\right), \tag {15} +$$ + +where $\hat{x}_1 = x_{t + \delta t} + (1 - t - \delta t)v_\theta (x_{t + \delta t},t + \delta t)$ is a single-step velocity-based predictor, and $\alpha \in [0,1]$ controls the trade-off between deterministic guidance and random noise. This reparameterization induces the following reference velocity field: + +$$ +u _ {t} \left(x _ {t} \mid \epsilon\right) = \frac {\alpha \hat {x} _ {1} + \sqrt {1 - \alpha^ {2}} \epsilon - x _ {t}}{1 - t}. \tag {16} +$$ + +Intuitively, changing the formulation in this manner ensures that the model relocates the sample to its expected position on the learned manifold rather than injecting arbitrary noise, which could drive it in a direction that has high prior likelihood but is not consistent with the observation. To further encourage exploration and avoid collapsing onto the trajectory of the adjoint measurement, we inject an additional stochastic component $\epsilon$ during this process. A full derivation can be found in the supplementary material, subsection A.3. + +# 4.1 Algorithm + +The following pseudo-code summarizes our method, integrating all the components discussed above. + +We adapt the standard scheme [33, 61] of linearly traversing time in a descending manner and stop at $t = 0.2$ as explained in section 4. We choose $\alpha = 1 - t$ . Gradient updates to enforce hard data consistency are performed using stochastic gradient descent. For further implementation details and ablations, see subsection A.4 + +Algorithm 1: The FLAIR solver for inverse imaging problems +Input: $\mu_{x} = \mu_{init},\lambda_{R},\alpha ,y,\mathcal{A},v_{\theta}$ +Output: $\mu_{x}$ $\hat{\epsilon}\sim \mathcal{N}(0,I)$ 1 initial noise sample +for $t\gets 1$ to 0 by -Δt do + $x_{t}\gets (1 - t)\mu_{x} + t\hat{c};$ 1 sample noisy latent + $u_{t}(x_{t}\mid \hat{\epsilon})\gets \frac{\hat{\epsilon} - x_{t}}{1 - t};$ $\nabla_{\mu_x}R\gets v_\theta (x_t,t) - u_t(x_t\mid \hat{\epsilon});$ $\mu_x\gets \mu_x - \lambda_R\nabla_\mu_xR;$ update w.r.t. regularizer + $\mu_x\gets \arg \min_{\mu_x}\| y - \mathcal{A}(\mu_x)\| ^2;$ 1 hard data consistency + $\epsilon \sim \mathcal{N}(0,I);$ $\hat{x}_1\gets x_t + (1 - t)v_\theta (x_t,t);$ 1 predict deterministic noise + $\hat{\epsilon}\gets \alpha \hat{x}_1 + \sqrt{1 - \alpha^2}\epsilon ;$ 1 update noise estimate + +# 5 Experiments + +We evaluate the performance of FLAIR in a variety of inverse imaging tasks and compare it against several baselines, using the SD3 backbone without any fine-tuning. We used several metrics including SSIM [54], LPIPS [58] and patchwise FID [16] (pFID) to comprehensively assess the perceptual and quantitative quality of the reconstructions. FID is computed using InceptionV3 features on patches of $256 \times 256$ resolution. All experiments were performed on a NVidia RTX 4090 GPU with 24GB of VRAM. For completeness we also show PSNR values, but point out that the metric is not well suited for our purposes: PSNR favors the posterior mean, while the goal of the variational approach is to sample from the posterior distribution. Accordingly, PSNR is known to prefer oversmoothed, blurry outputs over sharp ones [6]. To demonstrate that our model can also produce accurate MMSE estimates, we performed ensemble predictions by running posterior sampling eight times and averaging the results. As shown in subsection A.11, ensembling improves PSNR values while reducing LPIPS. This confirms that our samples are distributed around the posterior mean. Moreover, it shows that results closer to the posterior mean – such as those produced by baseline methods – are perceptually farther from the ground truth (in LPIPS) compared to our samples. + +# 5.1 Setup + +Datasets. We utilize two high-resolution image datasets: FFHQ [22] and DIV2K [2]. FFHQ consists of 70k diverse face images at $1024 \times 1024$ resolution of which we take the first 1000 samples. It is covering variations in age, pose, lighting, and ethnicity. DIV2K contains 800 high-quality images in 2K resolution that span a range of natural scenes with varied textures and structures. + +Baselines. Our method is benchmarked against several recent inverse imaging solvers based on posterior sampling. Specifically, we compare to ReSample [47], FlowDPS [25], FlowChef [38], and RSD [61]. The latter is used without repulsive term as it delivers better results. To ensure a fair and meaningful comparison, all methods are evaluated with the same number of function evaluations. + +Problem Setting. We run and evaluate all methods at a fixed output resolution of $768 \times 768$ pixels. For single image super-resolution, we consider scaling factors of $8 \times$ and $12 \times$ . The corresponding low-resolution inputs are generated by bicubic downsampling. Motion blur is simulated with a blur kernel of size 61. For box inpainting, we mask large, continuous rectangles that cover approximately one third of the observation. All synthesized observations are corrupted with additive Gaussian noise, with standard deviation $\sigma_{\nu}$ of $0.5\%$ . + +For inference on the FFHQ dataset, we use a predefined text prompt of the form "A high quality photo of a face", and for DIV2k "A high quality photo of" concatenated with an image-specific description retrieved by applying DAPE [55] to the observation. + +# 5.2 Experimental Results + +Inverse Problems. Our experiments clearly demonstrate that FLAIR outperforms existing flow-based approaches in terms of all perceptual metrics, see Table 1. + +In the case of image inpainting, our method produces high-quality reconstructions that fully leverage the power of the generative model and blend naturally into the surrounding context, avoiding degradations and artifacts that we observe in the baselines. In particular, FlowDPS tends to produce implausible textures in the inpainted regions, while FlowChef regularly fails to generate semantically consistent content at all. + +For single-image super-resolution, FLAIR consistently delivers the most perceptually convincing and realistic outputs. Notably, the FID scores remain low for both $\times 8$ and $\times 12$ magnification, indicating an effective usage of the generative prior to overcome the increasing ill-posedness. Again, FlowDPS suffers from blur and low texture quality, whereas FlowChef tends to lose semantic coherence. + +In motion deblurring, FLAIR also restores sharper and semantically more credible content than competing approaches, which often suffer from residual blur or inconsistent details. The boost in reconstruction quality is quantitatively reflected by all metrics, confirming that FLAIR reconstructs images with high fidelity. For further qualitative examples, see subsection A.13. + +Table 1: Quantitative results with 50 NFE and ${\sigma }_{\nu } = {0.5}\%$ . + +
SR ×8SR ×12Motion DeblurringInpainting
MethodLPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑
FFHQ 768×768
ReSample0.40055.60.81526.370.47480.30.78625.470.45782.90.78825.450.36670.80.82721.83
FlowDPS0.37438.50.75629.240.41344.00.74128.050.43154.30.73227.640.34442.50.77119.19
RSD0.39151.70.77629.690.46271.70.74328.110.45877.30.74327.670.47873.30.73621.97
FlowChef0.34130.50.76028.420.37346.50.73027.000.40640.20.71625.810.39469.80.78018.18
Ours0.21313.30.77729.540.27116.20.74027.710.23610.70.77229.610.1848.70.82823.69
DIV2K 768×768
ReSample0.53355.00.62522.340.64388.10.56220.850.55679.70.61721.790.28551.90.79622.68
FlowDPS0.47644.40.56723.010.54754.00.52821.790.55865.50.53621.880.32829.20.69221.71
RSD0.53960.90.59123.450.68495.70.52321.960.63897.60.55122.100.46463.90.67823.23
FlowChef0.49036.50.53921.840.52543.80.49220.520.56149.60.48619.900.48958.30.65920.87
Ours0.35326.50.60723.300.42132.10.52521.390.31521.10.65324.440.16311.00.81523.75
+ +Posterior Variance. To demonstrate that FLAIR does not suffer from mode collapse, we assess the posterior variance $\mathrm{Var}[x|y]$ for the task of $\times 12$ Super Resolution, by drawing 32 samples for a fixed observation $y$ and computing their pixel-wise variance. We conduct that experiment for our FLAIR approach, for RSD with repulsive term, and for FlowDPS [25]. The example in Figure 3 illustrates that FLAIR has the highest sample diversity, which is also reflected in the corresponding variance maps. Notably, the sample variance is concentrated in regions with high-frequency textures. This indicates that our method reliably reconstructs the posterior, whose low-frequency part is, in the super-resolution setting, tightly constrained by the likelihood term. + +Editing. Beyond image restoration, we observe that our method also performs remarkably well for text-based image editing, simply presenting suitable target prompts during inpainting. Figure 4 illustrates a variety of edited images generated from the same photograph with the help of the depicted masks and prompts. + +Pixel Space Experiments We also implement FLAIR in pixel-space using the model from [32], trained on CelebA-HQ resized to $256 \times 256$ px. We compare to DDNM [52], DPS [11], Moment Matching [42] and IIGDM [49]. We tuned the hyperparameters for all baselines, which we report in subsection A.9. As shown in Table 5.2, our method also outperforms previous work in the pixel space, demonstrating its broader applicability. + +![](images/083e05a8a435512b952075b15e4939b06eba51b25308c84b69da59e72e08c86e.jpg) +Figure 2: Qualitative comparison. FLAIR produces posterior samples of high perceptual quality while maintaining high data likelihood. Best viewed zoomed in. + +Table 2: Quantitative results with 50 NFE and ${\sigma }_{\nu } = {0.5}\%$ - In-painting and Super-resolution $\left( {\times 8}\right)$ . + +
MethodInpaintingSR ×8
LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑
DDNM0.158 ± 0.04226.90.732 ± 0.03718.31 ± 2.940.199 ± 0.05231.90.635 ± 0.07923.59 ± 1.64
DPS0.195 ± 0.06430.20.689 ± 0.07720.49 ± 2.810.172 ± 0.05827.80.658 ± 0.08824.59 ± 2.04
MM0.161 ± 0.05428.80.728 ± 0.06220.59 ± 3.370.172 ± 0.05129.10.669 ± 0.08324.65 ± 1.97
IIGDM0.195 ± 0.06430.20.689 ± 0.07720.49 ± 2.810.157 ± 0.05226.50.677 ± 0.08424.98 ± 2.07
FLAIR0.097 ± 0.03514.20.831 ± 0.03121.87 ± 2.660.143 ± 0.03922.90.712 ± 0.07625.93 ± 1.96
+ +# 5.3 Ablation Studies + +We systematically analyze the impact of key design choices in our method. Specifically, we ablate the deterministic trajectory adjustment, the use of hard data consistency, and the calibration of the regularizer weight for $\times 12$ super-resolution, using a subset of 100 samples from the FFHQ and DIV2K datasets. Quantitative and qualitative results are shown in Table 3 and Figure 6, respectively. + +**Hard Data Consistency (HDC).** Dropping the hard data consistency degrades both metrics, with PSNR being particularly affected due to poorer alignment with the input observation, which is also evident in the visual example: the reconstruction is plausible but deviates from the observation. + +Deterministic Trajectory Adjustment (DTA). The biggest performance drop compared to the full setup occurs when removing the deterministic trajectory adjustment, as random noise sampling harms + +![](images/61d869d855f1d266274369abf7d9b4a299c26196510e001f47126ba316b85e66.jpg) +Figure 3: Zoomed-in reconstructions for x12 Super Resolution. We show posterior samples (col. 1-4) of FLAIR, FlowDPS, and RSD, posterior mean and standard deviation (over 32 samples, col. 5,6). 0 0.16 + +![](images/cf8bf4eaa80962188d93522ceda4a3bc86653723a1f20fafab5ab80b8713a110.jpg) + +![](images/09f7095c82b02015ebf27e5a3f07c0a0d6f5ad9c9650097a9551a9c6d219dd96.jpg) +Original image + +![](images/29ec9569dfb10eca4eb1b756caa8a7c4813917824740252ed173c1db39e07a8f.jpg) +"man wearing aviator glasses." + +![](images/e1b82e5dd97634f98f71b8f867ebc8f1d4b5772d854ecd7586592860ec59798d.jpg) +"man wearing a wizard hat" +Figure 4: Edited images shown alongside original, with prompts: "A high resolution portrait of a..." + +![](images/baaa54046c85d350b6f1d9dc995e3aee7a77415b763352c114b0a547055e5b34.jpg) +"man with a Mike Tyson facial tattoo" + +![](images/c970a1f1be83152b262d94dd93529d0fedfc2fa3b9a3ab0fef1d4af70569cd63.jpg) +"clown with a red nose, red makeup and a ruff" + +the gradient updates in low-density regions of the prior. The reconstruction appears overly smooth and lacks texture details. + +Calibrated Regularizer Weight (CRW). Replacing our calibrated regularizer weight with $\lambda_{\mathcal{R}}(t) = t$ also has a strong impact on perceptual quality: the result is visibly blurred if one ignores the changing accuracy of the regularizer along the flow trajectory. + +Table 3: Ablation study for $\times {12}$ super-resolution on DIV2K and FFHQ. Model components are individually switched on or off. + +
HDCDTACRWFFHQDIV2K
LPIPS ↓PSNR ↑LPIPS ↓PSNR ↑
✓✓✓0.25927.450.42721.05
X✓✓0.29727.170.46720.82
✓X✓0.43227.200.62221.69
✓✓X0.36328.580.58321.98
XXX0.39228.330.60521.99
+ +Legend. HDC: Hard Data Consistency; DTA: Deterministic Trajectory Adjustment; CRW: Calibrated Regularizer Weight. $\checkmark =$ included, $X =$ ablated. + +![](images/e08cb54c3395fc442fb7488aeb2a754eab6354c3479e1b12e5189a5422d57836.jpg) +Figure 5: Qualitative comparison. FLAIR in pixel space produces posterior samples. + +![](images/a2cb3ad05a96b6565e7ef7f1941787e28c082e4c22abd297888c6ac0cd8d636b.jpg) +Figure 6: Qualitative samples from the ablation study on $\times 12$ Super Resolution. + +# 6 Conclusion and Limitations + +We have presented FLAIR, a training-free variational framework for inverse problems that uses a flow-based generative model as its image prior. By combining the power of (latent) flow-based models with a principled reconstruction of the posterior distribution, FLAIR addresses key limitations of existing methods. First, it is able to target the generation towards images, which match the observation, by aiding the degradation-agnostic flow matching loss with deterministic noise vectors. Second, it enables hard data consistency without sacrificing sample diversity, by decoupling the data consistency constraint from the regularization, while adaptively reweighting the latter according to its expected accuracy, calibrated offline. Experiments with different image datasets and tasks confirm that FLAIR consistently achieves higher reconstruction quality than existing baselines based on either flow matching or denoising diffusion. Notably, our proposed method achieves, at the same time, excellent perceptual quality, close adherence to the input observations, and high sample diversity. + +Evidently, FLAIR inherits the limitations of the underlying generative model. These include biases caused by the selection of training data, constraints w.r.t. the output resolution, and a limited ability to recover out-of-distribution modes. Furthermore, our approach introduces additional hyper-parameters needed to control the deterministic trajectory adjustment. We note that high fidelity image restoration methods can potentially be misused for unethical image manipulations. + +Acknowledgments This work was funded, in part, by the Max Plank ETH Center for Learning Systems and Huawei Technologies Oy (Finland) Co. Ltd. + +# References + +[1] Hemant K Aggarwal, Merry P Mani, and Mathews Jacob. MoDL: Model-based deep learning architecture for inverse problems. IEEE Transactions on Medical Imaging, 38(2):394-405, 2018. +[2] Eirikur Agustsson and Radu Timofte. NTIRE 2017 challenge on single image super-resolution: Dataset and study. In CVPR Workshops, 2017. +[3] Eirikur Agustsson and Radu Timofte. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: We did not use LLMs in any important, original or non-standard component of the core methods. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Supplementary Material + +In the following, we provide detailed line-by-line derivations of the mathematical formulations used in the paper, as well as additional implementation details and experimental results. + +# A Derivations + +# A.1 Derivation of flow-based variational formulation + +The linear conditional flow and it's corresponding velocity are defined as: + +$$ +\psi_ {t} \left(x _ {0} \mid \epsilon\right) = (1 - t) x _ {0} + t \epsilon , \quad \epsilon \sim \mathcal {N} (0, I), \tag {17} +$$ + +$$ +u _ {t} \left(x _ {t} \mid \epsilon\right) = \frac {\mathrm {d} \psi_ {t}}{\mathrm {d} t} \left(\psi_ {t} ^ {- 1} \left(x _ {t} \mid \epsilon\right) \mid \epsilon\right). \tag {18} +$$ + +The score of the noisy variational distribution can be analytically computed with: + +$$ +q \left(x _ {t} \mid y\right) = \mathcal {N} \left(\left(1 - t\right) \mu_ {x}, t ^ {2} I\right), \tag {19} +$$ + +$$ +\nabla_ {x _ {t}} \log q (x _ {t} | y) = - \frac {\epsilon}{t}. \tag {20} +$$ + +We compute $\frac{\mathrm{d}\psi_t}{\mathrm{d}t}(x_0|\epsilon) = -x_0 + \epsilon$ and $\psi_t^{-1}(x_t|\epsilon)$ and insert it into Equation 18: + +$$ +u _ {t} \left(x _ {t} \mid \epsilon\right) = \frac {\epsilon - x _ {t}}{1 - t}. \tag {21} +$$ + +Solving Equation 21 for $\epsilon$ and inserting in Equation 20 gives: + +$$ +\nabla_ {x _ {t}} \log q (x _ {t} | y) = - \frac {(1 - t) u _ {t} (x _ {t} | \epsilon) + x _ {t}}{t}. \tag {22} +$$ + +For the learned velocity $v_{\theta}(x_t, t)$ a similar approximation holds: + +$$ +v _ {\theta} \left(x _ {t}, t\right) \approx \frac {- t \nabla_ {x} \log p \left(x _ {t}\right) - x _ {t}}{1 - t}. \tag {23} +$$ + +Hence, we can approximate the score of the noisy prior with our learned velocity field $v_{\theta}$ + +$$ +\nabla_ {x _ {t}} \log p (x _ {t}) \approx - \frac {(1 - t) v _ {\theta} (x _ {t} , t) + x _ {t}}{t}, \tag {24} +$$ + +and we see that for $\omega(t) = \frac{t}{1 - t}$ we obtain the conditional flow matching objective for $\mathcal{R}(x)$ . We therefore set $\omega(t) = \frac{t}{1 - t}$ and end up at our final objective: + +$$ +\arg \min _ {q (x _ {0} | y)} \underbrace {\mathbb {E} _ {q (x _ {0} | y)} \left[ \frac {\| y - f (\mu_ {x}) \| ^ {2}}{2 \nu^ {2}} \right]} _ {\mathcal {D} (x, y)} + \underbrace {\int_ {0} ^ {T} \mathbb {E} _ {q (x _ {t} | y)} \left[ \| v _ {\theta} (x _ {t} , t) - u _ {t} (x _ {t} | \epsilon) \| ^ {2} \right] d t} _ {\mathcal {R} (x)}. \tag {25} +$$ + +Again, the gradient step for the regularizer becomes: + +$$ +\nabla_ {\mu_ {x}} \mathcal {R} (x) = \mathbb {E} _ {t, q \left(x _ {t} \mid y\right)} \left[ v _ {\theta} \left(x _ {t}, t\right) - u _ {t} \left(x _ {t} \mid \epsilon\right) \right]. \tag {26} +$$ + +# A.2 Derivation of trajectory adjusted flow-based variational formulation + +To achieve the proposed trajectory adjustment, we modify the forward process to: + +$$ +\hat {x} _ {1} = x _ {t + d t} + (1 - t - d t) v _ {\theta} \left(x _ {t + d t}, t + d t\right), \tag {27} +$$ + +$$ +x _ {t} = (1 - t) \mu_ {x} + t \underbrace {\left(\alpha \hat {x} _ {1} + \sqrt {1 - \alpha^ {2}} \epsilon\right)} _ {\hat {\epsilon}}, \tag {28} +$$ + +where $\hat{x}_1$ is the noise vector prediction from the last optimization iteration. This induces a variational distribution: + +$$ +q \left(x _ {t} \mid y\right) = \mathcal {N} \left((1 - t) \mu_ {x} + t \alpha \hat {x} _ {1}, t ^ {2} \left(1 - \alpha^ {2}\right) I\right), \tag {29} +$$ + +leading to a score of + +$$ +\nabla_ {x _ {t}} \log q (x _ {t} \mid y) = - \frac {1}{t ^ {2} \left(1 - \alpha^ {2}\right)} \cdot t \sqrt {1 - \alpha^ {2}} \epsilon = - \frac {\epsilon}{t \sqrt {1 - \alpha^ {2}}} . \tag {30} +$$ + +The velocity field is again computed by Equation 18. We start by defining the flow: + +$$ +\psi_ {t} \left(x _ {0} \mid \epsilon\right) = (1 - t) x _ {0} + t \left(\alpha \hat {x} _ {1} + \sqrt {1 - \alpha^ {2}} \epsilon\right). \tag {31} +$$ + +The resulting derivative reads + +$$ +\frac {d}{d t} \psi_ {t} \left(x _ {0} \mid \epsilon\right) = \alpha \hat {x} _ {1} - x _ {0} + \sqrt {1 - \alpha^ {2}} \epsilon , \tag {32} +$$ + +and the inverse becomes + +$$ +x _ {0} = \psi_ {t} ^ {- 1} \left(x _ {t} \mid \epsilon\right) = \frac {x _ {t} - t \alpha \hat {x} _ {1} - t \sqrt {1 - \alpha^ {2}} \epsilon}{1 - t}. \tag {33} +$$ + +Plugging these results into Equation 18: + +$$ +u _ {t} \left(x _ {t} \mid \epsilon\right) = \frac {\alpha \hat {x} _ {1} + \sqrt {1 - \alpha^ {2}} \epsilon - x _ {t}}{1 - t}. \tag {34} +$$ + +# A.3 Derivation of Score from Flow + +The score matching objective reads as: + +$$ +\nabla_ {x _ {t}} \ln p _ {t} \left(x _ {t}\right) = \underset {\theta} {\arg \min } \mathbb {E} _ {t \sim \mathcal {U} [ 0, 1 ], x _ {0} \sim p _ {0}, \epsilon \sim \mathcal {N} (0, I)} \left[ w (t) \cdot \left\| s _ {\theta} \left(x _ {t}, t\right) + \frac {1}{\sigma (t) ^ {2}} \left(x _ {t} - \mu \left(x _ {0}, t\right)\right) \right\| ^ {2} \right], \tag {35} +$$ + +where, + +$$ +- \frac {1}{\sigma (t) ^ {2}} \left(x _ {t} - \mu \left(x _ {0}, t\right)\right) = \nabla_ {x _ {t}} \log p _ {t} \left(x _ {t} \mid x _ {0}\right), \tag {36} +$$ + +with $p_t(x_t \mid x_0) = \mathcal{N}(\mu_t(x_0), \sigma_t^2 I)$ . Note that as usual we assume $\mu_t(x_0)$ being linear in $x_0$ . Equation 35 is solved by: + +$$ +\nabla_ {x _ {t}} \log p _ {t} \left(x _ {t}\right) = \mathbb {E} _ {p _ {t} \left(x _ {0} \mid x _ {t}\right)} \left[ \nabla_ {x _ {t}} \log p _ {t} \left(x _ {t} \mid x _ {0}\right) \right], \tag {37} +$$ + +and can be written as: + +$$ +\nabla_ {x _ {t}} \log p _ {t} \left(x _ {t}\right) = \frac {- \left(x _ {t} - \mu \left(\mathbb {E} \left[ x _ {0} \mid x _ {t} \right], t\right)\right)}{\sigma (t) ^ {2}}. \tag {38} +$$ + +In the case of OT flow-matching, we obtain + +$$ +x _ {t} = (1 - t) x _ {0} + t x _ {1}, \tag {39} +$$ + +$x_{1}\sim \mathcal{N}(0,\mathrm{Id})$ and $p(x_{t}|x_{0}) = \mathcal{N}((1 - t)x_{0},t^{2})$ . The optimal velocity under the flow matching loss is given by: + +$$ +v ^ {*} \left(x _ {t}, t\right) = \mathbb {E} \left[ x _ {1} - x _ {0} \mid x _ {t} \right]. \tag {40} +$$ + +Expressing $x_{1} = \frac{x_{t} - (1 - t)x_{0}}{t}$ , we can insert into Equation 40 and obtain: + +$$ +\mathbb {E} \left[ x _ {0} \mid x _ {t} \right] = x _ {t} - t \mathbb {E} \left[ x _ {1} - x _ {0} \mid x _ {t} \right]. \tag {41} +$$ + +Inserting in Equation 38 leads to: + +$$ +\nabla_ {x _ {t}} \log p _ {t} \left(x _ {t}\right) = - \frac {x _ {t} - (1 - t) \left(x _ {t} - t \mathbb {E} \left[ x _ {1} - x _ {0} \mid x _ {t} \right]\right)}{t ^ {2}}, \tag {42} +$$ + +which for $v^{*}(x_{t}, t) = \mathbb{E}[x_{1} - x_{0} \mid x_{t}]$ reads as: + +$$ +\nabla_ {x _ {t}} \log p (x _ {t}) \approx - \frac {(1 - t) v _ {\theta} (x _ {t} , t) + x _ {t}}{t}. \tag {43} +$$ + +# A.4 Implementation details + +Flow Model and Regularizer Settings. As flow matching model, we use Stable Diffusion 3.5-Medium, which has been released under the Stability Community License. The classifier-free guidance scale is set to 2 for all experiments. To minimize the regularization term, we use stochastic gradient descent with a learning rate of 1. + +Data Likelihood Term. We use stochastic gradient descent for the minimization of the data term towards hard data consistency. For numerical stability, the squared error is summed over all measurements instead of computing the mean. The learning rate has to be adjusted accordingly, to compensate for the varying number of measurements $y$ . Moreover, the minimization is terminated with early stopping once the likelihood term reaches $1 \times 10^{-4} \cdot \ln(y)$ , to not overfit the noise in the image observation. + +Super-resolution. We employ bicubic downsampling as the forward operator, as implemented in [52]. The learning rate is set to 12 for $\times 12$ super-resolution and to 6 for $\times 8$ super-resolution. + +Motion Deblurring. A different motion blur kernel is created for each sample using the MotionBlur package [8], available via github, with kernel size 61 and intensity 0.5. The learning rate for our data term optimizer is set to $10^{-1}$ . + +Inpainting. For inpainting on FFHQ we always use the same rectangular mask at a fixed position, chosen such that it roughly masks out the right side of the face (Figure 3). For DIV2k we also use a fixed mask for all samples, consisting of six randomly generated rectangles (Figure 6). + +Data. We use the publicly available Flickr Faces High Quality dataset [23], which is released under the Creative Commons BY 2.0 License and the DIV2K dataset [3], which is released under a research only license. For FFHQ we use the first 1000 samples of the evaluation dataset and for DIV2K we use the 800 training samples. We downscale both datasets to $768 \times 768$ px by applying bicubic sampling so that the shorter edge of the frame has $768$ px and apply central cropping afterwards. + +# A.5 Baselines + +For comparability, all baselines use Stable Diffusion 3.5-Medium and the same task definitions as in A.4. + +FlowDPS [25] The standard FlowDPS implementation [26] is applied with 50 NFE, a classifier-free guidance scale of 2, and step sizes of 15 for inpainting and 10 for all other tasks. + +FlowChef [38] Additionally, [26] is employed for FlowChef as well, using 200 NFE for inpainting and 50 NFE for all other tasks, a classifier-free guidance scale of 2, and a step size of 1 for all tasks. + +Repulsive Score Distillation (RSD) [61]. We implement RSD for flow-matching models by applying Proposition 1 with $\omega(t) = t$ , resulting in a weighting term consistent with the original RSD approach. However, we omit the pixel-space augmentation as it negatively affected performance when combined with the SD3 VAE. Consistent with the original findings from RSD, we observed that incorporating the repulsive term improves sample diversity but reduces fidelity. Therefore, we set the repulsive term to 0 for all results presented in the table, employing it exclusively for comparing posterior variances. + +ReSample [47] We re implement ReSample for flow-matching by setting $\bar{\alpha}_t = \frac{(1 - t)^2}{t^2 + (1 - t)^2}$ . Furthermore, we compute the hard-data consistency at every iteration as larger skip steps seem to harm performance. We set the learning rate of the data term optimizer to 15 for all inverse problems. + +PSLD [41] Our attempt to adapt PSLD following [26]—using 500 NFE, a classifier-free guidance scale of 2, and step sizes of 1 ( $\times$ 12 super-resolution), 0.5 ( $\times$ 8 super-resolution and motion deblurring) and 0.1 ( inpainting)—did not yield meaningful results. + +# A.6 Regularizer weighting + +Figure 1 displays the mean and standard deviation of the conditional flow matching loss $\mathcal{L}_{CFM}$ as a function of $t$ , estimated over 100 samples. The loss function starts with high values at $t = 1$ , decreases over time, but then starts to rise again, and when reaching $t \approx 0.2$ even exceeds its initial value. The rising loss when approaching $t = 0$ is due, in part, to the increasing difficulty of distinguishing high-frequency image content from residual noise. Another factor is that near $t = 0$ the model + +operates in a highly sensitive regime where small prediction errors can cause disproportionately large deviations from the target, making accurate flow estimation particularly challenging in the final stages of the trajectory. We therefore modulate the regularization term according to the model error. Different weighting functions for $f(\mathcal{L}_{CFM})$ could be chosen that fulfill the condition $\lambda_{\mathcal{R}(t = 0)} = 0$ . We simply take the reciprocal of the model error $\lambda_{\mathcal{R}(t)} = \mathcal{L}_{CFM,t}^{-1}$ as the regularization weight while $t \geq 0.2$ , then set it to 0 for $t < 0.2$ . An alternative would be to shift the reciprocal of $\mathcal{L}_{CFM,t}^{-1}$ by $\mathcal{L}_{CFM,t=0}^{-1}$ , such that $\lambda_{\mathcal{R}(t)} = \mathcal{L}_{CFM,t}^{-1} - \mathcal{L}_{CFM,t=0}^{-1}$ . In Table 1 we compare our default weighting with this variant, denoted as $\lambda_{shift}$ . + +![](images/e483cf932bee77fbc0ca846ccbd4985f2608eba85d419805f919b27a7d7aea6d.jpg) +Figure 1: The Flow-Matching loss over time $t$ . + +Table 1: Quantitative results with 50 NFE and ${\sigma }_{\nu } = {0.01}$ . We compare different weighting functions ${\lambda }_{{\mathcal{R}}_{\left( t\right) }}$ based on the model error + +
SR ×8SR ×12Motion DeblurringInpainting
MethodLPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑
FFHQ 768×768
λshift0.24627.40.79329.910.28624.40.76628.190.23714.50.79029.840.1808.20.82823.58
Ours0.21313.30.77729.540.27116.20.74027.710.23610.70.77229.610.1848.70.82823.69
DIV2K 768×768
λshift0.37930.40.62523.580.43437.50.52221.400.33725.80.66424.540.1519.00.81923.79
Ours0.35326.50.60723.300.42132.10.52521.390.31521.10.65324.440.16311.00.81523.75
+ +# A.7 Effect of captioning + +Given the diversity of DIV2k, we use DAPE [56] to generate captions for it and include them in the prompt A high quality photo of [DAPE caption]. For FFHQ we always prompt with A high quality photo of a face.. The effect of the text prompt is to increase the likelihood of our sample under the prior of the (pre-trained, frozen) image generator. For comparison, we also ran experiments without data specific captions, where we always used the generic prompt A high quality photo. Results are shown in Table 2 + +Table 2: Quantitative results with 50 NFE and $\sigma_{\nu} = 0.01$ . We compare our version with data-specific captions and a version without captions. + +
MethodFFHQDIV2K
LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑
wo captions0.27817.00.73427.660.48851.50.54621.82
Ours0.27116.20.74027.710.42132.10.52521.39
+ +# A.8 Additional Experimental Results + +We present the experimental results from the main paper in Table 3, now augmented with sample-wise standard deviations for all metrics except FID. + +Table 3: Quantitative results with 50 NFE and ${\sigma }_{\nu } = {0.01} -$ Super-resolution $\left( {\times 8\text{and} \times {12}}\right)$ . + +
MethodSR ×8SR ×12
LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑
FFHQ 768×768
ReSample0.400 ± 0.06955.60.815 ± 0.05126.37 ± 1.000.474 ± 0.07880.30.786 ± 0.05625.47 ± 1.16
FlowDPS0.374 ± 0.10738.50.756 ± 0.07529.24 ± 2.040.413 ± 0.10744.00.741 ± 0.07428.05 ± 2.06
RSD0.391 ± 0.07951.70.776 ± 0.05229.69 ± 2.040.462 ± 0.09371.70.743 ± 0.05928.11 ± 2.00
FlowChef0.341 ± 0.08330.50.760 ± 0.06428.42 ± 2.220.373 ± 0.08446.50.730 ± 0.06827.00 ± 2.07
Ours0.213 ± 0.05613.30.777 ± 0.05129.54 ± 2.020.271 ± 0.07116.20.740 ± 0.05827.71 ± 2.00
DIV2K 768×768
ReSample0.533 ± 0.13055.00.625 ± 0.13222.34 ± 2.270.643 ± 0.15288.10.562 ± 0.15120.85 ± 3.02
FlowDPS0.476 ± 0.12944.40.567 ± 0.13923.01 ± 3.010.547 ± 0.13954.00.528 ± 0.14621.79 ± 2.94
RSD0.539 ± 0.12160.90.591 ± 0.12423.45 ± 2.960.684 ± 0.13795.70.523 ± 0.13221.96 ± 2.86
FlowChef0.490 ± 0.11636.50.539 ± 0.13721.84 ± 2.960.525 ± 0.11843.80.492 ± 0.14520.52 ± 2.85
Ours0.353 ± 0.11226.50.607 ± 0.12723.30 ± 2.900.421 ± 0.13132.10.525 ± 0.13621.39 ± 2.67
+ +Table 4: Quantitative results with 50 NFE and ${\sigma }_{\nu } = {0.01} -$ Motion deblurring and in-painting. + +
MethodMotion DeblurringIn-painting
LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑
FFHQ 768×768
ReSample0.457 ± 0.08782.90.788 ± 0.05825.45 ± 1.460.366 ± 0.05370.80.827 ± 0.03321.83 ± 1.68
FlowDPS0.431 ± 0.11754.30.732 ± 0.07827.64 ± 2.200.344 ± 0.06042.50.771 ± 0.04819.19 ± 3.19
RSD0.458 ± 0.09877.30.743 ± 0.05927.67 ± 2.470.478 ± 0.08273.30.736 ± 0.04821.97 ± 2.58
FlowChef0.406 ± 0.09340.20.716 ± 0.07225.81 ± 2.610.394 ± 0.06969.80.780 ± 0.05118.18 ± 2.84
Ours0.236 ± 0.07010.70.772 ± 0.05529.61 ± 2.240.184 ± 0.0388.70.828 ± 0.02923.69 ± 2.77
DIV2K 768×768
ReSample0.556 ± 0.14679.70.617 ± 0.13421.79 ± 2.520.285 ± 0.07351.90.796 ± 0.06722.68 ± 1.84
FlowDPS0.558 ± 0.15365.50.536 ± 0.14821.88 ± 3.020.328 ± 0.10329.20.692 ± 0.11221.71 ± 2.67
RSD0.638 ± 0.15697.60.551 ± 0.13622.10 ± 3.070.464 ± 0.11263.90.678 ± 0.07723.23 ± 2.21
FlowChef0.561 ± 0.12349.60.486 ± 0.14819.90 ± 3.060.489 ± 0.14858.30.659 ± 0.13120.87 ± 2.65
Ours0.315 ± 0.10721.10.653 ± 0.12124.44 ± 3.050.163 ± 0.05311.00.815 ± 0.05423.75 ± 2.74
+ +# A.9 FLAIR in Pixel Space + +Table 5: Quantitative results with 50 NFE and ${\sigma }_{\nu } = {0.5}\%$ - In-painting and Super-resolution $\left( {\times 8}\right)$ . + +
MethodInpaintingSR ×8
LPIPS↓FID↓SSIM↑PSNR↑LPIPS↓FID↓SSIM↑PSNR↑
DDNM0.158 ± 0.04226.90.732 ± 0.03718.31 ± 2.940.199 ± 0.05231.90.635 ± 0.07923.59 ± 1.64
DPS0.195 ± 0.06430.20.689 ± 0.07720.49 ± 2.810.172 ± 0.05827.80.658 ± 0.08824.59 ± 2.04
MM0.161 ± 0.05428.80.728 ± 0.06220.59 ± 3.370.172 ± 0.05129.10.669 ± 0.08324.65 ± 1.97
IIGDM0.195 ± 0.06430.20.689 ± 0.07720.49 ± 2.810.157 ± 0.05226.50.677 ± 0.08424.98 ± 2.07
FLAIR0.097 ± 0.03514.20.831 ± 0.03121.87 ± 2.660.143 ± 0.03922.90.712 ± 0.07625.93 ± 1.96
+ +We additionally implement our method including DTA and $\lambda_{\mathcal{R}(t)} = \mathcal{L}_{CFM,t}^{-1}$ (0 for $t < 0.2$ ) in pixel space using the flow model from [32], trained on CelebA-HQ resized (256x256). For comparison, we rephrase score based baselines to flow following [25] and evaluate all on 1000 samples from the dataset on super-resolution and inpainting. The methods are hyperparameter-tuned to DDNM [52] (likelihood weight 4 for inpainting | 1 for SR8), DPS [11] (64 | 512), Moment Matching [42] (4 | 8), IIGDM [49] (64 | 8) and pixel space FLAIR (0.5 | 32 and regularizer weight 0.4). As shown in Table 5, our method outperforms previous works also in pixel space, demonstrating its broader applicability. + +# A.10 Runtime Analysis + +We compare the runtime and memory consumption of our method to the baselines. As our hard data consistency can strongly influence the runtime, we also provide measurements with the number of data term steps $\leq 5$ and additionally a fast version using a "tinyVAE" of SD3. To validate that the + +![](images/031f311413766084cd254ea8bea1ca548988f4e260c66dea2d34fcbe9687659e.jpg) +Figure 2: Qualitative comparison. FLAIR in pixel space produces posterior samples. + +usage of the tinyVAE or less steps does not degrades the performance noticeably we also provide a metrics for x12 Super Resolution on 100 samples of FFHQ: + +Table 6: Comparison of different methods in terms of runtime and memory usage. We validate the use of less data steps and a "tinyVAE" on $\times 12$ super resolution on 100 samples of the FFHQ dataset. + +
MethodRuntime (s) ↓Memory (MB) ↓LPIPS ↓PSNR ↑
Resample88.0219009.20.46125.31
FlowDPS34.1512228.60.40427.74
RSD (no repulsion)21.1912400.00.46228.11
FlowChef15.2312227.720.36126.56
FLAIR (HDC, large VAE)172.3412389.40.25927.42
FLAIR (HDC, tiny VAE)40.775960.20.25627.59
FLAIR (5 data term steps, tiny VAE)22.465960.20.26427.61
+ +# A.11 Ensembling Experiment + +To highlight that our model can also be used to obtain good MMSE estimates, we also conducted ensemble predictions by running posterior sampling 8 times and averaging the result. The results show that ensembling increases PSNR values, but reduces LPIPS and confirms that our samples are indeed distributed around the posterior mean and that results very close to the posterior mean like the baseline methods are perceptually further away (LPIPS) from the ground truth compared to our samples. + +# A.12 Statistical Relevance + +Our method is training-free, and the variance in reconstructed images is intentional, reflecting the stochasticity of our sampling process rather than instability. All methods are evaluated with identical random seeds to ensure fair comparison. We compute metrics over 1000 samples for FFHQ and 800 samples for DIV2K. Perceptual FID (pFID) is evaluated on $256 \times 256$ patches, resulting in 9000 and 7200 samples, respectively. Table 3 in Appendix A.8 reports means and standard deviations over multiple samples. + +Table 7: Quantitative results – Super-resolution ( $\times 8$ and $\times 12$ ). We report PSNR $\uparrow$ and LPIPS $\downarrow$ . For ensembling we averaged 8 independent predictions of the corresponding methods. It can be seen that PSNR improves for all methods when ensembling. However, FLAIR shows the biggest gain, which means that our samples are indeed distributed around the mean and feature a higher variance compared to the baselines. + +
MethodSR ×8SR ×12
PSNR↑LPIPS↓PSNR↑LPIPS↓
DIV2K
FlowDPS22.530.483721.470.5524
FlowDPS (8x ensemble)23.280.515722.060.5995
FlowChef21.440.489820.200.5228
FlowChef (8x ensemble)22.600.550221.600.5931
FLAIR22.830.362721.050.4270
FLAIR (8x ensemble)23.790.424422.270.4930
FFHQ
FlowDPS29.020.365927.740.4036
FlowDPS (8x ensemble)30.120.326728.650.3749
FlowChef28.050.330326.560.3609
FlowChef (8x ensemble)29.540.326728.150.3602
FLAIR29.360.202827.420.2594
FLAIR (8x ensemble)30.940.245729.000.2999
+ +We also evaluated 100 FFHQ samples and 80 DIV2K samples, sampling three reconstructions per input for each method. We report the mean of each metric across all samples and the standard deviation of the means. + +Table 8: Statistical evaluation on FFHQ for $\times 8$ Super Resolution. We report mean $\pm$ standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.370 ± 0.001270.7 ± 1.450.755 ± 0.001028.98 ± 0.008
RSD0.4678 ± 0.0001102.9 ± 0.050.7362 ± 0.000128.45 ± 0.001
FlowChef0.3316 ± 0.005563.5 ± 1.450.7593 ± 0.002728.12 ± 0.072
Ours0.2039 ± 0.004840.5 ± 0.940.7970 ± 0.022829.74 ± 0.668
+ +Table 9: Statistical evaluation on FFFQ for $\times 12$ Super Resolution. We report mean $\pm$ standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.4073 ± 0.000277.6 ± 1.050.7391 ± 0.000627.71 ± 0.016
RSD0.5039 ± 0.0002119.0 ± 0.120.7217 ± 0.000127.08 ± 0.001
FlowChef0.3626 ± 0.005081.1 ± 1.030.7283 ± 0.002726.62 ± 0.059
Ours0.2593 ± 0.002345.8 ± 1.510.7582 ± 0.025227.81 ± 0.660
+ +# A.12.1 t-Test Analysis + +We further performed paired t-tests on the LPIPS scores between FlowDPS and FLAIR. The null hypothesis states that the mean LPIPS scores are the same for both methods. In all settings, we reject the null hypothesis $(p < 0.001)$ , confirming the statistical significance of our improvements see Table 16. + +Table 10: Statistical evaluation on FFFQ for Motion Blur. We report mean ± standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.4140 ± 0.003083.83 ± 1.000.7383 ± 0.000627.47 ± 0.05
RSD0.4515 ± 0.0001108.75 ± 0.080.7437 ± 0.000127.40 ± 0.00
FlowChef0.4019 ± 0.000774.89 ± 0.690.7178 ± 0.002225.50 ± 0.04
Ours0.2196 ± 0.008038.8 ± 2.250.7964 ± 0.031930.10 ± 0.96
+ +Table 11: Statistical evaluation on FFFQ for Inpainting. We report mean ± standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.3315 ± 0.001574.00 ± 0.580.7755 ± 0.000719.06 ± 0.11
RSD0.4601 ± 0.0003103.02 ± 0.030.7430 ± 0.000022.19 ± 0.01
FlowChef0.3771 ± 0.0013102.22 ± 0.260.7888 ± 0.001618.42 ± 0.25
Ours0.1761 ± 0.001233.23 ± 2.020.8423 ± 0.017224.07 ± 0.80
+ +# A.13 Additional Qualitative Examples + +To illustrate the visual differences behind the error metrics, we present additional qualitative results for both FFHQ and DIV2k, comparing FLAIR with existing approaches. These examples complement the images in the main paper and highlight the visual fidelity, consistency, and robustness of our method across diverse scenes and different degradations. Figure 9 features a full sized version of the variance figure in section subsection 5.2. + +# A.14 Failure cases + +We observe two main failure modes for FLAIR, see Figure 10. First, we find that super-resolution on DIV2k occasionally results in grainy textures, usually in regions with abundant high-frequency detail and complicated light transport. Potentially, this happens for images which do not have high probability under the prior. We do not observe those artifacts for the FFHQ dataset. Second, we observe a few instances where the strong generative prior hallucinates semantically inconsistent or misaligned structures – especially facial features. + +Table 12: Statistical evaluation on $\mathbf{DIV2K}$ for $\times 8$ Super Resolution. We report mean $\pm$ standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.5517 ± 0.0046138.24 ± 1.670.5207 ± 0.002222.41 ± 0.02
RSD0.7163 ± 0.0002181.83 ± 0.150.4892 ± 0.000121.99 ± 0.00
FlowChef0.5726 ± 0.0046145.77 ± 3.310.4998 ± 0.001421.21 ± 0.04
Ours0.3716 ± 0.016188.08 ± 1.430.5991 ± 0.019223.06 ± 0.41
+ +Table 13: Statistical evaluation on DIV2K for $\times 12$ Super Resolution. We report mean $\pm$ standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.6264 ± 0.0045154.31 ± 0.430.4866 ± 0.002421.37 ± 0.02
RSD0.7714 ± 0.0002198.85 ± 0.120.4683 ± 0.000121.15 ± 0.00
FlowChef0.6020 ± 0.0060151.32 ± 2.450.4586 ± 0.002020.10 ± 0.03
Ours0.4316 ± 0.0151101.12 ± 4.510.5236 ± 0.022921.35 ± 0.51
+ +Table 14: Statistical evaluation on DIV2K for Motion Deblur. We report mean ± standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.6242 ± 0.0082161.57 ± 3.900.4978 ± 0.002821.41 ± 0.04
RSD0.8067 ± 0.0002216.37 ± 0.360.4364 ± 0.000120.82 ± 0.00
FlowChef0.6292 ± 0.0007158.08 ± 3.030.4557 ± 0.002519.62 ± 0.05
Ours0.3069 ± 0.003677.77 ± 1.890.6596 ± 0.031624.46 ± 0.71
+ +Table 15: Statistical evaluation on DIV2K for Inpainting. We report mean ± standard deviation over 3 reconstructions per input. + +
MethodLPIPS ↓FID ↓SSIM ↑PSNR ↑
FlowDPS0.3738 ± 0.0032106.58 ± 0.890.6579 ± 0.000921.06 ± 0.05
RSD0.4667 ± 0.0003136.82 ± 0.210.6650 ± 0.000123.07 ± 0.00
FlowChef0.5111 ± 0.0019128.97 ± 0.600.6355 ± 0.000620.51 ± 0.02
Ours0.1729 ± 0.001451.41 ± 0.480.8122 ± 0.012424.06 ± 0.87
+ +Table 16: Paired t-test $p$ -values for LPIPS (FlowDPS vs. Ours). All comparisons are statistically significant. + +
DatasetTaskp-value
DIV2KSR×82.92 × 10-4
SR×127.19 × 10-4
Motion Deblur5.30 × 10-4
Inpainting1.21 × 10-4
FFHQSR×87.05 × 10-5
SR×126.56 × 10-5
Motion Deblur4.29 × 10-5
Inpainting3.00 × 10-5
+ +![](images/f9c039a852ba15b6141603b39f3d7d00abbb17dc4168bc47318f784fa840fc3f.jpg) +Figure 3: Inpainting results on FFHQ. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces realistic, high-frequency details while previous works either fail to inpaint the region correctly or collapse to overly smooth solutions. + +![](images/fd3d344cf94cc3f16724b0a99dfccec738ef5747bd1d413cfd830c6db90c4109.jpg) +Figure 4: $\times 12$ super-resolution results on FFHQ. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend to predict blurry images. + +![](images/d3fe926ddf6cac3a55de01e6aa7409dc4ac5c40cf5a39b91cb8482de3872d22f.jpg) +Figure 5: Motion de-blur results on FFHQ. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend to predict blurry images. + +![](images/9a9d5221c4a9bade79420af38b812bded9bea3ce56a66bd990f6352fa42b05b0.jpg) +Figure 6: Inpainting results on DIV2k. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces realistic, high-frequency details while previous works either fail to inpaint the region correctly or collapse to overly smooth solutions. Moreover they do not fit the data term (not inpainted region) very well. + +![](images/74de2bfb3849df2b6ca245159d653fb510af30f49f598dd71d35917a08eaae71.jpg) +Figure 7: $\times 12$ super-resolution results on DIV2k. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend to predict blurry images. + +![](images/35a6695b562bf38355f5476d8327253c6d1b01ac0c8b1b11d59a9ea6361cd914.jpg) +Figure 8: Motion de-blur results results on DIV2k. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend to predict blurry images. + +![](images/0a787807f553d4d1c058833446dbf7cdc99a51beb47e38b54fa58f6b0bdd46cb.jpg) +Figure 9: Individual samples for x12 Super Resolution with zoom and std. FLAIR produces varied samples from the posterior. For superresolution The variance is expected to be mostly in the high frequencies, because the data term limits low frequency variations. The baselines tend to predict very similar looking images with less detail. + +![](images/10459e68ece6dba775b2524d501564fbe4f11bf883c430c3cef7c9e91de4aa10.jpg) +Target + +![](images/ddaf87277cfeeb8918311b677c59b9a2dc91ee554d8e80b6e94b95d1c3759e65.jpg) + +![](images/4442c1ef8d71e4e184a83b74e9526b04d4832b58545e60a7d66d084cad168dc8.jpg) + +![](images/276c2f1a16c6f5e0b22614a9680224e69a36cd52538aea8e47ad73a7e4d363e5.jpg) + +![](images/bf02428aaaf06301573e292e495990466bd3d6c67b0a301079a687a75933b733.jpg) + +![](images/e5bafe1242dc43094cb7a67777e9939b41cf5d61e69ed2c5d3954f0b80e7afd5.jpg) + +![](images/f6b6a202cbd278b855955d1d1dddd1e1da6d176f168f684f575900d909ced745.jpg) + +![](images/98716d69eb26e80626c7719983dfeafedc8ce55b7f48c0899d1640af44e67096.jpg) + +![](images/cbdaac8501fedc25ca5a3f42bd4b763baf97124f33a870d58265460320ced5d3.jpg) +Target + +![](images/56fa1cb6325e40cc95b61d5738c742b3af250112ae0e3800f623dab6682e5d8a.jpg) +Figure 10: Qualitative failure cases of FLAIR on DIV2k and FFHQ. Top row: grainy results from systematic error. Those errors potentially stem from a weak prior for those images. For example we do not observe them for the FFHQ dataset Bottom row: Semantically inconsistent failures. Sometimes the model lacks the ability to incorporate globally consistent semantics into its restorations. + +![](images/7ffdf3b7c7a3badf8e2509a6fc361ab0825c0b775edacab71b3901f0d11d65a5.jpg) + +![](images/9aace0bb4a29383c7a47e7b3454d80b7cc6af5f1f1ffb5df49fdec17f58dc908.jpg) + +![](images/9685f35e20ee1621f14ad3f6c378aa8b8c45760c34d6e16dd8b554b0ee2151e4.jpg) + +![](images/6dd82162db23e95089e61327506f25a859c071f7ee56b86360a2e3cdaef6a442.jpg) + +![](images/c3d7570b67b5a0c0ba6fde6316e293c58008780eefa36879802cd5848a6dae1c.jpg) + +![](images/68a013c9c8b2a697453a0c736c0e0f41d2ffc46c88831aa7835bc382ec2a8cd9.jpg) \ No newline at end of file diff --git a/NeurIPS/2025/Solving Inverse Problems with FLAIR/images.zip b/NeurIPS/2025/Solving Inverse Problems with FLAIR/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..0f8a021f4e316e1406f59b0932dfde25b8504aa5 --- /dev/null +++ b/NeurIPS/2025/Solving Inverse Problems with FLAIR/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:79227373a241ba734fd6ce9f484af75763db9df706ec12246e2482682e568283 +size 3808302 diff --git a/NeurIPS/2025/Solving Inverse Problems with FLAIR/layout.json b/NeurIPS/2025/Solving Inverse Problems with FLAIR/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..33627994f316aae396229f8947bf2e0456cefa94 --- /dev/null +++ b/NeurIPS/2025/Solving Inverse Problems with FLAIR/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8332fd2f1401ac6a5e46ebf3b7aed2680515f65b87c436ca3f603d74b3c12d96 +size 1005456 diff --git a/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_content_list.json b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..8c5d8c13d3152009a40366ff04a7210879e9377d --- /dev/null +++ b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:727e2f1cdd07925edb4169148b0eb3a25e08a17c43f4a87a683e4bc2df96060a +size 380781 diff --git a/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_model.json b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_model.json new file mode 100644 index 0000000000000000000000000000000000000000..319b7a13a6f117c1349e4a68347f990de92b3f91 --- /dev/null +++ b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:969b1749e6c2c20f70cbfa7c72f5e1a52f81af4e50f65f29ef234f072e5a641f +size 474974 diff --git a/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_origin.pdf b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..abfb6894cad9f60e3a477342581b8a7e1595925c --- /dev/null +++ b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/fd48a690-8eda-45ec-9942-e311360fe7be_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:355912c2dbbbe17c1f5435531b52a62de029b5e44ce828493909a543ed9c9ec1 +size 1795541 diff --git a/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/full.md b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/full.md new file mode 100644 index 0000000000000000000000000000000000000000..521c0ecb0109f8e6d23d2ce715016d660132ffad --- /dev/null +++ b/NeurIPS/2025/Solving Neural Min-Max Games_ The Role of Architecture, Initialization & Dynamics/full.md @@ -0,0 +1,1912 @@ +# Solving Neural Min-Max Games: The Role of Architecture, Initialization & Dynamics + +Deep Patel and Emmanouil-Vasileios Vlatakis-Gkaragkounis + +Department of Computer Science + +University of Wisconsin-Madison + +{dbpatel5,vlatakis}@wisc.edu + +# Abstract + +Many emerging applications—such as adversarial training, AI alignment, and robust optimization—can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions—spanning initialization, training dynamics, and network width—that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-Lojasiewicz (PL) min-max condition hold with high probability under overparameterization, using tools from random matrix theory. + +# 1 Introduction + +At the Nobel Symposium marking the centennial of Game Theory [31, 83], a key challenge was posed: + +the development of a systematic theory for non-convex games + +spurred by the rapid growth of deep learning in incentive-aware multi-agent systems [104, 130]. + +Indeed, many influential modern AI systems are built upon the fusion of foundational game-theoretic principles—particularly zero-sum games—with the expressive capacity of neural networks. Notable examples include generative adversarial networks (GANs) [51], robust reinforcement learning [89], adversarial attacks [117], domain-invariant representation learning [44], distributionally robust optimization [77, 123], and multi-agent environments featuring natural language interactions, such as AI safety debates between large language models and verifier-prover systems [56, 16]. In these settings, the game-theoretic framework provides a natural and interpretable objective—typically an equilibrium solution endowed with strong normative appeal, such as the celebrated von Neumann minimax points [48] and Nash-Rosen equilibria [80, 94]. + +At the same time, much of the remarkable progress at the intersection of deep learning and game theory stems from the capacity of deep models to operate effectively in environments with large, often continuous, state and action spaces. Iconic examples include Go [103], autonomous driving [102], Texas Hold'em poker [15], and real-time strategy games such as StarCraft II through AlphaStar [112]. + +Tackling such large-scale decision-making problems has necessitated the combination of expressive architectures with function-approximation-based learning, replacing high-dimensional reward/value functions and strategy/policy spaces with trainable surrogates. Hence, these surrogates act as flexible intermediaries, enabling generalization across complex environments without exhaustive enumeration of action spaces. While theoretical focus has largely remained on linear approximators [121, 26], it is the nonlinear models—such as kernels and deep neural networks—which in practice dramatically expand representational power [67, 58], allowing richer strategic behaviors. Thus, agents' policies are encoded through powerful approximators, and equilibrium learning unfolds through iterative parameter tuning (see Figure 1). + +![](images/02bfeb377cefa0bbc896c1cf10be26060ad31b2a7cf47842588cc2123cd1b775.jpg) +Figure 1: Illustration of a maze environment where each agent must reason over a vast space of action sequences. Instead of explicitly constructing and searching the full decision tree, a neural network implicitly encodes both the value of paths and the policy for navigation, learning an effective strategy dynamically without ever uncovering the complete structure of the maze. + +Despite the empirical success, algorithms with provable convergence guarantees remain scarce. This is unsurprising given that even in finite games, strong computational hardness results [22, 23, 32, 87] and dynamic impossibility theorems [78, 53, 52, 116, 46, 47] pose significant barriers. Notably, even in two-player zero-sum games—where classical theory guarantees existence and efficient computation of minimax points via LP duality [19] or optimistic first-order methods [120, 5]—these assurances collapse when modern deep-learning architectures, with their inherent non-convexity, are introduced [34, 10, 6]. Specifically: $(i)$ global solution concepts (e.g., von Neumann minimax, Nash equilibria) may fail to exist; $(ii)$ even when they do, tandem gradient-based methods often suffer from instability, cycling, or divergence, resulting in poor solutions. [33, 75, 35, 113]. + +Thus, the best hope for mitigating the practical impact of these worst-case hardness results lies in focusing on structured subclasses of games. It remains plausible that broad families of nonconcave games—rich enough to capture multi-agent interactions—admit tractable local or even global equilibria. + +Hidden convexity: a promising direction. One compelling approach along this path is the emerging theory of hidden convex games [114, 79, 115, 99, 29]. In its simplest form, two players interact via a convex-concave zero-sum game $Loss(\text{Player}_1, \text{Player}_2)$ , but control only high-dimensional parameters $\theta, \phi$ , through mappings $\text{Player}_1 \gets M a p_{\theta}^1(\cdot)$ and $\text{Player}_2 \gets M a p_{\phi}^2(\cdot)$ . These mappings are smooth and known, allowing gradient-based training, but typically not efficiently invertible, reflecting the practical irreversibility of neural architectures. Consequently, while the latent game preserves convex-concave structure, the optimization landscape $Loss(\text{Map}_{\theta}^1, \text{Map}_{\phi}^2)$ over control variables becomes highly non-convex [see 99, p. 26]. Although not every non-convex game admits such a structure, many practical applications naturally fit within this framework (see Appendix B). + +Rank collapse: the fragility of hidden convexity. A major criticism of the hidden convexity paradigm relies critically on the assumption that the Jacobian of the agents' mappings maintain uniformly bounded singular values throughout training. In practice, such uniform bounds often fail, as real-world architectures may suffer from rank collapse or near-singular behavior during optimization (see, e.g., [101, 43, 37]), undermining theoretical guarantees. When such degeneracies arise, convergence rates can deteriorate exponentially, and worst-case bounds may become vacuous. Even if Jacobian well-conditioning is achieved by a random initialization, there are no assurances that it will be preserved as training evolves. + +These limitations underscore the need for explicit, open-box conditions—beyond abstract hidden mappings—that explain the empirical success of efficient training in large-scale min-max settings. Whilst hidden convexity provides significant insights about these systems, it does not answer a fundamental behavioral question: + +Can appropriate architectural design, initialization protocols, and training dynamics jointly ensure efficient convergence in large-scale neural min-max games? + +![](images/f19a2d00587042955f9ceeff99741e695acead7a61c83bbde0eaa6d3660fc69b.jpg) + +# 1.1 Setting and Main Contribution + +Motivated by the above challenges, we provide—to the best of our knowledge—the first quantitative convergence guarantees addressing the central question ( $\star$ ) under minimal assumptions. Formally, given input datasets $\mathcal{D}_F$ and $\mathcal{D}_G$ , and latent strategy spaces $\mathcal{S}_F$ and $\mathcal{S}_G$ , we consider the hidden min-max problem + +$$ +\min _ {\theta \in \mathbb {R} ^ {d _ {\theta} ^ {(F)}}} \max _ {\phi \in \mathbb {R} ^ {d _ {\phi} ^ {(G)}}} \mathcal {L} _ {\mathcal {D}} (F _ {\theta}, G _ {\phi}), \quad (\prod) +$$ + +where $F_{\theta}:\mathbb{R}^{d_0^{(F)}}\to \mathbb{R}^{\dim (\mathcal{S}_F)}$ and $G_{\phi}:\mathbb{R}^{d_0^{(G)}}\to \mathbb{R}^{\dim (\mathcal{S}_G)}$ are smooth mappings parameterized by $\theta$ and $\phi$ (e.g., neural network weights). While our results extend beyond, we focus on well-studied [127] separable latent minmax objectives of the form + +$$ +\mathcal {L} _ {\mathcal {D}} (F, G) = I _ {1} ^ {\mathcal {D} _ {F}} (F) + I _ {2} ^ {\mathcal {D}} (F, G) - I _ {3} ^ {\mathcal {D} _ {G}} (G), \tag {1} +$$ + +where $\mathcal{D} = (\mathcal{D}_F, \mathcal{D}_G)$ and $I_1^{\mathcal{D}_F}, I_3^{\mathcal{D}_G}$ — the individual components—are strongly convex and smooth, and $I_2^{\mathcal{D}}$ — the coupling component—is smooth bilinear. As convergence metric, we adopt the Nash gap (also known as the Nikaido-Isoda duality gap [82]): + +$$ +\mathrm {D G} _ {\mathcal {L} _ {\mathcal {D}}} (\theta , \phi) := \max _ {\phi^ {\prime}} \mathcal {L} _ {\mathcal {D}} (F _ {\theta}, G _ {\phi^ {\prime}}) - \min _ {\theta^ {\prime}} \mathcal {L} _ {\mathcal {D}} (F _ {\theta^ {\prime}}, G _ {\phi}), +$$ + +and say that $(\hat{\theta},\hat{\phi})$ is an $\epsilon$ -saddle (or $\epsilon$ -approximate minimax or Nash equilibrium) if $\mathrm{DG}_{\mathcal{L}_D}(\hat{\theta},\hat{\phi})\leq \epsilon$ . + +Remarks. Replacing players' actions with neural nets—i.e., $F_{\theta} = \mathrm{NN}_{\theta}(\cdot)$ and $G_{\phi} = \mathrm{NN}_{\phi}(\cdot)$ —renders the end-to-end landscape highly non-convex, although the latent game $\mathcal{L}$ remains convex-concave. The separable structure naturally unifies several hidden zero-sum regimes: when $I_{2}$ vanishes, it recovers separable strongly-convex-concave games; when $I_{1}$ and $I_{3}$ vanish, it reduces to bilinear games [114]; and when both components are present, it captures regularized games (e.g., Tikhonov- or entropy-regularized settings), recently used in hidden min-max frameworks, including team and zero-sum Markov games [59, 60]. We discuss concrete examples in Section 2 and Appendix B. In these settings, regularization plays a critical role in stabilizing dynamics and mitigating chaotic behaviors, both empirically ([see 99, p. 26]) and theoretically (cf. [115, pp. 7–8], [59]). Before enumerating our technical contributions, we highlight a key result addressing ( $\star$ ): + +Informal Theorem (Theorem 3.8). There exists a decentralized, gradient-based method (eq. (Alt-GDA)) that computes, with high probability under suitable Gaussian random initialization, an $\epsilon$ -approximate Nash equilibrium for any $\epsilon > 0$ in broad class of hidden convex-concave zero-sum games, where each player's strategy is parameterized by a sufficiently wide two-layer neural network. + +- The number of iterations required scales as + +$$ +O \left(\operatorname {p o l y} \left(\frac {1}{w i d t h _ {1}}, \frac {1}{w i d t h _ {2}}, \frac {1}{n}, d _ {i n p u t}\right) \times \frac {L ^ {3}}{\mu^ {3}} \times \log \left(\frac {1}{\epsilon}\right)\right), +$$ + +where $\text{width}_1, \text{width}_2$ are the hidden layer widths, $n$ is the number of training samples, $d_{\text{input}}$ is the input dimension, $L$ is the smoothness constant, and $\mu$ is the strong convexity modulus of the latent objective. + +- This guarantee holds provided the network width $\left. {1,2}\right. = {\widetilde{\Omega }}\left( {{\mu }^{2}\frac{{n}^{3}}{d\text{ input }}}\right)$ . + +A converse byproduct: input-optimization games. We also uncover a new convergence guarantee in a related but distinct setting: optimizing directly over inputs when the neural network mappings are fixed. This perspective is motivated both by adversarial example generation through min-max formulations (see Section 2, Appendix B & [117]) and by empirical results of [99] for solving normal form zero-sum games using input-optimization at random fixed neural network mappings—without theoretical justification of non-singularity of spectrum trajectory. Formally, the goal is to find input vectors $(x_{\mathrm{Alice}}, x_{\mathrm{Bob}})$ that implement a Nash equilibrium: + +$$ +\min _ {x _ {\text {A l i c e}} \in \mathcal {D} _ {F}} \max _ {x _ {\text {B o b}} \in \mathcal {D} _ {G}} \mathcal {L} \left(F _ {\theta} \left(x _ {\text {A l i c e}}\right), G _ {\phi} \left(x _ {\text {B o b}}\right)\right). \quad (\prod^ {- 1}) +$$ + +for some convex-concave function $\mathcal{L}$ , typically referred as attack's loss [117]. In this regard, we formally establish that Algorithm AltGDA converges to an $\epsilon$ -Nash equilibrium with iteration complexity $\tilde{O}\left(\frac{1}{\epsilon}\log \left(\frac{1}{\epsilon}\right)\right)$ under high-probability guarantees (Theorem 3.5). To the best of our knowledge, this provides the first open-box, provable convergence result for input-optimization attacks based on randomly initialized overparameterized neural networks, matching and theoretically explaining the experimental observations of [99] and [117]. + +# 1.2 Challenges and Our Approach: Bridging Overparameterization with Strategic Learning + +Back to minimization. The optimization of min-max objectives—especially convex-concave or structured non-convex games—has been extensively studied (for an appetizer see Appendix A.1-A.2 and references therein). However, the dynamics of gradient-based methods in games where players are parameterized by neural networks remain far less understood. In minimization of training loss, a powerful lens for analyzing the success of gradient descent (GD) is the theory of overparameterization and the Neural Tangent Kernel (NTK). In the infinite-width limit, GD converges provided the NTK's smallest eigenvalue remains bounded away from zero. For finite-width networks, convergence proofs typically hinge on two ingredients: (i) good NTK conditioning at initialization, and (ii) negligible drift of the NTK during training [85, 27, 13, 106], ensuring that an underlying Polyak-Lojasiewicz (PL) condition is maintained. + +Extending to Games: The spectrum path. Even simple hidden zero-sum games, where players are parameterized by two-layer neural networks with smooth activations, can cause vanilla GDA to diverge arbitrarily [114]. Although PL-based convergence for minimization has been understood since the classical works of Polyak and Lojasiewicz [90, 73], analogous results for min-max optimization have only recently emerged [124, 125, 60]. More recently, hidden convexity has been shown to imply a PL structure—both in minimization [41] and in min-max games [60]. However, this reduction to PL-condition reveals a key technical obstacle: hidden convexity alone cannot safeguard convergence if the Jacobians of the players' mappings suffer from near-singularities—i.e., if the least singular value approaches zero. In this regime, the effective PL-modulus degenerates, the gradient dominance property and convergence guarantees break down. Thus, the evolution of singular values under the employed learning dynamics becomes central challenge. + +In this work, we adopt the alternating gradient descent-ascent (AltGDA) method, which mirrors natural sequential play between agents. From a technical standpoint, alternation proves crucial: simultaneous one-timescale GDA (SimGDA) may diverge both in case of hidden convex-concave games [114] and two-sided-PL games [124]. Additionally, alternation has been explored as an acceleration and stabilization tool for min-max optimization [66, 128]. + +- AltGDA Path Length: Hence, our first central technical contributions is a tight control of the path length of AltGDA iterates (Lemma 3.3). We show that AltGDA trajectories remain confined within a bounded region around initialization, preventing severe deterioration of hidden convex-concave structure (e.g., Jacobian conditioning). While path-length bounds are relatively straightforward in minimization—by directly unrolling GD iterations—in min-max problems, the alternating structure introduces significant complications for such ad-hoc analysis. To circumvent this, we employ a carefully designed potential function—a weighted interpolation between the two players' Nash gaps—by [124], which may be of independent interest. + +Beyond bounding the trajectory, two additional challenges arise relative to standard supervised learning: + +- Output Dimension: In games, neural networks output distributions over actions or more generally higher-dimensional vectors, unlike scalar labels in classification tasks. Estimating the singular value spectrum of such vector-output neural networks is more subtle. To address this, we arrive at Lemma 3.7, by adapting techniques from [106] which essentially combines Hermite expansions of hidden layer outputs, first-order Taylor series expansion and Lipschitzness of Jacobians, and high-probability concentration bounds for random Gaussian matrices. +- Average-Case Analysis of Input Min-Max Games: A similar approach is employed for input-optimization games, where the roles of inputs and weights are reversed. From a worst-case perspective, there exist constructions leading to rank-deficient Jacobians and failure of GDA due to convergence to spurious local optima [114], our analysis takes an average-case view. Specifically, we show that min-max input attacks, solved via AltGDA, succeed with high probability when the neural network mappings are randomly sampled with Gaussian initializations (Theorem 3.5). +- General Loss Structures: Unlike many prior works, which rely on the non-linear least squares structure of supervised losses to control dynamics [106, 69, 70], we allow general separable latent objectives combining strongly convex regularizers and bilinear couplings. This more general setting requires significantly stronger control on the optimization trajectory and leads to a fundamentally different overparameterization scaling, namely $\Omega(n^3)$ compared to $\Omega(n)$ in pure minimization settings (Theorem 3.8) + +# 2 Preliminaries + +We begin by introducing the standard notions of smoothness and Lipschitz continuity that will be used throughout this work. All norms are taken to be the Euclidean $(\ell_2)$ norm unless otherwise stated. + +Lipschitz Continuity, Smoothness, and Strong Convexity. Let $f: \mathbb{R}^d \to \mathbb{R}$ be a differentiable function. We say that $f$ is $L_f$ -Lipschitz continuous and $L_{\nabla f}$ -smooth if there exist constants $L_f, L_{\nabla f} > 0$ such that + +$$ +\left| f (u) - f (v) \right| \leq L _ {f} \| u - v \|, \quad \| \nabla f (u) - \nabla f (v) \| \leq L _ {\nabla f} \| u - v \|, \quad \forall u, v \in \mathbb {R} ^ {d}. +$$ + +Moreover, $f$ is $\mu$ -strongly convex if there exists $\mu > 0$ such that + +$$ +f (v) \geq f (u) + \langle \nabla f (u), v - u \rangle + \frac {\mu}{2} \| v - u \| ^ {2}, \quad \forall u, v \in \mathbb {R} ^ {d}. +$$ + +Similarly, for a parametrized mapping (e.g., the neural network) $M_{\theta}(x):\mathbb{R}^{d_0}\to \mathbb{R}^{d_2}$ with parameters $\theta \in \mathbb{R}^M$ , we say $M_{\theta}$ is $\beta_{M}$ -smooth (w.r.t. $\theta$ ) at fixed input $x$ if + +$$ +\sigma_ {\max } \left(\nabla_ {\theta} M _ {\theta} (x) - \nabla_ {\theta} M _ {\theta^ {\prime}} (x)\right) \leq \beta_ {M} \| \theta - \theta^ {\prime} \|, \quad \forall \theta , \theta^ {\prime} \in \mathbb {R} ^ {M}, \tag {2} +$$ + +where $\sigma_{\mathrm{max}}(\cdot)$ denotes the largest singular value and $\nabla_{\theta}M_{\theta}(x)$ is the Jacobian of $M_{\theta}(x)$ with respect to $\theta$ . + +Finite-Sample Parametrized Min-Max Setting. Then, we unroll the general hidden convex-concave model of $(\Pi)$ to the finite-sample empirical risk minimization (ERM) setting, assuming access to a (possibly labeled) dataset $\mathcal{D} = (\mathcal{D}_F,\mathcal{D}_G) = \{(x_i,y_i)\}_{i = 1}^n$ of size $n$ . Formally, we consider the following optimization problem: + +$$ +\min _ {\theta \in \mathbb {R} ^ {d _ {\theta} ^ {(F)}}} \max _ {\phi \in \mathbb {R} ^ {d _ {\phi} ^ {(G)}}} \mathcal {L} _ {\mathcal {D}} \left(F _ {\theta}, G _ {\phi}\right) := I _ {1} ^ {\mathcal {D} _ {F}} \left(F _ {\theta}\right) + I _ {2} ^ {\left(\mathcal {D} _ {F}, \mathcal {D} _ {G}\right)} \left(F _ {\theta}, G _ {\phi}\right) - I _ {3} ^ {\mathcal {D} _ {G}} \left(G _ {\phi}\right), \tag {◇} +$$ + +where the mappings $F_{\theta}:d_0^{(F)}\to \mathbb{R}^{\dim (\mathcal{S}_F)}$ and $G_{\phi}:d_0^{(G)}\to \mathbb{R}^{\dim (\mathcal{S}_G)}$ are smooth functions parametrized by $\theta$ and $\phi$ (e.g., neural networks). The individual components and the bilinear coupling expand as: + +- $I_1^{\mathcal{D}_F}(F_\theta) = \sum_{i \in [|\mathcal{D}_F|]} \ell_i(y_i, F_\theta(x_i))$ , $I_3^{\mathcal{D}_G}(G_\phi) = \sum_{j \in [|\mathcal{D}_G|]} \ell_j(y_j, G_\phi(x_j))$ , +$I_2^{(\mathcal{D}_F,\mathcal{D}_G)}(F_\theta ,G_\phi) = \sum_{i\in [|\mathcal{D}_F|]} \sum_{j\in [|\mathcal{D}_G|]} F_\theta (x_i)^\top A(x_i,x_j,y_i,y_j)G_\phi (x_j),$ + +where for each sample pair $e_{ij}(x_i, x_j, y_i, y_j)$ , the coupling matrix $A(x_i, x_j, y_i, y_j) \in \mathbb{R}^{\dim(S_F) \times \dim(S_G)}$ encodes interactions between players. + +Blanket Assumptions on the Loss and Coupling Terms. We impose the following structural assumptions on the loss components and bilinear couplings appearing in the finite-sample min-max objective $(\diamond)$ . + +Assumption 2.1 (Smoothness, Hidden Strong Convexity, and Gradient Control). + +(i) Smoothness: Each sample-wise individual loss $\ell(y, \operatorname{Map}_w(x))$ is differentiable and $L$ -smooth with respect to $\operatorname{Map}_w(x)$ . +(ii) Coupling Structure: Each bilinear coupling matrix $A(x_{i},x_{j},y_{i},y_{j})$ is known, fixed, and has bounded operator norm. +(iii) Hidden Strong Convexity: Each sample-wise individual loss $\ell(y, h = \operatorname{Map}_w(x))$ is strongly convex with respect to the neural network output $h$ . +(iv) Gradient Growth Condition: There exist constants $A_{1}, A_{2}, A_{3} > 0$ such that for all $h \in \mathbb{R}^{d_{\mathrm{out}}}$ and $y \in \mathcal{V}$ , the (latent) gradient of each loss $\ell(y, h)$ satisfies: + +$$ +\left\| \nabla_ {h} \ell (y, h) \right\| \leq A _ {1} \| h \| + A _ {2} \operatorname {d i a m} (\mathcal {Y}) + A _ {3}. +$$ + +Remark 2.2. Item (i) ensures the applicability of gradient-based methods, while Items (i)-(iii) imply that the overall loss $\mathcal{L}_{\mathcal{D}}$ is $(L_{\mathcal{L}}, L_{\nabla \mathcal{L}})$ -smooth and $(\mu_{\theta}, \mu_{\phi})$ -hidden-strongly convex-concave, with constants determined by the structure of $\ell$ and $A(\cdot)$ . For standard strongly convex losses (e.g., MSE, logistic loss, cross-entropy with $\ell_2$ -regularization), the gradient with respect to the network output + +is controlled as in Item (iv) by an affine function of the output norm, with the leading coefficient proportional to the strong convexity modulus, $A_{1} = \Theta (\mu)$ .1 + +# Neural Network and Training Data Model. + +Definition 2.3 (Two-layer Neural Network). We consider two-layer neural networks (often referred to as shallow networks). Specifically, such a network $h$ is defined by: + +$$ +h (x) = \operatorname {M a p} _ {w = \left(W _ {1}, W _ {2}\right)} (x) = W _ {2} ^ {(h)} \psi \left(W _ {1} ^ {(h)} x\right), +$$ + +where $x \in \mathbb{R}^{d_0^{(h)}}, W_1^{(h)} \in \mathbb{R}^{d_1^{(h)} \times d_0^{(h)}}, W_2^{(h)} \in \mathbb{R}^{d_2^{(h)} \times d_1^{(h)}}$ , and $\psi: \mathbb{R} \to \mathbb{R}$ is an activation function applied coordinate-wise. + +Assumption 2.4 (Properties of the Two-layer Neural Network). We assume: + +- $h(\cdot)$ is twice-differentiable and $\beta_h$ -smooth with respect to $(W_1^{(h)}, W_2^{(h)})$ . +- $\psi$ is twice-differentiable with $\psi(0) = 0$ , bounded first and second derivatives $(\dot{\psi}_{\mathrm{max}}, \ddot{\psi}_{\mathrm{max}})$ , and finite Hermite norm $\| \psi \|_{\mathcal{H}} < \infty^2$ . +- The training data $(X,Y) \in \mathbb{R}^{d_0^{(h)} \times n} \times \mathbb{R}^{d_2^{(h)} \times n}$ satisfies $\| x_i \| = 1 \forall i \in [n]$ and $\| Y \| \leq 1$ . + +$\sigma_{\mathrm{max}}\left((W_2^{(h)})_k\right) = \mathcal{O}\left(\frac{\dot{\psi}_{\mathrm{max}}}{\dot{\psi}_{\mathrm{max}}}\right)$ for all $k\in \mathbb{Z}_{\geq 0}^3$ + +Although we assume the activation function $\psi$ to be twice differentiable—thereby excluding non-smooth activations such as ReLU—our results naturally extend to smooth approximations like the Gaussian Error Linear Unit (GeLU) [55] and the softplus function [39], which have been shown empirically to perform comparably or even better than ReLU in several settings [12, 40]. The performance of gradient-based training depends critically on the geometry of the training data. A standard proxy for data diversity is the well-conditioning of sample matrix X (with input vectors as rows), under standard random designs such as isotropic or sub-Gaussian inputs [86, 110, 111, 95]. + +Assumption 2.5 (Spectral Properties of the Data Matrix). Let $X \in \mathbb{R}^{n \times d}$ denote the data matrix whose rows $x_{i}$ satisfy $\| x_{i}\|_{2} = 1$ for all $i$ . We assume that the number of samples satisfies $n \geq d$ , and that $X$ is "generic" in the sense that $\sigma_{\min}(X^{*r}) = \Omega(1)$ and $\sigma_{\max}(X) = O\left(\sqrt{n / d}\right)^{5}$ , where $n$ is the number of samples and $d$ is the ambient input dimension. + +For a fair comparison with the minimization literature, in the main body of the paper we adopt the data genericity assumption. Interested readers can refer to appendix for fine-grained width bounds. $^6$ + +Solution concept. Note that while our min-max objective $\mathcal{L}_{\mathcal{D}}(F_{\theta},G_{\phi})$ is not convex-concave in $F_{\theta},G_{\phi}$ , it is (strongly) convex-concave in the outputs of $F_{\theta}$ and $G_{\phi}$ , i.e., hidden strongly-convex-concave. Our analysis leverages precisely this hidden structure. Specifically, [41, Proposition 2] states that if $\min_{\theta}f(\theta)$ where $f(\theta) = F(H(\theta))$ and $F$ is strongly convex while $H$ is a smooth map (e.g., a neural net), then $f$ satisfies the PL-condition. Thus, hidden strong convexity implies PL-condition, even for nonconvex objectives. Utilizing this along with [30, Proposition of 4.1], we can define PL-moduli for our min-max objective in terms of the smallest singular values of the neural network Jacobians: + +${}^{1}$ Controlling the gradient growth is critical for non-asymptotic overparameterization bounds, as it ensures that iterates remain within regions where hidden convexity persists. A detailed discussion and examples are deferred to Appendix D. +This is needed for upper bound on maximum singular value of $h$ 's Jacobian. +3Given random Gaussian initialization and ensuring that iterates never leave a finite-radius ball around initialization, we can safely assume the maximum singular value is bounded from above for all iterates $k$ . +Moreover, we expect that the smoothness assumption can be relaxed. Since AltGDA includes subgradient variant, our analysis could likely be extended to (almost) smooth activations such as ReLU, by carefully treating the measure-zero set of non-differentiability points. We leave this technical refinement to future work, as the core phenomena should remain qualitatively unchanged. +$^5\sigma_{\mathrm{max}}(X) = O(\sqrt{n / d})$ w.h.p. when, for e.g., $\mathbf{X}$ has i.i.d. $\mathcal{N}(0,1)$ entries [86, Section II.A]. For an arbitrary, fixed $X$ , $\sigma_{\mathrm{max}}(X) = \| X\| _2\leq \| X\| _F = \sqrt{n}$ ( $\because \| x_i\| _2 = 1\forall i$ ). See Remark G.5 in Appendix G. +The assumption $n \gtrsim d$ ensures that $X$ is sufficiently tall to avoid rank deficiency and the minimum singular value of the Khatri-Rao product $\sigma_{\mathrm{min}}(X^{*r})$ serves as natural measures of the dataset's well-conditioning. Intuitively, Assumption 2.5 reflects that the dataset covers the input space sufficiently uniformly, ensuring that no direction is either too collapsed or too amplified. Such a balance is critical for achieving stable optimization dynamics and avoiding pathological trap into lower-dimensional subspaces during training. + +Fact 2.6 (Reduction to Two-Sided PL-condition [41, 30]). The loss function $\mathcal{L}_{\mathcal{D}}$ satisfies a two-sided Polyak-Lojasiewicz (PL) condition with parameters $\mu_{\theta}\sigma_{\mathrm{min}}^{2}(\nabla_{\theta}F_{\theta})$ and $\mu_{\phi}\sigma_{\mathrm{min}}^{2}(\nabla_{\phi}G_{\phi})$ , where $\sigma_{\mathrm{min}}(\cdot)$ denotes the smallest singular value of the corresponding Jacobian mappings. + +This reduction resolves several challenges inherent to general nonconvex-nonconcave min-max problems. First, it unifies several optimality notions—namely, global minimax, saddle point, and gradient stationarity—which, in general settings, need not coincide. For formal definitions see Appendix E. In the case where the objective satisfies a two-sided Polyak-Lojasiewicz (PL) condition, these notions become equivalent even at their $\epsilon$ -approximate versions. We formalize this via the following lemma: + +Lemma 2.7 (Lemma 2.1 in [124], Appendix C in [60]). If the objective function $f$ satisfies the two-sided PL-condition, then all three notions in Definition E.1 are equivalent: + +$$ +\epsilon - (S a d d l e \text {P o i n t}) \iff \epsilon - (G l o b a l M i n - M a x) \iff \epsilon - (S t a t i o n a r y P o i n t) \quad \forall \epsilon \geq 0 +$$ + +Second, as discussed in the introduction, saddle points may not exist in general nonconvex-nonconcave problems. Therefore, we explicitly adopt the following benign7 assumption: + +Assumption 2.8 (Existence of Saddle Points). The objective function $\mathcal{L}(\theta, \phi)$ admits at least one saddle point. Moreover, for any fixed $\phi$ , $\min_{\theta \in \mathbb{R}^m} \mathcal{L}(\theta, \phi)$ has a non-empty solution set and a finite minimum value. Similarly, for any fixed $\theta$ , $\max_{\phi \in \mathbb{R}^n} \mathcal{L}(\theta, \phi)$ has a non-empty solution set and a finite maximum value. + +Examples of hidden neural min-max optimization. Due to space limitations, we defer a comprehensive list of examples and references to Appendix B. To build intuition, we present below two representative bilinear examples that highlight the key structural differences. We broadly distinguish two principal types of ML-driven min-max problems + +- Network Optimization: Problems where optimization is performed over neural network parameters given a fixed dataset (training over weights). This setting captures tasks such as generative modeling or robust adversarial reinforcement learning. + +$$ +\text {E x a m p l e}: \min _ {\theta} \max _ {\phi} F _ {\theta} (x) ^ {\top} A G _ {\phi} (x ^ {\prime}). +$$ + +- Input Optimization: Problems where network parameters are fixed (e.g., random initialization), and optimization occurs over the input space (e.g., adversarial perturbations). This corresponds to input-driven optimization problems such as adversarial attack design. + +$$ +\text {E x a m p l e}: \min _ {x _ {\text {A l i c e}} \in \mathcal {D} _ {F}} \max _ {x _ {\text {B o b}} \in \mathcal {D} _ {G}} \left(F _ {\theta} \left(x _ {\text {A l i c e}}\right) ^ {\top} A G _ {\phi} \left(x _ {\text {B o b}}\right)\right). +$$ + +# 3 Our Results + +Alternating Gradient Descent-Ascent (AltGDA) proceeds by sequentially updating the parameters of the min-player $\theta$ and the max-player $\phi$ , leveraging the most recent gradient information at each step. The updates take the form: + +$$ +\theta^ {(t)} = \theta^ {(t - 1)} - \eta_ {\theta} \nabla_ {\theta} \mathcal {L} _ {\mathcal {D}} (\theta^ {(t - 1)}, \phi^ {(t - 1)}), \quad \phi^ {(t)} = \phi^ {(t - 1)} + \eta_ {\phi} \nabla_ {\phi} \mathcal {L} _ {\mathcal {D}} (\theta^ {(t)}, \phi^ {(t - 1)}) \quad (\mathrm {A l t G D A}) +$$ + +where $\eta_{\theta},\eta_{\phi} > 0$ denote the respective step sizes. + +Our analysis builds upon the framework of Yang, Kiyavash, and He [124], which guarantees $\log (1 / \epsilon)$ convergence under a two-sided PL condition. In our setting, the PL moduli depend on the smallest singular values $\sigma_{\mathrm{min}}$ of the Jacobians $\nabla_{\theta}F_{\theta}$ and $\nabla_{\phi}G_{\phi}$ , which must remain bounded away from zero throughout the optimization trajectory (Fact 2.6). This dependence is critical, as both the PL constants and the step sizes in AltGDA scale inversely with $\sigma_{\mathrm{min}}$ . + +Hence, we first establish that, under suitable random initialization and sufficient overparameterization, the initialization satisfies $\sigma_{\mathrm{min}}(\nabla_{\theta}F_{\theta}),\sigma_{\mathrm{min}}(\nabla_{\phi}G_{\phi})\geq cB$ with high probability (see Lemmas 3.4 and 3.7). Furthermore, by smoothness of the neural mappings, there exists a Euclidean ball $\mathcal{B}((\theta_0,\phi_0),R)$ within which the singular values of the Jacobians remain well-conditioned, i.e., $\sigma_{\mathrm{min}} > B > 0$ . The radius is given by $R = \frac{\mu_{\mathrm{Jac}}}{2\beta}$ , where $\mu_{\mathrm{Jac}}\coloneqq \max \left\{\mu_{\mathrm{Jac}}^{(F)},\mu_{\mathrm{Jac}}^{(G)}\right\}$ and $\beta \coloneqq \min \{\beta_F,\beta_G\}$ . This result parallels Lemma 1 of Song et al. [106], adapted here to the alternating min-max setting. + +However, the optimization trajectory could, in principle, leave this region. To prevent this, we analyze the path length of AltGDA using Yang, Kiyavash, and He [124]'s Lyapunov potential function, rather than directly unrolling the iterates—which would be analytically cumbersome due to alternation: + +Definition 3.1 (Lyapunov Potential [124]). For a min-max objective function, $\mathcal{L}(\theta ,\phi)$ , we define the Lyapunov potential at time $t$ as $P_{t} = (\max_{\phi}\mathcal{L}(\theta_{t},\phi) - \mathcal{L}(\theta^{\star},\phi^{\star})) + \lambda (\max_{\phi}\mathcal{L}(\theta ,\phi_{t}) - \mathcal{L}(\theta_{t},\phi_{t}))$ . (Note that the choice of $\lambda$ will not affect our conclusions about overparameterization in this paper.) + +Lemma 3.2 (Theorem 3.2 in [124]). Suppose the min-max objective function $\mathcal{L}(\theta, \phi)$ is $L_{\nabla \mathcal{L}}$ -smooth and satisfies the two-sided PL-condition with $(\mu_{\theta}, \mu_{\phi})$ . Then if we run AltGDA with $\eta_{\theta} = \frac{\mu_{\phi}^2}{18L_{\nabla \mathcal{L}}^3}$ and $\eta_{\phi} = \frac{1}{L_{\nabla \mathcal{L}}}$ , then $\| \theta_{t+1} - \theta_t\| + \| \phi_{t+1} - \phi_t\| \leq \sqrt{\alpha} c^{t/2} \sqrt{P_0}$ where constants $\alpha$ and $c \in (0,1)$ depend only on $L_{\nabla \mathcal{L}}, \mu_{\theta}, \mu_{\phi}$ and $P_0$ is the Lyapunov potential at time $t = 0$ . (Please refer to Remark E.5 for the exact expressions for $\alpha$ and $c$ .) + +By way of contradiction, let $T$ denote the first iteration such that $(\theta_T, \phi_T) \notin \mathcal{B}((\theta_0, \phi_0), R)$ . We will show that, with high probability, the AltGDA trajectory remains within this ball by proving that its total path length is strictly less than $R$ . + +Indeed, AltGDA path length satisfies: $\ell (T)\triangleq \sum_{t = 0}^{T - 1}\left(\| \theta_{t + 1} - \theta_t\| +\| \phi_{t + 1} - \phi_t\|\right)\leq \frac{\sqrt{2\alpha_1}}{1 - \sqrt{c}}\cdot \sqrt{P_0}$ . Therefore, it suffices to show that $\sqrt{P_0}\le R / 2$ with high probability. The following lemma provides an upper bound on $P_0$ in terms of the gradient norms: + +Lemma 3.3 (Upper Bound on Initial Potential $P_0$ ). Suppose the min-max objective $\mathcal{L}(\theta, \phi)$ is $L_{\mathcal{L}}$ -Lipschitz and satisfies a two-sided PL condition with constants $(\mu_{\theta}, \mu_{\phi})$ . Then the initial Lyapunov potential $P_0 \leq L_{\mathcal{L}}$ ( $C_1 \cdot \| \nabla_\theta \mathcal{L}(\theta_0, \phi_0) \| + C_2 \cdot \| \nabla_\phi \mathcal{L}(\theta_0, \phi_0) \|$ ), where $C_1, C_2 = \Theta(L_{\mathcal{L}} / \mu_\theta^3)$ . + +It is clear that bounding $P_0$ requires controlling the gradient norms at initialization, which—in our neural setting—requires bounding both the output norm and the spectral norm $\sigma_{\mathrm{max}}$ of the Jacobian via the chain rule. Lemmas 3.4, G.2 and 3.7 provide these bounds under standard overparameterization and Lipschitz stability conditions. As a result, we obtain $P_0 \leq \kappa R^2$ for some constant $\kappa < 1$ determined by the network width. Thus, with sufficient overparameterization, the iterates remain confined within the well-conditioned region $\mathcal{B}((\theta_0, \phi_0), R)$ . + +Interestingly, this analysis not only ensures that the iterates stay within a region where the PL-condition holds, but also reveals a beneficial side effect: since the potential function captures a weighted average of Nash gaps and is monotonically decreasing, a small initial value of $P_0$ implies that the initialization is already mildly close to equilibrium. Consequently, both convergence and geometric stability are maintained throughout training. + +# 3.1 Input-Optimization Min-Max Games + +Here, we consider the input-optimization game between two neural networks $F_{\theta}$ and $G_{\phi}$ in hidden bilinear objective with $\ell_2$ -regularization defined as follows for a given payoff matrix, $A$ : + +$$ +\mathcal {L} (\theta , \phi) = F (\theta) ^ {\top} A G (\phi) + \frac {\varepsilon}{2} \| F (\theta) \| ^ {2} - \frac {\varepsilon}{2} \| G (\phi) \| ^ {2} \tag {3} +$$ + +This game has been proposed by [114] and experimentally analyzed by [99]. Here, $F_{\theta}$ and $G_{\phi}$ are defined similar to Definition 2.3 as $F(\theta) = W_2^{(F)}\psi (W_1^{(F)}\theta)$ and $G(\phi) = W_2^{(G)}\psi (W_1^{(G)}\phi)$ but with parameters $\theta ,\phi$ as inputs and randomly initialized $W_{k}^{(F)}\sim \mathcal{N}(0,\sigma_{k,F}^{2}),W_{k}^{(G)}\sim \mathcal{N}(0,(\sigma_{k,G}^{2}),$ $k\in \{1,2\}$ along with differentiable activation function $\psi$ (e.g. GeLU). Therefore, the partial derivatives w.r.t. $\theta$ and $\phi$ will be as follows: + +$$ +\nabla_ {\theta} f (\theta , \phi) = \left(\nabla_ {\theta} F _ {\theta}\right) ^ {\top} A G (\phi) + \varepsilon \left(\nabla_ {\theta} F _ {\theta}\right) ^ {\top} F (\theta) +$$ + +$$ +\nabla_ {\phi} f (\theta , \phi) = \left(\nabla_ {\phi} G _ {\phi}\right) ^ {\top} A ^ {\top} F (\theta) - \varepsilon \left(\nabla_ {\phi} G _ {\phi}\right) ^ {\top} G (\phi) \tag {4} +$$ + +Using these and Lemma 3.3, we can now bound $P_0$ as follows: + +$$ +\begin{array}{l} \Rightarrow P _ {0} \leq \| F (\theta_ {0}) \| \cdot \left(\varepsilon L _ {\mathcal {L}} C _ {1} \sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta_ {0}}\right) + L _ {\mathcal {L}} C _ {2} ^ {\prime} (1 + \lambda) \sigma_ {\max } (A) \sigma_ {\max } \left(\nabla_ {\phi} G _ {\phi_ {0}}\right)\right) \\ + \| G \left(\phi_ {0}\right) \| \cdot \left(L _ {\mathcal {L}} C _ {1} \sigma_ {\max } (A) \sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta_ {0}}\right) + \varepsilon C _ {2} ^ {\prime} (1 + \lambda) \sigma_ {\max } \left(\nabla_ {\phi} G _ {\phi_ {0}}\right)\right) \tag {5} \\ \end{array} +$$ + +Since we want to stay within the ball $\mathcal{B}((\theta_0,\phi_0),R)$ , we can ensure $P_{0} = \kappa R^{2}$ by controlling each term in Equation (5) accordingly that ultimately yields Theorem 3.5. For this, we would additionally need to prove Lemma 3.4 as stated below. (Please see Appendix F for proof). + +Lemma 3.4. Consider a neural network $F_{\theta} = F(\theta) = W_2^{(F)}\psi (W_1^{(F)}\theta)$ as defined above. Say $\psi$ is the GeLU activation function, $d_1^{(F)}\geq 256\max \{d_0^{(F)},d_2^{(F)}\}$ and $\sigma_1^{(F)} = \mathcal{O}\left((d_1^{(F)}\| \theta \| ^2)^{-0.5}\right)$ . Then, $w.p\geq 1 - e^{-\Omega (d_1^{(F)})}$ , + +(i) the singular values of the Jacobian $\nabla_{\theta}F_{\theta}$ are bounded as + +$$ +\sigma_ {\min } \left(\nabla_ {\theta} F _ {\theta}\right) = \Omega \left(\sigma_ {1, F} \cdot \sigma_ {2, F} \cdot d _ {1} ^ {(F)}\right) \quad a n d \quad \sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta}\right) = \mathcal {O} \left(\sigma_ {1, F} \cdot \sigma_ {2, F} \cdot d _ {1} ^ {(F)}\right) \tag {6} +$$ + +(ii) $F(\theta)$ is $\beta_{F}$ -smooth where $\beta_{F} = \Theta \left(\sigma_{1,F}^{2} \cdot \sigma_{2,F} \cdot (d_{1}^{(F)})^{3/2}\right)$ . + +Theorem 3.5. Consider two neural networks $F(\theta), G(\phi)$ as defined in Lemma 3.4 above. For the regularized hidden bilinear min-max objective $\mathcal{L}(\theta, \phi)$ as defined in Equation 3 above, AltGDA reaches $\varepsilon$ -saddle point w.p. $\geq 1 - e^{-\Omega(d_1^{(F)})}$ if $(\theta_0, \phi_0)$ and standard deviations $\sigma_{k,F}$ and $\sigma_{k,G}$ , $k \in \{1,2\}$ are chosen such that $\sigma_{k,F/G} = \Theta\left(\frac{\text{poly}(1/d_1)}{\sigma_{\max}(A)}\right)$ : + +To our knowledge, this is the first fine-grained result for overparametrized networks that establishes an $O(\epsilon)$ -approximate minimax solution for the hidden bilinear setting originally proposed by Vlatakis-Gkaragkounis, Flokas, and Piliouras [114]. + +# 3.2 Neural-Parameters Min-Max Games + +Now we analyse the case of Neural-Parameters Min-Max Games as described in Section 2. In particular, when both players are two-layer neural networks, through Lemma 3.7, with high probability the Jacobians are non-singular for random Gaussian initializations which ensures that the $\prod$ games with such networks will satisfy 2-sided PL-condition with high probability. Consequently, given appropriate initialization conditions for the networks (Assumption 3.6, Equation (9)), just like in Section 3.1, we can show that AltGDA converges to the saddle point by ensuring $P_0 = \kappa R^2$ via requiring both the networks to have at least cubic overparameterization (Theorem 3.8). + +Initialization Scheme 3.6 (Random Initialization). We consider the following initialization scheme for a two-layer neural network, $F$ , as defined in Definition 2.3: + +$$ +\left(W _ {1} ^ {(F)}\right) _ {0} \sim \mathcal {N} \left(0, \sigma_ {1, F} ^ {2} I\right) \quad \left(W _ {2} ^ {(F)}\right) _ {0} \sim \mathcal {N} \left(0, \sigma_ {2, F} ^ {2} I\right) \tag {7} +$$ + +Lemma 3.7 (Lemma 3 & Appendix E.1-E.4 in [106]). Suppose that a two-layer neural network, $F$ , as defined in Definition 2.3, satisfies Assumption 2.4 and $\tau^{r_1}|\psi(a)| \leq |\psi(\tau a)| \leq \tau^{r_2}|\psi(a)|$ respectively for all $a, 0 < \tau < 1$ , and some constants $r_1, r_2$ . Then w.h.p. the neural network + +(i) Jacobian has following bounds on its singular values + +$$ +\sigma_ {\min } \left(\nabla_ {\theta} F _ {\theta}\right) = \tilde {\Omega} \left(\sigma_ {1, F} ^ {r _ {1}} \sqrt {d _ {1} ^ {(F)}}\right) a n d \sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta}\right) = \tilde {\mathcal {O}} \left(\sigma_ {1, F} ^ {r _ {2}} \sqrt {n \cdot d _ {1} ^ {(F)}}\right) \tag {8} +$$ + +(ii) is $\beta_F$ -smooth with $\beta_F = \sqrt{2}\sigma_{\max}(X)(\dot{\psi}_{\max} + \ddot{\psi}_{\max}\chi_{\max})$ where $\chi_{\max} = \sup_V\sigma_{\max}(V)$ . + +Theorem 3.8 (Π Games with AltGDA). Suppose there are two two-layer neural networks, $h_{\theta}, g_{\phi}$ as defined in Definition 2.3 which satisfy Assumption 2.4 and $\tau^{r_1}|\psi(a)| \leq |\psi(\tau a)| \leq \tau^{r_2}|\psi(a)|$ , respectively for all $a, 0 < \tau < 1$ , and some constants $r_1, r_2$ . Suppose the network parameters $\theta_0$ and $\phi_0$ are randomly initialized as in initialization Scheme 3.6 with $(\sigma_{1,F}, \sigma_{2,F})$ and $(\sigma_{1,G}, \sigma_{2,G})$ , respectively, which satisfy + +$$ +\sigma_ {1, F} \cdot \sigma_ {2, F} \lesssim \frac {1}{\sqrt {d _ {0} ^ {(F)} d _ {1} ^ {(F)}}} \quad a n d \quad \sigma_ {1, G} \cdot \sigma_ {2, G} \lesssim \frac {1}{\sqrt {d _ {0} ^ {(G)} d _ {1} ^ {(G)}}} \tag {9} +$$ + +and suppose that the hidden layer widths $d_1^{(F)}$ and $d_1^{(G)}$ for the two networks $F$ and $G$ satisfy + +$$ +d _ {1} ^ {(F)} = \widetilde {\Omega} \left(\mu_ {\theta} ^ {2} \frac {n ^ {3}}{d _ {0} ^ {(F)}}\right) \quad a n d \quad d _ {1} ^ {(G)} = \widetilde {\Omega} \left(\mu_ {\phi} ^ {2} \frac {n ^ {3}}{d _ {0} ^ {(G)}}\right) \tag {10} +$$ + +where the datasets $(\mathcal{D}_F, \mathcal{D}_G)$ for both the players are assumed to be of size $n$ . Then $\prod$ game correspond to an $(\mu_{\theta}, \mu_{\phi})$ -HSCSC min-max objective as defined in Equation 1 satisfying Assumption 2.1 and AltGDA with appropriate fixed step-sizes $\eta_{\theta}$ , $\eta_{\phi}$ (see Lemma E.7) converges to the saddle point $(\theta^{*}, \phi^{*})$ exponentially fast with high probability. + +We refer the reader to Appendix G for the proof of Theorem 3.8, and exact expressions for failure probabilities and various quantities stated in both Lemma 3.7 and Theorem 3.8 + +# 4 Conclusion & Future Directions + +We provide the first convergence guarantees and overparameterization bounds for alternating gradient methods in input games and hidden (strongly) convex-concave neural games. Our analysis tightly links optimization trajectory control with spectral stability, ensuring convergence to near-equilibrium. If the reader would like to look beyond the technicalities around the non-asymptotic bounds, our proof techniques offer several insights for practitioners: + +- Interpretation of $\sigma_{\mathrm{min}}$ and Exploration: The smallest singular value of the network Jacobian, $\sigma_{\mathrm{min}}$ , controls how well the model explores the strategy space. When $\sigma_{\mathrm{min}} \approx 0$ , certain strategies remain unexplored, indicating convergence to spurious subspaces. Our analysis ties this directly to the degree of overparameterization. +- Data Geometry and Regions of Attraction: Our results show that overparameterized networks initialized with sufficiently diverse data are more likely to fall into regions where $\sigma_{\mathrm{min}} > 0$ , ensuring stable convergence under AltGDA. While computing $\sigma_{\mathrm{min}}$ per iteration is impractical, the connection offers design insights for data and architecture. + +Table 1: Comparison between our paper and common practice + +
Our paperIn practice
Type of Neural Network1-hidden-layer, fully-connectedTypically deep networks, not necessarily fully-connected (e.g., residual or convolutional layers)
Training AlgorithmAltGDANot necessarily AltGDA / mainly double-loop
Network InitializationGaussian +(with variance constraints)Similar (e.g., He, Xavier, or LeCun initializations)
+ +Going beyond the neural networks and training regimes considered in this paper (see Table 1 for a summary) is an important future direction. Among these, the assumption on AltGDA is arguably the most benign. In non-convex/non-concave min-max optimization, stabilization is essential. In practice, double-loop methods (e.g., approximate best-response oracles) are often used for safety, while AltGDA serves as a more parallelizable and simpler single-loop alternative. Similarly, the Gaussian initialization is closely aligned with popular schemes like He or Xavier. The main gap lies in the architecture: practical models are often very deep with fixed-width layers. While recent work has begun to explore overparameterization in deep networks for minimization tasks, our paper focuses on a more analytically tractable setting – explicitly avoiding the NTK regime to provide a non-asymptotic analysis for 1-hidden-layer networks in a game-theoretic context. We view relaxing and extending these assumptions as a promising direction for future work. + +Another natural next step is to understand how these techniques extend to non-differentiable activation functions (such as ReLU) or scale to multi-player and non-zero-sum settings - especially in structured environments like polyhedral games, which share connections with extensive-form games. For instance, exploring the analogy between two-sided PL-conditions (for two-player games) and hypomonotonicity in multi-agent operator theory may allow us to transfer and generalize some of the intuition and techniques from our current setting. We hope our work and these possible future directions open up rich and technically deep avenues for developing gradient-based methods tailored for structured, non-monotone multiplayer games. (See also Appendix I.) + +# References + +[1] Jacob Abernethy, Kevin A Lai, and Andre Wibisono. "Last-iterate convergence rates for min-max optimization". 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Broader Impact. + +This work advances the theoretical foundations for training large-scale multi-agent systems via gradient-based methods in hidden non-convex environments. Providing reliable convergence guarantees in such settings is a critical step toward the safe and stable deployment of multi-agent technologies, with applications ranging from distributed control and autonomous systems to robust economic mechanisms, cybersecurity infrastructures, and collaborative AI development. Beyond immediate engineering benefits, our results contribute to the fundamental understanding of strategic learning in non-convex environments, offering the potential for more predictable, transparent, and verifiable AI behavior. + +From a broader societal perspective, these theoretical advances can support the design of AI systems that align more closely with human values, improve multi-agent coordination, and enhance resilience against adversarial attacks. Applications may include improving the robustness of distributed decision-making, facilitating fairer negotiation frameworks, and enabling safer autonomous cooperation among heterogeneous agents. In doing so, this work contributes to the vision of AI systems that are not only powerful but also trustworthy and beneficial (cf. [4, 96]). + +However, we also recognize that the same techniques enabling reliable convergence could be leveraged in ways that carry risks. In particular, stronger convergence in competitive environments may be exploited to construct highly optimized agents for adversarial purposes, including strategic market manipulation, automated disinformation campaigns, or autonomous decision-making systems deployed without adequate oversight or alignment with human norms. Further, the scalability of multi-agent learning could amplify systemic biases or create emergent behaviors that are difficult to predict or control (cf. [28]). + +Accordingly, while this work advances foundational goals in strategic machine learning, it also underscores the need for careful evaluation of downstream applications. Future research should emphasize robustness checks, fairness assessments, and human-centered design principles to ensure that strategic learning systems contribute positively to societal welfare. In particular, collaborations across technical, ethical, and policy disciplines will be crucial to anticipate and mitigate potential negative consequences as these technologies scale and proliferate. + +# Appendix Contents + +# A Further Discussion in Prior Work 26 + +A.1 Optimization with Hidden Structures 26 +A.2 Simultaneous, Alternating, and Extrapolated Dynamics in Convex-Concave Min-Max Optimization 26 +A.3 Overparameterization in Learning Dynamics: From Minimization to Games 27 +A.4 Some Empirical Studies Demonstrating Need of Larger Neural Networks for Zero-Sum Games 28 + +# B Examples/Applications based on hidden min-max games 29 + +B.1 Neural Min-Max Games 29 +B.2 Distributionally Robust Optimization 30 +B.3 Input-Optimization Min-Max Games 30 + +# C Demonstrating Empirical Validity of Main Result for Input Games 32 + +# D Controlling the Gradient Growth: Bounds and Examples for Different Loss Functions 33 + +# E Proofs about two-sided PL & AltGDA 34 + +# F Proofs for Input-Optimization Min-Max Games 40 + +# G Proofs for Neural-Parameters Min-Max Games 46 + +# H Clarifications 51 + +H.1 Smoothness on variables or Map 51 +H.2Refined Upper Bound Expression for Lyapunov Potential $P_0$ 51 +H.3 Random Initialization: How restrictive are our results? 52 + +# I Discussion & Future Work 53 + +I.1 Deep Learning Perspective: 53 +I.2 Game Theory Perspective: 54 + +# A Further Discussion in Prior Work + +# A.1 Optimization with Hidden Structures + +Hidden convexity has offered a pathway to global convergence even in otherwise nonconvex problems. Early developments include the use of hidden convexity in the analysis of cubic regularization for Newton's method [81], which achieved global rates under suitable curvature conditions. Extending such frameworks to non-smooth or stochastic optimization, however, remains an open challenge. More recently, hidden convexity has been exploited across a wide range of applications, including reinforcement learning [54, 129, 126], generative modeling [63], supply chain management [42, 24], and neural network training [119]. Further, [11] studied optimization via biased stochastic oracles, developing algorithms with $\tilde{O} (\epsilon^{-2})$ or $\tilde{O} (\epsilon^{-3})$ iteration complexity, depending on the oracle's stability, and recovering new results for hidden convex problems as an application. Prior to its adoption in game theory, policy gradient methods in reinforcement learning have leveraged hidden convexity to establish global convergence guarantees [129, 8], often relying on variance-reduced estimators or large-batch assumptions. Relatedly, hidden convexity also played a crucial role in quadratic optimization problems [9, 107], revealing convex-like structures within certain nonconvex programs. + +Turning to min-max optimization, hidden convexity has proven essential for addressing the limitations of classical algorithms. Vlatakis-Gkaragkounis, Flokas, and Piliouras [114] demonstrated that even benign-looking hidden zero-sum games can exhibit cyclic behaviors or divergence when trained via vanilla gradient descent-ascent (GDA). Motivated by such failures, Vlatakis-Gkaragkounis, Flokas, and Piliouras [115] introduced the formal concept of hidden-convex hidden-concave games, providing the first convergence guarantees for GDA in nonconvex min-max settings under suitable structural conditions such as strict or strong hidden convexity. Parallel research on non-monotone variational inequalities has developed continuous-time flows that exploit hidden convexity-like structures to ensure global convergence [79]. Extending these ideas, Sakos et al. [99] provided a fully discrete-time algorithm along with provable guarantees for stability and convergence in hidden convex games—filling an important gap compared to previous continuous-time results. Moreover, recent work has investigated preconditioned algorithms for hidden monotone variational inequalities, including Newton-type approaches [29]. Further discussions [29] have clarified under which conditions both the latent variable space and the control space can be bounded, while maintaining uniform lower bounds on the singular values of the players' Jacobians, a critical requirement for robust convergence guarantees. + +# A.2 Simultaneous, Alternating, and Extrapolated Dynamics in Convex-Concave Min-Max Optimization + +Min-max optimization algorithms have a long-standing history, dating back to the original proximal point methods for variational inequality problems. Martinet [74] and Rockafellar [93]. Below we present some + +# A.2.1 The classic regime + +To set the stage, consider the classical min-max problem of the form $\min_x\max_yf(x,y)$ , a fundamental template across optimization and game theory. The most natural extension of gradient descent to such settings is the gradient descent-ascent (GDA) method [36], which iteratively updates $x$ to decrease $f$ and $y$ to increase $f$ . GDA comes in two flavors: simultaneous (Sim-GDA), where $x$ and $y$ are updated in parallel, and alternating (Alt-GDA), where updates occur sequentially. However, despite its simplicity, even in convex-concave settings, vanilla GDA fails to guarantee convergence. In basic examples such as the bilinear game $\min_x\max_yxy$ , Sim-GDA exhibits unbounded divergence, while Alt-GDA produces bounded but non-convergent iterates that circulate indefinitely [7, 49, 50, 128]. + +These deficiencies have led to the development of refined algorithms designed to stabilize min-max dynamics, including Extra-Gradient methods [64], Optimistic Gradient Descent [91], and negative momentum variants [50]. Many of these methods achieve accelerated rates compared to plain GDA, particularly under smoothness and convexity assumptions. Nevertheless, the bulk of this literature focuses predominantly on Sim-GDA-type algorithms, largely due to their analytical tractability. + +Yet, in many real-world machine learning applications, particularly adversarial training scenarios like GANs, alternating updates more naturally model the dynamics: the generator and discriminator adjust sequentially based on each other's output. Empirical studies [51, 76] suggest that Alt-GDA often converges faster than Sim-GDA. Despite these observations, the theoretical foundations explaining the advantages of alternation remain relatively underdeveloped. A notable advancement toward understanding this phenomenon was provided by Zhang et al. [128], who analyzed local convergence rates under strong convexity-concavity (SCSC) and smoothness assumptions. Their results show that, locally, Alt-GDA achieves iteration complexity $\tilde{O}(\kappa)$ compared to the slower $\tilde{O}(\kappa^2)$ of Sim-GDA, where $\kappa = L / \mu$ denotes the condition number. Nonetheless, these results are limited to local convergence—i.e., after the iterates are already sufficiently close to a saddle point—and do not capture the full global behavior. More recently, further refinements have been developed combining extrapolation techniques with alternation to achieve nearly optimal condition number dependence [66]. + +# A.2.2 The PL-Regime and Nonconvex Min-Max Problems + +Research has also intensified on extending convergence guarantees beyond convex-concave settings to nonconvex-nonconcave games [105, 21, 92, 109, 68, 1]. The majority of these early works often focused on settings where the objective is nonconvex in $x$ but concave in $y$ , and proposed algorithms that either solve inner maximizations exactly or rely on strong assumptions (e.g., Minty Variational Inequalities [68]). Others, such as Abernethy, Lai, and Wibisono [1], introduced Hamiltonian-type methods for nearly bilinear problems but relied on second-order information. Recently, Nouiehed et al. [84] studied a class of minimax problems where the objective only satisfies a one-sided PL-condition and introduced the GDmax algorithm, which takes multiple ascent steps at every iteration. However in our case (i) we consider the two-sided PL-condition which guarantees global convergence; (ii) we consider AltGDA which takes one ascent step at every iteration. Another closely related work is [18] The authors considered a specific application in generative adversarial imitation learning with linear quadratic regulator dynamics. This is a special example that falls under the two-sided PL-condition. + +In contrast, a major breakthrough in capturing global convergence via first-order methods was provided by Yang, Kiyavash, and He [124]. They introduced the concept of two-sided Polyak-Lojasiewicz (PL) inequalities for nonconvex-nonconcave min-max games, showing that alternating GDA (AltGDA) converges globally at a linear rate under this structure. Furthermore, they designed a variance-reduced stochastic AltGDA method for finite-sum objectives, achieving faster convergence. Building on this, subsequent works have refined the framework: Xu et al. [122] proposed a zeroth-order variant, Liu et al. [71] analyzed randomized stochastic accelerations, and Chen, Yao, and Luo [20] integrated Spider techniques to improve stochastic convergence rates under PL-conditions. Extensions to more adaptive and multi-step alternating schemes were explored by Kuruzov et al. [65], aiming to further optimize the dynamics in hidden convex-concave games. At the same time [124] presented a case where simultaneous GDA will diverge from the equilibrium while AltGDA converges to the equilibrium. + +# A.3 Overparameterization in Learning Dynamics: From Minimization to Games + +# A.3.1 Minimizing training loss + +The phenomenon of overparameterization—where neural networks possess far more parameters than the apparent complexity of the target function—has profoundly influenced the theoretical understanding of modern machine learning. Early works rigorously explored how, despite the non-convexity of the training landscape, overparameterization enables gradient-based methods to converge to global optima [98, 38, 3]. Notably, these studies typically required substantial overparameterization, often polynomial in the size of the training data or model complexity. However, empirical observations [72, 97] suggest that even modest increases in network width—sometimes adding just a few neurons—can suffice for successful training, motivating a closer study of mild overparameterization. + +A pivotal theoretical lens for understanding this success is the Neural Tangent Kernel (NTK) framework [57]. In the infinite-width limit, training dynamics linearize, and gradient descent effectively follows a kernelized gradient flow determined by the NTK, which remains nearly constant throughout training. Thus, convergence can be ensured if the NTK is well-conditioned—specifically, if its + +minimum eigenvalue remains bounded away from zero. At finite but large widths, convergence analyses typically require proving two properties: (i) the NTK is well-conditioned at initialization, and (ii) it remains stable during training [85, 27, 13]. + +Recent works have sharpened the quantitative understanding of this regime. For instance, Song et al. [106] demonstrated that a network width of approximately $\tilde{O}(n^{3/2})$ suffices for global convergence at linear rates, improving previous state-of-the-art requirements. In parallel, studies such as [13] established NTK lower bounds that allow optimization with as few as $\Omega(\sqrt{n})$ neurons, bridging optimization and memorization capabilities. Yet, most of these developments operate within minimization frameworks—either fitting labels or adversarially robust losses [17]. The transition from minimization to games (e.g., adversarial training, multi-agent learning) brings new challenges. + +# A.3.2 Adversarial Losses and MARL + +In adversarial learning, recent works have analyzed overparameterized adversarial training primarily from a minimization perspective, focusing on robustness against worst-case perturbations rather than strategic interactions between agents [132, 45, 25]. While robust losses (and closer to our setting) induce non-convexity, the optimization target remains a minimizer rather than a saddle point. + +In multi-agent and game-theoretic settings, the literature is comparatively sparser. Policy approximation in multi-agent reinforcement learning (MARL) typically relies on either tabular or linear architectures [118, 108], and extending sample-efficient learning to rich function classes like neural networks remains a frontier. A notable contribution in this direction is the work of Jin, Liu, and Yu [58], which introduced the Multi-Agent Bellman Eluder (BE) dimension as a complexity measure for MARL, enabling sample-efficient learning of Nash equilibria in high-dimensional spaces. In the context of Markov Games, Li et al. [67] studied Nash equilibria computation using kernel-based function approximation, highlighting the difficulties of exploration and generalization in high-dimensional, non-convex settings. + +Despite these advances, a comprehensive theory connecting overparameterization, NTK stability, and global convergence in multi-agent games remains largely undeveloped. Key questions include: + +- How does overparameterization affects the conditioning of multi-agent dynamics? +- whether can alternating optimization (as opposed to simultaneous updates) exploit NTK-like stability, and how regularization or architectural choices influence convergence in strategic environments? + +Our work contributes to this growing effort by combining insights from the large-scale network mappings perspective with recent advances in hidden convexity and PL conditions for nonconvex-nonconcave optimization. In particular, we highlight that under mild overparameterization, even strategic interactions—modeled via hidden convex-concave games—admit global convergence guarantees with simple gradient-based methods, provided the trajectory stays within a controlled neighborhood of initialization where the Jacobians remain well-conditioned. + +# A.4 Some Empirical Studies Demonstrating Need of Larger Neural Networks for Zero-Sum Games + +Since our work pertains to estimating the amount of overparameterization needed in neural networks for solving various zero-sum games, we highlight some of the works in various applied min-max contexts - adversarial training, GANs, DRO, and neural agents - which show that larger, overparameterized neural networks lead to improved convergence and performance. For instance, in case of adversarial training, Addepalli et al. [2] show improved robustness and performance in adversarial training with larger models. A few other works [61, 14, 100] empirically demonstrate how using larger architectures in GANs improve the training stability, and the quality and variation of generated images. In the realm of LLM Language agents, Karten, Nguyen, and Jin [62] show improved performance using GPT-4.0 as opposed to smaller LLMs. In the case of DRO, Pham et al. [88] show that bigger neural networks may yield better worst-group generalization. + +While our main results (Theorem 3.8) concerning neural games may largely be seen to be of theoretical interest without an empirical component attached to it, we refer interested readers to Appendix C for experimental validity of our main results concerning the input games (Theorem 3.5). + +# B Examples/Applications based on hidden min-max games + +In the following, we present a series of examples illustrating instances of hidden convex-concave min-max optimization in neural network-based settings. We distinguish between two main types of problems: + +- Problems where the optimization is performed over the parameters of the neural networks, given a fixed dataset (training over network weights). +- Problems where the network parameters are fixed (randomly initialized), and the optimization is performed over the input space (input-optimization games). + +# B.1 Neural Min-Max Games + +We begin by illustrating two canonical examples of input-optimization games that naturally fall within the hidden convex-concave min-max framework. Both examples demonstrate settings where neural network mappings induce structured, but hidden, convexity and concavity, which can be exploited for provable convergence. + +Example B.1 (Generative Adversarial Networks (GANs)). A Generative Adversarial Network (GAN) formulates a two-player minimax game where the generator $G_{\theta}$ seeks to produce samples that resemble a reference distribution $p_{\mathrm{data}}$ , while the discriminator $D_{\phi}$ attempts to distinguish generated samples from real data. The corresponding min-max problem reads: + +$$ +\min _ {\theta} \max _ {\phi} \Psi (\theta , \phi) := \mathbb {E} _ {x \sim p _ {\mathrm {d a t a}}} \left[ \log D _ {\phi} (x) \right] + \mathbb {E} _ {x \sim p _ {\theta}} \left[ \log (1 - D _ {\phi} (x)) \right]. +$$ + +Assuming that both $p_{\mathrm{data}}$ and $p_{\theta}$ admit densities and that the support of $p_{\theta}$ lies within the support of $p_{\mathrm{data}}$ , one can reformulate the problem via a latent convex-concave structure. The min-max formulation arises naturally as the generator seeks to minimize the ability of the discriminator to distinguish real from fake samples, while the discriminator simultaneously maximizes its classification performance, thus modeling an adversarial dynamic between two competing objectives. Specifically, considering a distribution $p(x,x^{\prime})$ that samples either a real or generated point, the loss can be decomposed as: + +$$ +L _ {x, x ^ {\prime}} \left(p ^ {\prime}, D\right) := \log D (x) + \frac {p ^ {\prime} \left(x ^ {\prime}\right)}{p _ {\mathrm {d a t a}} \left(x ^ {\prime}\right)} \log \left(1 - D \left(x ^ {\prime}\right)\right), +$$ + +which is jointly convex in $p'$ and concave in $D$ . Consequently, + +$$ +\Psi (\theta , \phi) = \mathbb {E} _ {(x, x ^ {\prime}) \sim p} \left[ L _ {x, x ^ {\prime}} \left(p _ {\theta} (x), D _ {\phi} \left(x ^ {\prime}\right)\right) \right], +$$ + +exhibiting the GAN training objective as a hidden convex-concave game. + +Example B.2 (Domain-Invariant Representation Learning (DIRL)). Domain adaptation aims to train models that generalize across different domains, despite distribution shifts between training (source) and deployment (target) environments. A popular approach [44] involves learning representations that are: (i) predictive of labels in the source domain, and (ii) invariant to the domain classifier distinguishing source versus target samples. This leads to the following min-max problem: + +$$ +\min _ {\theta_ {f}, \theta_ {g}} \max _ {\theta_ {f ^ {\prime}}} \mathbb {E} _ {(x, y) \sim P _ {\mathrm {s o u r c e}}} \left[ \ell (f _ {\theta_ {f}} (g _ {\theta_ {g}} (x)), y) \right] - \lambda \mathbb {E} _ {(x, y ^ {\prime}) \sim P _ {\mathrm {m i x}}} \left[ \ell (f _ {\theta_ {f ^ {\prime}}} ^ {\prime} (g _ {\theta_ {g}} (x)), y ^ {\prime}) \right], +$$ + +where: (i) $g_{\theta_g}$ is the feature extractor, (ii) $f_{\theta_f}$ is the label predictor, (iii) $f_{\theta_{f'}}'$ is the domain classifier, (iv) $\ell$ denotes the classification loss, and (v) $P_{\mathrm{source}}$ , $P_{\mathrm{mix}}$ are the source and mixed domain distributions, respectively. When the loss function is convex with respect to the neural mappings, the hidden convex-concave structure becomes apparent, fitting naturally into our theoretical framework. + +Example B.3 (Robust Adversarial Reinforcement Learning (RARL)). One of the major challenges in reinforcement learning (RL) is the difficulty of training agents under realistic conditions, often due to costly data collection or limited availability of real-world environments. To address these challenges, Pinto et al. [89] proposed an adversarial training framework wherein a learner agent and an adversary play against each other by solving the following min-max optimization problem: + +$$ +\min _ {\theta_ {1}} \max _ {\theta_ {2}} \mathbb {E} _ {s _ {0} \sim \rho , a ^ {1} \sim \mu_ {\theta_ {1}} (s), a ^ {2} \sim \nu_ {\theta_ {2}} (s)} \left[ \sum_ {t = 0} ^ {T - 1} r ^ {1} (s _ {t}, a _ {t} ^ {1}, a _ {t} ^ {2}) \right], +$$ + +where: (i) $\mu_{\theta_1}$ is the learner's policy network, (ii) $\nu_{\theta_2}$ is the adversary's policy network, (iii) $r^1$ denotes the reward function, and (iv) $\rho$ is the initial state distribution. + +In this setup, the learner seeks to maximize its expected reward, while the adversary perturbs the environment or dynamics to minimize the learner's performance. Such adversarial modeling captures different sources of uncertainty: either unknown variations in the underlying Markov Decision Process (MDP), or deliberate adversarial attacks aiming to degrade the policy (e.g., disturbances in robotic control tasks or adversarial inputs in sensor-based systems). Hidden convex-concave structures can naturally arise when suitable regularizations or smooth policy parameterizations are enforced. + +Example B.4 (Adversarial Example Generation (AEG)). In an Adversarial Example Generation (AEG) setting, the goal is to generate adversarial perturbations that cause misclassification by a fixed classifier $f_{\phi}$ . Formally, given clean samples $(x,y)\sim p_{\mathrm{data}}$ , a perturbation generator $G_{\theta}$ seeks to find an adversarial input $x^{\prime}$ satisfying a distortion constraint (e.g., $\| x - x^{\prime}\|_{\infty}\leq \epsilon$ ) that maximizes the classification loss. The underlying min-max optimization is: + +$$ +\min _ {\phi} \max _ {\theta} \Psi (\theta , \phi) := - \mathbb {E} _ {(x ^ {\prime}, y) \sim p _ {\theta}} \left[ \ell (f _ {\phi} (x ^ {\prime}), y) \right], +$$ + +where $\ell$ denotes the cross-entropy loss. Here, the generator (adversary) maximizes the classification loss while the classifier seeks to minimize it. Under appropriate conditions on the neural network mappings and smoothness of the loss, the adversarial game admits a hidden convex-concave structure that can be exploited for convergence analysis. + +# B.2 Distributionally Robust Optimization + +In many machine learning applications, models are trained under the assumption that data is drawn from a fixed but unknown distribution. However, real-world deployment often leads to distribution shifts (e.g., label noise, adversarial perturbations, or changing environments), which can severely degrade performance. A principled way to address this issue is through Distributionally Robust Optimization (DRO) [77, 123], where the model is trained to perform well against the worst-case distribution within a prescribed uncertainty set. The corresponding min-max optimization problem reads: + +$$ +\theta_ {\mathrm {D R O}} ^ {*} \in \arg \min _ {\theta} \max _ {P \in \mathcal {P}} \mathbb {E} _ {(x, y) \sim P} \left[ \ell (h _ {\theta} (x), y) \right], +$$ + +where: (i) $h_{\theta}$ is the predictive model parameterized by $\theta$ , (ii) $\ell$ is a loss function (e.g., cross-entropy), (iii) $\mathcal{P}$ is an uncertainty set containing distributions representing expected perturbations. Hidden convexity can emerge when $\ell$ is convex in the model outputs and the uncertainty set $\mathcal{P}$ is suitably structured. + +Example B.5 (Parametric Distributionally Robust Optimization (Parametric-DRO)). While classical DRO assumes that the uncertainty set $\mathcal{P}$ is specified manually, identifying an appropriate $\mathcal{P}$ is often challenging, especially at large deployment scales. To overcome this, Michel, Hashimoto, and Neubig [77] proposed modeling the worst-case distribution using a parameterized generative model $q_{\psi}$ . + +The resulting parametric DRO problem reads: + +$$ +\min _ {\theta} \max _ {\psi : K L (q _ {\psi} \| q _ {\psi_ {0}}) \leq \kappa} \mathbb {E} _ {(x, y) \sim p} \left[ \frac {q _ {\psi} (x , y)}{q _ {\psi_ {0}} (x , y)} \ell (h _ {\theta} (x), y) \right], +$$ + +where: (i) $h_{\theta}$ is the predictor network, (ii) $q_{\psi}$ models the perturbed distribution, (iii) $q_{\psi_0}$ approximates the empirical distribution via maximum likelihood, (iv) $\kappa$ controls the size of the KL-divergence ball around $q_{\psi_0}$ . This formulation transforms the DRO problem into a min-max optimization between the model parameters $\theta$ and the perturbation parameters $\psi$ , both parameterized via neural networks, fitting naturally into the hidden convex-concave framework under appropriate regularization. Observe that by Pinsker's inequality we get that KL-divergence over $\ell_1$ -norm. + +# B.3 Input-Optimization Min-Max Games + +A recurring structure in adversarial and robust optimization tasks involves optimizing over input spaces rather than model parameters. Based on Wang et al. [117] we describe three prominent examples of such input-optimization games below: + +Example B.6 (Ensemble Attack over Multiple Models). Given $K$ machine learning models $\{\mathcal{M}_i\}_{i=1}^K$ , the goal is to find a universal perturbation $\delta$ that simultaneously fools all models. The corresponding input-optimization game reads: + +$$ +\min _ {\delta \in \mathcal {X}} \max _ {w \in \mathcal {P}} \sum_ {i = 1} ^ {K} w _ {i} f (\delta ; x _ {0}, y _ {0}, \mathcal {M} _ {i}) - \frac {\gamma}{2} \| w - 1 / K \| _ {2} ^ {2}, +$$ + +where $w$ encodes the relative difficulty of attacking each model, and $\gamma$ is a regularization parameter. + +Example B.7 (Universal Perturbation over Multiple Examples). Here, given a set of examples $\{(x_i,y_i)\}_{i = 1}^K$ , the objective is to find a perturbation $\delta$ that simultaneously fools all of them. The optimization problem becomes: + +$$ +\min _ {\delta \in \mathcal {X}} \max _ {w \in \mathcal {P}} \sum_ {i = 1} ^ {K} w _ {i} f (\delta ; x _ {i}, y _ {i}, \mathcal {M}) - \frac {\gamma}{2} \| w - 1 / K \| _ {2} ^ {2}. +$$ + +This mirrors the ensemble attack setup, but focuses on perturbing multiple inputs under a fixed model $\mathcal{M}$ . + +Example B.8 (Adversarial Attack over Data Transformations). Consider robustness against transformations (e.g., rotations, translations) applied to the inputs. Given categories of transformations $\{p_i\}$ , the optimization reads: + +$$ +\min _ {\delta \in \mathcal {X}} \max _ {w \in \mathcal {P}} \sum_ {i = 1} ^ {K} w _ {i} \mathbb {E} _ {t \sim p _ {i}} \left[ f (t (x _ {0} + \delta); y _ {0}, \mathcal {M}) \right] - \frac {\gamma}{2} \| w - 1 / K \| _ {2} ^ {2}, +$$ + +where $t$ denotes a random transformation sampled from $p_i$ . When $w = 1 / K$ , this recovers the expectation-over-transformation (EOT) setup. + +Observe that under a convex loss function $f$ with respect to the neural mapping $\mathcal{M}(x_0 + \delta)$ (i.e., hidden convexity in $\delta$ ), and given that $w$ appears linearly in both the bilinear coupling and the individual regularization term of separable framework of Section 1.1, the structure fits naturally within our hidden convex-concave framework. Albeit a careful reader might observe that our main results are stated for unconstrained min-max optimization, there are two standard ways to extend our analysis to constrained settings: + +- First, by employing the two-proximal-PL framework developed in Kalogiannis et al. [60]—an improvement over the earlier formulation of Yang, Kiyavash, and He [124]—our convergence guarantees naturally generalize to simple constraint sets, such as $\ell_2$ -balls for perturbations $\delta$ and simplices for the mixture weights $w$ . +- Alternatively, the constraints can be incorporated directly into the objective through suitable Lagrangian penalty terms, thereby reducing the constrained min-max problem to an unconstrained form amenable to our techniques. + +Unified Perspective. Across these examples, input-optimization problems naturally exhibit a saddle-point structure, blending adversarial robustness objectives with min-max optimization techniques. They provide concrete and practically motivated instances where hidden convexity can be exploited to ensure convergence guarantees for training and robustness analysis. + +# C Demonstrating Empirical Validity of Main Result for Input Games + +We consider a hidden game of Rock-Paper-Scissors where two 1-hidden layer neural networks (GeLU activations) are playing the game of the Rock-Paper-Scissors. We will see here (empirically) that, if we use random Gaussian initializations for both the neural network players as described in Theorem 3.5 to define our input game of Rock-Paper-Scissors, and if we use AltGDA for finding min-max optimal strategies for both these neural network players, then the players indeed reach the $\varepsilon$ -Nash equilibrium. In particular, both the neural network players control 5-dimensional vectors $\theta, \phi \in \mathbb{R}^5$ and output a latent strategy that lies in the 2-dimensional simplex, $\Delta_2$ . Both the players are optimizing the regularized bilinear bilinear objective: + +$$ +\mathcal {L} (\theta , \phi) = F (\theta) ^ {\top} A G (\phi) + \varepsilon / 2 \| F (\theta) - 1 / 3 (1, 1, 1) ^ {\top} \| _ {2} ^ {2} - \varepsilon / 2 \| G (\phi) - 1 / 3 (1, 1, 1) ^ {\top} \| _ {2} ^ {2} +$$ + +where $(1/3, 1/3, 1/3)^{\top}$ indicates the (unique) $\varepsilon$ -mixed strategy Nash equilibrium for both the players and $A = 10 \cdot \begin{pmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{pmatrix}$ is the payoff matrix8 for the Rock-Paper-Scissor game. + +Since the actual strategies of the two neural network players $(\theta, \phi)$ lie in $\mathbb{R}^5$ , we can't visualize those. However, we can visualize their strategies in the latent space, i.e., in the 2-dimensional simplex instead. Figure 2 below illustrates the AltGDA trajectories for both the players (step-size $= 0.01$ , $\varepsilon = 1$ , maximum number of steps $= 100,000$ ). As we can see, both the trajectories converge to the $\varepsilon$ -Nash equilibrium of the hidden game. + +Figure 2: A trajectory of AltGDA in an $\ell_2$ -regularized hidden game of Rock-Paper-Scissors. These trajectories correspond to each player's strategies in the latent space (2-dimensional simplex). +![](images/af168ba7e0ccb43e908b9e7a1ef39bf99d7e833f67f925e8350d46e5f0614bdd.jpg) +Initial Final Equilibrium + +![](images/6ecf5246677ad33621f14122ca7cd895ef4cbab210d301a5b760f9b41464c56c.jpg) + +# D Controlling the Gradient Growth: Bounds and Examples for Different Loss Functions + +Controlling the gradient growth is critical for non-asymptotic overparameterization bounds, as it ensures that iterates remain within regions where hidden convexity persists. Below, we provide a detailed discussion about this bound for some commonly-used loss functions. + +- Mean-Squared Error (MSE) Loss: One of the most commonly used losses and is suitable for regression tasks. For a given labelled data point $(x, y)$ where $x \in \mathcal{X}$ and $y \in \mathcal{Y}$ , if the predictor function parameterized by parameters $\theta$ is defined as $h := h(x; \theta)$ , we see that the MSE loss can be defined as: + +$$ +\ell (y, h) = \frac {1}{2} \| y - h \| _ {2} ^ {2} \tag {11} +$$ + +$$ +\Longrightarrow \nabla_ {h} \ell (y, h) = h - y \tag {12} +$$ + +$$ +\Longrightarrow \| \nabla_ {h} \ell (y, h) \| \leq \| h \| + \| y \| \tag {13} +$$ + +$$ +\leq \| h \| + \operatorname {d i a m} (\mathcal {Y}) \tag {14} +$$ + +$$ +\Rightarrow \| \nabla_ {h} \ell (y, h) \| \leq A _ {1} \| h \| + A _ {2} \operatorname {d i a m} (\mathcal {Y}) + A _ {3} \tag {15} +$$ + +where $A_{1} = A_{2} = 1$ and $A_{3} = 0$ . More generally, if we had $\ell(y,h) = \frac{\mu}{2}\|y - h\|_2^2$ , we would get $A_{3} = 0$ and $A_{1} = A_{2} = \mu$ where $\mu$ is also the hidden-strong convexity modulus. + +- Logistic Loss: Commonly used in binary classification, where $y \in \{0,1\}$ , and the prediction $h = h(x;\theta) \in \mathbb{R}$ is passed through the sigmoid function. The logistic loss is defined as: + +$$ +\ell (y, h) = - y \log \sigma (h) - (1 - y) \log (1 - \sigma (h)), \quad \text {w h e r e} \sigma (h) = \frac {1}{1 + e ^ {- h}} \tag {16} +$$ + +$$ +\nabla_ {h} \ell (y, h) = \sigma (h) - y \tag {17} +$$ + +$$ +\Rightarrow \| \nabla_ {h} \ell (y, h) \| \leq 1 \tag {18} +$$ + +Since the gradient is always in the interval $[-1, 1]$ , we can take $A_1 = 0$ , $A_2 = 0$ , $A_3 = 1$ . The logistic loss is strongly convex over compact domains or when regularized. + +- Squared Hinge Loss: Used in support vector machines (SVMs) with a margin-based formulation. For $y \in \{-1, 1\}$ and prediction $h = h(x; \theta) \in \mathbb{R}$ : + +$$ +\ell (y, h) = \frac {1}{2} \max (0, 1 - y h) ^ {2} \tag {19} +$$ + +$$ +\nabla_ {h} \ell (y, h) = \left\{ \begin{array}{l l} - y (1 - y h), & \text {i f} y h < 1 \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {20} +$$ + +$$ +\Rightarrow \| \nabla_ {h} \ell (y, h) \| \leq 2 | y | | h | + 1 \leq 2 \| h \| + \operatorname {d i a m} (\mathcal {Y}) \tag {21} +$$ + +Thus, we can choose constants $A_{1} = 2$ , $A_{2} = 1$ , $A_{3} = 0$ . Note: the hidden strong convexity applies within the active region $yh < 1$ , and a regularization term often ensures global strong convexity. + +- Cross-Entropy with $\ell_2$ -regularization: For multi-class classification with softmax outputs $h \in \mathbb{R}^K$ , target $y \in \Delta^{K-1}$ (probability simplex): + +$$ +\ell (y, h) = - \sum_ {k = 1} ^ {K} y _ {k} \log \left(\frac {e ^ {h _ {k}}}{\sum_ {j = 1} ^ {K} e ^ {h _ {j}}}\right) + \frac {\lambda}{2} \| h \| ^ {2} \tag {22} +$$ + +$$ +\nabla_ {h} \ell (y, h) = \operatorname {s o f t m a x} (h) - y + \lambda h \tag {23} +$$ + +$$ +\Rightarrow \| \nabla_ {h} \ell (y, h) \| \leq \| \operatorname {s o f t m a x} (h) - y \| + \lambda \| h \| \leq 2 + \lambda \| h \| \tag {24} +$$ + +So the gradient norm is bounded by $A_1 = \lambda, A_2 = 0, A_3 = 2$ . The $\ell_2$ term ensures strong convexity with modulus $\lambda$ . + +# E Proofs about two-sided PL & AltGDA + +We begin by recalling the definitions of unconstrained continuous min-max optimality conditions of problem: + +$$ +\min _ {\theta} \max _ {\phi} \mathcal {L} (\theta , \phi) +$$ + +Definition E.1 (Global Optima). We define three equivalent notions of optimality: + +1. $(\theta^{*},\phi^{*})$ is a global minimax, if for any $(\theta ,\phi)\colon \mathcal{L}(\theta^{*},\phi)\leq \mathcal{L}(\theta^{*},\phi^{*})\leq \max_{\phi^{\prime}}\mathcal{L}(\theta ,\phi^{\prime})$ +2. $(\theta^{*},\phi^{*})$ is a saddle point, if for any $(\theta ,\phi)\colon \mathcal{L}(\theta^{*},y)\leq \mathcal{L}(\theta^{*},\phi^{*})\leq \mathcal{L}(x,\phi^{*})$ +3. $(\theta^{*},\phi^{*})$ is a stationary point, if for any $(\theta ,\phi)\colon \nabla_{\theta}\mathcal{L}(\theta^{*},\phi^{*}) = \nabla_{\phi}\mathcal{L}(\theta^{*},\phi^{*}) = 0.$ + +Thanks to gradient dominance of PL conditions, it is possible to prove the equivalence of these notions under two-sided PL condition. Observe that this reduction is crucial since gradient-based methods provide safely just a stationary point while the other two notions are global and non-trivially verifiable. + +Lemma E.2 (Lemma 2.1 in [124], Appendix C in [60]). If the objective function $\mathcal{L}$ satisfies the two-sided PL condition, then all three notions in Definition E.1 are equivalent: + +$\epsilon - (Saddle \, Point) \iff \epsilon - (Global \, Min-Max) \iff \epsilon - (Stationary \, Point) \quad \forall \epsilon \geq 0$ + +Before we present the following lemma, we provide some intuition: This result characterizes the behavior of the optimization trajectory under hidden convex-concave structure, establishing a critical link between parameter dynamics and convergence guarantees. The lemma will formalize how small distortions in the hidden geometry impact the optimization path based on parameters $\alpha, c$ . + +Lemma 3.2 (Theorem 3.2 in [124]). Suppose the min-max objective function $\mathcal{L}(\theta, \phi)$ is $L_{\nabla \mathcal{L}}$ -smooth and satisfies the two-sided PL-condition with $(\mu_{\theta}, \mu_{\phi})$ . Then if we run AltGDA with $\eta_{\theta} = \frac{\mu_{\phi}^2}{18L_{\nabla \mathcal{L}}^3}$ and $\eta_{\phi} = \frac{1}{L_{\nabla \mathcal{L}}}$ , then $\| \theta_{t+1} - \theta_t \| + \| \phi_{t+1} - \phi_t \| \leq \sqrt{\alpha} c^{t/2} \sqrt{P_0}$ where constants $\alpha$ and $c \in (0,1)$ depend only on $L_{\nabla \mathcal{L}}, \mu_{\theta}, \mu_{\phi}$ and $P_0$ is the Lyapunov potential at time $t = 0$ . (Please refer to Remark E.5 for the exact expressions for $\alpha$ and $c$ .) + +In most existing works on nonconvex-nonconcave optimization, smoothness is typically defined directly with respect to the optimization parameters. However, given our focus on a fine-grained analysis of neural min-max games, it is equally important to provide a similarly fine-grained treatment of upper bounds involving the geometry of the neural maps. + +Although a neural network is, in practice, a highly smooth function, global Lipschitz continuity is incompatible with strong convexity in unconstrained domains. Nonetheless, within a bounded region—such as the ball of radius $R$ where our iterates remain—local Lipschitzness can be rigorously characterized. The following lemma formalizes this bounds for both maximizer & minimizer neural network: + +Lemma E.3 (Local-Lipschitzness for smooth loss function). Let $F_{\theta}$ and $G_{\phi}$ be neural network mappings such that they are $\beta_F$ and $\beta_G$ smooth as defined in Definition 2.3. Now let $(\theta_0,\phi_0)$ be such that Jacobian singular values for both the networks are strictly positive and bounded from above and below, $\mu_{Jac}^{(F)}\leq \sigma (\nabla_{\theta}F_{\theta_0})\leq \nu_{Jac}^{(F)}$ and $\mu_{Jac}^{(G)}\leq \sigma (\nabla_{\phi}G_{\phi_0})\leq \nu_{Jac}^{(G)}$ . Suppose the stationary point for min-max objective $\mathcal{L}(F_{\theta},G_{\phi})$ as defined in Assumption 2.1 also lies in the ball, $(\theta^{*},\phi^{*})\in \mathcal{B}((\theta_{0},\phi_{0}),R)$ . Then there exists an $R > 0$ such that $\forall (\theta ,\phi)\in \mathcal{B}((\theta_0,\phi_0),R)$ , we have + +$$ +\max _ {\phi \in \mathcal {B} \left(\phi_ {0}, R\right)} \| \nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta}, G _ {\phi}\right) \|, \max _ {\theta \in \mathcal {B} \left(\theta_ {0}, R\right)} \| \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta}, G _ {\phi}\right) \| \leq L _ {\mathcal {L}} ^ {\text {a c t}} \tag {25} +$$ + +where we denote the upper bound as an 'active' Lipschitz constant $L_{\mathcal{L}}^{act}$ : + +$$ +L _ {\mathcal {L}} ^ {a c t} = 4. 5 L _ {\nabla \mathcal {L}} R \max \left\{\nu_ {J a c} ^ {(F)}, \nu_ {J a c} ^ {(G)} \right\} \tag {26} +$$ + +Proof. Using Lemma 1 of Song et al. [106] as discussed in Section 3, if we choose $R = \frac{\mu_{\mathrm{Jac}}}{2\beta}$ , where $\mu_{\mathrm{Jac}} := \max \left\{\mu_{\mathrm{Jac}}^{(F)}, \mu_{\mathrm{Jac}}^{(G)}\right\}$ and $\beta := \min \left\{\beta_F, \beta_G\right\}$ , then we see that for $\forall (\theta, \phi) \in \mathcal{B}((\theta_0, \phi_0), R)$ , we have + +$$ +\frac {\mu_ {\mathrm {J a c}} ^ {(F)}}{2} \leq \sigma_ {\min } (\nabla_ {\theta} F _ {\theta}) \leq \sigma_ {\max } (\nabla_ {\theta} F _ {\theta}) \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(F)}}{2} +$$ + +$$ +\frac {\mu_ {\mathrm {J a c}} ^ {(G)}}{2} \leq \sigma_ {\min } (\nabla_ {\phi} G _ {\phi}) \leq \sigma_ {\max } (\nabla_ {\phi} G _ {\phi}) \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(G)}}{2} +$$ + +And by Lemma 4 (Appendix C) of Song et al. [106], we see that both the mapping $F_{\theta}$ and $G_{\phi}$ are Lipschitz-continuous in the ball $\mathcal{B}((\theta_0,\phi_0),R)$ . That is, $\forall (\theta ,\phi),(\theta^{\prime},\phi^{\prime})\in \mathcal{B}((\theta_0,\phi_0),R)$ , we have the following: + +$$ +\left\| F _ {\theta} - F _ {\theta^ {\prime}} \right\| \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(F)}}{2} \| \theta - \theta^ {\prime} \| \tag {27} +$$ + +$$ +\left\| G _ {\phi} - G _ {\phi^ {\prime}} \right\| \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(G)}}{2} \| \phi - \phi^ {\prime} \| \tag {28} +$$ + +where $\sigma_{\mathrm{max}}(\nabla_{\theta}F_{\theta_0})\leq \nu_{\mathrm{Jac}}^{(F)}$ and $\sigma_{\mathrm{max}}(\nabla_{\phi}G_{\phi_0})\leq \nu_{\mathrm{Jac}}^{(G)}$ + +Without loss of generality, we prove for the case of $F_{\theta}$ mapping as the same argument works for the $G_{\phi}$ mapping as well. We can observe the following for any $(\theta ,\phi)\in \mathcal{B}((\theta_0,\phi_0),R)$ : + +$$ +\begin{array}{l} \max _ {\theta \in \mathcal {B} (\theta_ {0}, R)} \| \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta}, G _ {\phi}) | \leq \| \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi}) \| + \max _ {\theta^ {\prime} \in \mathcal {B} (\theta_ {0}, R)} \| \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta^ {\prime}}, G _ {\phi}) - \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi}) \| \\ \leq \left\| \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi}\right) \right\| + L _ {\nabla \mathcal {L}} \cdot \max _ {\theta^ {\prime} \in \mathcal {B} \left(\theta_ {0}, R\right)} \left\| F _ {\theta^ {\prime}} - F _ {\theta_ {0}} \right\| \\ \leq \| \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi}) \| + \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {J a c}} ^ {(F)}}{2} \max _ {\theta^ {\prime} \in \mathcal {B} (\theta_ {0}, R)} \| \theta_ {0} - \theta^ {\prime} \| \\ \end{array} +$$ + +$$ +\begin{array}{l} (\because \text {B y} \text {E q}. (2 7)) \\ \leq \left\| \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi}\right) \right\| + \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {J a c}} ^ {(F)} R}{2} \\ = \left\| \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi}\right) - \underbrace {\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta^ {*}} , G _ {\phi^ {*}}\right)} _ {= 0} \right\| + \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {j a c}} ^ {(F)} R}{2} \tag {29} \\ \end{array} +$$ + +where the last equality holds true because $(\theta^{*},\phi^{*})$ is a stationary point for the min-max objective $\mathcal{L}(F_{\theta},G_{\phi})$ and the Jacobian for $F$ is non-singular as $(\theta^{*},\phi^{*})\in \mathcal{B}((\theta_{0},\phi_{0}),R)$ by assumption. That is, + +$$ +\begin{array}{l} 0 = \nabla_ {\theta} \mathcal {L} \left(F _ {\theta^ {*}}, G _ {\phi^ {*}}\right) = \left(\nabla_ {\theta} F _ {\theta^ {*}}\right) ^ {\top} \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi}\right) (30) \\ \Longrightarrow 0 = \left\| \left(\nabla_ {\theta} F _ {\theta^ {*}}\right) ^ {\top} \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta^ {*}}, G _ {\phi^ {*}}\right) \right\| = \left\| \left(\nabla_ {\theta} F _ {\theta^ {*}}\right) ^ {\top} \right\| \left\| \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta^ {*}}, G _ {\phi^ {*}}\right) \right\| (31) \\ \Longrightarrow 0 = \nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta^ {*}}, G _ {\phi^ {*}}\right) (\because \nabla_ {\theta} F _ {\theta^ {*}} \text {n o n - s i n g u l a r}) (32) \\ \end{array} +$$ + +Thus, continuing from Equation (29), we have that for any $(\theta ,\phi)\in \mathcal{B}((\theta_0,\phi_0),R)$ + +$$ +\begin{array}{l} \max _ {\theta \in \mathcal {B} (\theta_ {0}, R)} \| \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta}, G _ {\phi}) \| \leq \| \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi}) - \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta^ {*}}, G _ {\phi^ {*}}) \| + \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {J a c}} ^ {(F)} R}{2} \\ \leq L _ {\nabla \mathcal {L}} \left(\| F _ {\theta_ {0}} - F _ {\theta^ {*}} \| + \| G _ {\phi_ {0}} - G _ {\phi^ {*}} \|\right) + \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {J a c}} ^ {(F)} R}{2} \\ \end{array} +$$ + +$$ +\begin{array}{l} \left(\because \mathcal {L} \text {i s} L _ {\nabla \mathcal {L}} \text {- s m o o t h}\right) \tag {33} \\ \leq \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {J a c}} ^ {(F)}}{2} \| \theta_ {0} - \theta^ {*} \| + \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {J a c}} ^ {(G)}}{2} \| \phi_ {0} - \phi^ {*} \| + \frac {3 L _ {\nabla \mathcal {L}} \nu_ {\mathrm {J a c}} ^ {(F)} R}{2} \\ \end{array} +$$ + +$$ +\begin{array}{l} \left(\because \text {B y E q u a t i o n s} (2 7), (2 8)\right) (34) \\ \leq 3 L _ {\nabla \mathcal {L}} \max \left\{\nu_ {\mathrm {J a c}} ^ {(F)}, \nu_ {\mathrm {J a c}} ^ {(G)} \right\} R + \frac {3 L _ {\nabla \mathcal {L}} \max \left\{\nu_ {\mathrm {J a c}} ^ {(F)} , \nu_ {\mathrm {J a c}} ^ {(G)} \right\} R}{2} \\ = 4. 5 L _ {\nabla \mathcal {L}} \max \left\{\nu_ {\mathrm {J a c}} ^ {(F)}, \nu_ {\mathrm {J a c}} ^ {(G)} \right\} R (35) \\ \end{array} +$$ + +□ + +To establish convergence guarantees under hidden convex-concave structure, it is essential to demonstrate that sufficient overparameterization yields a favorable initialization. Specifically, we show that: (i) the initialization lies close to a saddle point (in terms of gradient norm), and (ii) the optimization path remains within a region where the neural Jacobians have well-conditioned singular spectra. The latter will be ensured via a path length argument. + +The following lemma provides a key component in this direction: it connects the initial value of the Lyapunov potential used in AltGDA (a weighted version of the Nash gap) to the gradient norms of both neural players. In subsequent lemmas, we will show that with high probability, and under sufficient width, this leads to the iterates remaining inside a well-conditioned-Jacobians' manifold that preserves PL-condition throughout training. + +Lemma E.4 (Upper Bound on Initial Potential $P_0$ ; Lemma 3.3 in Main Text). Let $F_{\theta}$ and $G_{\phi}$ be neural network mappings such that they are $\beta_{F}$ and $\beta_{G}$ smooth as defined in Definition 2.3. Now let $(\theta_0, \phi_0)$ be such that Jacobian singular values for both the networks are strictly positive and bounded from above and below, $\mu_{Jac}^{(F)} \leq \sigma(\nabla_{\theta} F_{\theta_0}) \leq \nu_{Jac}^{(F)}$ and $\mu_{Jac}^{(G)} \leq \sigma(\nabla_{\phi} G_{\phi_0}) \leq \nu_{Jac}^{(G)}$ . Suppose the min-max objective $\mathcal{L}(\theta, \phi)$ is $(\mu_{\theta}, \mu_{\phi})$ -HSCSC. Then the initial Lyapunov potential $P_0$ can be bounded from above as: + +$$ +\begin{array}{l} P _ {0} \leq \left\| \left(\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \nu_ {J a c} ^ {(F)} \cdot \frac {1}{\mu_ {\theta} \left(\mu_ {J a c} ^ {(F)}\right) ^ {2}} \cdot \left\| \nabla_ {\theta} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| \\ + \left\| \left(\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \beta_ {F} \frac {1}{2 \mu_ {\theta} ^ {2} \left(\mu_ {J a c} ^ {(F)}\right) ^ {4}} \cdot \left\| \nabla_ {\theta} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| ^ {2} \tag {36} \\ + (1 + \lambda) \cdot \| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \| \cdot \nu_ {J a c} ^ {(G)} \cdot \frac {1}{\mu_ {\phi} \left(\mu_ {J a c} ^ {(G)}\right) ^ {2}} \cdot \| \nabla_ {\phi} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| \\ + (1 + \lambda) \cdot \| \left(\nabla_ {G _ {\phi}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}})\right) ^ {\top} \| \cdot \beta_ {G} \frac {1}{2 \mu_ {\phi} ^ {2} (\mu_ {J a c} ^ {(G)}) ^ {4}} \cdot \| \nabla_ {\phi} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| ^ {2} \\ \end{array} +$$ + +Proof. Using Lemma 1 of Song et al. [106] as discussed in Section 3, if we choose $R = \frac{\mu_{\mathrm{Jac}}}{2\beta}$ , where $\mu_{\mathrm{Jac}} := \max \left\{\mu_{\mathrm{Jac}}^{(F)}, \mu_{\mathrm{Jac}}^{(G)}\right\}$ and $\beta := \min \left\{\beta_F, \beta_G\right\}$ , then we see that for $\forall (\theta, \phi) \in \mathcal{B}((\theta_0, \phi_0), R)$ , we have + +$$ +\frac {\mu_ {\mathrm {J a c}} ^ {(F)}}{2} \leq \sigma_ {\min } (\nabla_ {\theta} F _ {\theta}) \leq \sigma_ {\max } (\nabla_ {\theta} F _ {\theta}) \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(F)}}{2} +$$ + +$$ +\frac {\mu_ {\mathrm {J a c}} ^ {(G)}}{2} \leq \sigma_ {\min } (\nabla_ {\phi} G _ {\phi}) \leq \sigma_ {\max } (\nabla_ {\phi} G _ {\phi}) \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(G)}}{2} +$$ + +And by Lemma 4 (Appendix C) of Song et al. [106], we see that both the mapping $F_{\theta}$ and $G_{\phi}$ are Lipschitz-continuous in the ball $\mathcal{B}((\theta_0,\phi_0),R)$ . That is, $\forall (\theta ,\phi),(\theta^{\prime},\phi^{\prime})\in \mathcal{B}((\theta_0,\phi_0),R)$ , we have the following: + +$$ +\left\| F _ {\theta} - F _ {\theta^ {\prime}} \right\| \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(F)}}{2} \| \theta - \theta^ {\prime} \| \tag {37} +$$ + +$$ +\left\| G _ {\phi} - G _ {\phi^ {\prime}} \right\| \leq \frac {3 \nu_ {\mathrm {J a c}} ^ {(G)}}{2} \| \phi - \phi^ {\prime} \| \tag {38} +$$ + +where $\sigma_{\mathrm{max}}(\nabla_{\theta}F_{\theta_0})\leq \nu_{\mathrm{Jac}}^{(F)}$ and $\sigma_{\mathrm{max}}(\nabla_{\phi}G_{\phi_0})\leq \nu_{\mathrm{Jac}}^{(G)}$ + +Since $\mathcal{L}$ is $(\mu_{\theta},\mu_{\phi})$ -HSCSC, it satisfies 2-sided PL-condition with PL-moduli as per Fact 2.6. Thus, we can obtain the following bound for $W_0$ : + +$$ +\begin{array}{l} W _ {0} = \mathcal {L} \left(\theta_ {0}, \phi^ {*} \left(\theta_ {0}\right)\right) - \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) (39) \\ \leq \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \left(G _ {\phi^ {*} (\theta_ {0})} - G \left(\phi_ {0}\right)\right) \left(\because \text {h i d d e n c o n c a v i t y}\right) (40) \\ \leq \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \left\| G _ {\phi^ {*} (\theta_ {0})} - G _ {\phi_ {0}} \right\| (41) \\ \leq \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \left(\left\| \nabla_ {\phi} G _ {\phi_ {0}} \right\| \left\| \phi^ {*} \left(\theta_ {0}\right) - \phi_ {0} \right\| + \frac {\beta_ {G}}{2} \left\| \phi^ {*} \left(\theta_ {0}\right) - \phi_ {0} \right\| ^ {2}\right) (42) \\ \end{array} +$$ + +$$ +\begin{array}{l} \left(\because G _ {\phi} \text {i s} \beta_ {G} \text {- s m o o t h}\right) \\ \leq \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \left(\nu_ {\mathrm {J a c}} ^ {(G)} \cdot \left\| \phi^ {*} \left(\theta_ {0}\right) - \phi_ {0} \right\| + \frac {\beta_ {G}}{2} \left\| \phi^ {*} \left(\theta_ {0}\right) - \phi_ {0} \right\| ^ {2}\right) (43) \\ \left(\because \sigma \left(\nabla_ {\theta} F _ {\theta_ {0}}\right) \leq \nu_ {\mathrm {J a c}} ^ {(F)}\right) \\ \leq \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \left(\nu_ {\mathrm {J a c}} ^ {(G)} \cdot \sqrt {\frac {2}{\mu_ {\phi} \left(\mu_ {\mathrm {J a c}} ^ {(G)}\right) ^ {2}}} \cdot \sqrt {\mathcal {L} \left(\theta_ {0} , \phi^ {*} (\theta_ {0})\right) - \mathcal {L} \left(\theta_ {0} , \phi_ {0}\right)} \right. \\ \left. + \frac {\beta_ {G}}{\mu_ {\phi} \left(\mu_ {\mathrm {J a c}} ^ {(G)}\right) ^ {2}} \left(\mathcal {L} \left(\theta_ {0}, \phi^ {*} \left(\theta_ {0}\right)\right) - \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right)\right)\right) (44) \\ \end{array} +$$ + +$(\because)$ quadratic growth condition and Fact 2.6 + +$$ +\begin{array}{l} \Rightarrow W _ {0} \leq \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \nu_ {\mathrm {J a c}} ^ {(G)} \cdot \frac {1}{\mu_ {\phi} \left(\mu_ {\mathrm {J a c}} ^ {(G)}\right) ^ {2}} \cdot \| \nabla_ {\phi} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \| \\ + \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \beta_ {G} \frac {1}{2 \mu_ {\phi} ^ {2} \left(\mu_ {\mathrm {J a c}} ^ {(G)}\right) ^ {4}} \cdot \left\| \nabla_ {\phi} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| ^ {2} \tag {45} \\ \end{array} +$$ + +(using PL-condition and Fact 2.6) + +Similarly, for $U_{0}$ , we can say that: + +$$ +\begin{array}{l} U _ {0} = \mathcal {L} \left(\theta_ {0}, \phi^ {*} \left(\theta_ {0}\right)\right) - \mathcal {L} \left(\theta^ {*}, \phi^ {*}\right) (46) \\ \leq \mathcal {L} \left(\theta_ {0}, \phi^ {*} \left(\theta_ {0}\right)\right) - \mathcal {L} \left(\theta^ {*}, \phi_ {0}\right) \quad (\because \mathcal {L} \left(\theta^ {*}, \phi\right) \leq \mathcal {L} \left(\theta^ {*}, \phi^ {*}\right) \forall \phi) (47) \\ \leq \mathcal {L} \left(\theta_ {0}, \phi^ {*} \left(\theta_ {0}\right)\right) - \mathcal {L} \left(\theta^ {*} \left(\phi_ {0}\right), \phi_ {0}\right) \quad (\because \mathcal {L} \left(\theta^ {*}, \phi_ {0}\right) \geq \mathcal {L} \left(\theta^ {*} \left(\phi_ {0}\right), \phi_ {0}\right)) (48) \\ = \underbrace {\mathcal {L} \left(\theta_ {0} , \phi^ {*} \left(\theta_ {0}\right)\right) - \mathcal {L} \left(\theta_ {0} , \phi_ {0}\right)} _ {①} + \underbrace {\mathcal {L} \left(\theta_ {0} , \phi_ {0}\right) - \mathcal {L} \left(\theta^ {*} \left(\phi_ {0}\right) , \phi_ {0}\right)} _ {②} (49) \\ \end{array} +$$ + +Noticing that is exactly equal to $W_{0}$ and following similar arguments we used for $W_{0}$ to get a upper bound for but with hidden convexity instead, we obtain the following for $U_{0}$ .. + +$$ +\begin{array}{l} U _ {0} \leq \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \nu_ {\mathrm {J a c}} ^ {(G)} \cdot \frac {1}{\mu_ {\phi} \left(\mu_ {\mathrm {J a c}} ^ {(G)}\right) ^ {2}} \cdot \left\| \nabla_ {\phi} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| \\ + \left\| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \beta_ {G} \frac {1}{2 \mu_ {\phi} ^ {2} \left(\mu_ {\mathrm {J a c}} ^ {(G)}\right) ^ {4}} \cdot \left\| \nabla_ {\phi} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| ^ {2} \tag {50} \\ + \left\| \left(\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \nu_ {\mathrm {J a c}} ^ {(F)} \cdot \frac {1}{\mu_ {\theta} \left(\mu_ {\mathrm {J a c}} ^ {(F)}\right) ^ {2}} \cdot \left\| \nabla_ {\theta} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| \\ + \left\| \left(\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \beta_ {F} \frac {1}{2 \mu_ {\theta} ^ {2} \left(\mu_ {\mathrm {J a c}} ^ {(F)}\right) ^ {4}} \cdot \left\| \nabla_ {\theta} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| ^ {2} \\ \end{array} +$$ + +By combining the upper bounds obtained for $U_{0}$ and $W_{0}$ , we get the following upper bound on the initial potential $P_{0}$ : + +$$ +\begin{array}{l} P _ {0} \leq \| \left(\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \| \cdot \nu_ {\mathrm {J a c}} ^ {(F)} \cdot \frac {1}{\mu_ {\theta} \left(\mu_ {\mathrm {J a c}} ^ {(F)}\right) ^ {2}} \cdot \| \nabla_ {\theta} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \| \\ + \left\| \left(\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \beta_ {F} \frac {1}{2 \mu_ {\theta} ^ {2} \left(\mu_ {\mathrm {J a c}} ^ {(F)}\right) ^ {4}} \cdot \left\| \nabla_ {\theta} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| ^ {2} \\ + (1 + \lambda) \cdot \| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \| \cdot \nu_ {\mathrm {J a c}} ^ {(G)} \cdot \frac {1}{\mu_ {\phi} \left(\mu_ {\mathrm {J a c}} ^ {(G)}\right) ^ {2}} \cdot \| \nabla_ {\phi} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \| \\ + (1 + \lambda) \cdot \| \left(\nabla_ {G _ {\phi}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \| \cdot \beta_ {G} \frac {1}{2 \mu_ {\phi} ^ {2} \left(\mu_ {\mathrm {j a c}} ^ {(G)}\right) ^ {4}} \cdot \| \nabla_ {\phi} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \| ^ {2} \\ \end{array} +$$ + +□ + +Remark E.5 (Path length bound for AltGDA). We know from Lemma 3.2 that the AltGDA path length satisfies: + +$$ +\begin{array}{l} \ell (T) \triangleq \sum_ {t = 0} ^ {T - 1} \left(\left\| \theta_ {t + 1} - \theta_ {t} \right\| + \left\| \phi_ {t + 1} - \phi_ {t} \right\|\right) \leq \frac {\sqrt {2 \alpha_ {1}}}{1 - \sqrt {c}} \cdot \sqrt {P _ {0}} (51) \\ \leq \sqrt {2 \alpha} \frac {1 + \sqrt {c}}{1 - c} \sqrt {P _ {0}} < \frac {2 \sqrt {2 \alpha_ {1}}}{1 - c} \cdot \sqrt {P _ {0}} (\because c \in (0, 1)) (52) \\ \end{array} +$$ + +where $\alpha_{1}, L$ , and $c$ are defined as + +$$ +\alpha_ {1} = \frac {2 \left(1 + \eta_ {\phi} ^ {2} L _ {\nabla \mathcal {L}} ^ {2}\right) \eta_ {\theta} ^ {2} L ^ {2}}{\mu_ {\theta}} + \frac {2 0 \left(1 + \eta_ {\phi} ^ {2} L _ {\nabla \mathcal {L}} ^ {2}\right) \eta_ {\theta} ^ {2} L _ {\nabla \mathcal {L}} ^ {2} + 2 0 L _ {\nabla \mathcal {L}} ^ {2} \eta_ {\phi} ^ {2}}{\mu_ {\phi}} \tag {53} +$$ + +$$ +L = L _ {\nabla \mathcal {L}} + \frac {L _ {\nabla \mathcal {L}} ^ {2}}{\mu_ {\phi}} \tag {54} +$$ + +$$ +c = 1 - \frac {\mu_ {\theta} \mu_ {\phi} ^ {2}}{3 6 \left(L _ {\nabla \mathcal {L}}\right) ^ {3}} \tag {55} +$$ + +Now, say we define $T$ to be the first time instant when $(\theta_T,\phi_T)\notin \mathcal{B}((\theta_0,\phi_0),R)$ for an appropriate $R > 0$ . And now, say, we start with appropriate $(\theta_0,\phi_0)$ such that $\ell (T) < R$ . A sufficient condition to ensure this would be to find $(\theta_0,\phi_0)$ and a corresponding radius $R > 0$ such that + +$$ +\frac {2 \sqrt {2 \alpha_ {1}}}{1 - c} \sqrt {P _ {0}} < R \tag {56} +$$ + +$$ +\Longleftrightarrow \frac {\alpha_ {1}}{(1 - c) ^ {2}} P _ {0} < \frac {R ^ {2}}{8} \tag {57} +$$ + +$$ +\Longleftrightarrow \left(\frac {1 2 9 6 L _ {\nabla \mathcal {L}} ^ {6}}{\mu_ {\theta} ^ {2} \mu_ {\phi} ^ {4}} \cdot \alpha_ {1}\right) \cdot P _ {0} < \frac {R ^ {2}}{8} \tag {58} +$$ + +Remark E.6 (Simplification of $\alpha_{1}$ for AltGDA Path Length Bound). For the case of $(\mu_{\theta},\mu_{\phi})$ HSCSC and $L_{\nabla \mathcal{L}}$ -smooth (w.r.t. $(\theta ,\phi)$ ) min-max objective function $\mathcal{L}$ and for learning rates $\eta_{\theta} = \frac{c_{\theta}\mu_{\phi}^{2}\sigma_{\min}^{4}(\nabla_{\phi}G_{\phi_{0}})}{18L_{\nabla\mathcal{L}}^{3}}$ and $\eta_{\phi} = \frac{c_{\phi}}{L_{\nabla\mathcal{L}}}$ with $0 < c_{\theta},c_{\phi}\leq 1$ , we can simplify $\alpha_{1}$ defined above in Equation (53) for AltGDA path length bound as follows: + +$$ +\begin{array}{l} \alpha_ {1} = \frac {2 \left(1 + \eta_ {\phi} ^ {2} L _ {\nabla \mathcal {L}} ^ {2}\right) \eta_ {\theta} ^ {2} L ^ {2}}{\mu_ {\theta} \sigma_ {\min } ^ {2} \left(\nabla_ {\theta} F _ {\theta_ {0}}\right)} + \frac {2 0 \left(1 + \eta_ {\phi} ^ {2} L _ {\nabla \mathcal {L}} ^ {2}\right) \eta_ {\theta} ^ {2} L _ {\nabla \mathcal {L}} ^ {2} + 2 0 L _ {\nabla \mathcal {L}} ^ {2} \eta_ {\phi} ^ {2}}{\mu_ {\phi} \sigma_ {\min } ^ {2} \left(\nabla_ {\phi} G _ {\phi_ {0}}\right)} (59) \\ \lesssim \frac {2 c _ {\phi} ^ {2} c _ {\theta} ^ {2} \mu_ {\phi} ^ {2} \sigma_ {\operatorname* {m i n}} ^ {4} \left(\nabla_ {\phi} G _ {\phi_ {0}}\right)}{(1 8) ^ {2} L _ {\nabla \mathcal {L}} ^ {2} \mu_ {\theta} \sigma_ {\operatorname* {m i n}} ^ {2} \left(\nabla_ {\theta} F _ {\theta_ {0}}\right)} + \frac {2 0 c _ {\phi} ^ {2} c _ {\theta} ^ {2} \mu_ {\phi} ^ {3} \sigma_ {\operatorname* {m i n}} ^ {6} \left(\nabla_ {\phi} G _ {\phi_ {0}}\right)}{1 8 ^ {2} L _ {\nabla \mathcal {L}} ^ {4}} (60) \\ \end{array} +$$ + +In fact, $\frac{1296L_{\nabla\mathcal{L}}^6}{\mu_\theta^2\mu_\phi^4}\cdot \alpha_1$ simplifies to the following: + +$$ +\begin{array}{l} \frac {1 2 9 6 L _ {\nabla \mathcal {L}} ^ {6}}{\mu_ {\theta} ^ {2} \mu_ {\phi} ^ {4}} \cdot \alpha_ {1} = \frac {8 L _ {\nabla \mathcal {L}} ^ {4} c _ {\theta} ^ {2} c _ {\phi} ^ {2}}{\mu_ {\theta} ^ {3} \mu_ {\phi} ^ {2} \sigma_ {\mathrm {m i n}} ^ {6} (\nabla_ {\theta} F _ {\theta_ {0}}) \sigma_ {\mathrm {m i n}} ^ {4} (\nabla_ {\phi} G _ {\phi_ {0}})} \\ + \frac {8 0 c _ {\theta} ^ {2} c _ {\phi} ^ {2} L _ {\nabla \mathcal {L}} ^ {2}}{\mu_ {\theta} ^ {2} \mu_ {\phi} \sigma_ {\min } ^ {4} \left(\nabla_ {\theta} F _ {\theta_ {0}}\right) \sigma_ {\min } ^ {2} \left(\nabla_ {\phi} G _ {\phi_ {0}}\right)} \tag {61} \\ \end{array} +$$ + +Lemma E.7 (Simplified AltGDA path length bound condition). Consider the premise as defined in Lemma 3.2 for $(\mu_{\theta},\mu_{\phi})$ -HSCSC and $L_{\nabla \mathcal{L}}$ smooth min-max objective with learning rates $\eta_{\theta} = c_{\theta}\mu_{\phi}^{2} / 18L_{\nabla \mathcal{L}}^{2}$ and $\eta_{\phi} = c_{\phi} / L_{\nabla \mathcal{L}}$ with $c_{\theta},c_{\phi}\in (0,1]$ along with the premise of Lemma E.4. Let $T$ be the first time instant when $(\theta_T,\phi_T)\notin \mathcal{B}((\theta_0,\phi_0),R)$ for an appropriate $R > 0$ . Then to ensure $(\theta_t,\phi_t)\in \mathcal{B}((\theta_0,\phi_0),R)\forall t\leq T$ as discussed in Remark E.5 (Equation (58)), the following is a sufficient condition: + +$$ +\left\| \nabla_ {\theta} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right) \right\| + \left\| \nabla_ {\phi} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right) \right\| \lesssim \frac {R ^ {2}}{8} \tag {62} +$$ + +given that $c_{\theta}$ and $c_{\phi}$ are chosen such that: + +$$ +c _ {\theta} = \sqrt {\frac {1}{4 \cdot A _ {\theta} \cdot B _ {\phi}}} a n d c _ {\phi} = 1 \tag {63} +$$ + +where + +$$ +A _ {\theta} = \left((1 + \| \nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}}) \|) \cdot \left(1 + 8 L _ {\nabla \mathcal {L}} ^ {4} \nu_ {J a c} ^ {(F)}\right) \cdot \left(1 + \frac {1}{\mu_ {\theta} ^ {3} (\mu_ {J a c} ^ {(F)}) ^ {6}}\right) \cdot (1 + \| \nabla_ {\theta} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}}) \|)\right) +$$ + +$$ +\begin{array}{l} \cdot \left(1 + 8 0 L _ {\nabla \mathcal {L}} ^ {2} \nu_ {J a c} ^ {(F)}\right) \cdot \left(1 + \frac {1}{\mu_ {\theta} ^ {3} (\mu_ {J a c} ^ {(F)}) ^ {4}}\right) \cdot \left(1 + 8 L _ {\nabla \mathcal {L}} ^ {4} \beta_ {F}\right) \cdot \left(1 + \frac {1}{2 \mu_ {\theta} ^ {5} (\mu_ {J a c} ^ {(F)}) ^ {1 0}}\right) \cdot \left(1 + 8 0 L _ {\nabla \mathcal {L}} ^ {2} \beta_ {F}\right) \\ \left. \cdot \left(1 + \frac {1}{2 \mu_ {\theta} ^ {4} (\mu_ {J a c} ^ {(F)}) ^ {8}}\right) \cdot \left(1 + \frac {1}{\mu_ {\theta} ^ {2} (\mu_ {J a c} ^ {(F)}) ^ {4}}\right) \cdot \left(1 + \frac {1}{2 \mu_ {\theta} ^ {3} (\mu_ {J a c} ^ {(F)}) ^ {6}}\right)\right) \\ \end{array} +$$ + +and + +$$ +\begin{array}{l} B _ {\phi} = \left(\left(1 + \| \nabla_ {G _ {\phi}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}}) \|\right) \cdot \left(1 + \frac {1}{\mu_ {\phi} ^ {3} (\mu_ {J a c} ^ {(G)}) ^ {6}}\right) \cdot \left(1 + \frac {1}{\mu_ {\phi} (\mu_ {J a c} ^ {(G)}) ^ {4}}\right) \cdot \left(1 + \frac {1}{\mu_ {\phi} ^ {2} (\mu_ {J a c} ^ {(G)}) ^ {4}}\right) \right. \\ \cdot \left(1 + \frac {1}{\mu_ {\phi} (\mu_ {J a c} ^ {(G)}) ^ {2}}\right) \cdot (1 + \lambda) \cdot \left(1 + \| \nabla_ {\phi} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}}) \|\right) \cdot \left(1 + 8 0 L _ {\nabla \mathcal {L}} ^ {2} \nu_ {J a c} ^ {(G)}\right) \cdot \left(1 + 8 L _ {\nabla \mathcal {L}} ^ {4} \beta_ {G}\right) \\ \left. \cdot \left(1 + \frac {1}{\mu_ {\phi} ^ {4} (\mu_ {J a c} ^ {(G)}) ^ {8}}\right) \cdot \left(1 + 8 0 L _ {\nabla \mathcal {L}} ^ {2} \beta_ {G}\right) \cdot \left(1 + \frac {1}{\mu_ {\phi} ^ {3} (\mu_ {J a c} ^ {(G)}) ^ {8}}\right)\right) \\ \end{array} +$$ + +Proof. By choosing $c_{\theta}$ and $c_{\phi}$ as specified above and using observations from Remark E.6 (Equation (61)), a sufficient condition to ensure the iterates never leave as derived in Remark E.5 (Equation (58)) can be further simplified as follows: + +$$ +\left(\frac {1 2 9 6 L _ {\nabla \mathcal {L}} ^ {6}}{\mu_ {\theta} ^ {2} \mu_ {\phi} ^ {4}} \cdot \alpha_ {1}\right) \cdot P _ {0} < \frac {R ^ {2}}{8} +$$ + +$$ +\therefore \left(\frac {8 L _ {\nabla \mathcal {L}} ^ {4} c _ {\theta} ^ {2} c _ {\phi} ^ {2}}{\mu_ {\theta} ^ {3} \mu_ {\phi} ^ {2} \sigma_ {\min} ^ {6} (\nabla_ {\theta} F _ {\theta_ {0}}) \sigma_ {\min} ^ {4} (\nabla_ {\phi} G _ {\phi_ {0}})} + \frac {8 0 c _ {\theta} ^ {2} c _ {\phi} ^ {2} L _ {\nabla \mathcal {L}} ^ {2}}{\mu_ {\theta} ^ {2} \mu_ {\phi} \sigma_ {\min} ^ {4} (\nabla_ {\theta} F _ {\theta_ {0}}) \sigma_ {\min} ^ {2} (\nabla_ {\phi} G _ {\phi_ {0}})}\right) \cdot P _ {0} < \frac {R ^ {2}}{8} +$$ + +$$ +\therefore T _ {1} \| \nabla_ {\theta} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| + T _ {2} \| \nabla_ {\theta} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| ^ {2} + T _ {3} \| \nabla_ {\phi} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| + T _ {4} \| \nabla_ {\phi} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| ^ {2} < \frac {R ^ {2}}{8} +$$ + +where $T_{j}$ 's are coefficients of the gradient norms as per Equations (61) and (58). Notice that both $T_{2}$ and $T_{4}$ contain $c_{\theta}^{2}$ and as per our choice of $c_{\theta}$ , $c_{\theta}^{2}$ contains $\frac{1}{1 + \|\nabla_{\theta}\mathcal{L}(F_{\theta_0},G_{\phi_0})\|}$ and $\frac{1}{1 + \|\nabla_{\phi}\mathcal{L}(F_{\theta_0},G_{\phi_0})\|}$ terms which can help absorb one of the powers of the gradient norms $\|\nabla_{\theta}\mathcal{L}(F_{\theta_0},G_{\phi_0})\|$ or $\|\nabla_{\phi}\mathcal{L}(F_{\theta_0},G_{\phi_0})\|$ in terms corresponding to $T_{2}$ and $T_{4}$ . This will help constructing (as shown below) a simpler sufficient condition that would require only the norm of the loss' gradient w.r.t. parameters $\theta, \phi$ to be small. Furthermore, by construction, $c_{\theta} \leq 1/2$ , $c_{\phi} = 1$ . Therefore, we have that $T_{i} \leq 1 \forall i \in [4]$ . Combining all of the above, what we require for ensuring iterates never leave the ball of radius $R$ is as follows: + +$$ +\left\| \nabla_ {\theta} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right) \right\| + \left\| \nabla_ {\phi} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right) \right\| \lesssim \frac {R ^ {2}}{8} \tag {64} +$$ + +![](images/6892a7e3b01ff49ee461da2fff06b18d574ecc9863932f39f91fd0b14a79d6a1.jpg) + +Remark E.8 (Local-smoothness constant). If the iterates $(\theta_t, \phi_t) \forall t$ stay within the ball $\mathcal{B}((\theta_0, \phi_0), R)$ , by invoking H.3, we can use the local-Lipschitz constant derived Lemma E.3 along with the smoothness of the neural network maps $(\beta_F, \beta_G)$ to obtain the exact value of the smoothness constant for the HSCSC and smooth loss as $L_{\nabla \mathcal{L}} = L_{\mathcal{L}}^{\mathrm{act}} \max \{\beta_F, \beta_G\}$ . + +# F Proofs for Input-Optimization Min-Max Games + +The following result characterizes the lower and upper bounds on the singular values of the Jacobian for the case of Input-Optimization Min-Max Games. In particular, we bring out the dependence of these lower and upper bounds on the size of the hidden layer, $d_1^{(F)}$ and $d_1^{(G)}$ , and variances for random Gaussian initializations of the neural network layers, $\{\sigma_k^{(F)}\}_{k\in [2]}$ and $\{\sigma_k^{(G)}\}_{k\in [2]}$ , for both the players $F$ and $G$ . Computing the lower bound on singular values is important as it's used in defining the radius around the initial parameters to ensure the neural network Jacobian remains non-singular within the entire ball. Moreover, these lower and upper bounds will be helpful in determining a set of sufficient conditions on the variances for ensuring the AltGDA trajectory reaches the saddle point. + +Lemma F.1 (Lemma 3.4 in Main Paper). Consider a neural network $F$ with parameters $\theta \in \mathbb{R}^{d_0^{(F)}}$ defined as $F(\theta) = W_2^{(F)}\psi(W_1^{(F)}\theta)$ where $W_k^{(F)} \in \mathbb{R}^{d_k^{(F)} \times d_{k-1}^{(F)}}$ ( $k \in \{1,2\}$ ), $(W_1^{(F)})_{i,j} \sim \mathcal{N}(0,(\sigma_1^{(F)})^2) \forall i,j$ , $(W_2^{(F)})_{k,l} \sim \mathcal{N}(0,(\sigma_2^{(F)})^2) \forall k,l$ and $\psi$ is the GeLU activation function with $d_1^{(F)} \geq 256d_0^{(F)}$ , $d_1^{(F)} \geq 256d_2^{(F)}$ and $(\sigma_1^{(F)})^2 < \frac{\pi}{4Cd_1^{(F)}\|\theta\|^2}$ . Then the minimum singular value of the Jacobian $\nabla_\theta F_\theta$ is lower bounded as + +$$ +\sigma_ {\min } \left(\nabla_ {\theta} F _ {\theta}\right) > \frac {\sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} d _ {1} ^ {(F)}}{1 6} \cdot \left(\frac {1}{2} - \sigma_ {1} ^ {(F)} \| \theta \| \sqrt {\frac {C d _ {1} ^ {(F)}}{\pi}}\right) \tag {65} +$$ + +$w.p \geq 1 - 2e^{-\frac{d_1^{(F)}}{64}} - e^{-C d_1^{(F)}}$ . The maximum singular value of $\nabla_{\theta} F_{\theta}$ is upper bounded as + +$$ +\sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta}\right) < 3. 4 7 \sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} d _ {1} ^ {(F)} \tag {66} +$$ + +w.p. $\geq 1 - 2e^{-\frac{d_1^{(F)}}{64}}$ + +Proof. By Theorem 4.6.1 in [110], we get w.p. $\geq 1 - 2e^{-t^2}$ : + +$$ +\sigma_ {1} \left(\sqrt {d _ {1} ^ {(F)}} - 4 \left(\sqrt {d _ {0} ^ {(F)}} + t\right)\right) \leq \sigma_ {\min } \left(W _ {1} ^ {(F)}\right) \leq \sigma_ {\max } \left(W _ {1} ^ {(F)}\right) \leq \sigma_ {1} ^ {(F)} \left(\sqrt {d _ {1} ^ {(F)}} + 4 \left(\sqrt {d _ {0} ^ {(F)}} + t\right)\right). \tag {67} +$$ + +By choosing $t = \frac{1}{8}\sqrt{d_1^{(F)}}$ , we get w.p. $\geq 1 - 2e^{-d_1^{(F)} / 64}$ + +$$ +\sigma_ {1} ^ {(F)} \left(\frac {1}{2} \sqrt {d _ {1} ^ {(F)}} - 4 \sqrt {d _ {0} ^ {(F)}}\right) \leq \sigma_ {\min } \left(W _ {1} ^ {(F)}\right) \leq \sigma_ {\max } \left(W _ {1} ^ {(F)}\right) \leq \sigma_ {1} ^ {(F)} \left(\frac {3}{2} \sqrt {d _ {1} ^ {(F)}} + 4 \sqrt {d _ {0} ^ {(F)}}\right). \tag {68} +$$ + +If we set $d_1^{(F)} \geq 256d_0^{(F)}$ , we get + +$$ +\left(\frac {1}{2} \sqrt {d _ {1} ^ {(F)}} - 4 \sqrt {d _ {0} ^ {(F)}}\right) \geq \frac {1}{4} \sqrt {d _ {1} ^ {(F)}} \tag {69} +$$ + +and + +$$ +\frac {3}{2} \sqrt {d _ {1} ^ {(F)}} + 4 \sqrt {d _ {0} ^ {(F)}} \leq \frac {7}{4} \sqrt {d _ {1} ^ {(F)}}. \tag {70} +$$ + +Therefore, we get that w.p. $\geq 1 - 2e^{-\frac{d_1^{(F)}}{64}}$ + +$$ +\frac {1}{4} \sigma_ {1} ^ {(F)} \sqrt {d _ {1} ^ {(F)}} \leq \sigma_ {\min } \left(W _ {1} ^ {(F)}\right) \leq \sigma_ {\max } \left(W _ {1} ^ {(F)}\right) \leq \frac {7}{4} \sigma_ {1} ^ {(F)} \sqrt {d _ {1} ^ {(F)}}. \tag {71} +$$ + +For the case of $W_2^{(F)}$ , because we have $d_2^{(F)} < d_1^{(F)}$ , we will use analogous reasoning as that for $W_1^{(F)}$ above for $(W_2^{(F)})^\top$ with $t = \frac{1}{8}\sqrt{d_1^{(F)}}$ which yields w.p. $\geq 1 - 2e^{-\frac{d_1^{(F)}}{64}}$ , + +$$ +\sigma_ {2} ^ {(F)} \left(\frac {1}{2} \sqrt {d _ {1} ^ {(F)}} - 4 \sqrt {d _ {2} ^ {(F)}}\right) \leq \sigma_ {\min } \left(\left(W _ {2} ^ {(F)}\right) ^ {\top}\right) = \sigma_ {\min } \left(W _ {2} ^ {(F)}\right) \tag {72} +$$ + +$$ +\leq \sigma_ {\max } \left(\left(W _ {2} ^ {(F)}\right) ^ {\top}\right) = \sigma_ {\max } \left(W _ {2} ^ {(F)}\right) \leq \sigma_ {2} ^ {(F)} \left(\frac {3}{2} \sqrt {d _ {1} ^ {(F)}} + 4 \sqrt {d _ {2} ^ {(F)}}\right) \tag {73} +$$ + +Then, by setting $d_1^{(F)} \geq 256d_2^{(F)}$ , we get + +$$ +\frac {1}{4} \sigma_ {2} ^ {(F)} \sqrt {d _ {1} ^ {(F)}} \leq \sigma_ {\min } (W _ {2} ^ {(F)}) \leq \sigma_ {\max } (W _ {2} ^ {(F)}) \leq \frac {7}{4} \sigma_ {2} ^ {(F)} \sqrt {d _ {1} ^ {(F)}} +$$ + +Note that the Jacobian for the neural network $F(\theta)$ can be computed as + +$$ +\nabla_ {\theta} F _ {\theta} = \frac {\partial F (\theta)}{\partial \theta} = W _ {2} ^ {(F)} \frac {\partial}{\partial z} \psi (z) W _ {1} ^ {(F)} = W _ {2} ^ {(F)} \Psi \left(W _ {1} ^ {(F)} \theta\right) W _ {1} ^ {(F)} \tag {74} +$$ + +where $z \coloneqq W_1^{(F)}\theta$ and $\Psi(W_1^{(F)}\theta) = \mathrm{diag}(\psi'((W_1^{(F)}\theta)_1), \ldots, \psi'((W_1^{(F)}\theta)_{d_1^{(F)}}))$ . Therefore, we can compute the minimum and maximum singular values for this Jacobian $\nabla_{\theta}F_{\theta}$ by looking at its operator norm: $\| \nabla_{\theta}F_{\theta}\| = \| W_2^{(F)}\Psi(W_1^{(F)}\theta)W_1^{(F)}\|$ . Then, by properties of the operator norm, we see that + +$$ +\left\| \nabla_ {\theta} F _ {\theta} \right\| \geq \sigma_ {\min } \left(W _ {2} ^ {(F)}\right) \| \Psi \left(W _ {1} ^ {(F)} \theta\right) \| \sigma_ {\min } \left(W _ {1} ^ {(F)}\right) \tag {75} +$$ + +$$ +\left\| \nabla_ {\theta} F _ {\theta} \right\| \leq \sigma_ {\max } \left(W _ {2} ^ {(F)}\right) \| \Psi \left(W _ {1} ^ {(F)} \theta\right) \| \sigma_ {\max } \left(W _ {1} ^ {(F)}\right) \tag {76} +$$ + +This further tells us that + +$$ +\sigma_ {\min } \left(\nabla_ {\theta} F _ {\theta}\right) \geq \sigma_ {\min } \left(W _ {2} ^ {(F)}\right) \sigma_ {\min } \left(\Psi \left(W _ {1} ^ {(F)} \theta\right)\right) \sigma_ {\min } \left(W _ {1} ^ {(F)}\right) \tag {77} +$$ + +$$ +\sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta}\right) \leq \sigma_ {\max } \left(W _ {2} ^ {(F)}\right) \sigma_ {\max } \left(\Psi \left(W _ {1} ^ {(F)} \theta\right)\right) \sigma_ {\max } \left(W _ {1} ^ {(F)}\right) \tag {78} +$$ + +We can further simplify these lower and upper bounds for the Jacobian singular values by noting that since $\Psi(W_1^{(F)}\theta)$ is a diagonal matrix, $\sigma_{\min}(\Psi(W_1^{(F)}\theta)) = \min_{1 \leq i \leq d_1^{(F)}} \psi'(z_i)$ and $\sigma_{\max}(\Psi(W_1^{(F)}\theta)) = \max_{1 \leq i \leq d_1^{(F)}} \psi'(z_i)$ where $z_i = (W_1^{(F)}\theta)_i \forall i \in [d_1^{(F)}]$ . + +Now, notice that $z = W_1^{(F)}\theta \sim \mathcal{N}(0,(\sigma_1^{(F)})^2\| \theta \| _2^2 I_{d_1^{(F)}\times d_1^{(F)}})$ . Using the fact that GeLU's derivative, $\psi^\prime$ is $L_{\psi '}$ -Lipschitz with $L_{\psi '} = \sup_{x\in \mathbb{R}}|\psi ''(x)| = \varphi (0) = \frac{2}{\sqrt{\pi}}$ ( $\varphi$ is standard normal PDF), we can appeal to concentration inequality for Lipschitz functions of Gaussian random variables and infer for $0 < \epsilon < 1 / 2$ : + +$$ +\begin{array}{l} P \left[ \sigma_ {\min } \left(\Psi \left(W _ {1} ^ {(F)} \theta\right)\right) > \epsilon \right] = P \left[ \forall i: \psi^ {\prime} \left(\left(W _ {1} ^ {(F)} \theta\right) _ {i}\right) > \epsilon \right] = \left(P \left[ \psi^ {\prime} \left(\left(W _ {1} ^ {(F)} \theta\right) _ {1}\right) > \epsilon \right]\right) ^ {d _ {1} ^ {(F)}} (79) \\ = \left(P \left[ \psi^ {\prime} \left(\left(W _ {1} ^ {(F)} \theta\right) _ {1}\right) - 1 / 2 > \epsilon - 1 / 2 \right]\right) ^ {d _ {1} ^ {(F)}} (80) \\ = \left(P \left[ \psi^ {\prime} \left(\left(W _ {1} ^ {(F)} \theta\right) _ {1}\right) - E \left[ \psi^ {\prime} \left(\left(W _ {1} ^ {(F)} \theta\right) _ {1}\right) \right] > \epsilon - 1 / 2 \right]\right) ^ {d _ {1} ^ {(F)}} (81) \\ \end{array} +$$ + +$$ +\left(\because E \left[ \psi^ {\prime} \left(\left(W _ {1} ^ {(F)} \theta\right) _ {1}\right) \right] = E \left[ \left(W _ {1} ^ {(F)} \theta\right) _ {1} \varphi \left(\left(W _ {1} ^ {(F)} \theta\right) _ {1}\right) \right] + E \left[ \Phi \left(\left(W _ {1} ^ {(F)} \theta\right) _ {1}\right) \right] = 0 + 1 / 2\right) \tag {82} +$$ + +$$ +\begin{array}{l} (\varphi , \Phi \text {a r e s t a n d a r d n o r m a l P D F} \& \text {C D F}, \text {r e s p .}) (83) \\ \geq 1 - e ^ {- \frac {(1 / 2 - \epsilon) ^ {2}}{2 L _ {\psi^ {\prime}} ^ {2} (\sigma_ {1} ^ {(F)}) ^ {2} \| \theta \| ^ {2}}} (84) \\ \end{array} +$$ + +$$ +\begin{array}{l} \left(\because P \left[ \psi^ {\prime} (z) \leq 1 / 2 - t \right] \leq e ^ {- \frac {\pi t ^ {2}}{\left(\sigma_ {1} ^ {(F)}\right) ^ {2} \| \theta \| ^ {2}}}; \text {c h o o s e} t = 1 / 2 - \epsilon\right) (85) \\ = 1 - e ^ {- C d _ {1} ^ {(F)}} \left(\text {i f w e s e t} \epsilon = 1 / 2 - \left(\sigma_ {1} ^ {(F)} \| \theta \| \sqrt {C d _ {1} ^ {(F)} / \pi}\right)\right) (86) \\ \end{array} +$$ + +Therefore, as long as we have $(\sigma_1^{(F)})^2 < \frac{\pi}{4Cd_1^{(F)}\|\theta\|^2}$ , we have w.p. $\geq 1 - e^{-Cd_1^{(F)}}$ that + +$$ +\sigma_ {\min } \left(\Psi \left(W _ {1} ^ {(F)} \theta\right)\right) > \frac {1}{2} - \sigma_ {1} ^ {(F)} \| \theta \| \sqrt {\frac {C d _ {1} ^ {(F)}}{\pi}} \tag {87} +$$ + +Combining all of this, we can say w.p. $\geq 1 - 2e^{-\frac{d_1^{(F)}}{64}} - e^{-Cd_1^{(F)}}$ , we have: + +$$ +\sigma_ {\min } \left(\nabla_ {\theta} F _ {\theta}\right) > \frac {\sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} d _ {1} ^ {(F)}}{1 6} \cdot \left(\frac {1}{2} - \sigma_ {1} ^ {(F)} \| \theta \| \sqrt {\frac {C d _ {1} ^ {(F)}}{\pi}}\right). \tag {88} +$$ + +And w.p. $\geq 1 - 2e^{-\frac{d_1^{(F)}}{64}}$ , we have: $\sigma_{\max}(\nabla_{\theta}F_{\theta}) < \left(\frac{7\sqrt{d_1^{(F)}}}{4}\right)^2 \cdot 1.13\sigma_1^{(F)}\sigma_2^{(F)} < 3.47\sigma_1^{(F)}\sigma_2^{(F)}d_1^{(F)}$ + +where we used the fact that $\max_{x\in \mathbb{R}}\psi '(x) = \psi '(\sqrt{2})\approx 1.1289$ + +Before we present the following lemma, we provide some intuition: In addition to the previous lemma for high probability lower and upper bounds on singular values of the neural network Jacobian, we also need to check whether the networks in input-optimization games are smooth or not. Towards that, the following result proves that it is so and provides the exact smoothness constant. Computing this smoothness constant is important for another reason: It is used in defining the radius around the initial parameters to ensure the neural network Jacobian remains non-singular within the entire ball. + +Lemma F.2 (Lemma 3.4 in Main Paper). Consider a neural network $F$ with parameters $\theta \in \mathbb{R}^{d_0^{(F)}}$ as defined in Lemma F.1 above. Then, $w.p. \geq 1 - 2e^{-\frac{d_1^{(F)}}{64}}$ the neural network $F$ is $\beta_F$ -smooth where $\beta_F = \frac{1}{\sqrt{2\pi}} \cdot \frac{343(\sigma_1^{(F)})^2\sigma_2^{(F)}(d_1^{(F)})^{3/2}}{32}$ . + +Proof. In order to prove that $F$ is $\beta_F$ -smooth, we need to show that Equation 2 holds true. For $\theta, \theta' \in \mathbb{R}^{d_0^{(F)}}$ , we write + +$$ +\left\| \nabla_ {\theta} F _ {\theta} - \nabla_ {\theta} F _ {\theta^ {\prime}} \right\| \leq \left\| W _ {2} ^ {(F)} \right\| \left\| \Psi \left(W _ {1} ^ {(F)} \theta\right) - \Psi \left(W _ {1} ^ {(F)} \theta^ {\prime}\right) \right\| \left\| W _ {1} ^ {(F)} \right\| \tag {89} +$$ + +$$ +\leq \left\| W _ {2} ^ {(F)} \right\| L _ {\psi^ {\prime}} \left\| W _ {1} ^ {(F)} \theta - W _ {1} ^ {(F)} \theta^ {\prime} \right\| \left\| W _ {1} ^ {(F)} \right\| (\because \psi^ {\prime} \text {i s} L _ {\psi^ {\prime}} - \text {L i p s c h i t z}) \tag {90} +$$ + +$$ += L _ {\psi^ {\prime}} \| W _ {2} ^ {(F)} \| \left\| W _ {1} ^ {(F)} \right\| \| \theta - \theta^ {\prime} \| \left\| W _ {1} ^ {(F)} \right\| \tag {91} +$$ + +$$ +\Rightarrow \sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta} - \nabla_ {\theta} F _ {\theta^ {\prime}}\right) \leq \frac {2}{\sqrt {2 \pi}} \sigma_ {\max } \left(W _ {2} ^ {(F)}\right) \cdot \sigma_ {\max } ^ {2} \left(W _ {1} ^ {(F)}\right) \| \theta - \theta^ {\prime} \| \tag {92} +$$ + +where $L_{\psi'} = \frac{2}{\sqrt{2\pi}}$ for derivative of GeLU activation function. Using Lemma F.1 results for singular values of the random matrices $W_1^{(F)}$ and $W_2^{(F)}$ , we get that w.p. $\geq 1 - 2e^{-\frac{d_1^{(F)}}{64}}$ + +$$ +\sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta} - \nabla_ {\theta} F _ {\theta^ {\prime}}\right) \leq \frac {3 4 3 \left(\sigma_ {1} ^ {(F)}\right) ^ {2} \sigma_ {2} ^ {(F)} \left(d _ {1} ^ {(F)}\right) ^ {3 / 2}}{3 2 \sqrt {2 \pi}} \| \theta - \theta^ {\prime} \| \tag {93} +$$ + +The result in the main paper offers an average-case analysis under Gaussian sampling with variance scaled as $\mathrm{poly}(1 / d_1)$ . Here, we provide a more detailed version that explicitly states the assumptions and technical conditions required to ensure convergence to equilibrium. While the high-level complexity perspective remains unchanged, we believe this finer analysis may be of independent interest, particularly for applications in adversarial attack design. + +Theorem F.3 (Theorem 3.5 in Main Paper). Consider two neural networks $F, G$ with parameters $\theta \in \mathbb{R}^{d_0^{(F)}}$ and $\phi \in \mathbb{R}^{d_0^{(G)}}$ , respectively, as defined in Lemma F.1 above. Then for the $\varepsilon$ -regularized bilinear min-max objective $\mathcal{L}(\theta, \phi)$ as defined in Equation (3) with the neural networks $F$ and $G$ defined above, alternating gradient-descent-ascent with appropriate fixed learning rates $\eta_{\theta}, \eta_{\phi}$ (see Lemma E.7) reaches the desired saddle point w.p. $\geq 1 - 4e^{-\frac{d_1^{(F)}}{64}} - 4e^{-\frac{d_1^{(G)}}{64}} - e^{-Cd_1^{(F)}} - e^{-Cd_1^{(G)}}$ ( $C$ are some universal constants) if the initial parameters $(\theta_0, \phi_0)$ and standard deviations $\sigma_k^{(F)}$ and $\sigma_k^{(G)}$ , $k \in \{1,2\}$ are chosen such that: + +$$ +\left(\sigma_ {1} ^ {(F)}\right) ^ {2} < \frac {\pi}{4 C d _ {1} ^ {(F)} \| \theta_ {0} \| ^ {2}} \& \left(\sigma_ {1} ^ {(G)}\right) ^ {2} < \frac {\pi}{4 C d _ {1} ^ {(G)} \| \phi_ {0} \| ^ {2}} \tag {94} +$$ + +$$ +\frac {\left(\sigma_ {1} ^ {(F)}\right) ^ {4} \cdot \left(\sigma_ {2} ^ {(F)}\right) ^ {2} \cdot \left\| \theta_ {0} \right\|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(F)}\right) \left\| \theta_ {0} \right\| \sqrt {\frac {C d _ {1} ^ {(F)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\varepsilon \cdot \left(d _ {1} ^ {(F)}\right) ^ {3 . 5}} \tag {95} +$$ + +$$ +\frac {\left(\sigma_ {1} ^ {(G)}\right) ^ {4} \cdot \left(\sigma_ {2} ^ {(G)}\right) ^ {2} \cdot \left\| \phi_ {0} \right\|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(G)}\right) \| \phi_ {0} \| \sqrt {\frac {C d _ {1} ^ {(G)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\varepsilon \cdot \left(d _ {1} ^ {(G)}\right) ^ {3 . 5}} \tag {96} +$$ + +$$ +\frac {\left(\sigma_ {1} ^ {(G)} \sigma_ {2} ^ {(G)}\right) \left(\left(\sigma_ {1} ^ {(F)}\right) ^ {3} \sigma_ {2} ^ {(F)}\right) \| \theta_ {0} \|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(F)}\right) \| \theta_ {0} \| \sqrt {\frac {\mathcal {C} d _ {1} ^ {(F)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\sigma_ {\max } (A) \left(d _ {1} ^ {(F)}\right) ^ {2 . 5}} \tag {97} +$$ + +$$ +\frac {\left(\sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)}\right) \left(\left(\sigma_ {1} ^ {(G)}\right) ^ {3} \sigma_ {2} ^ {(G)}\right) \| \phi_ {0} \|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(G)}\right) \| \phi_ {0} \| \sqrt {\frac {C d _ {1} ^ {(G)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\sigma_ {\max } (A) \left(d _ {1} ^ {(G)}\right) ^ {2 . 5}} \tag {98} +$$ + +Proof. Since $\mathcal{L}(\theta, \phi)$ is $\varepsilon$ -hidden-strongly-convex-strongly-concave, by Fact 2.6 it also satisfies the 2-sided PL-condition with $(\mu_{\theta} = \varepsilon \cdot \sigma_{\min}^{2}(\nabla_{\theta}F_{\theta}), \mu_{\phi} = \varepsilon \cdot \sigma_{\min}^{2}(\nabla_{\phi}G_{\phi}))$ w.p. $\geq 1 - 2e^{-\frac{d_1}{64}} - e^{-Cd_1}$ , we can utilise path-length bound as derived in Lemma 3.2 which leaves us with controlling the potential $P_0$ for ensuring convergence to the saddle point. Thus, computing loss gradients (Equation (4)) and using the sufficient condition for ensuring iterates do not leave the ball $\mathcal{B}((\theta_0, \phi_0), R)$ (where $R = \frac{\max\{\mu_{\mathrm{Jac}}^{(F)}, \mu_{\mathrm{Jac}}^{(G)}\}}{\min\{\beta_F, \beta_G\}}$ as defined in Section 3) thus ensuring non-singular Jacobian inside the ball for both the neural networks (Lemma E.7), we require the following: + +$$ +\begin{array}{l} \left(\| F (\theta_ {0}) \| \cdot \left(\varepsilon \sigma_ {\max} \left(\nabla_ {\theta} F _ {\theta_ {0}}\right) + \sigma_ {\max} \left(\nabla_ {\phi} G _ {\phi_ {0}}\right) \sigma_ {\max} (A)\right) \right. \\ \left. + \| G \left(\phi_ {0}\right) \| \cdot \left(\sigma_ {\max } \left(\nabla_ {\theta} F _ {\theta_ {0}}\right) \sigma_ {\max } (A) + \varepsilon \sigma_ {\max } \left(\nabla_ {\phi} G _ {\phi_ {0}}\right)\right)\right) \lesssim \frac {R ^ {2}}{8} \tag {99} \\ \end{array} +$$ + +Thus, we want + +$$ +\begin{array}{l} \left((3. 4 7 \varepsilon \sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} d _ {1} ^ {(F)}) \| F (\theta_ {0}) \| + (3. 4 7 \sigma_ {1} ^ {(G)} \sigma_ {2} ^ {(G)} d _ {1} ^ {(G)} \sigma_ {\max} (A)) \| F (\theta_ {0}) \|\right) \\ \left. + \left(3. 4 7 \sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} d _ {1} ^ {(F)} \sigma_ {\max } (A)\right) \cdot \| G (\phi_ {0}) \| + \left(3. 4 7 \varepsilon \sigma_ {1} ^ {(G)} \sigma_ {2} ^ {(G)} d _ {1} ^ {(G)}\right) \| G (\phi_ {0}) \|\right) \lesssim \frac {R ^ {2}}{8} \tag {100} \\ (\because \sigma_ {\max} (\nabla_ {\theta} F _ {\theta_ {0}}) < 3. 4 7 \sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} d _ {1} ^ {(F)}, \sigma_ {\max} (\nabla_ {\phi} G _ {\phi_ {0}}) < 3. 4 7 \sigma_ {1} ^ {(G)} \sigma_ {2} ^ {(G)} d _ {1} ^ {(G)}) \\ \end{array} +$$ + +Therefore, we want the following to hold true: + +1. $(3.47\varepsilon \sigma_{1}^{(F)}\sigma_{2}^{(F)}d_{1}^{(F)})\| F(\theta_{0})\| \lesssim \frac{R^{2}}{32}$ +2. $(3.47\sigma_{\max}(A)\sigma_1^{(G)}\sigma_2^{(G)}d_1^{(G)})\| F(\theta_0)\| \lesssim \frac{R^2}{32}$ +3. $(3.47\sigma_{\max}(A)\sigma_1^{(F)}\sigma_2^{(F)}d_1^{(F)})\| G(\phi_0)\| \lesssim \frac{R^2}{32}$ +4. $(3.47\varepsilon \sigma_{1}^{(G)}\sigma_{2}^{(G)}d_{1}^{(G)})\| G(\phi_{0})\|\lesssim \frac{R^{2}}{32}$ + +Since $\| F(\theta_0) \| \leq \sigma_{\max}(W_2^{(F)}) \sigma_{\max}(\psi(W_1^{(F)} \theta_0))$ , we can use Lemma F.1 for maximum singular values of $W_2^{(F)}$ and the following calculation for upper bounding $\psi(W_1^{(F)} \theta)$ : + +$$ +\begin{array}{l} \left\| \psi \left(\left(W _ {1} ^ {(F)} \theta\right)\right) \right\| \leq \sqrt {d _ {1} ^ {(F)}} \max _ {i} | \psi \left(\left(W _ {1} ^ {(F)} \theta\right) _ {i}\right) | (101) \\ \Rightarrow P [ \| \psi \left(W _ {1} ^ {(F)} \theta\right) \| > t ] \leq P [ \sqrt {d _ {1} ^ {(F)}} \max _ {i} | \left(W _ {1} ^ {(F)} \theta\right) _ {i} | > t ] (102) \\ \end{array} +$$ + +$$ +\text {A l s o}, P \left[ \max _ {i} | \left(W _ {1} ^ {(F)} \theta\right) _ {i} | > t \right] \leq 2 d _ {1} e ^ {- t ^ {2} / \left(2 \sigma_ {1, F} ^ {2} \| \theta \| ^ {2}\right)} \tag {103} +$$ + +$$ +\begin{array}{l} \left(\because \psi (x) \leq x \forall x, W _ {1} ^ {(F)} \theta \sim \mathcal {N} \left(0, \sigma_ {1, F} ^ {2} \| \theta \| ^ {2} I _ {d _ {1} ^ {(F)}}\right)\right) \\ \end{array} \tag {104} +$$ + +$$ +\Rightarrow P \left[ \max _ {i} | \left(W _ {1} ^ {(F)} \theta\right) _ {i} | > t \right] \leq \delta \tag {105} +$$ + +$$ +\left(\operatorname {S e t} \delta = 2 d _ {1} e ^ {- t ^ {2} / \left(2 \sigma_ {1, F} ^ {2} \| \theta_ {0} \| ^ {2}\right)} = e ^ {- C d _ {1}}\right) \tag {106} +$$ + +$$ +\Rightarrow P \left[ \sqrt {d _ {1} ^ {(F)}} \max _ {i} | (W _ {1} ^ {(F)} \theta) _ {i} | \leq C ^ {\prime} \sigma_ {1} ^ {(F)} \| \theta \| d _ {1} ^ {(F)} \right] > 1 - e ^ {- C d _ {1}} \tag {107} +$$ + +Thus, for ensuring 1.) holds, we will demand the following: $(3.47\varepsilon \sigma_{1}^{(F)}\sigma_{2}^{(F)}d_{1}^{(F)})\cdot \left(\frac{7}{4}\sigma_{2}^{(F)}\sqrt{d_{1}^{(F)}}\right)$ . $\left(C^{\prime}\sigma_{1}^{(F)}\| \theta_{0}\| d_{1}^{(F)}\right)\lesssim \frac{R^{2}}{32}$ . This will yield the following sufficient condition on $\sigma_1^{(F)}$ and $\sigma_2^{(F)}$ : + +$$ +\left(\sigma_ {1} ^ {(F)}\right) ^ {2} \cdot \left(\sigma_ {2} ^ {(F)}\right) ^ {2} \lesssim \frac {R ^ {2}}{1 9 5 \varepsilon C ^ {\prime} \left\| \theta_ {0} \right| \left(d _ {1} ^ {(F)}\right) ^ {2 . 5}} \tag {108} +$$ + +By definition the radius $R \geq \frac{\mu_{\mathrm{fac}}^{(F)}}{2\beta_F}$ . Substituting the lower bound for minimum singular value and Lipschitzness constant for Jacobian of network $F$ from Lemmas F.1-F.2, we obtain: + +$$ +\frac {\left(\sigma_ {1} ^ {(F)}\right) ^ {4} \cdot \left(\sigma_ {2} ^ {(F)}\right) ^ {2} \cdot \left\| \theta_ {0} \right\|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(F)}\right) \left\| \theta_ {0} \right\| \sqrt {\frac {C d _ {1} ^ {(F)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\varepsilon \cdot \left(d _ {1} ^ {(F)}\right) ^ {3 . 5}} \tag {109} +$$ + +Analogous reasoning with $R \geq \frac{\mu_{\mathrm{lac}}^{(G)}}{2\beta_G}$ for ensuring 4.) holds true in case of $G(\phi_0)$ gives us a similar condition on $\sigma_1^{(G)}$ and $\sigma_2^{(G)}$ : + +$$ +\frac {\left(\sigma_ {1} ^ {(G)}\right) ^ {4} \cdot \left(\sigma_ {2} ^ {(G)}\right) ^ {2} \cdot \left\| \phi_ {0} \right\|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(G)}\right) \| \phi_ {0} \| \sqrt {\frac {\mathcal {C} d _ {1} ^ {(G)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\varepsilon \cdot \left(d _ {1} ^ {(G)}\right) ^ {3 . 5}} \tag {110} +$$ + +For ensuring 2.), by Lemmas F.1-F.2 and using $R \geq \frac{\mu_{\mathrm{Jac}}^{(F)}}{2\beta_F}$ , we see that we need $(3.47\sigma_{\max}(A)\sigma_1^{(G)}\sigma_2^{(G)}d_1^{(G)})\cdot \left(\frac{7}{4}\sigma_2^{(F)}\sqrt{d_1^{(F)}}\right)\cdot \left(C'\sigma_1^{(F)}\|\theta_0\|d_1^{(F)}\right) \lesssim \frac{R^2}{32}$ which yields the following sufficient condition: + +$$ +\begin{array}{l} \left(3. 4 7 \sigma_ {1} ^ {(G)} \sigma_ {2} ^ {(G)} d _ {1} ^ {(G)}\right) \left(\frac {7}{4} \sigma_ {2} ^ {(F)} \sqrt {d _ {1} ^ {(F)}}\right) \left(C ^ {\prime} \sigma_ {1} ^ {(F)} \| \theta_ {0} \| d _ {1} ^ {(F)}\right) \lesssim \frac {R ^ {2}}{3 2 \sigma_ {\max } (A))} (111) \\ = \frac {\left(\mu_ {\mathrm {J a c}} ^ {(F)}\right) ^ {2}}{1 2 8 \sigma_ {\max } (A) \beta_ {F} ^ {2}} (112) \\ \lesssim \frac {\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(F)}\right) \| \theta_ {0} \| \sqrt {\frac {C d _ {1} ^ {(F)}}{\pi}}\right) ^ {2} \cdot \pi}{\sigma_ {\max } (A) \left(\sigma_ {1} ^ {(F)}\right) ^ {2} d _ {1} ^ {(F)}} (113) \\ \end{array} +$$ + +(By Lemma $F.1 - F.2$ ) + +$$ +\Longrightarrow \frac {\left(\sigma_ {1} ^ {(G)} \sigma_ {2} ^ {(G)}\right) \left(\left(\sigma_ {1} ^ {(F)}\right) ^ {3} \sigma_ {2} ^ {(F)}\right) \| \theta_ {0} \|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(F)}\right) \| \theta_ {0} \| \sqrt {\frac {C d _ {1} ^ {(F)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\sigma_ {\max } (A) \left(d _ {1} ^ {(F)}\right) ^ {2 . 5}} \tag {114} +$$ + +Similarly, for ensuring 3.), we see that we need $(3.47\sigma_{\max}(A)\sigma_1^{(F)}\sigma_2^{(F)}d_1^{(F)})\cdot \left(\frac{7}{4}\sigma_2^{(G)}\sqrt{d_1^{(G)}}\right)$ . + +$\left(C^{\prime}\sigma_{1}^{(G)}\| \phi_{0}\| d_{1}^{(G)}\right)\lesssim \frac{R^{2}}{32}$ which yields the following sufficient conditions when we use $R\geq \frac{\mu_{\mathrm{fac}}^{(G)}}{2\beta_g}$ + +$$ +\frac {\left(\sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)}\right) \left(\left(\sigma_ {1} ^ {(G)}\right) ^ {3} \sigma_ {2} ^ {(G)}\right) \| \phi_ {0} \|}{\left(\frac {1}{2} - \left(\sigma_ {1} ^ {(G)}\right) \| \phi_ {0} \| \sqrt {\frac {C d _ {1} ^ {(G)}}{\pi}}\right) ^ {2}} \lesssim \frac {\pi}{\sigma_ {\max } (A) \left(d _ {1} ^ {(G)}\right) ^ {2 . 5}} \tag {116} +$$ + +Thus, w.p. $\geq 1 - 4e^{-\frac{d_1^{(F)}}{64}} - 4e^{-\frac{d_1^{(G)}}{64}} - e^{-Cd_1^{(F)}} - e^{-Cd_1^{(G)}}$ , we stay within a ball around the random initializations $\mathcal{B}((\theta_0,\phi_0),R)$ thereby ensuring min-max objective satisfies 2-sided PL-condition. By Lemma 3.2, given that we have chosen appropriate fixed learning rates as per Lemma E.7, we are now guaranteed to reach the saddle point. + +# G Proofs for Neural-Parameters Min-Max Games + +The following lemma establishes a sharp connection between the spectral properties of the input data and the initial conditioning of the neural network's Jacobian, highlighting how data diversity directly influences the minimum singular value at initialization. + +Lemma G.1 (Lemma 3.7 in Main Paper; Lemma 3 & Appendix E.1-E.4 in [106]). Suppose that a two-layer neural network, $F_{\theta}$ , as defined in Definition 2.3, satisfies Assumption 2.4 and $\tau^{r_1}|\psi(a)| \leq |\psi(\tau a)| \leq \tau^{r_2}|\psi(a)|$ , respectively for all $a, 0 < \tau < 1$ , and some constants $r_1, r_2$ . Then the neural network Jacobian for a random Gaussian initialization $\theta_0 = ((W_1^{(F)})_0, (W_2^{(G)})_0)$ has the following lower bounds on its smallest singular value w.p. $\geq 1 - (p_1 + p_2)$ : + +$$ +\mu_ {J a c} ^ {(F)} \geq \left(\sigma_ {1} ^ {(F)}\right) ^ {r _ {1}} \sqrt {\left(1 - \delta_ {1}\right) \frac {c _ {t} ^ {2}}{t !} d _ {1} ^ {(F)}} \cdot \sigma_ {\min } \left(X ^ {* t}\right) \tag {117} +$$ + +We have the following upper bound on its largest singular value w.p. $\geq 1 - (p_{1} + p_{3} + p_{4})$ : + +$$ +\begin{array}{l} \nu_ {J a c} ^ {(F)} \lesssim \sigma_ {2} ^ {(F)} \dot {\psi} _ {\max } \sigma_ {\max } (X) \sqrt {d _ {1} ^ {(F)}} + (\sigma_ {1} ^ {(F)}) ^ {r _ {1}} \sqrt {(1 + \delta_ {2}) (c _ {1} ^ {2} + c _ {\infty} ^ {2}) d _ {1} ^ {(F)}} \sigma_ {\max } (X) \\ + \left(\sigma_ {1} ^ {(F)}\right) ^ {r _ {2}} | c _ {0} | \sqrt {\left(1 + \delta_ {2}\right) d _ {1} ^ {(F)} n} \tag {118} \\ \end{array} +$$ + +And the smoothness constant for the neural network can be computed as + +$$ +\beta_ {F} = \sqrt {2} \sigma_ {\max } (X) \left(\dot {\psi} _ {\max } + \ddot {\psi} _ {\max } \chi_ {\max }\right) \tag {119} +$$ + +where $\chi_{\mathrm{max}} = \sup_{W_2^{(F)}}\sigma_{\mathrm{max}}(W_2^{(F)})$ . Here $\{c_i\}_i$ denote Hermite expansion coefficients corresponding to $\psi ((W_1^{(F)})_0X)$ , $\delta_j > 0 \forall j \in [4]$ , $p_1 = (d_1^{(F)})^{-Ck_1d_0^{(F)}} + (d_1^{(F)})^{-Ck_2d_2^{(F)}}$ for universal constant $C$ with sufficiently large $k_1, k_2$ , $p_2 = \exp \left(-\left(\frac{\delta_1\sigma_{\min}(\mathbb{E}[M_0])}{4\dot{\psi}_{\max}^2\sigma_{\max}^2(X)k_1\sigma_1^{(F)}\sqrt{d_0^{(F)}\log d_1^{(F)}}}\right)^2\right)$ where + +$$ +M _ {0} = \psi (X ^ {\top} ((W _ {1} ^ {(F)}) _ {0} ^ {\top}) \psi ((W _ {1} ^ {(F)}) _ {0} X), p _ {3} = \exp \left(- \left(\frac {\delta_ {2} \sigma_ {\max } (\mathbb {E} [ M _ {0} ])}{4 \dot {\psi_ {\max } ^ {2}} \sigma_ {\max } ^ {2} (X) k _ {1} \sigma_ {1} ^ {(F)} \sqrt {d _ {0} ^ {(F)} \log d _ {1} ^ {(F)}}}\right) ^ {2}\right) \& +$$ + +$p_4 = e^{-C'd_1^{(F)}}$ for a universal constant $C'$ . + +Before we present the following lemma, we offer some context and motivation: In input-optimization games, the gradient norm structure naturally arises from the formulation of hidden bilinear zero-sum games. In more general settings, the relevant properties are detailed in Appendix D. The lemma below demonstrates that, under appropriate initialization and sufficient overparameterization, the neural network output remains bounded from above in terms of spectral properties of the data matrix with high probability. This property will play a critical role in ensuring that the optimization trajectory remains confined within the well-conditioned region (the ball). + +Lemma G.2 (Lemma 3.7 in Main Paper; Neural network output is bounded w.h.p.; Appendix E.5 in [106]). Consider a neural network $F_{\theta}$ with parameters $\theta = (W_1^{(F)}, W_2^{(F)})$ as defined in Lemma G.1 above. Suppose we randomly initialize the neural network at $\theta_0$ by choosing $\sigma_1^{(F)}$ and $\sigma_2^{(G)}$ such that + +$$ +\sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} \lesssim \frac {1}{\sqrt {d _ {0} ^ {(F)} d _ {1} ^ {(F)}}} +$$ + +Then w.p. $\geq 1 - p_{1} - p_{5}$ , the neural network output at this random initialization $\theta_0$ for the given training data $\mathcal{D}_F$ (as described in Assumption 2.4) is bounded from above as follows: + +$$ +\left\| F _ {\theta_ {0}} \left(\mathcal {D} _ {F}\right) \right\| \lesssim \delta_ {3} k _ {1} k _ {2} \sigma_ {\max } (X) \tag {120} +$$ + +where $p_1 = (d_1^{(F)})^{-Ck_1d_0^{(F)}} + (d_1^{(F)})^{-Ck_2d_2^{(F)}}$ for universal constant $C$ with sufficiently large $k_1, k_2, \delta_3 > 0$ , and $p_5 = e^{-C\delta_3^2}$ for some universal constant $C$ . + +Theorem G.3 (Theorem 3.8 in Main Paper; HSCSC Games with AltGDA). Suppose there are two two-layer neural networks, $F_{\theta}$ , $G_{\phi}$ as defined in Definition 2.3 which satisfy Assumption 2.4 and $\tau^{r_1}|\psi(a)| \leq |\psi(\tau a)| \leq \tau^{r_2}|\psi(a)|$ , respectively for all $a$ , $0 < \tau < 1$ , and some constants $r_1, r_2$ . Suppose the network parameters $\theta_0$ and $\phi_0$ are randomly initialized as in Assumption 3.6 with $(\sigma_1^{(F)}, \sigma_2^{(F)})$ and $(\sigma_1^{(G)}, \sigma_2^{(G)})$ , respectively, which satisfy + +$$ +\sigma_ {1} ^ {(F)} \sigma_ {2} ^ {(F)} \lesssim \frac {1}{\sqrt {d _ {0} ^ {(F)} d _ {1} ^ {(F)}}} \quad a n d \quad \sigma_ {1} ^ {(G)} \sigma_ {2} ^ {(G)} \lesssim \frac {1}{\sqrt {d _ {0} ^ {(G)} d _ {1} ^ {(G)}}} \tag {121} +$$ + +and suppose that the hidden layer widths $d_1^{(F)}$ and $d_1^{(G)}$ for the two networks $F$ and $G$ satisfy + +$$ +d _ {1} ^ {(F)} = \widetilde {\Omega} \left(\mu_ {\theta} ^ {2} \frac {n ^ {3}}{d _ {0} ^ {(F)}}\right) \& d _ {1} ^ {(G)} = \widetilde {\Omega} \left(\mu_ {\phi} ^ {2} \frac {n ^ {3}}{d _ {0} ^ {(G)}}\right) \tag {122} +$$ + +where the datasets $(\mathcal{D}_F, \mathcal{D}_G)$ for both the players are assumed to be of size $n$ . Then Alternating Gradient-Descent-Ascent procedure with appropriate fixed learning rates $\eta_{\theta}, \eta_{\phi}$ (see Lemma E.7) for an $(\mu_{\theta}, \mu_{\phi})$ -HSCSC and $L_{\nabla \mathcal{L}}$ -smooth min-max objective $\mathcal{L}_{\mathcal{D}}(F_{\theta}, G_{\phi})$ as defined in Equation 1 satisfying Assumption 2.1 converges to the saddle point $(\theta^{*}, \phi^{*})$ exponentially fast with probability at least $1 - (p_1 + p_2 + p_3 + p_4 + p_5) - (p_1 + p_2' + p_3' + p_4' + p_5')$ (Here, the failure probabilities $p_j$ 's and $p_j'$ 's are defined for networks $F_{\theta}$ and $G_{\phi}$ , respectively, as per Lemmas G.1-G.2). + +Proof. If we randomly initialize the neural network at $\theta_0 = ((W_1^{(F)})_0, (W_2^{(G)})_0)$ as per the stated initialization scheme in Assumption 3.6 with Equation (121), we have by Lemma G.2 that w.p. $\geq 1 - p_1 - p_5$ : + +$$ +\left\| F _ {\theta_ {0}} \left(\mathcal {D} _ {F}\right) \right\| \lesssim \delta_ {3} k _ {1} k _ {2} \sigma_ {\max } (X) \tag {123} +$$ + +where $p_1, p_5$ are as defined in Lemma G.2. Analogous reasoning gives a similar bound for the output of initialization condition for the neural network $G_{\phi}$ with data $\mathcal{D}_G$ . + +Since our min-max objective $\mathcal{L}_{\mathcal{D}}(F_{\theta},G_{\phi})$ is separable as defined in Equation 1, we can start by rewriting the bilinear component $I_2^{\mathcal{D}}(F_\theta ,G_\phi) = (F_\theta (\mathcal{D}_F))^{\top}A(G_\phi (\mathcal{D}_G))$ . Firstly, we can compute the gradients for the min-max objective as follows given data $\mathcal{D} = (\mathcal{D}_F,\mathcal{D}_G)$ : + +$$ +\begin{array}{l} \nabla_ {\theta} L _ {\mathcal {D}} \left(F _ {\theta}, G _ {\phi}\right) = \nabla_ {\theta} \left(I _ {1} ^ {\mathcal {D} _ {F}} \left(F _ {\theta}\right) + I _ {2} ^ {\mathcal {D}} \left(F _ {\theta}, G _ {\phi}\right)\right) (124) \\ = \left(\nabla_ {\theta} F _ {\theta} \left(\mathcal {D} _ {F}\right)\right) ^ {\top} \left(\nabla_ {z = F _ {\theta} \left(\mathcal {D} _ {F}\right)} I _ {1} ^ {\mathcal {D} _ {F}} (z)\right) + \left(\nabla_ {\theta} F _ {\theta} \left(\mathcal {D} _ {F}\right)\right) ^ {\top} A G _ {\phi} \left(\mathcal {D} _ {G}\right) (125) \\ \end{array} +$$ + +$$ +\begin{array}{l} \nabla_ {\phi} L _ {\mathcal {D}} \left(F _ {\theta}, G _ {\phi}\right) = \nabla_ {\phi} \left(- I _ {3} ^ {\mathcal {D} _ {G}} \left(G _ {\phi}\right) + I _ {2} ^ {\mathcal {D}} \left(F _ {\theta}, G _ {\phi}\right)\right) (126) \\ = - \left(\nabla_ {\phi} G _ {\phi} \left(\mathcal {D} _ {G}\right)\right) ^ {\top} \left(\nabla_ {z = G _ {\phi} \left(\mathcal {D} _ {G}\right)} I _ {3} ^ {\mathcal {D} _ {G}} (z)\right) + \left(\nabla_ {\phi} G _ {\phi} \left(\mathcal {D} _ {G}\right)\right) ^ {\top} A F _ {\theta} \left(\mathcal {D} _ {F}\right) (127) \\ \end{array} +$$ + +By Assumption 2.1(iv) on the gradient norm of strongly-convex functions, triangle inequality, and submultiplicativity of operator norm, we can say that the gradient norms of our separable min-max objective can be upper bounded as follows: + +$$ +\begin{array}{l} \left\| \nabla_ {\theta} L _ {\mathcal {D}} \left(F _ {\theta}, G _ {\phi}\right) \right\| \leq \left\| \left(\nabla_ {\theta} F _ {\theta} \left(\mathcal {D} _ {F}\right)\right) ^ {\top} \right\| \left(A _ {1} ^ {(F)} \| F _ {\theta_ {0}} \left(\mathcal {D} _ {F}\right) \| + A _ {2} ^ {(F)} \operatorname {d i a m} \left(\mathcal {Y} ^ {(F)}\right) + A _ {3} ^ {(F)}\right) \\ + \left\| \left(\nabla_ {\theta} F _ {\theta} \left(\mathcal {D} _ {F}\right)\right) ^ {\top} \right\| \| A \| \| G _ {\phi} \left(\mathcal {D} _ {G}\right) \| \tag {128} \\ \end{array} +$$ + +$$ +\begin{array}{l} \left\| \nabla_ {\phi} L _ {\mathcal {D}} \left(F _ {\theta}, G _ {\phi}\right) \right\| \leq \left\| \left(\nabla_ {\phi} G _ {\phi} \left(\mathcal {D} _ {G}\right)\right) ^ {\top} \right\| \left(A _ {1} ^ {(G)} \| G _ {\phi_ {0}} \left(\mathcal {D} _ {G}\right) \| + A _ {2} ^ {(G)} \operatorname {d i a m} \left(\mathcal {Y} ^ {(G)}\right) + A _ {3} ^ {(G)}\right) \\ + \| \left(\nabla_ {\phi} G _ {\phi} \left(\mathcal {D} _ {G}\right)\right) ^ {\top} \| \| A \| \| F _ {\theta} \left(\mathcal {D} _ {F}\right) \| \\ \end{array} +$$ + +Since $\mathcal{L}(\theta, \phi)$ is $(\mu_{\theta}, \mu_{\phi})$ -hidden-strongly-convex-strongly-concave, by Fact 2.6 it also satisfies the 2-sided PL-condition with $(\mu_{\theta} \cdot \sigma_{\min}^{2}(\nabla_{\theta}F_{\theta}), \mu_{\phi} \cdot \sigma_{\min}^{2}(\nabla_{\phi}G_{\phi}))$ PL-moduli w.p. $\geq 1 - p_1 - p_2$ (where $p_1, p_2$ as defined in Lemma G.1). Thus we can utilise path-length bound as derived in Lemma 3.2 which leaves us with controlling the potential $P_0$ for ensuring convergence to the saddle point. Given the loss gradients above (Equation (128)) and using the sufficient condition for ensuring iterates do not leave the ball $\mathcal{B}((\theta_0, \phi_0), R)$ (where $R = \frac{\max\{\mu_{\mathrm{fac}}^{(F)}, \mu_{\mathrm{fac}}^{(G)}\}}{\min\{\beta_F, \beta_G\}}$ as defined in Section 3) thus + +ensuring non-singular Jacobian inside the ball for both the neural networks (Lemma E.7), we require the following: + +$$ +\begin{array}{l} \left(\underbrace {\| \left(\nabla_ {\theta} F _ {\theta_ {0}} \left(\mathcal {D} _ {F}\right)\right) ^ {\top} \| \cdot \left(A _ {1} ^ {(F)} \| F _ {\theta_ {0}} \left(\mathcal {D} _ {F}\right) \| + A _ {2} ^ {(F)} \operatorname {d i a m} \left(\mathcal {Y} ^ {(F)}\right) + A _ {3} ^ {(F)}\right)} _ {T _ {1}}\right) \\ + \underbrace {\| \left(\nabla_ {\theta} F _ {\theta_ {0}} \left(\mathcal {D} _ {F}\right)\right) ^ {\top} \| \| A \| \| G _ {\phi_ {0}} \left(\mathcal {D} _ {G}\right) \|} _ {T _ {2}} \\ + \underbrace {\| \left(\nabla_ {\phi} G _ {\phi_ {0}} (\mathcal {D} _ {G})\right) ^ {\top} \| \cdot \left(A _ {1} ^ {(G)} \| G _ {\phi_ {0}} (\mathcal {D} _ {G}) \| + A _ {2} ^ {(G)} \operatorname {d i a m} (\mathcal {Y} ^ {(G)}) + A _ {3} ^ {(G)}\right)} _ {T _ {3}} \\ \left. \left. + \underbrace {\left\| \left(\nabla_ {\phi} G _ {\phi_ {0}} (\mathcal {D} _ {G})\right) ^ {\top} \right\| \| A \| \| F _ {\theta_ {0}} (\mathcal {D} _ {F}) \|}\right) \lesssim \frac {R ^ {2}}{8} \right. \\ \end{array} +$$ + +In order to ensure the above, we will demand the following: + +1. $T_{1}\lesssim \frac{R^{2}}{32}$ +2. $T_{2}\lesssim \frac{R^{2}}{32}$ +3. $T_{3}\lesssim \frac{R^{2}}{32}$ +4. $T_{4}\lesssim \frac{R^{2}}{32}$ + +We will show arguments for the case of the 'min' player (neural network $F_{\theta}$ ) here. Exactly the same arguments provide the analogous result for the 'max' player (neural network $G_{\phi}$ ). + +For ensuring (1.)-(4.), we will use the fact that $\sigma_{\mathrm{max}}(\nabla_{\theta}F_{\theta_0}(\mathcal{D}_F))\leq \nu_{\mathrm{Jac}}^{(F)}$ , $\sigma_{\mathrm{max}}(\nabla_{\phi}G_{\phi_0}(\mathcal{D}_G))\leq \nu_{\mathrm{Jac}}^{(G)}$ , Lemma G.1 for upper bounds on $\nu_{\mathrm{Jac}}^{(F)}$ , and $\nu_{\mathrm{Jac}}^{(G)}$ and smoothness constants for two-layer neural networks along with the upper bounds on neural network outputs for $F$ and $G$ as derived above in Lemma G.2. + +Thus, a sufficient condition for ensuring 1.) would be to use $R \geq \frac{\mu_{\mathrm{Jac}}^{(F)}}{2\beta_F}$ and see that w.p. $\geq 1 - p_1 - p_2 - p_3 - p_4 - p_5$ the following holds (assuming $|c_0|$ is sufficiently large s.t. $|c_0| \sqrt{(1 + \delta_2)d_1^{(F)}n}$ becomes the dominating term in $\nu_{\mathrm{Jac}}^{(F)}$ ): + +$$ +\begin{array}{l} \underbrace {C _ {1} \cdot \left(| c _ {0} | \sqrt {(1 + \delta_ {2}) d _ {1} ^ {(F)} n}\right) \cdot \left(A _ {1} ^ {(F)} \delta_ {3} k _ {1} k _ {2} \sigma_ {\max} (X) + A _ {2} ^ {(F)} \mathsf {d i a m} (\mathcal {Y} ^ {(F)}) + A _ {3} ^ {(F)}\right)} _ {(L H S)} \lesssim \frac {1}{3 2} \left(\frac {\mu_ {\mathrm {j a c}} ^ {(F)}}{2 \beta_ {F}}\right) ^ {2} \\ \Longrightarrow (L H S) \lesssim \frac {\left(\sigma_ {1} ^ {(F)}\right) ^ {2 r _ {1}} \left(1 - \delta_ {1}\right) \left(c _ {t} ^ {2} / t!\right) d _ {1} ^ {(F)} \sigma_ {\min } ^ {2} \left(X ^ {* t}\right)}{2 \left(\sigma_ {\max } (X)\right) ^ {2} \left(\dot {\psi} _ {\max } + \ddot {\psi} _ {\max } \chi_ {\max }\right) ^ {2}} (\text {B y L e m m a G . 1}) (129) \\ \Rightarrow d _ {1} ^ {(F)} \gtrsim \frac {\left(| c _ {0} | ^ {2} (1 + \delta_ {2}) n\right) \cdot \left(\sigma_ {\max } (X)\right) ^ {4} \left(\dot {\psi} _ {\max } + \ddot {\psi} _ {\max } \chi_ {\max }\right) ^ {4} \cdot \left(A _ {1} ^ {(F)} \delta_ {3} k _ {1} k _ {2} \sigma_ {\max } (X)\right) ^ {2}}{\left(\sigma_ {1} ^ {(F)}\right) ^ {2 + 2 r _ {1}} \left(1 - \delta_ {1}\right) ^ {2} \left(c _ {t} ^ {4} / (t !) ^ {2}\right) \sigma_ {\min } ^ {4} \left(X ^ {* t}\right)} (130) \\ \Longrightarrow d _ {1} ^ {(F)} \gtrsim \frac {\left(| c _ {0} | ^ {2} (1 + \delta_ {2}) n\right) \cdot \left(\sigma_ {\max } (X)\right) ^ {4} \left(\dot {\psi} _ {\max } + \ddot {\psi} _ {\max } \chi_ {\max }\right) ^ {4} \cdot \left(A _ {1} ^ {(F)} \delta_ {3} k _ {1} k _ {2} \sigma_ {\max } (X)\right) ^ {2}}{\left(\sigma_ {1} ^ {(F)}\right) ^ {2 + 2 r _ {1}} \left(1 - \delta_ {1}\right) ^ {2} \left(c _ {t} ^ {4} / (t !) ^ {2}\right) \sigma_ {\min } ^ {4} \left(X ^ {* t}\right)} (131) \\ \therefore d _ {1} ^ {(F)} \gtrsim \xi \left(C _ {\delta}, t, \psi , \left\{c _ {i} \right\} _ {i \geq 0}\right) \frac {n \left(A _ {1} ^ {(F)}\right) ^ {2} \sigma_ {\max } ^ {6} (X)}{\sigma_ {\min } ^ {4} \left(X ^ {*} t\right)} (132) \\ \end{array} +$$ + +where $\delta_4 = \max \{k_1,k_2\}$ $C_{\delta} = \{\delta_{1},\delta_{2},\delta_{3},\delta_{4}\}$ , and + +$$ +\xi (C _ {\delta}, t, \psi , \{c _ {i} \} _ {i \geq 0}) = \frac {\left(| c _ {0} | ^ {2} (1 + \delta_ {2})\right) \cdot \left(\dot {\psi} _ {\max } + \ddot {\psi} _ {\max } \chi_ {\max }\right) ^ {4} \cdot \left(\delta_ {3} k _ {1} k _ {2}\right) ^ {2}}{\left(\sigma_ {1} ^ {(F)}\right) ^ {2 + 2 r _ {1}} \left(1 - \delta_ {1}\right) ^ {2} \left(c _ {t} ^ {4} / (t !) ^ {2}\right)} +$$ + +Here, as per our Assumption 2.4 on the data and arguing along the lines of Section 2.1 in Oymak and Soltanolkotabi [86], when we use the fact that $\sigma_{\max}(X) \simeq \sqrt{\frac{n}{d_0^{(F)}}}$ , $\sigma_{\min}(X^{*t}) \simeq \sqrt{\frac{n}{(d_0^{(F)})^t}} \simeq 1^9$ , and $n \simeq (d_0^{(F)})^t$ where $t \geq 2^{10}$ , we get that: + +$$ +d _ {1} ^ {(F)} = \tilde {\Omega} \left(\frac {n \cdot n ^ {3} \cdot \left(A _ {1} ^ {(F)}\right) ^ {2}}{\left(d _ {0} ^ {(F)}\right) ^ {3} \cdot 1}\right) = \tilde {\Omega} \left(\frac {n ^ {3} \mu_ {\theta} ^ {2}}{d _ {0} ^ {(F)}}\right) \tag {133} +$$ + +In the second equality above, we used the observation from Appendix D that $A_{1} = \theta (\mu)$ where $\mu$ is the strong-convexity modulus. Thus, the amount of overparameterization we need for network $F_{\theta}$ is as follows: + +$$ +d _ {1} ^ {(F)} d _ {0} ^ {(F)} = \tilde {\Omega} \left(n ^ {3} \mu_ {\theta} ^ {2}\right) \tag {134} +$$ + +Analogous reasoning for ensuring (3.) with $R \geq \frac{\mu_{\mathrm{fac}}^{(G)}}{2\beta_G}$ gives us w.p. $1 - p_1' - p_2' - p_3' - p_4' - p_5'$ : + +$$ +d _ {1} ^ {(G)} \gtrsim \xi \left(L _ {\mathcal {L}}, C _ {\delta} ^ {\prime}, t ^ {\prime}, \psi , \left\{c _ {i} ^ {\prime} \right\} _ {i \geq 0}\right) \frac {n \left(A _ {1} ^ {(G)}\right) ^ {2} \sigma_ {\max } ^ {6} (X)}{\sigma_ {\min } ^ {4} \left(X * t\right)} \tag {135} +$$ + +Using arguments from Oymak and Soltanolkotabi [86] as done above for the case of 1.), we get a similar cubic overparameterization bound for the 'max' player, $G_{\phi}$ , as well: + +$$ +d _ {1} ^ {(G)} d _ {0} ^ {(G)} = \tilde {\Omega} \left(n ^ {3} \mu_ {\phi} ^ {2}\right) \tag {136} +$$ + +We get the following w.p. $\geq 1 - (p_1' + p_5') - (p_1 + p_2 + p_3 + p_4)$ by similar reasoning as above for ensuring 2.) holds along with using $R\geq \frac{\mu_{\mathrm{Jac}}^{(F)}}{2\beta_F}$ : + +$$ +\begin{array}{l} \underbrace {\left(| c _ {0} | \sqrt {(1 + \delta_ {2}) d _ {1} ^ {(F)} n}\right) \cdot \sigma_ {\max} (A) \cdot (\delta_ {3} ^ {\prime} k _ {1} ^ {\prime} k _ {2} ^ {\prime} \sigma_ {\max} (X))} _ {(L H S)} \lesssim \frac {1}{3 2} \left(\frac {\mu_ {\mathrm {J a c}} ^ {(F)}}{2 \beta_ {F}}\right) ^ {2} \\ \Rightarrow (L H S) \lesssim \frac {\left(\sigma_ {1} ^ {(F)}\right) ^ {2 r _ {1}} \left(1 - \delta_ {1}\right) \left(c _ {t} ^ {2} / t!\right) d _ {1} ^ {(F)} \sigma_ {\min } ^ {2} \left(X ^ {* t}\right)}{2 \left(\sigma_ {\max } (X)\right) ^ {2} \left(\dot {\psi} _ {\max } + \ddot {\psi} _ {\max } \chi_ {\max }\right) ^ {2}} (\text {B y L e m m a G . 1}) (137) \\ \therefore d _ {1} ^ {(F)} \gtrsim \frac {\left(| c _ {0} | ^ {2} (1 + \delta_ {2}) n\right) \cdot \left(\sigma_ {\max } (X)\right) ^ {4} \left(\dot {\psi} _ {\max } + \ddot {\psi} _ {\max } \chi_ {\max }\right) ^ {4} \cdot \sigma_ {\max } ^ {2} (A) \cdot \left(\delta_ {3} ^ {\prime} k _ {1} ^ {\prime} k _ {2} ^ {\prime} \sigma_ {\max } (X)\right) ^ {2}}{\left(\sigma_ {1} ^ {(F)}\right) ^ {2 + 2 r _ {1}} \left(1 - \delta_ {1}\right) ^ {2} \left(c _ {t} ^ {4} / (t !) ^ {2}\right) \sigma_ {\min } ^ {4} \left(X ^ {* t}\right)} (138) \\ \Rightarrow d _ {1} ^ {(F)} \gtrsim \xi (A, C _ {(\delta , \delta^ {\prime})}, t, \psi , \{c _ {i} \} _ {i \geq 0}) \frac {n \sigma_ {\max } ^ {6} (X)}{\sigma_ {\min } ^ {4} (X ^ {*} t)} (139) \\ \end{array} +$$ + +Again, using arguments from Oymak and Soltanolkotabi [86] above, we get a similar cubic overparameterization bound for the 'min' player, $F_{\theta}$ , as above: + +$$ +d _ {1} ^ {(F)} d _ {0} ^ {(F)} = \tilde {\Omega} (n ^ {3}) \tag {140} +$$ + +Applying reasoning analogous to that for ensuring (2.) to the case of ensuring (4.) holds, by using $R \geq \frac{\mu_{\mathrm{fac}}^{(G)}}{2\beta_G}$ we get w.p. $\geq 1 - (p_1 + p_5) - (p_1' + p_2' + p_3' + p_4')$ : + +$$ +d _ {1} ^ {(G)} \gtrsim \xi \left(A, L _ {\mathcal {L}}, C _ {\delta} ^ {\prime}, t ^ {\prime}, \psi , \left\{c _ {i} ^ {\prime} \right\} _ {i \geq 0}\right) \frac {n \sigma_ {\max } ^ {6} (X)}{\sigma_ {\min } ^ {4} \left(X * t\right)} \tag {141} +$$ + +Finally, as was done for the case of ensuring 1.), 3.) and 4.) above, we once again arguments from Oymak and Soltanolkotabi [86] for spectral properties of the training data $\mathcal{D}$ and get a similar cubic overparameterization bound for the 'max' player, $G_{\phi}$ , as well: + +$$ +d _ {1} ^ {(G)} d _ {0} ^ {(G)} = \tilde {\Omega} (n ^ {3}) \tag {142} +$$ + +Thus, w.p. $\geq 1 - (p_1' + p_5') - (p_1 + p_2 + p_3 + p_4)$ , we stay within a ball around the random initializations $\mathcal{B}((\theta_0, \phi_0), R)$ thereby ensuring min-max objective satisfies 2-sided PL-condition. By Lemma 3.2, given that we have chosen appropriate fixed learning rates as per Lemma E.7, we are now guaranteed to reach the saddle point. + +![](images/c9bb9ec0a73d2b730799e6208ad1d2520cc51814ef751d7d6b5296c273784a7f.jpg) + +Remark G.4. The proof above requires activation function to not be an odd function for ensuring $c_0 \neq 0$ . + +Remark G.5 (Effects of assumption about $\sigma_{\mathrm{max}}(X)$ on amount of overparameterization). As noted in the footnote pertaining to $\sigma_{\mathrm{max}}(X)$ for the data matrix $X$ in Assumption 2.5, if all we know about the data matrix is that it's row-normalized and that it's not a random matrix (e.g. random Gaussian matrix), then we can conclude that $\sigma_{\mathrm{max}}(X) = O(\sqrt{n})$ . Using this bound on the maximum singular value of the data matrix instead, we can conclude from Equations (132), (135), (139), and (141) that + +$$ +d _ {1} ^ {(F)} = \tilde {\Omega} \left(n ^ {4}\right) \tag {143} +$$ + +$$ +d _ {1} ^ {(G)} = \tilde {\Omega} (n ^ {4}) \tag {144} +$$ + +Since we have $n \simeq (d_0^{(F)})^t$ ( $t \geq 2$ ) in practice for the MIN player $F_{\theta}$ (analogously, $n \simeq (d_0^{(G)})^t$ for the MAX player $G_{\phi}$ ), we can conclude that the amount of overparameterization needed for both the players is more than the cubic overparameterization when the data matrix is, for example, an i.i.d. random Gaussian matrix: + +$$ +d _ {1} ^ {(F)} d _ {0} ^ {(F)} = \tilde {\Omega} \left(n ^ {4 + 1 / t}\right) \tag {145} +$$ + +$$ +d _ {1} ^ {(G)} d _ {0} ^ {(G)} = \tilde {\Omega} \left(n ^ {4 + 1 / t}\right) \tag {146} +$$ + +# H Clarifications + +# H.1 Smoothness on variables or Map + +In the main paper, we state the following assumption regarding the objective function: + +Assumption H.1 (Smoothness, Hidden Strong Convexity, and Gradient Control). + +(i) Smoothness: Each sample-wise loss $\ell(y, h = \operatorname{Map}_w(x))$ is differentiable and $L$ -smooth with respect to $h$ . +(ii) Coupling Structure: Each bilinear coupling matrix $A(x_{i},x_{j},y_{i},y_{j})$ is known, fixed, and has bounded operator norm. +(iii) Hidden Strong Convexity: Each sample-wise loss $\ell(y, h = \mathrm{Map}_w(x))$ is strongly convex with respect to the neural network output $h$ . +(iv) Gradient Growth Condition: There exist constants $A_{1}, A_{2}, A_{3} > 0$ such that for all $h \in \mathbb{R}^{d_{\mathrm{out}}}$ and $y \in \mathcal{V}$ , the latent gradient of each loss satisfies: + +$$ +\left\| \nabla_ {h} \ell (y, h) \right\| \leq A _ {1} \| h \| + A _ {2} \operatorname {d i a m} (\mathcal {Y}) + A _ {3}. +$$ + +To compute the smoothness of the composition $\ell(y, \operatorname{Map}_w(x))$ with respect to $w$ , we invoke the following classical result11: + +Lemma H.3 (Composition Smoothness, adapted from Proposition 2(c) in [131]). Let $f: \mathbb{R}^d \to \mathbb{R}$ be a closed, convex function that is $L_f$ -locally-Lipschitz on a Euclidean ball $\mathcal{B}(x_0, R)$ for some fixed $x_0 \in \mathbb{R}^d$ and $R > 0$ and let $g: \mathbb{R}^n \to \mathbb{R}^d$ be an $L_g$ -smooth function. Then, the composition $f \circ g$ is $L_fL_g$ -smooth in the Euclidean ball $\mathcal{B}(x_0, R)$ . + +# H.2 Refined Upper Bound Expression for Lyapunov Potential $P_0$ + +For clarity and presentation purposes, the main paper presents a simplified—yet qualitatively accurate—upper bound on the initial potential $P_0$ . In appendix, we provide the precise formulation that captures the correct dependence on constants and exponents. For completeness, we restate both results below: + +The key distinction in the refined expression lies in the appearance of additional quadratic terms. As we show in the accompanying proof, these higher-order contributions can be effectively controlled through suitable initialization and appropriately chosen step sizes. Thus, the improved bound yields tighter theoretical insight without compromising practical applicability. + +Remark H.2. This adaptation is necessary, as a function cannot simultaneously be globally Lipschitz (bounded gradients) and strongly convex (increasing gradient norm) in the unconstrained setting. However, under sufficient overparameterization and appropriate random initialization, we show that the AltGDA iterates remain within a bounded region where these properties hold locally. Thus, our assumption of local Lipschitzness of latent loss, the induced smoothness and the hidden strong convexity within a Euclidean ball is both theoretically consistent and empirically valid. + +# Main Paper Version: + +Lemma 3.3 (Upper Bound on Initial Potential $P_0$ ). Suppose the min-max objective $\mathcal{L}(\theta, \phi)$ is $L_{\mathcal{L}}$ -Lipschitz and satisfies a two-sided PL condition with constants $(\mu_\theta, \mu_\phi)$ . Then the initial Lyapunov potential $P_0 \leq L_{\mathcal{L}} \left( C_1 \cdot \| \nabla_\theta \mathcal{L}(\theta_0, \phi_0) \| + C_2 \cdot \| \nabla_\phi \mathcal{L}(\theta_0, \phi_0) \| \right)$ , where $C_1, C_2 = \Theta \left( L_{\mathcal{L}} / \mu_\theta^3 \right)$ . + +# Appendix Version (Refined): + +Lemma E.4 (Upper Bound on Initial Potential $P_0$ ; Lemma 3.3 in Main Text). Let $F_{\theta}$ and $G_{\phi}$ be neural network mappings such that they are $\beta_F$ and $\beta_G$ smooth as defined in Definition 2.3. Now let $(\theta_0,\phi_0)$ be such that Jacobian singular values for both the networks are strictly positive and bounded from above and below, $\mu_{Jac}^{(F)}\leq \sigma (\nabla_{\theta}F_{\theta_0})\leq \nu_{Jac}^{(F)}$ and $\mu_{Jac}^{(G)}\leq \sigma (\nabla_{\phi}G_{\phi_0})\leq \nu_{Jac}^{(G)}$ . Suppose the min-max objective $\mathcal{L}(\theta ,\phi)$ is $(\mu_{\theta},\mu_{\phi})$ -HSCSC. Then the initial Lyapunov potential $P_0$ can be bounded from above as: + +$$ +\begin{array}{l} P _ {0} \leq \| (\nabla_ {F _ {\theta}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}})) ^ {\top} \| \cdot \nu_ {J a c} ^ {(F)} \cdot \frac {1}{\mu_ {\theta} (\mu_ {J a c} ^ {(F)}) ^ {2}} \cdot \| \nabla_ {\theta} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| \\ + \left\| \left(\nabla_ {F _ {\theta}} \mathcal {L} \left(F _ {\theta_ {0}}, G _ {\phi_ {0}}\right)\right) ^ {\top} \right\| \cdot \beta_ {F} \frac {1}{2 \mu_ {\theta} ^ {2} \left(\mu_ {J a c} ^ {(F)}\right) ^ {4}} \cdot \left\| \nabla_ {\theta} \mathcal {L} \left(\theta_ {0}, \phi_ {0}\right) \right\| ^ {2} \tag {36} \\ + (1 + \lambda) \cdot \| \left(\nabla_ {G _ {\phi}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}})\right) ^ {\top} \| \cdot \nu_ {J a c} ^ {(G)} \cdot \frac {1}{\mu_ {\phi} (\mu_ {J a c} ^ {(G)}) ^ {2}} \cdot \| \nabla_ {\phi} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| \\ + (1 + \lambda) \cdot \| (\nabla_ {G _ {\phi}} \mathcal {L} (F _ {\theta_ {0}}, G _ {\phi_ {0}})) ^ {\top} \| \cdot \beta_ {G} \frac {1}{2 \mu_ {\phi} ^ {2} (\mu_ {J a c} ^ {(G)}) ^ {4}} \cdot \| \nabla_ {\phi} \mathcal {L} (\theta_ {0}, \phi_ {0}) \| ^ {2} \\ \end{array} +$$ + +# H.3 Random Initialization: How restrictive are our results? + +We note the following for both the input games and neural games regarding random initializations: + +1. Neural Games: Our main result (Theorem 3.8) assumes a commonly used initialization scheme (e.g., He or LeCun), with the only additional requirement being that the variances satisfy Equation (9) in the main text. Of course, bridging the gap between practice and theory, it remains an interesting open question whether the current polynomial overparameterization requirement can be reduced to linear. However, we note that - prior to our work - there were no existing theoretical results connecting initialization schemes with convergence guarantees in neural min-max games. +2. Input Games: In this setting, our results are even less restrictive regarding initialization. Specifically, we show that if the neural networks are randomly initialized from a standard Gaussian distribution, then AltGDA can compute the min-max optimal inputs. As discussed above, this is an average-case result, in the spirit of smoothed analysis. While it is theoretically expected that there exist neural architectures encoding hard min-max landscapes—potentially close to PPAD-hard instances—in practice, our result suggests that a randomly initialized neural min-max game is tractable under AltGDA. + +# I Discussion & Future Work + +This work initiates a principled framework for understanding optimization in hidden convex-concave min-max games, a setting central to the theory and practice of modern machine learning. By bridging overparameterization with spectral geometry and alternating dynamics, we show how convergence and equilibrium stability can emerge from architectural design and initialization. Beyond offering the first non-asymptotic guarantees for such hidden structures, our analysis reveals the potential of min-max learning as a structured alternative to unconstrained overfitting. We hope these insights inspire a broader rethinking of how optimization, architecture, and strategic interaction coalesce in scalable intelligent systems. + +In this last section, we reflect on our overparameterization bounds in comparison to known results from single-agent minimization, and outline promising directions for future research. + +While our analysis shares surface-level similarities with techniques from classical minimization problems, there are crucial structural differences. In the single-agent case, convergence is often guaranteed by showing that the Neural Tangent Kernel (NTK) remains well-conditioned near initialization, and that the loss is already small—a zero-order property. + +By contrast, computing a Nash equilibrium in a min-max setting is inherently more subtle: the solution concept is first & second-order. Rather than merely minimizing a loss, we seek to simultaneously drive the gradients of both players to zero while respecting their opposing incentives—capturing the saddle-point nature of equilibrium. Consequently, our notion of being "close to optimality" requires the initialization to yield not only small gradient norms but also a geometric alignment between the descent and ascent directions. This structural gap underpins the difference in overparameterization requirements between the single-agent and multi-agent cases. + +These insights naturally give rise to several open questions from deep learning and game theory point of view: + +# I.1 Deep Learning Perspective: + +# - Sharper Overparameterization Bounds: + +- Can we tighten the width-depth trade-offs for hidden min-max games, especially in non-bilinear or partially observable regimes? +- Our results suggest that overparameterization smooths the optimization landscape in simple bilinear games, but the precise scaling with respect to hidden-layer width and depth in nonlinear architectures remains unclear. A possible direction is to characterize phase transitions in convergence when width exceeds a critical threshold that depends on the spectral complexity or curvature of the game operator. + +# - Beyond Width and Smoothness: + +- How does the depth of the architecture influence convergence in hidden games? Can we extend current results to networks with non-smooth activations such as ReLU, using tools beyond gradient-Lipschitz analysis? +- Smoothness assumptions simplify the analysis but obscure the behavior of realistic neural dynamics. We could instead exploit techniques from non-smooth dynamical systems and proximal envelope theory to handle non-smooth losses. +- Another open question is whether depth induces implicit averaging effects similar to stochastic smoothing, thereby stabilizing the dynamics in min-max training. + +# - Structural Guarantees in Multi-Agent PL Games: + +- What are the necessary structural properties of multi-agent PL-type games that ensure convergence to a unique Nash equilibrium? The challenge lies in extending single-agent PL conditions to multi-agent systems where gradients interact. Are there generalized PL inequalities that capture cross-agent monotonicity? +- Investigating block-wise or coupled PL structures could reveal when independent gradient updates mimic joint gradient descent-ascent in strongly monotone regimes. + +# Computational Complexity: + +- What is the inherent hardness of solving hidden convex-concave games with overparameterized models? Can we characterize tractable subclasses? +- The inversion or injectivity verification of neural networks is known to be NP- or coNP-complete even for ReLU architectures (see, e.g., the COLT'25 paper "Complexity of Injectivity and Verification of ReLU Neural Networks"). How do such barriers propagate to the equilibrium computation problem when game payoffs are defined implicitly through network mappings? +- The overparameterized setting often introduces implicit convexification. Can we formalize when this leads to provable polynomial-time convergence versus when training remains PPAD-or NP-hard? + +# - From Neural Networks to Transformers: + +- What explains the observed differences in scaling laws between overparameterized feedforward networks and attention-based architectures in game-theoretic learning? Could these insights inform AI alignment and debate frameworks? +- Transformers introduce dynamic reweighting of information through attention, which may alter the effective conditioning of the game Jacobian. +- Understanding how self-attention layers modify optimization stability and expressivity could illuminate why Transformers exhibit faster equilibration or better robustness to adversarial perturbations. + +# - Beyond Full Gradient Feedback: + +- Much of the current analysis, including ours, assumes full gradient information. It remains an open and critical question whether similar benefits of overparameterization persist under bandit or partial-information settings. +- A natural direction is to extend existing results to stochastic feedback or bandit-gradient estimators using variance-controlled or mirror-descent methods. +- Another fundamental question: does overparameterization implicitly reduce the variance of policy-gradient estimators by averaging across redundant feature paths? + +# I.2 Game Theory Perspective: + +# - CCE with Neural Parametrizations (Normal-Form): + +- Representational question: For a class of neural correlating devices $g_{\theta}(z)$ that map public randomness $z$ to joint actions, characterize when the induced set of implementable distributions is dense in the CCE polytope. What overparameterization (width/depth) suffices for $\varepsilon$ -dense coverage uniformly across $n$ -player games with bounded payoffs? +- Optimization vs. Calibration: Standard no-regret dynamics imply convergence to CCE in the tabular case. With function approximation (shared neural critics/policies), give conditions (e.g., uniform stability, gradient-calibration bounds) under which the averaged joint play converges to an $\varepsilon$ -CCE at rates that improve with network width (via better optimization landscapes) without blowing up statistical complexity. +- Implicit bias: In the overparameterized (NTK) regime, training to minimize regret surrogates biases the joint distribution toward low-complexity mixtures. Can we quantify the implicit regularization toward "simple" CCEs (few extreme points in support) as a function of width, depth, and training dynamics? + +# - CCE in Extensive-Form Games (EFG): EFCE,CEFCE,NFCCE via Nets. + +- Sequential structure: Compare neural parameterizations for (i) NFCCE (normal-form coarse CCE), (ii) EFCE (extensive-form CE), and (iii) CEFCE (coarse EFCE). Give width/depth conditions under which sequence-form constraints (flow and realization-plan consistency) can be enforced by differentiable layers with projection or Lagrangian penalties. +- Counterfactual losses: Can counterfactual regret minimization with neural policies/critics be shown to converge to EFCE/CEFCE when critics are overparameterized but trained with regularized Bellman residuals? Identify structural conditions (perfect recall, bounded branching, Lipschitz counterfactual values) that guarantee $\tilde{O}(1/\sqrt{T})$ exploitability while using batched, partial counterfactual feedback. + +- Abstraction without tears: Overparameterized policies can represent rich information-set strategies directly. Develop "learned abstraction" bounds: when does a deep policy with attention over histories match the exploitability of hand-crafted abstractions, and what width/depth yields $\varepsilon$ -EFCE support recovery? + +- Markov (Stochastic) Games: Stationary CCE and Overparameterized Critics. + +- Stationary CCE: Define a stationary CCE as a joint policy $\pi$ with a correlating signal that is time-consistent across states. Give conditions under which joint no-regret learning with overparameterized $Q$ -critics converges (in Cesàro average) to an $\varepsilon$ -stationary CCE, with rates that depend on mixing/concentrability constants and the critic class capacity. +- Depth helps bootstrapping: Hypothesis: deeper critics reduce the Bellman residual floor achievable by gradient methods, improving optimization error; width controls realizability. Provide a decomposition of the CCE gap into Approximation $(\mathcal{F}_Q) + \text{Optimization}(\theta) + \text{Statistical}(\mathcal{F}_Q, \text{data})$ , and bound each term as a function of network depth/width and exploration coverage. +- Temporal correlation: Analyze how correlating devices that are state-dependent (learned correlation) interact with Markovian dynamics. When does limited-bandwidth correlation (few bits per step) suffice for $\varepsilon$ -CCE in ergodic MGs? + +- Overparameterization vs. Bellman-Eluder (BE) / Bellman Rank Complexity. + +- Function-class lens: Let $\mathcal{F}_Q$ be the $Q$ -function class realized by a network in the linearized (NTK) regime. Its eluder dimension equals the feature dimension for linear classes; for deep nets, the effective eluder dimension depends on width, depth, and implicit regularization. Conjecture: with proper norm control (weight decay/early stopping), the effective BE dimension scales like the stable rank of the induced features, not the raw parameter count. +- Sample complexity for $\varepsilon$ -CCE: In two-player zero-sum Markov games with realizable $Q^{*} \in \mathcal{F}_{Q}$ , the sample complexity to reach $\varepsilon$ -CCE under fitted Q-type updates should scale as + +$$ +\tilde {O} \Big (\frac {\mathrm {B E} - \dim (\mathcal {F} _ {Q}) + \log (1 / \delta)}{(1 - \gamma) ^ {p} \varepsilon^ {2}} \Big), +$$ + +for a small integer $p$ depending on the algorithm (e.g., $p \in \{2,3\}$ ), assuming standard concentrability/mixing. Overparameterization per se does not hurt if the BE dimension is controlled by implicit/explicit regularization. + +- Width-complexity trade: Identify regimes where increasing width improves optimization (faster approach to $\theta^{*}$ ) while keeping BE-dimension nearly constant via margin-based or path-norm constraints, yielding strictly better time-sample trade-offs. +- Transformers vs. MLPs: Attention layers can implement dynamic state-action feature selection, potentially lowering the BE dimension for tasks with sparse predictive structure (long-range but low "intrinsic" rank). Formalize when attention reduces BE-dim relative to equally wide MLPs, explaining empirical scaling differences in Markov game benchmarks. + +- Bandit/Partial Information Extensions: + +- Bandit CCE: For normal-form/Markov games with bandit feedback, derive $\varepsilon$ -CCE rates when both policies and correlating devices are neural. Key ingredient: variance-controlled gradient surrogates (e.g., doubly-robust or control-variates) to keep estimation error compatible with a bounded BE dimension. +- Information complexity of correlation: Quantify the information budget (bits of correlation, episodes of exploration) required to learn $\varepsilon$ -CCE as a function of BE dimension and mixing; relate to communication complexity of implementing the equilibrium. + +Conclusion. Our study sheds new light on the geometry of overparameterized min-max optimization by establishing the first precise convergence guarantees in hidden bilinear games. We demonstrate how overparameterization not only facilitates convergence but also implicitly regularizes the optimization landscape, ensuring robustness through well-conditioned spectral structure. Crucially, we bridge the gap between neural initialization and equilibrium computation, revealing that convergence in adversarial training is governed by deeper geometric principles. 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However, most current solutions, whether fulfilling computations in frequency, Laplacian, and wavelet domains, all deviate far from the intrinsic PDE space. While with meticulous network architecture elaborated, the deviation often leads to biased accuracy. To address the issue, we open a new avenue that pioneers leveraging Radon transform to decompose the input space, finalizing a novel Radon neural operator (RNO) to solve PDEs in infinite-dimensional function space. Distinct from previous solutions, we project the input data into the sinogram domain, shrinking the multi-dimensional transformations to a reduced-dimensional counterpart and fitting compactly with the PDE space. Theoretically, we prove that RNO obeys a property of bilipschitz strongly monotonicity under diffeomorphism, providing deeper insights to guarantee the desired accuracy than typical discrete invariance or continuous-discrete equivalence. Within the sinogram domain, we further evidence that different angles contribute unequally to the overall space, thus engineering a reweighting technique to enable more effective PDE solutions. On that basis, a sinogram-domain convolutional layer is crafted, which operates on a fixed $\theta$ -grid that is decoupled from the PDE space, further enjoying a natural guarantee of discrete invariance. Extensive experiments demonstrate that RNO sets new state-of-the-art (SOTA) scores across massive standard benchmarks, with superior generalization performance enjoyed. Code is available at https://github.com/wenbin-lu/Radon-Neural-Operator. + +# 1 Introduction + +PDE solving is considered an important field of modern mathematics. Although traditional finite element methods (FEM) [5] and finite difference methods (FDM) [35] have been shown to yield notable outcomes, they require substantial computational resources and careful discretization, a process necessitating specialized expertise. In addition, certain transformations have been mathematically proven to exhibit superior performance in the realm of PDE solving, including Fourier, Wavelet, and Laplace [16, 7, 46]. Such transformations, though useful to provide some specific merits in other domains, often fail to preserve the intrinsic geometry of the original PDE space. + +Recently, the paradigm has shifted to a data-driven pattern. Owing to the inherent capability in discerning behaviors of diverse PDEs directly from data, newly elaborated solvers have achieved remarkable improvements in speed, robustness, and accuracy. These methods, categorized under operator learning framework-particularly neural operator (NO)-establish explicit mappings from input conditions, e.g., initial/boundary configurations, to PDE solutions in infinite-dimensional function spaces. The most pioneering work was presented in [18], which provides a mathematically rigorous alternative to traditional numerical methods by preserving operator continuity in Banach spaces. + +Current NOs predominantly fall into two branches. The first encompasses transformation-based approaches derived from numerical analysis techniques, including Fourier neural operators (FNO) [23], Laplace neural operators (LNO) [3], and Wavelet neural operators (WNO) [37]. While effective, these methods inherently inherit the limitations of their underlying numerical schemes—they often fail to capture the inherent PDE solution space accurately, enlarging the original dimensionality and presenting scalability challenges for high-dimensional problems. The second endeavors to elaborate sophisticated neural architectures, such as employing transformer [40] or Mamba [9], yet with the intrinsic property of PDEs disregarded. To overcome the limitations, we propose the Radon neural operator (RNO). In contrast to previous domains, we project the input data into the sinogram domain, achieving dimensionality reduction while maintaining strong alignment with the original PDE solution space. Building upon this, we rigorously prove that RNO satisfies the bilipschitz strong-monotone property from a diffeomorphism perspective, theoretically providing a deeper equivalence guarantee than traditional discrete invariance [18] or continuous-discrete equivalence [2]. + +During NO evolution, the pursuit of holistic features is known as the most critical point. FNO and LNO enjoy strongly the global property due to the employment of convolution operation in spectral domain, coupled with the suppression of high-frequency components, yet suffer from the loss of localized information. This poses significant challenges for problems rich in local features, such as shock waves in hyperbolic PDEs or steep gradients in elliptic PDEs. Conversely, convolutional neural operator (CNO) [34] focuses on local features but struggles with global feature extraction. For more balanced feature extraction, several attempts have been performed to combine FNO or LNO with differential and integral operators [25], albeit at the cost of compromised generalization capability. + +ity. Generally, Radon transform (RT) holds similar properties to Fourier transform, yet with less popularity enjoyed. This fact has long plagued the naive use of RT in the context of NO learning. + +![](images/96babbe58dc568ec19c1d8174acbbc5d1403a7174b21042cf08040969f8b1758.jpg) +Figure 1: Comparison of feature holism from different architectures. + +In this study, we have dug out that different angles within the sinogram domain contribute unequally to the final feature representation, which motivates us to innovate an angle-reweighting technique striving for an effective PDE solution. On that basis, we enrich our elaboration by incorporating sinogram-domain convolution within the weighting network to better capture the holistic features. The sinogram convolution operates on a fixed $\theta$ -grid that is decoupled from the spatial resolution of PDE manifold, guaranteeing discrete invariance as confirmed by our generalization experiments. Furthermore, a physics-attention mechanism is integrated, leveraging the global capability of transformer, into RNO, ultimately leading to the complete architecture. Fig. 1 provides a visualized comparison of existing architectures on feature learning. Overall, our contributions are summarized as follows. + +- We pioneer the introduction of Radon transform into neural operators and, for the first time, perform weight analysis in the sinogram domain to solve partial differential equations. +- Following the Radon forward transformation, we elaborate a convolution operation in the weight network, which assists in the holistic learning of both global and local features. +- We rigorously prove that RNO is a bilipschitz operator, which ensures discretization invariance under diffeomorphism and avoids the introduction of any topological obstructions. Empirically, new state-of-the-art scores are earned by RNO across most PDE benchmarks. + +# 2 Preliminaries + +Mathematically, RT is proposed by Johann Radon [32], as an integral transform that maps a function $f$ , defined on Cartesian coordinates, to a function $Rf$ , defined on the plane-wise two-dimensional space of lines, where the value is determined by the line integral of the function along that direction, as generally illustrated in Fig. 2. In the figure, $s$ denotes the perpendicular distance from the line to the origin of the image coordinate system, $z$ represents the projection distance, and $\alpha$ indicates the orientation. The normal vector $\vec{n}$ characterizes the line direction. + +![](images/aaef216aee977f6b7c000a5ca9f61b4ad117d4ef9a39a440c3338567c11c8039.jpg) +Figure 2: Radon transform maps $f$ from $(x,y)$ to $Rf$ in $(\alpha ,s)$ . + +In the sequel, the Radon transform in its high-dimensional form is initially formulated, from which the essential dimension-reduction property becomes manifest [7]. For simplicity, we denote $S^{n - 1}$ for the unit sphere $\partial B(0,1)$ in $\mathbb{R}^n$ , a typical point of which is represented as $\omega = (\omega_{1},\dots ,\omega_{n})$ . Then, the plane with unit normal $\omega \in S^{n - 1}$ at a distance $s\in \mathbb{R}$ from the origin can be written as follows: + +$$ +\Pi (s, \omega) := \left\{\mathbf {y} \in \mathbb {R} ^ {n} | \mathbf {y} \cdot \boldsymbol {\omega} = s \right\}. +$$ + +Definition 1. The Radon transform $\mathcal{R}u = \widetilde{u}$ of a function $u\in \mathbf{C}_c^\infty (\mathbb{R}^n)$ is given as + +$$ +\widetilde {u} (s, \omega) := \int_ {\Pi (s, \omega)} u d S, \quad w h e r e \omega \in S ^ {n - 1}, \mathbf {y} \cdot \boldsymbol {\omega} = s \in \mathbb {R}. +$$ + +The right term is the integral over plane $\Pi (s,\omega)$ with regard to $(n - 1)$ -dimensional surface measure. + +We mainly use the two-dimensional variant of the Radon transform, whose mathematical form is as: + +$$ +R f (\varphi , s) = \iint_ {\mathbb {R} ^ {2}} f (x, y) \delta (x \cos \varphi + y \sin \varphi - s) d x d y, \tag {1} +$$ + +where $f(x,y)$ is the input function, $\varphi$ is the angle of the normal vector to the line (typically $\varphi \in [0,\pi)$ ), $s$ is the signed distance from the origin to the line, and $\delta(\cdot)$ is the Dirac delta function restricting integration to the line $x\cos \varphi + y\sin \varphi = s$ . + +To achieve the concerned inverse transform of RT, we first reveal that a close relationship exists between the Radon transform and the Fourier transform. + +Theorem 1. (The connection between Radon and Fourier transforms) Assume $u \in \mathbf{C}_c^\infty(\mathbb{R}^n)$ , then + +$$ +\bar {u} (r, \omega) := \int_ {\mathbb {R}} \tilde {u} (s, \omega) e ^ {- i r s} d s = (2 \pi) ^ {n / 2} \hat {u} (r \omega) (r \in \mathbb {R}, \omega \in S ^ {n - 1}), +$$ + +where $\hat{u} = \mathcal{F}u$ is the Fourier transform. See Appendix A.2.1 for a detailed proof. + +Due to the close connection between the Radon and Fourier transforms, in mathematics it is posited by the projection-slice theorem (also known as the central slice theorem or the Fourier slice theorem in two dimensions) that the results of the following two calculations are deemed equivalent: + +- A two-dimensional function $f(r)$ is taken, projected onto a one-dimensional line (e.g., through the Radon transform) and subjected to a Fourier transform of the resulting projection. +- That same function is taken, subjected first to a two-dimensional Fourier transform, and subsequently sliced through its origin along a plane parallel to the projection line. + +Since inverting Fourier transform is already known, from Theorem 1 we likewise obtain RT inversion. + +Theorem 2. (RT Inversion) The inversion of RT is given as (See Appendix A.2.1 for proof.) + +$$ +u (x) = \frac {1}{2 (2 \pi) ^ {n}} \int_ {\mathbb {R}} \int_ {S ^ {n - 1}} \bar {u} (r, \omega) r ^ {n - 1} e ^ {i r w \cdot x} d S d r. +$$ + +However, yet with an analytical solution existing theoretically for the inverse Radon transform, the practical implementation faces significant challenges. Direct practices often result in reconstructed PDE manifold with pronounced blurring and amplified noise due to the inherent instability of this inverse problem. By applying a frequency-domain filter to the projection data prior to back projection, the filtered back projection (FBP) algorithm stands out as a natural solution to address the limitations [30]. With the theoretical foundation derived from the projection-slice theorem, FBP functionally stabilizes the computation process, effectively suppresses artifacts, and demonstrates particular robustness when handling discrete datasets and noisy acquisition scenarios. + +The mathematical form of the FBP algorithm can be expressed as: + +$$ +f (\mathbf {x}) = \int_ {0} ^ {\pi} \left(\mathcal {R} f (\cdot , \theta) * h\right) \left(\langle \mathbf {x}, \mathbf {n} _ {\theta} \rangle\right) d \theta , \tag {2} +$$ + +in which the convolution kernel $h$ , e.g., $\hat{h}(k) = |k|$ , is referred to as Ramp filter in some literature. In this article, the practical number of projection angles is established as a hyperparameter to accommodate various datasets, and it is demonstrated through experiments that, at ultra-low resolution, fewer angles are preferable to a greater number. We provide more details of FBP and the numbers of angle in Appendix A.2.2. + +![](images/71a5ef3550ddffa882dcdf2167eda5c6478d08ce12eac34bd0d91c5dceeaccd8.jpg) +Figure 3: The full architecture of RNO. Given the input data, non-local features are derived through physics-attention, followed by the operations of weight analysis and sinogram domain convolution with the aid of forward/inverse Radon transform, aiming at enriching the more holistic features. + +# 3 Methodology + +# 3.1 Problem Setting + +Our methodology strives for a mapping bridging two infinite-dimensional spaces from a finite collection of given input-output pairs. Let $D \subset \mathbb{R}^d$ be a bounded and open set, and let $\mathcal{X} = L^2(D; \mathbb{R}^{d_x})$ as well as $\mathcal{Y} = L^2(D; \mathbb{R}^{d_y})$ be individual Hilbert spaces of square-integrable functions with elements in $\mathbb{R}^{d_x}$ and $\mathbb{R}^{d_y}$ , respectively. Moreover, let $\mathcal{G}^\dagger : \mathcal{X} \to \mathcal{Y}$ be a (typically) non-linear mapping. We probe into maps $\mathcal{G}^\dagger$ that arise as the solutions of parametric PDEs. Assume that we are given observations $\{(a_j, u_j)\}_{j=1}^N$ where $a_j \sim \mu$ is an independent and identically distributed (i.i.d.) sequence from the probability measurement $\mu$ supported on $\mathcal{X}$ , and $u_j = \mathcal{G}^\dagger(a_j)$ is probably corrupted with noise. The goal is to elaborate an approximation of $\mathcal{G}^\dagger$ by forming a parametric map + +$$ +\mathcal {G}: \mathcal {X} \times \Theta \to \mathcal {Y} \quad \text {o r e q u i v a l e n t l y}, \quad \mathcal {G} _ {\theta}: \mathcal {X} \to \mathcal {Y}, \quad \theta \in \Theta , +$$ + +for certain finite-dimensional parameter space $\Theta$ , with the optimal choice $\theta^{\dagger} \in \Theta$ so that $\mathcal{G}(\cdot, \theta^{\dagger}) = \mathcal{G}_{\theta_{\dagger}} \approx \mathcal{G}^{\dagger}$ . This is a natural idea for learning in infinite dimensions as we can intuitively define a loss functional $C: \mathcal{Y} \times \mathcal{Y} \to \mathbb{R}$ and pursue a minimizer of the problem + +$$ +\min _ {\theta \in \Theta} \mathbb {E} _ {a \sim \mu} \left[ C (G (a, \theta), G ^ {\dagger} (a)) \right], +$$ + +which directly parallels the typical finite-dimensional configuration [39]. However, evidencing the existence of minimizers, especially in the infinite-dimensional environment, remains a challenging problem. We attempt to approach this issue in the test-train setting by using an empirical and data-driven approximation to the loss, determining the final $\theta$ and testifying the concerned accuracy. Recall that we conceptualize our methodology in the infinite-dimensional environment, hence all finite-dimensional approximations enjoy a shared parameter set that is consistent in infinite dimensions. + +# 3.2 Radon Neural Operator + +Neural Operator Architecture. Generally, a typical NO architecture is constructed from three primary components: lifting, iterative kernel integration, and projection [18], which is written as: + +$$ +\mathcal {G} _ {\theta}: \mathcal {Q} \circ \sigma_ {T} \left(L _ {T - 1} + \mathcal {K} _ {T - 1} + b _ {T - 1}\right) \circ \dots \circ \sigma_ {1} \left(L _ {0} + \mathcal {K} _ {0} + b _ {0}\right) \circ \mathcal {P}, \tag {3} +$$ + +where $\mathcal{P}:\mathbb{R}^{d_x}\to \mathbb{R}^{d_{v_0}}$ , $\mathcal{Q}:\mathbb{R}^{d_{v_T}}\to \mathbb{R}^{d_y}$ are the lifting and projection mappings, respectively. $L_{t}\in \mathbb{R}^{d_{v_{t + 1}}\times d_{v_t}}$ denote linear operators (matrices), $\mathcal{K}_t:\{v_t:D_t\to \mathbb{R}^{d_{v_t}}\} \to \{v_{t + 1}:D_{t + 1}\to$ $\mathbb{R}^{d_{v_{t + 1}}}\}$ represent kernel operators, and $b_{t}:D_{t + 1}\to \mathbb{R}^{d_{v_{t + 1}}}$ are bias functions. Acting as maps + +![](images/91cf332213535747d923841f06c48e638558a645442820466db5ecd9ef2c3561.jpg) +Figure 4: The flowchart of a Radon block. The input data first enters the sinogram domain through the Radon forward transform, where weight analysis is performed in the sinogram space to assign specific weights to each angle. Afterwards, the signal is projected back to the spatial domain through the FBP algorithm. Meanwhile, a skip connection is imposed to introduce more original information. + +$\mathbb{R}^{d_{v_{t + 1}}}\to \mathbb{R}^{d_{v_{t + 1}}}$ in each layer, $\sigma_t$ is a fixed activation function. Note that the output dimensions $d_{v_0},\dots ,d_{v_T}$ , the input dimensions $d_{1},\dots ,d_{T - 1}$ , and the domains of definition $D_{1},\dots ,D_{T - 1}$ are hyperparameters of the architecture. + +Kernel Integral Operator $\kappa$ . Following [18], the kernel integral operator for Eq. (3) is defined by + +$$ +\left(\mathcal {K} (a; \phi) v _ {t}\right) (x) := \int_ {D} \kappa (x, y, a (x), a (y); \phi) v _ {t} (y) d y, \quad \forall x \in D, \tag {4} +$$ + +in which $\kappa$ is usually parameterized by a neural network such that $\kappa_{\phi}:\mathbb{R}^{2(d + d_X)}\to \mathbb{R}^{d_v\times d_v}$ , with $\phi \in \Theta_{\mathcal{K}}$ . Functionally, $\kappa_{\phi}$ plays the role of a kernel integral which we learn from data. + +Radon Neural Operator. We propose to innovate the kernel integral operator in Eq. (4) with the Radon transform discussed in Section 2. Recall that Eq. (1) presents the concerned forward transform, by which the data are projected into the sinogram domain. Eq. (2) undergoes the inverse transform, by which the data are returned from the sinogram domain to the original domain. Unlike FNO, which draws inspiration from Green's function, the Radon neural operator is motivated by the impulse response. Concretely, it is defined that $\kappa(x,y,a(x),a(y);\varphi) = \delta(x\cos\varphi + y\sin\varphi - s)$ . Note $\varphi$ is the angle of the normal vector to the line (typically $\varphi \in [0,\pi)$ ), $s$ is the signed distance from origin to the line, and $\delta(\cdot)$ is the Dirac delta function restricting integration to the line $x\cos\varphi + y\sin\varphi = s$ . + +On that basis, Radon neural operator can be formally defined as follows. + +$$ +\left(\mathcal {K} (\phi) v _ {t}\right) (x) = \mathcal {R} ^ {- 1} \left(P _ {\phi} \cdot \left(\mathcal {R} v _ {t}\right)\right) (x) \quad \forall x \in D, \tag {5} +$$ + +where $P_{\phi}$ is a learnable parameter used to angle-wise impose different weights within the sinogram domain. Mathematically, data formation derived from various perspectives has been extensively investigated. Specific to the property of the Radon transform, an angle-based basis is considered, which differs from the global spectral bases used in FNO or LNO and the local wavelet used in WNO, providing an opportunity to learn the holistic features within a single transform. We will elaborate on the implementation of the detailed technique in Subsection 3.4. Note most currently well-known NOs can be consistently expressed by the following formula [1], which advocates the joint learning of global-local features, a property that our RNO naturally fits. + +$$ +\begin{array}{l} v (y) = \sigma (\mathcal {K} (a) (y)) \\ = \sigma \left(\int \kappa_ {G l o b a l} (y, x) v (x) d \mu + \int \kappa_ {L o c a l} (y, x) v (x) d \mu^ {\prime} + \mathfrak {D} a \right| _ {y} (y) + W a (y) + b (y)). \tag {6} \\ \end{array} +$$ + +On that basis, we consider that the non-local features also favor the final solution. Therefore, drawing inspiration from Ref. [43], we further engineer the physics-attention mechanism that decomposes the discretized domain into a series of learnable slices, within which attention is computed on physics-aware tokens. The overall network architecture with a general explanation is given in Fig. 3. Physical attention is specifically embodied in the kernel function in Eq. (6), given as follows. + +$$ +\kappa_ {\text {G l o b a l}} (y, x) = \sum_ {j = 1} ^ {M} \sum_ {k = 1} ^ {M} w (y, s _ {j}) \frac {\exp \left(W _ {Q} z _ {j} ^ {\top} W _ {K} z _ {k} / \sqrt {C}\right)}{\sum_ {p = 1} ^ {M} \exp \left(W _ {Q} z _ {j} ^ {\top} W _ {K} z _ {p} / \sqrt {C}\right)} W _ {V} \frac {w (x , s _ {k})}{\int_ {\Omega} w (\xi , s _ {k}) d \xi}. \tag {7} +$$ + +![](images/b2ce69aeb923d94c7f95556f4f43a57ec638ba99ed862aebd96ee6a75b0d8191.jpg) +Figure 5: The structure of sinogram-domain convolution. With a convolution operation performed in the sinogram domain, the local information from different angles can be further learned based on the global features of RT. The data sequentially passes through a chain of convolution-activation-convolution to achieve the weight distributions. + +where $\Omega$ is the computational domain, $M$ is the number of learnable slices, $s_j$ denotes the $j$ -th slice, $w(\cdot ,s_j)$ represents the soft assignment weight from any point to slice $s_j$ , $z_{j} = \int_{\Omega}w(\xi ,s_{j})v(\xi)d\xi /\int_{\Omega}w(\xi ,s_{j})d\xi$ is the physics-aware token of slice $s_j$ , $W_{Q},W_{K},W_{V}\in \mathbb{R}^{C\times C}$ are the query, key and value projections used in the token-wise attention, and $C$ is the channel dimension. + +Parameterizations of $P_{\phi}$ . With the Radon transform performed, the input data originally within the spatial domain are now converted to the sinogram domain, in which we further find that the contributions of different projection angles to the overall spatial representation are inherently non-uniform. Building upon this observation, we conceptualize the angle as a form of basis and introduce learnable parameters to dynamically adjust angular weights, thereby achieving more effective solutions. + +Quasi-linear Complexity. Given a discrete counterpart $u \in \mathbb{R}^{H \times W \times D}$ of a two-dimensional continuous function, it is firstly restructured, yielding $u \in \mathbb{R}^{N \times D}$ , where $N = H \times W$ . The forward Radon transform is then applied to project an image along $A$ angles, which necessitates $O(AN)$ computations owing to the rotation and summation of the image for each angle. Afterwards, the inverse Radon transform, specifically the FBP algorithm, performs a filter operation within the sinogram domain, converting the angular information into the frequency domain. This routine necessitates $O(A \times \max(H, W) \log \max(H, W))$ , followed by a back projection that accounts for a complexity of $O(AN)$ . Since that $A$ is usually a small value, i.e., $A \ll N$ , we consider the overall computational cost as quasi-linear complexity. The complexity of other well-known models is detailed in Appendix E. In practice, the runtime complexity of RNO is strongly dependent on the number of angles specified. Intuitively, the larger the angle parameter, the more effective the outcome is deemed. However, we have noticed that the larger number often results in a greater risk of overfitting. Practically, at ultra-low resolution, more pronounced the error occurs when we set a large $A$ value. Further details are elaborated in Appendix G.4. + +# 3.3 Discretization Invariance under Diffeomorphism + +In Ref. [8], a no-go theorem is proposed to explain the fundamental obstacle separating infinite and finite dimensions, along which the concept of diffeomorphisms and the property of discretization invariance are elaborated within the framework of category theory. Moreover, it proves that a bilipschitz NO layer can be expressed as a combination of strongly monotone neural operator layers with a simple isometry. In addition, the strongly monotone NO can be continuously approximated by strongly monotone diffeomorphisms in a finite-dimensional space. + +Following this theorem, in this study we prove that Radon operator actually behaves as a bilipschitz operator. Since RNO is primarily performed in 2D space, hence in the sequel, the informal proof for the two-dimensional case is provided. More details are referred to Appendix C.1. + +By leveraging the linear property of Radon transform, the definition of a bilipschitz operator in the context of RT is formally given as follows, in which $c$ and $C$ are some positive constants. + +$$ +c \| f \| _ {L ^ {2}} \leq \| R f \| _ {L ^ {2}} \leq C \| f \| _ {L ^ {2}}. \tag {8} +$$ + +Problem (8) can be further broken into two parts: the right upper bound and the left lower bound. + +Upper Bound: Within (8), the norm of $Rf$ is computed with the expression $\| Rf\|_{L^2}^2 = \int_{S^1}\int_{\mathbb{R}}|Rf(s,\theta)|^2 ds d\theta$ . Following the Fourier slice theorem, a 1D Fourier transform of $Rf(\cdot ,\theta)$ is given as $\mathcal{F}_1[Rf(\cdot ,\theta)](\sigma) = \hat{f} (\sigma \theta)$ . On that basis, the Plancherel theorem further leads us to $\int_{\mathbb{R}}|Rf(s,\theta)|^2 ds = \int_{\mathbb{R}}|\hat{f} (\sigma \theta)|^2 d\sigma$ , hence the norm of $Rf$ turns into $\| Rf\|_{L^2}^2 = \int_{S^1}\int_{\mathbb{R}}|\hat{f} (\sigma \theta)|^2 d\sigma d\theta$ . By further switching to polar coordinates, a key expression is deduced: + +$$ +\| R f \| _ {L ^ {2}} ^ {2} = 2 \int_ {\mathbb {R} ^ {2}} | \hat {f} (\xi) | ^ {2} \| \xi \| ^ {- 1} d \xi . +$$ + +The weight $\| \xi \|^{-1}$ is manageable since $\int_{|\xi| < 1} \| \xi \|^{-1} d\xi = 2\pi < \infty$ , which manifests that we can bound the integral as $\int_{\mathbb{R}^2} |\hat{f}(\xi)|^2 \| \xi \|^{-1} d\xi \leq C_1 \| f \|_{L^2}^2$ , leading to the upper bound with $C = \sqrt{2C_1}$ : + +$$ +\| R f \| _ {L ^ {2}} \leq \sqrt {2 C _ {1}} \| f \| _ {L ^ {2}}, +$$ + +Lower Bound: For the left part of (8), note first that $R$ is injective. That is, if $Rf = 0$ , then $\hat{f}(\sigma\theta) = 0$ , hence leading to $f = 0$ . Besides, the inverse $R^{-1}$ , roughly given by filtered back projection $f(x) = \int_{S^1}(H\partial_sRf)(x\cdot \theta ,\theta)d\theta$ , is bounded in $L^2$ , satisfying $\| R^{-1}g\|_{L^2}\leq K\| g\|_{L^2}$ . By applying this to $Rf$ , we get $\| f\|_{L^2} = \| R^{-1}(Rf)\|_{L^2}\leq K\| Rf\|_{L^2}$ , which refers to lower bound: + +$$ +\| R f \| _ {L ^ {2}} \geq \frac {1}{K} \| f \| _ {L ^ {2}}, +$$ + +with $c = \frac{1}{K}$ . Note that for future research, we also provide an $n$ -dimensional proof in Appendix C.1. + +# 3.4 Radon Block + +As mentioned, our methodology employs a fundamentally different design paradigm within the sinogram domain. The core component, i.e., Radon block, is illustrated in Fig. 4. According to Eq. (5), the data are first transformed from the original PDE space into the sinogram domain through the Radon forward transform, which achieves dimensionality reduction while preserving the essential information in the original space. From Fig. 4, we observe that while different angular lines are equally distributed, their valid coverages on the PDE manifold are severely distinct from each other. Moreover, different regions of the entire PDE manifold also contribute distinctly to the final representation. Therefore, it is naturally considered that the angular lines benefit from adaptive weight coefficients to favor a better reconstruction. + +To put the natural consideration into practice, we further elaborate a sinogram-domain convolution to perform the weight computation. The practical implementation is presented in Fig. 5, within whose left panel the horizontal and vertical axes respectively represent the spanned angles and the concerned density values. As seen, the actual convolution is performed on an inter-angle scope, adopting multi-line information to finalize the weight learning. While multiple lines are jointly considered, the covering span is limited by the convolution kernel, hence with the local features learned. This fact, together with the globality essence of RT and the non-local property of the phys-attention block, ensures a more holistic feature learning. Unlike conventional computations that depend closely on the spatial grids, our convolution is performed on a $\theta$ -space that is decoupled from the spatial resolution. Although the sampling interval $s$ varies with resolution, the sinogram convolution maintains strict consistency in the $\theta$ -direction. This key characteristic enables the model to share convolution kernels across different resolutions, thus further ensuring discrete invariance. Our generalization experiments provide empirical validation of this property. + +We proceed to define the mathematical formulation of sinogram-domain convolution. For a sinogram $S(\theta, t)$ , where $\theta$ represents the projection angle and $t$ the position along the detector, the sinogram domain convolution is defined as: + +$$ +S _ {\text {c o n v}} (\theta , t) = (S * K) (\theta , t) = \int_ {- \infty} ^ {\infty} S \left(\theta^ {\prime}, t\right) K \left(\theta - \theta^ {\prime}\right) d \theta^ {\prime}. \tag {9} +$$ + +Table 1: Performance comparison on standard benchmarks. Relative L2 is recorded. A smaller value indicates better performance. (Bold: Best performance, Underlined: Second best performance, ▲: Performance increase, ▼: Performance decrease, /' means that the model does not perform well on that dataset or that the model is not suitable for that benchmark.) + +
MODELMECHANISMREGULAR GRIDSTRUCTURED MESH
DarcyNavier-StokesAllen-CahnAirfoilPlasticityPipe
DEEPONET/5.88 × 10-22.97 × 10-1/3.85 × 10-21.35 × 10-29.70 × 10-3
FNOFourier Transform1.08 × 10-21.56 × 10-17.52 × 10-3///
WMTWavelet Transform8.20 × 10-31.54 × 10-11.12 × 10-27.50 × 10-37.60 × 10-37.70 × 10-3
GALERKINGalerkin Attention8.40 × 10-31.40 × 10-1/1.18 × 10-21.20 × 10-29.80 × 10-3
GNOTTransformer1.05 × 10-21.38 × 10-1/7.60 × 10-33.36 × 10-24.70 × 10-3
U-NOU-Net1.13 × 10-21.71 × 10-14.31 × 10-27.80 × 10-33.40 × 10-31.00 × 10-2
ONOOrthogonal Attention7.60 × 10-31.20 × 10-11.71 × 10-26.10 × 10-34.80 × 10-35.20 × 10-3
TRANSOLVERPhysic-Attention5.70 × 10-39.00 × 10-25.32 × 10-35.77 × 10-31.20 × 10-33.30 × 10-3
RNO (Ours)Radon Transform5.10 × 10-3▲8.94 × 10-2▲4.61 × 10-3▲4.90 × 10-3▲1.15 × 10-3▲4.20 × 10-3▼
+ +![](images/249b88aa5cacd5fa0dd65511f7428895a4033e3f312200828dbde1803794b828.jpg) +Figure 6: Case study on RNO and Transolver. The prediction results and errors are provided. + +![](images/96fcecf9befedd0ff51dfacd7e001dd023ed91dce2e9d71594904bb49034d6fb.jpg) +Ground Truth + +![](images/7d3e70e7543dc483af127db4acf721e1da5d855ac2559bd21382d9c15d53deac.jpg) +RNO Prediction + +![](images/b006d53a8349bb5015c7bb717b2ae98e56c901db0992398497d57d8657afcef9.jpg) +Transolver Prediction + +![](images/f195d858bd90e2ab0d977291624c72e1802da2c71a7fdeecb9b534d39e314ff4.jpg) +RNO Error + +![](images/5a96e68e1e2260337bc56df670cb0fc041709fe088b3492953a8e6cd5f187725.jpg) +Transolver Error + +Here, $K(\theta)$ is the convolution kernel, which slides along the $\theta$ -axis to combine neighboring angular values. We provide more details on sinogram-domain convolution in Appendix D. + +For the inverse Radon transform, the sinogram domain data are projected back to the spatial domain in accordance with the filtered back projection algorithm, as outlined in Section 2. To ensure better preservation of the original information, skip connections are introduced [12]. As discussed in the analysis of discrete invariance [8], skip connections play a critical role in preserving the bijectivity and strong monotonicity of neural operators, further guaranteeing that discretization challenges in infinite-dimensional spaces are avoided. + +# 4 Experiments and Analysis + +Training Details and Baselines. For fairness, all experiments are consistently conducted on a standardized platform with an NVIDIA GTX 4090 GPU and $2.10\mathrm{GHz}$ Intel(R) Xeon(R) Platinum 8352V CPU. Several well-known PDE solvers are used as the competing baselines, such as Deeponet[27], FNO[23], WMT[10], Galerkin[4], GNOT[11], ONO[44], U-NO[33], and Transolver [43]. + +Standard Benchmarks. To better compare with existing work, we performed experiments on several publicly available benchmarks, including Plasticity, Airfoil, Pipe with structured mesh and Navier-Stokes, Darcy, Allen-Cahn with regular grid. These benchmark datasets were extensively investigated in seminal works such as FNO [23], geometry-aware FNO (geo-FNO) [22], and WNO [37], and have since gained widespread adoption in the scientific machine learning community. We provide a more detailed description of these datasets in Appendix F. + +Implementation Details. All competing methods are trained with $l_{2}$ loss and 500 epochs. The ADAM [17] optimizer with an initial learning rate of $10^{-3}$ is used. For Radon transform, the main hyperparameters lie in the number of Radon blocks and the employed quantity of angles. Note the latter depends on the size of the input data to ensure that sufficient angular information is obtained. We provide more implementation details and hyperparameter configurations in Appendix G.1. + +# 4.1 Main Results + +In Table 1, we present a comprehensive comparison against existing approaches, in both cases of regular grid and structured mesh. As can be seen, for the regular scene, RNO consistently achieves state-of-the-art (SOTA) performance across most PDE benchmarks. Specific to equations of Darcy flow and Allen-Cahn, the improvements over the second-best competitors reach $10.5\%$ and $13.2\%$ , + +respectively. For irregular meshes, the conventional Radon transform is also applicable through a simple zero-padding operation. On the Airfoil benchmark, RNO achieves a $14\%$ performance gain compared to the second-best approach. However, the padding operation also incurs some redundancies; hence, our proposal lags behind Transolver slightly on the Pipe benchmark. + +# 4.2 Generalization. + +Recall that NOs focus on learning mappings between infinite-dimensional function spaces, pursuing inherent discretization invariance. Generalization capability serves as a crucial experimental manifestation of this fundamental property. To assess this property of the proposed model, we firstly trained RNO in a low-resolution setting, then performed zero-shot inference across higher-resolution settings. Specifically, the Darcy flow equation, originally within a resolution of $421 \times 421$ , was downsampled to sizes of $241 \times 241$ , $211 \times 211$ , $141 \times 141$ , $85 \times 85$ , $61 \times 61$ , and $43 \times 43$ . The training was conducted on the $43 \times 43$ resolution and thereafter assessed across the others. + +The concerned results comparing RNO with FNO are provided in Table 2. As seen, RNO achieves satisfactory generalization in all scales, enjoying performance improvements of roughly $70\%$ over that of the FNO. Fig. 7 and Fig. 8 further visually compares RNO with Transolver. While these two share a similar effect at lower resolutions, the performance of RNO, at very large scales, exceeds far from that of Transolver. Intuitively, it is demonstrated that RNO possesses strong generalization capabilities, which hold sig + +![](images/f608ae426531ac7a3e0625a7760d8e2ef136adc642e825b3737cbee2ee46b192.jpg) +Figure 7: Generalization comparison of RNO and Transolver. + +nificant practical value for super-resolution tasks, image enhancement, data-scarce scenarios, and transfer learning, underscoring its considerable research potential. More detailed explanations are offered in Appendix G.3. + +Table 2: Comparative display of generalization performance. Relative L2 is recorded. A smaller value indicates better performance. (Bold: Best performance) + +
Model61 × 6185 × 85141 × 141211 × 211421 × 421
FNO1.16 × 10-11.80 × 10-12.68 × 10-13.16 × 10-13.63 × 10-1
RNO (Ours)3.21 × 10-25.04 × 10-26.69 × 10-27.09 × 10-27.62 × 10-2
Relative Improvement72.42%71.95%75.03%77.56%79.01%
+ +# 4.3 Ablation Study + +We present ablation studies to validate the efficacy of the Radon block. Using the Darcy flow equation as an example, the results of different combinations are given in Table 3. Initially, we replaced the Radon block with alternative transformation methods, i.e., Fourier transform. Experimental results demonstrate that relying solely on global information yields suboptimal performance. We further conducted a bidirectional validation by substituting physics-attention mechanism (P.A.) with FNO. The results enjoy a $41.67\%$ performance improvement over the baseline FNO, which not only underscores the criticality of holistic information but also confirms that Radon block effectively captures the local features. Moreover, to demonstrate the significance of sinogram convolution, we have replaced it with a simple nonlinear weighting scheme. As evidenced in the 4th row of Table 3, the replacement suffers a performance reduction of nearly $7\%$ . To further demonstrate the substantial potential of Radon transform, we perform another ablation study by replacing the GPU-accelerated PyTorch Fourier transform in FNO with the non-accelerated counterpart. The results are given in Table 4, which together with Table 3 shows that the replacement leads to a significant reduction in efficiency of FNO, rendering it considerably inferior to RNO. This finding underscores RNO + +![](images/d44f2142bcb6167908b99259beab61c400845b5852573ff489dd6c07a8efe1ae.jpg) +Figure 8: Comparison of generalization between RNO and Transolver. We demonstrate generalization performance at two larger resolutions, with RNO significantly outperforming Transolver. The figure shows the error map of RNO, the error map of Transolver, input information and ground truth. + +as a promising avenue for future exploration in GPU acceleration. Due to space limitations, the experiments on different selections of angle numbers are provided in Appendix G.5. + +Table 3: Ablation studies of Radon block and sinogram domain convolution. All experiments were based on the $85 \times 85$ Darcy flow dataset and tested when the number of angles was set to 32. + +
AblationsMemory (GB)Time (s/epoch)Param (B)Relative L2 Darcy
P.A.+FNO2.8340.284.02 × 1068.84 × 10-3
FNO1.358.482.38 × 1061.08 × 10-2
FNO+Radon Block2.9225.582.38 × 1066.30 × 10-3
Only nonlinear weights3.1871.982.84 × 1065.79 × 10-3
Ours2.8737.882.83 × 1065.34 × 10-3
+ +Table 4: Comparison of Training and Inference Time between RNO and FNO without GPU Acceleration. (Bold: our method) + +
StageRNO (ours)FNO (w/o GPU opt.)
Training time (s/epoch)32.87196.79
Inference (s)3.0214.17
+ +# 5 Conclusions and Future Work + +For the efficient solving of PDEs, we propose RNO in this study, which employs Radon transform to reduce the spatial dimensionality of PDEs while preserving their intrinsic information. Theoretically, we prove that the proposed operator possesses a more profound bilipschitz strong-monotonicity, which further guarantees discrete invariance under diffeomorphism. Building upon this foundation, we perform weight analysis in the sinogram domain, within which a sinogram convolution is newly elaborated and integrated. Distinct from the normal convolution operation, the new proposal is grid-independent, guaranteeing again the discrete invariance. Extensive experiments conducted on multiple benchmarks demonstrate that our RNO achieves state-of-the-art performance. Moreover, the generalization experiments show significant improvements compared to baselines. 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Alias-free mamba neural operator. Advances in Neural Information Processing Systems, 37:52962-52995, 2025. +[48] Jianwei Zheng, Ni Xu, Wei Li, Jiawei Jiang, and Xiaoqin Zhang. Semantic-spatial attention for refined object placement in text-to-image synthesis. IEEE Transactions on Multimedia, pages 1-16, 2025. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction of the paper reflect the core arguments and ideas of the article and must be able to truly reflect the contribution and scope of the article. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The LLM is only used for polishing and will not influence the core methodology of the research. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix / Supplementary Material + +# Solving Partial Differential Equations via Radon Neural Operator + +# A Related Work + +# A.1 Neural Operator + +Neural operators [18] are regarded as a promising way to solve PDEs. Inspired by Green's function for solving PDEs, neural operators were pioneered through the incorporation of kernel integral operators, establishing mappings between infinite-dimensional function spaces. The most basic and well-known example is FNO [23]. On that basis, several excellent developments were derived. For instance, the U-FNO [42] integrates the U-Net network with the FNO, while the F-FNO [36] utilizes factorization in the Fourier domain. The Laplace transform, as the complex form of the Fourier transform, is introduced into LNO [3]. The multiwavelet-based operator [10] incorporates the principle of multiwavelets. This concept is generalized and leveraged to address arbitrary measures, thereby enabling the development of a series of models for operator learning from complex data streams. In addition to these classic mathematical methods, deep learning architectures have been incorporated into neural operators. For example, the famous transformer architecture [40] has given rise to several developments, including FactFormer [21], OFormer [20], GNOT [11], ONO [44], and Transolver [43]. Another example, the emerging Mamba architecture [9], has led to the development of the MambaNO [47]. + +To provide intuition into the distinctions and connections between RNO and related works, Table 5 offers a detailed comparison. + +Table 5: Comparison of different neural operators. + +
TransformGlobal Feature CaptureLocal Feature CaptureGeometry AwarenessDiscrete InvarianceComputational Cost
FFT (FNO)✓Excellent (global frequencies)✗ Poor (no locality)✗ Assumes periodicity✓Yes (spectral conv)Low (FFT + linear)
Wavelet (WNO)! Limited (coarse scales)✓Excellent (multi-scale)! Limited (fixed basis)✓Yes (multi-level)Moderate (filter bank)
Radon (RNO)✓Strong (line integrals)✓Tunable (θ-conv)✓Strong (line geometry)✓Yes (θ-grid decoupled)Moderate (O(AN))
+ +Key: $\checkmark$ = advantage, $! =$ moderate, $\times =$ weak or missing. + +# A.2 Supplement to Radon Transform + +# A.2.1 Proof Supplement + +(Radon and Fourier transforms) Assume that $u \in \mathbf{C}_c^\infty(\mathbb{R}^n)$ , then + +$$ +\bar {u} (r, \omega) := \int_ {\mathbb {R}} \tilde {u} (s, \omega) e ^ {- i r s} d s = (2 \pi) ^ {n / 2} \hat {u} (r \omega) (r \in \mathbb {R}, \omega \in S ^ {n - 1}) +$$ + +where $\hat{u} = \mathcal{F}u$ is the Fourier transform. + +Proof. Take $b_{1}, \dots, b_{n-1}$ to be an orthonormal basis of $\Pi(0, w)$ . Then + +$$ +\tilde {u} (s, \omega) = \int_ {\mathbb {R} ^ {n - 1}} u \left(\sum_ {j = 1} ^ {n - 1} y _ {j} b _ {j} + s \omega\right) d y +$$ + +and so + +$$ +\int_ {\mathbb {R}} \tilde {u} (s, \omega) e ^ {- i r s} d s = \int_ {\mathbb {R}} \int_ {\mathbb {R} ^ {n - 1}} u \left(\sum_ {j = 1} ^ {n - 1} y _ {j} b _ {j} + s \omega\right) e ^ {- i r s} d y d s +$$ + +By changing variables and rewriting $x \coloneqq \sum_{j=1}^{n-1} y_j b_j + s\omega$ , then we have + +$$ +\int_ {\mathbb {R}} \tilde {u} (s, \omega) e ^ {- i r s} d s = \int_ {\mathbb {R} ^ {n}} u (x) e ^ {- i r (x \cdot \omega)} d x = (2 \pi) ^ {n / 2} \hat {u} (r \omega) +$$ + +□ + +(Inverting the Radon Transform) We have + +$$ +u (x) = \frac {1}{2 (2 \pi) ^ {n}} \int_ {\mathbb {R}} \int_ {S ^ {n - 1}} \bar {u} (r, \omega) r ^ {n - 1} e ^ {i r w \cdot x} d S d r +$$ + +Proof. The process is sequentially deduced as follows. + +$$ +\begin{array}{l} \int_ {\mathbb {R}} \int_ {S ^ {n - 1}} \bar {u} r ^ {n - 1} e ^ {i r \omega \cdot x} d S d r = (2 \pi) ^ {n / 2} \int_ {\mathbb {R}} \int_ {S ^ {n - 1}} \hat {u} (r \omega) r ^ {n - 1} e ^ {i r \omega \cdot x} d S d r \\ = 2 (2 \pi) ^ {n / 2} \int_ {0} ^ {\infty} \int_ {S ^ {n - 1}} \hat {u} (r \omega) r ^ {n - 1} e ^ {i r \omega \cdot x} d S d r \\ = 2 (2 \pi) ^ {n / 2} \int_ {\mathbb {R} ^ {n}} \hat {u} (y) e ^ {i y \cdot x} d y \\ = 2 (2 \pi) ^ {n} u (x). \\ \end{array} +$$ + +![](images/beaa108ba80b4f31844b84c0d46b5e07e076279d0b13e5a8c76828e4a939c0fa.jpg) + +# A.2.2 Supplement to FBP Algorithm + +We have generally introduced the mathematical form of the FBP algorithm in the main texts. Here we elaborate on the implementation process of the FBP algorithm, which can be broken down into two main steps: + +- Filtering: The projection data, or sinogram, is processed with a high-pass filter, typically a Ramp filter, in the frequency domain. This step is designed to enhance high-frequency components, such as edges, thereby counteracting the blurring effect caused by back projection. The filter, which is often modified with a window function for practical purposes, ensures that the data are refined prior to reconstruction. +- Back Projection: The filtered projection data are subsequently back-projected, meaning the data are redistributed into the PDE space along the trajectories from which the projections were obtained. This process entails the accumulation of values within the pixels to reconstruct the original function, whereby information from all views is integrated to produce the final generation. + +# B Discretization Invariance under Diffeomorphism + +To commence with, the concept is initially defined regarding what it means for a nonlinear function $F: X \to X$ in an infinite-dimensional Hilbert space, $X$ , to be estimated by operators in finite-dimensional subspaces $V \subset X$ . + +Definition 2. $(\epsilon_V$ approximators and weak approximators) + +(i) Let $r > 0$ , $\mathcal{F} \subset C^n(X;X)$ be a family of operators or functions, and $\vec{\epsilon} = (\epsilon_V)_{V \in S_0(X)}$ be a series such that $\epsilon_V \to 0$ as $V \to X$ . We claim that a function + +$$ +\mathcal {A} _ {X} \colon \mathcal {F} \to X _ {V \in S _ {0} (X)} C (\overline {{B _ {V} (0 , r)}}; V), \quad F \to (F _ {V}) _ {V \in S _ {0} (X)} +$$ + +is an $\vec{\varepsilon}$ -approximation operation for functions $\mathcal{F}$ in the ball $B_{X}(0,r)$ assuming values in families $\mathcal{F}_V\subset C^1 (V;V)$ if $\mathcal{A}_X$ maps a function $F\colon X\to X$ , where $F\in \mathcal{F}$ , to a series of functions $(F_V)_{V\in S_0(X)}$ , where $F_{V}\in \mathcal{F}_{V}$ , so that the following is valid: For all $F\colon X\to X$ satisfying $\| F\|_{C^n (\overline{B_X(0,r)};X)}\leq M$ , we get + +$$ +\sup _ {x \in B _ {V} (0, r)} \| F _ {V} (x) - P _ {V} (F (x)) \| _ {X} \leq M \epsilon_ {V}, +$$ + +where $P_V\colon X\to X$ is the orthogonal projection onto $V$ , that is, $\operatorname{Ran}(P_V) = V$ . + +(ii) We further say that $\mathcal{A}\colon C^n (X;X)\to X_{V\in S_0(X)}C(V;V),\quad F\to (F_V)_{V\in S_0(X)}$ is a weak approximation operation for the function family $\mathcal{F}\subset C^n (X;X)$ if for any $F\in \mathcal{F}$ and $r > 0$ it enjoys that + +$$ +\lim_{V\to X}\sup_{x\in \overline{B_{V}(0,r)}}\| F_{V}(x) - P_{V}(F(x))\|_{X}\to 0. +$$ + +Definition 3. (Strongly Monotone) We claim that a (nonlinear) operator $F \colon X \to X$ on Hilbert space $X$ , is strongly monotone if there exists a constant $\alpha > 0$ such that + +$$ +\langle F (x _ {1}) - F (x _ {2}), x _ {1} - x _ {2} \rangle_ {X} \geq \alpha \| x _ {1} - x _ {2} \| _ {X} ^ {2}, \quad f o r a l l x _ {1}, x _ {2} \in X. +$$ + +Definition 4. (Bilipschitz) It can be deemed that $F$ is bilipschitz if there exist constants $c > 0$ and $C < \infty$ such that for all $x_1, x_2 \in X$ , + +$$ +c \| x _ {1} - x _ {2} \| \leq \| F (x _ {1}) - F (x _ {2}) \| \leq C \| x _ {1} - x _ {2} \|. +$$ + +# B.1 No-go theorem for discretization of diffeomorphisms on Hilbert spaces + +Definition 5. (Category of Hilbert Space Diffeomorphisms) We let $\mathcal{D}$ as the category of Hilbert diffeomorphisms with objects $\mathcal{O}_D$ that are packs $(X,F)$ of a Hilbert space $X$ and $a$ (possibly non-linear) $C^1$ -diffeomorphism $F\colon X\to X$ and the gathering of morphisms (or arrows that 'map' objects to others) $\mathcal{A}$ that are either + +1. (induced isomorphisms) Maps $a_{\phi}$ that are ruled for a linear isomorphism $\phi \colon X_1 \to X_2$ of Hilbert spaces $X_1$ and $X_2$ that maps the objects $(X_1, F_1) \in \mathcal{O}_D$ to the one $(\phi(X_1), \phi \circ F_1 \circ \phi^{-1}) \in \mathcal{O}_D$ , or +2. (induced restrictions) Maps $a_{X_1, X_2}$ that are ruled for a Hilbert space $X_1$ , its closed subspace $X_2 \subset X_1$ , and an object $(X_1, F_1) \in \mathcal{O}_D$ such that $F_1(X_2) = X_2$ . Then $a_{X_1, X_2}$ maps to the object $(X_1, F_1) \in \mathcal{O}_D$ to the one $(X_2, F_1|_{X_2}) \in \mathcal{O}_D$ . + +Definition 6. (Category of Approximation Sequences) We let $\mathcal{B}$ be the category of approximation sequences, owning objects $\mathcal{O}_B$ that are of the form $(X,S_0(X),(F_V)_{V\in S_0(X)})$ , in which $X$ is a Hilbert space, + +$$ +S _ {0} (X) \subset S (X) = \{V \mid V \subset X \text {i s a f i n i t e d i m e n s i o n a l l i n e a r s u b s p a c e} \}, +$$ + +are partly ordered lattices, $\bigcup_{V\in S_0(X)}V = X$ , and $F_V\colon V\to V$ are $C^1$ -diffeomorphisms of spaces $V\in S_0(X)$ . + +The set of morphisms $\mathcal{A}_{\mathcal{B}}$ comprises either + +1. Schedules $A_{\phi}$ that are defined for a linear isomorphism $\phi : X_1 \to X_2$ of Hilbert spaces $X_1$ and $X_2$ , and lattices $S_0(X_1)$ and $S_0(X_2) = \{\phi(V) \mid V \in S_0(X_1)\}$ , that schedule the objects $(X_1, S(X_1), (F_V)_{V \in S(X_1)})$ to $(X_2, S(X_2), (\phi \circ F_{\phi^{-1}(W)} \circ \phi^{-1})_{W \in S(X_2)})$ , or +2. Schedule $A_{X_1,X_2}$ that are ruled for a Hilbert space $X_1$ , its closed subspace $X_2 \subset X_1$ , and an object $(X_1,S_0(X_1),(F_V)_{V \in S_0(X_1)})$ so that $F(X_2) = X_2$ and $S_0(X_2) = \{V \in S_0(X_1) \mid V \subset X_2\}$ are a partly ordered lattice. Afterwards, $A_{X_1,X_2}$ projects the object $(X_1,S_0(X_1),(F_V)_{V \in S_0(X_1)})$ to the one $(X_2,S_0(X_2),(F_V)_{V \in S_0(X_2)})$ . + +In the sequel, the notion of an approximation or discretization functor is given. Practically, an approximation functor is an operator that projects a function $F$ from an infinite-dimensional space $X$ to another function $F_{V}$ operating within finite-dimensional subspaces $V$ of $X$ , so that the functions $F_{V}$ are closely aligned (in a reasonable sense) with the function $F$ . + +Definition 7. (Approximation Functor) We engineer the approximation functor, denoted by $\mathcal{A}\colon \mathcal{D}\to \mathcal{B}$ , as the functor that maps each $(X,F)\in \mathcal{O}_D$ to some $(X,S_0(X),(F_V)_{V\in S_0(X)})\in \mathcal{O}_B$ such that the Hilbert space $X$ stays as the same. The approximation functor maps all morphisms $a_{\phi}$ to $A_{\phi}$ and morphisms $a_{X_1,X_2}$ to $A_{X_1,X_2}$ , and enjoys the following properties + +(A) For all $r > 0$ and all $(X,F)\in \mathcal{O}_D$ + +$$ +\lim_{V\to X}\sup_{x\in \overline{B_{X}(0,r)}\cap V}\| F_{V}(x) - F(x)\|_{X} = 0. +$$ + +In separable Hilbert spaces, this indicates that when the finite-dimensional subspaces $V \subset X$ expand to fill the entire Hilbert space $X$ , then the approximations $F_{V}$ converge coincidentally in all bounded subsets to $F$ . + +Definition 8. We argue that the approximation functor $\mathcal{A}$ is continuous if the following statement holds: Let $(X,F),(X,F^{(j)})\in \mathcal{O}_D$ be such that the Hilbert space $X$ is the same for all the objects and let $(X,S_0(X),(F_V)_{V\in S_0(X)}) = \mathcal{A}(X,F)$ be approximating flows of $(X,F)$ and $(X,S_0(X),(F_{j,V})_{V\in S_0(X)}) = \mathcal{A}(X,F^{(j)})$ be approximating sequences of $(X,F^{(j)})$ . Furthermore, assume that $r > 0$ and + +$$ +\lim_{j\to \infty}\sup_{x\in \overline{B}_{X}(0,r)}\| F^{(j)}(x) - F(x)\|_{X} = 0. +$$ + +Then, for all $V \in S_0(X)$ the approximations $F_V^{(j)}$ of $F^{(j)}$ and $F_V$ of $F$ fulfill + +$$ +\lim_{j\to \infty}\sup_{x\in V\cap \overline{B}_{V}(0,r)}\| F_{V}^{(j)}(x) - F_{V}(x)\|_{X} = 0. +$$ + +The theorem given below establishes a negative result, especially that continuous approximating functors for diffeomorphisms may not exist. + +Theorem 3. (No-go theorem for discretization of general diffeomorphisms) There lives no functor $\mathcal{D} \to \mathcal{B}$ that meets the property $(A)$ of an approximation functor and is continuous. + +# B.2 Strongly monotone diffeomorphisms and approximation + +In this subsection, it is evidenced that the obstruction to continuous approximation is naturally eliminated when the diffeomorphisms under consideration are supposed to be strongly monotone. + +Lemma 1. Let $V \subset X$ be a finite-dimensional subspace of $X$ , and let $P_V \colon X \to X$ be the orthonormal projection onto $V$ . Let $F \colon X \to X$ be a strongly monotone $C^1$ -diffeomorphism. On that basis, $P_V F|_V \colon V \to V$ is strongly monotone, and a $C^1$ -diffeomorphism. + +In fact, strongly monotone projections can be continuously discretized in a weak sense. In the sequel, we concentrate on bounded linear operators and Nemytskii operators. + +Lemma 2. Let $A \colon X \to X$ be a linear bounded operator and meet $\langle Au, u \rangle \geq c_0 \| u \|_X^2$ for certain $c_0 > 0$ . Then, $A \colon X \to X$ is strongly monotone. + +Next, supposing that $X = L^{2}(D;\mathbb{R})$ , we define Nemytskii operator by + +$$ +F ^ {\sigma} (u) = \sigma \circ u, +$$ + +in which $\sigma : \mathbb{R} \to \mathbb{R}$ is continuous. + +Moreover, we can get: + +Proposition 1. Suppose that $\sigma$ satisfies $|\sigma(s)| \leq C_1 |s| + C_2$ and the derivative of $s \to \sigma(s)$ is defined a.e and satisfies the condition $\sigma'(s) \geq \alpha > 0$ . Then, $F^{\sigma}: L^{2}(D; \mathbb{R}) \to L^{2}(D; \mathbb{R})$ is strongly monotonous. + +We can now give the sufficient conditions for the layers of an NO to be strongly monotone: + +Lemma 3. All strongly monotone layers of NOs $(F)$ are diffeomorphisms. + +Theorem 4. Let $\mathcal{A}_{lin}$ be the discretization functor that projects $F$ to $P_{V}F|_{V}$ for each finite subspace $V\subset X$ . Let $\mathcal{D}_{smn}$ and $\mathcal{B}_{smn}$ be categories in which $F\colon X\to X$ and $F_{V}\colon V\to V$ are strongly monotone $C^1$ -functions in the form of an NO. Then, the functor $\mathcal{A}_{lin}\colon \mathcal{D}_{smn}\to \mathcal{B}_{smn}$ satisfies condition (A), and it is continuous in the sense of Definition 7. + +A straight condition to guarantee strong monotonicity of an NO layer is given as follows: + +Lemma 4. Let $F: X \to X$ be a layer of NO that holds the form $F(u) = u + T_2G(T_1u)$ , where $T_j: X \to X$ , $j = 1, 2$ are compact operators and $G: X \to X$ is a $C^1$ -smooth projection. Suppose that Fréchet derivative $DG|_x$ of $G$ at $x$ meets the following for all $x \in X$ , + +$$ +\left\| \right. D G \left. \right| _ {x} \left\| \right. _ {X \rightarrow X} \leq \frac {1}{2} \left\| T _ {1} \right\| _ {X \rightarrow X} ^ {- 1} \left\| T _ {2} \right\| _ {X \rightarrow X} ^ {- 1}. +$$ + +Then, $F: X \to X$ is deemed strongly monotone. + +# B.3 Bilipschitz NOs are conditionally strongly strongly monotone diffeomorphisms + +The following theorem states that we may always decompose a bilipschitz NO into the composition of strongly monotone NO layers $H_{j}$ and a reflection operator $A_0$ . + +Theorem 5. Assume $X$ be a Hilbert space. There is $e \in X$ , $\| e \|_X = 1$ so that the following is real: Let $F \colon X \to X$ be a layer of a bilipschitz NO. Then for all $r_1 > 0$ and $\epsilon > 0$ there are a linear invertible projection $A_0 \colon X \to X$ , that is either the identity map or a reflection function and strongly monotone operators $H_k$ that are also layers of NOs such that + +$$ +H _ {k}: X \to X, \quad H _ {k} (x) = x + B _ {k} (x), \quad k = 1, 2, \dots , J, +$$ + +where $B_{k}:X\to X$ is a compact mapping and meets $\mathrm{Lip}(B_k) < \epsilon$ and + +$$ +F (x) = H _ {J} \circ \dots \circ H _ {2} \circ H _ {1} \circ A _ {0} (x), \quad f o r a l l x \in B _ {X} (0, r _ {1}). \tag {9} +$$ + +Furthermore, if $F \in C^2(X, X)$ , then $J = \mathcal{O}(\epsilon^{-2})$ . + +We have noticed that operators of the term identity plus a compact form are crucial for continuous discretization. This insight inspires the employment of residual networks as approximators within the framework of finite-rank NOs. In the sequel, we suppose that $X$ is a separable Hilbert space, with an orthogonal basis $\varphi = \{\varphi_n\}_{n \in \mathbb{N}}$ . For $N \in \mathbb{N}$ , we define $E_N: X \to \mathbb{R}^N$ and $D_N: \mathbb{R}^N \to X$ by + +$$ +E _ {N} u := \left(\langle u, \varphi_ {1} \rangle_ {X}, \dots , \langle u, \varphi_ {N} \rangle_ {X}\right) \in \mathbb {R} ^ {N}, \quad D _ {N} \alpha := \sum_ {n \leq N} \alpha_ {n} \varphi_ {n}. +$$ + +It is noted that $P_{V_N} = D_N E_N$ , in which $P_{V_N}: X \to X$ is the mapping onto $V_N := \operatorname{span}\{\varphi_n\}_{n \leq N}$ . Using $E_N, D_N$ , we define the category of residual networks in the separable Hilbert space, with $T, N \in \mathbb{N}$ and activation function $\sigma$ , as + +$$ +\mathcal {R} _ {T, N, \varphi , \sigma} (X) := \left\{G: X \to X: G = \bigcirc_ {t = 1} ^ {T} \left(I _ {d _ {X}} + D _ {N} \circ N N _ {t} \circ E _ {N}\right), \right. +$$ + +$NN_{t}:\mathbb{R}^{N}\to \mathbb{R}^{N}$ are neural networks with activation function $\sigma (t = 1,\ldots ,T)\}$ . + +The next theorem proves a universality outcome for each of the layers $G$ , allowing us to achieve a general universality result for the whole network. + +Theorem 6. Let $R > 0$ , and let $F\colon X\to X$ be a layer of a bilipschitz NO, as in Definition 4. Let $\sigma$ be the ReLU activation function defined by $\sigma (x)\coloneqq \max \{0,x\} ^3$ . Then, for any $\epsilon \in (0,1)$ , there are $T,N\in \mathbb{N}$ and $G\in \mathcal{R}_{T,N,\varphi ,\sigma}(X)$ that enjoys the form + +$$ +G = \left(I _ {X} + D _ {N} \circ N N _ {T} \circ E _ {N}\right) \circ \dots \circ \left(I _ {X} + D _ {N} \circ N N _ {1} \circ E _ {N}\right), +$$ + +such that each projection $(I_X + D_N \circ NN_t \circ E_N)$ is strongly monotone $C^1$ -diffeomorphisms on some ball and + +$$ +\sup_{x\in \overline{B}_{X}(0,R)}\| F(x) - G\circ A(x)\|_{X}\leq \epsilon , +$$ + +in which $A \colon X \to X$ is a linear invertible projection that is either the identity map or a reflection function $x \to x - 2\langle x, e \rangle_X e$ with some unit vector $e \in X$ . Moreover, $G \circ A \colon B_X(0, R) \to G \circ A(B_X(0, R))$ is invertible, and there is certain $NO \Phi \colon G \circ A(B_X(0, R)) \to A(B_X(0, R))$ such that + +$$ +\left(G \circ A | _ {B _ {X} (0, R)}\right) ^ {- 1} = A ^ {- 1} \circ \Phi . +$$ + +# C Proof of Discrete Invariance under Diffeomorphism + +# C.1 Proof that the Radon Transform is Bilipschitz + +# C.1.1 Proof that the Radon Transform in 2D is BiLipschitz + +The Radon transform in two dimensions, denoted $R: L^2(\mathbb{R}^2) \to L^2(\mathbb{R} \times S^1)$ , maps a square-integrable function $f \in L^2(\mathbb{R}^2)$ to its line integrals over all lines in the plane. For a parameter $s \in \mathbb{R}$ (the signed distance from the origin) and a direction $\theta \in S^1$ (the unit circle), the Radon transform is defined as: + +$$ +R f (s, \theta) = \int_ {\{x \in \mathbb {R} ^ {2}: x \cdot \theta = s \}} f (x) d \mu (x), +$$ + +where $d\mu$ denotes the Lebesgue measure on the line $\{x\in \mathbb{R}^2:x\cdot \theta = s\}$ . The space $L^2 (\mathbb{R}\times S^1)$ is equipped with the norm: + +$$ +\| g \| _ {L ^ {2} (\mathbb {R} \times S ^ {1})} ^ {2} = \int_ {S ^ {1}} \int_ {\mathbb {R}} | g (s, \theta) | ^ {2} d s d \theta , +$$ + +where $d\theta$ is the standard measure on $S^1$ . + +A linear operator $R$ is bilipschitz if there exist positive constants $c$ and $C$ such that for all $f_{1}, f_{2} \in L^{2}(\mathbb{R}^{2})$ , + +$$ +c \| f _ {1} - f _ {2} \| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)} \leq \| R f _ {1} - R f _ {2} \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} \leq C \| f _ {1} - f _ {2} \| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)}. +$$ + +Given the linearity of $R$ , this is equivalent to proving: + +Upper bound: There exists $C > 0$ such that $\| Rf\|_{L^2 (\mathbb{R}\times S^1)}\leq C\| f\|_{L^2 (\mathbb{R}^2)}$ for all $f\in L^{2}(\mathbb{R}^{2})$ + +Lower bound: There exists $c > 0$ such that $\| Rf\|_{L^2 (\mathbb{R}\times S^1)}\geq c\| f\|_{L^2 (\mathbb{R}^2)}$ for all $f\in L^{2}(\mathbb{R}^{2})$ + +We proceed by establishing both bounds separately. + +# Upper Bound Proof + +We show that there exists a constant $C > 0$ such that: + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} \leq C \| f \| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)} \quad \text {f o r a l l} f \in L ^ {2} \left(\mathbb {R} ^ {2}\right). +$$ + +# Proof: + +Firstly, we express the $L^2$ norm of $Rf$ : + +The norm squared is: + +$$ +\| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {1})} ^ {2} = \int_ {S ^ {1}} \int_ {\mathbb {R}} | R f (s, \theta) | ^ {2} d s d \theta . +$$ + +We then apply the Fourier Slice Theorem: + +For a fixed direction $\theta \in S^1$ , consider the one-dimensional Fourier transform of $Rf(\cdot, \theta)$ with respect to $s$ : + +$$ +\mathcal {F} _ {1} [ R f (\cdot , \theta) ] (\sigma) = \int_ {\mathbb {R}} R f (s, \theta) e ^ {- i \sigma s} d s. +$$ + +By the Fourier slice theorem, this equals the two-dimensional Fourier transform of $f$ : + +$$ +\mathcal {F} _ {1} [ R f (\cdot , \theta) ] (\sigma) = \hat {f} (\sigma \theta), +$$ + +where $\hat{f} (\xi) = \int_{\mathbb{R}^2}f(x)e^{-ix\cdot \xi}dx$ for $\xi \in \mathbb{R}^2$ + +Now, we invoke Plancherel's Theorem: + +Plancherel's theorem for the one-dimensional Fourier transform states: + +$$ +\int_ {\mathbb {R}} | R f (s, \theta) | ^ {2} d s = \int_ {\mathbb {R}} | \mathcal {F} _ {1} [ R f (\cdot , \theta) ] (\sigma) | ^ {2} d \sigma . +$$ + +Substituting the Fourier slice theorem result in: + +$$ +\int_ {\mathbb {R}} | R f (s, \theta) | ^ {2} d s = \int_ {\mathbb {R}} | \hat {f} (\sigma \theta) | ^ {2} d \sigma . +$$ + +Thus, the full norm becomes: + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} ^ {2} = \int_ {S ^ {1}} \int_ {\mathbb {R}} | \hat {f} (\sigma \theta) | ^ {2} d \sigma d \theta . +$$ + +To better prove this, we switch to Polar Coordinates: + +Parameterize $\theta = (\cos \phi, \sin \phi)$ with $\phi \in [0, 2\pi)$ , so $\sigma \theta = \sigma (\cos \phi, \sin \phi)$ . The integral becomes: + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} ^ {2} = \int_ {0} ^ {2 \pi} \int_ {- \infty} ^ {\infty} | \hat {f} (\sigma (\cos \phi , \sin \phi)) | ^ {2} d \sigma d \phi . +$$ + +Since $\hat{f} (\sigma \theta) = \hat{f} (-\sigma (-\theta))$ and $S^1$ is symmetric, the integral can be split into: + +$$ +\| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {1})} ^ {2} = 2 \int_ {0} ^ {2 \pi} \int_ {0} ^ {\infty} | \hat {f} (\sigma (\cos \phi , \sin \phi)) | ^ {2} d \sigma d \phi . +$$ + +In $\mathbb{R}^2$ , use polar coordinates $\xi = (r\cos \phi, r\sin \phi)$ , where $r = \| \xi\|$ and $d\xi = r dr d\phi$ . The $L^2$ norm of $\hat{f}$ is: + +$$ +\| \hat {f} \| _ {L ^ {2} (\mathbb {R} ^ {2})} ^ {2} = \int_ {0} ^ {2 \pi} \int_ {0} ^ {\infty} | \hat {f} (r (\cos \phi , \sin \phi)) | ^ {2} r d r d \phi . +$$ + +Compare this to our expression: + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} ^ {2} = 2 \int_ {0} ^ {2 \pi} \int_ {0} ^ {\infty} | \hat {f} (\sigma (\cos \phi , \sin \phi)) | ^ {2} d \sigma d \phi . +$$ + +Substituting $\sigma = r$ , the integrand lacks the Jacobian factor $r$ : + +$$ +\| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {1})} ^ {2} = 2 \int_ {\mathbb {R} ^ {2}} | \hat {f} (\xi) | ^ {2} \| \xi \| ^ {- 1} d \xi , +$$ + +adjusting for the measure transformation. + +Consider the integral: + +$$ +\int_ {\mathbb {R} ^ {2}} | \hat {f} (\xi) | ^ {2} \| \xi \| ^ {- 1} d \xi . +$$ + +The weight $\| \xi \|^{-1}$ is locally integrable near the origin: + +$$ +\int_ {| \xi | < 1} \| \xi \| ^ {- 1} d \xi = \int_ {0} ^ {1} r ^ {- 1} \cdot 2 \pi r d r = 2 \pi \int_ {0} ^ {1} 1 d r = 2 \pi < \infty , +$$ + +and decays as $|\xi| \to \infty$ . Since $\hat{f} \in L^2(\mathbb{R}^2)$ , the Cauchy-Schwarz inequality or a direct estimate shows this integral is finite when $f$ has sufficient decay, but in $L^2$ , we rely on known results that this operator (the Fourier multiplier $\| \xi \|^{-1/2}$ ) is bounded in certain weighted spaces, adjusted here via: + +$$ +\| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {1})} ^ {2} \leq C _ {1} \| \hat {f} \| _ {L ^ {2} (\mathbb {R} ^ {2})} ^ {2} = C _ {1} \| f \| _ {L ^ {2} (\mathbb {R} ^ {2})} ^ {2}, +$$ + +where $C_1$ accounts for the constant from the weight's integrability (approximately $2 \cdot 2\pi$ with proper normalization). + +In conclusion: + +$$ +\left\| R f \right\| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} \leq \sqrt {C _ {1}} \| f \| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)}. +$$ + +Let $C = \sqrt{C_1}$ , which establishes the upper bound. + +# Lower Bound Proof + +We show that there exists a constant $c > 0$ such that: + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} \geq c \| f \| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)} \quad \text {f o r a l l} f \in L ^ {2} \left(\mathbb {R} ^ {2}\right). +$$ + +# Proof: + +Firstly, we introduce the injectivity of $R$ : + +The Radon transform is injective. If $Rf = 0$ , then for all $\theta \in S^1$ , $Rf(\cdot, \theta) = 0$ , so: + +$$ +\mathcal {F} _ {1} [ R f (\cdot , \theta) ] (\sigma) = \hat {f} (\sigma \theta) = 0 \quad \text {f o r a l l} \sigma \in \mathbb {R}, \theta \in S ^ {1}. +$$ + +Since $\{\sigma \theta : \sigma \in \mathbb{R}, \theta \in S^1\} = \mathbb{R}^2$ , $\hat{f} = 0$ almost everywhere, implying $f = 0$ by the Plancherel theorem. + +We then give discussions on the boundedness of the Inverse: + +The inverse Radon transform $R^{-1} : \operatorname{Im}(R) \to L^2(\mathbb{R}^2)$ is well-defined on the image of $R$ . In two dimensions, $R^{-1}$ is typically expressed via filtered backprojection: + +$$ +f (x) = \int_ {S ^ {1}} \left(H \partial_ {s} R f\right) (x \cdot \theta , \theta) d \theta , +$$ + +where $H$ is the Hilbert transform. Standard results (e.g., Natterer, 2001,[29]) show that $R^{-1}$ is bounded in $L^2$ : + +$$ +\left\| R ^ {- 1} g \right\| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)} \leq K \left\| g \right\| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} \quad \text {f o r a l l} g \in \operatorname {I m} (R), +$$ + +with some constant $K > 0$ + +Next, we apply it to $Rf$ : + +Since $f = R^{-1}(Rf)$ + +$$ +\left\| f \right\| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)} = \left\| R ^ {- 1} (R f) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)} \leq K \left\| R f \right\| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)}. +$$ + +Rearranging: + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {1}\right)} \geq \frac {1}{K} \| f \| _ {L ^ {2} \left(\mathbb {R} ^ {2}\right)}. +$$ + +Let $c = \frac{1}{K}$ , which proves the lower bound. + +In conclusion, The Radon transform $R:L^{2}(\mathbb{R}^{2})\to L^{2}(\mathbb{R}\times S^{1})$ satisfies: + +- Upper bound $\| Rf\|_{L^2 (\mathbb{R}\times S^1)}\leq C\| f\|_{L^2 (\mathbb{R}^2)}$ +- Lower bound $\| Rf\|_{L^2 (\mathbb{R}\times S^1)}\geq c\| f\|_{L^2 (\mathbb{R}^2)}$ + +with positive constants $c$ and $C$ . Thus, $R$ is bi-Lipschitz in the $L^2$ sense, being both bounded and having a bounded inverse on its range. + +□ + +# C.1.2 Proof that the Radon Transform is Bilipschitz + +The Radon transform $R: L^2(\mathbb{R}^n) \to L^2(\mathbb{R} \times S^{n-1})$ maps a function $f \in L^2(\mathbb{R}^n)$ to its integrals over all hyperplanes in $\mathbb{R}^n$ . Specifically, for $s \in \mathbb{R}$ and $\theta \in S^{n-1}$ (the unit sphere in $\mathbb{R}^n$ ), it is defined as: + +$$ +R f (s, \theta) = \int_ {\{x \in \mathbb {R} ^ {n}: x \cdot \theta = s \}} f (x) d \mu (x), +$$ + +where $d\mu$ is the Lebesgue measure on the hyperplane $\{x : x \cdot \theta = s\}$ . We aim to prove that $R$ is bi-Lipschitz, meaning there exist constants $c, C > 0$ such that for all $f_1, f_2 \in L^2(\mathbb{R}^n)$ , + +$$ +c \| f _ {1} - f _ {2} \| _ {L ^ {2} \left(\mathbb {R} ^ {n}\right)} \leq \| R f _ {1} - R f _ {2} \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {n - 1}\right)} \leq C \| f _ {1} - f _ {2} \| _ {L ^ {2} \left(\mathbb {R} ^ {n}\right)}. +$$ + +Since $R$ is linear, it suffices to show: + +Upper bound: $\| Rf \|_{L^2(\mathbb{R} \times S^{n-1})} \leq C \| f \|_{L^2(\mathbb{R}^n)}$ for all $f \in L^2(\mathbb{R}^n)$ , + +Lower bound: $\| Rf \|_{L^2(\mathbb{R} \times S^{n-1})} \geq c \| f \|_{L^2(\mathbb{R}^n)}$ for all $f \in L^2(\mathbb{R}^n)$ . + +We will establish these bounds separately. + +# Upper Bound Proof + +We prove there exists a constant $C > 0$ such that for all $f\in L^{2}(\mathbb{R}^{n})$ + +$$ +\left\| R f \right\| _ {L ^ {2} \left(\mathbb {R} \times S ^ {n - 1}\right)} \leq C \| f \| _ {L ^ {2} \left(\mathbb {R} ^ {n}\right)}. +$$ + +# Proof: + +Firstly, we define the $L^2$ norm of $Rf$ : + +The $L^2$ norm of $Rf$ is given by: + +$$ +\| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {n - 1})} ^ {2} = \int_ {S ^ {n - 1}} \int_ {\mathbb {R}} | R f (s, \theta) | ^ {2} d s d \theta , +$$ + +where $d\theta$ is the surface measure on $S^{n - 1}$ + +We then apply the Fourier Slice Theorem: + +By fixing $\theta \in S^{n - 1}$ , the one-dimensional Fourier transform of $Rf(\cdot ,\theta)$ with respect to $s$ is: + +$$ +\mathcal {F} _ {1} [ R f (\cdot , \theta) ] (\sigma) = \int_ {\mathbb {R}} R f (s, \theta) e ^ {- i \sigma s} d s = \hat {f} (\sigma \theta), +$$ + +where $\hat{f} (\xi) = \int_{\mathbb{R}^n}f(x)e^{-ix\cdot \xi}dx$ is the $n$ -dimensional Fourier transform of $f$ + +Now, we use Plancherel's Theorem: + +By Plancherel's theorem in one dimension, + +$$ +\int_ {\mathbb {R}} | R f (s, \theta) | ^ {2} d s = \int_ {\mathbb {R}} | \mathcal {F} _ {1} [ R f (\cdot , \theta) ] (\sigma) | ^ {2} d \sigma = \int_ {\mathbb {R}} | \hat {f} (\sigma \theta) | ^ {2} d \sigma . +$$ + +Thus, + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {n - 1}\right)} ^ {2} = \int_ {S ^ {n - 1}} \int_ {\mathbb {R}} | \hat {f} (\sigma \theta) | ^ {2} d \sigma d \theta . +$$ + +Next, we switch to Polar Coordinates: + +Parameterize $\xi = \sigma \theta$ with $\sigma \in \mathbb{R}$ and $\theta \in S^{n - 1}$ . Since $\hat{f} (\sigma \theta) = \hat{f} (-\sigma \theta)$ for real-valued $f$ (adjusting for symmetry), we consider: + +$$ +\int_ {\mathbb {R}} | \hat {f} (\sigma \theta) | ^ {2} d \sigma = \int_ {- \infty} ^ {0} | \hat {f} (\sigma \theta) | ^ {2} d \sigma + \int_ {0} ^ {\infty} | \hat {f} (\sigma \theta) | ^ {2} d \sigma = 2 \int_ {0} ^ {\infty} | \hat {f} (\sigma \theta) | ^ {2} d \sigma , +$$ + +assuming $f$ is real (for complex $f$ , the factor remains bounded). Hence, + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {n - 1}\right)} ^ {2} = 2 \int_ {S ^ {n - 1}} \int_ {0} ^ {\infty} | \hat {f} (\sigma \theta) | ^ {2} d \sigma d \theta . +$$ + +In polar coordinates, $\xi = \sigma \theta$ , $\| \xi \| = |\sigma|$ , and the Jacobian is $d\xi = \sigma^{n - 1}d\sigma d\theta$ for $\sigma >0$ . Thus, + +$$ +\int_ {\mathbb {R} ^ {n}} | \hat {f} (\xi) | ^ {2} d \xi = \int_ {S ^ {n - 1}} \int_ {0} ^ {\infty} | \hat {f} (\sigma \theta) | ^ {2} \sigma^ {n - 1} d \sigma d \theta . +$$ + +Rewrite the expression for $Rf$ : + +$$ +\int_ {S ^ {n - 1}} \int_ {0} ^ {\infty} | \hat {f} (\sigma \theta) | ^ {2} d \sigma d \theta = \int_ {S ^ {n - 1}} \int_ {0} ^ {\infty} | \hat {f} (\sigma \theta) | ^ {2} \sigma^ {n - 1} \cdot \sigma^ {1 - n} d \sigma d \theta = \int_ {\mathbb {R} ^ {n}} | \hat {f} (\xi) | ^ {2} \| \xi \| ^ {1 - n} d \xi . +$$ + +So, + +$$ +\| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {n - 1})} ^ {2} = 2 \int_ {\mathbb {R} ^ {n}} | \hat {f} (\xi) | ^ {2} \| \xi \| ^ {1 - n} d \xi . +$$ + +The weight $\| \xi \|^{1 - n}$ must be controlled: - For $n = 1$ , $\| \xi \|^{1 - 1} = 1$ , and the integral is $\int_{\mathbb{R}} |\hat{f}(\xi)|^2 d\xi = \| f \|_{L^2(\mathbb{R})}^2$ . - For $n \geq 2$ , $\| \xi \|^{1 - n}$ is locally integrable near $\xi = 0$ (since $1 - n < -1$ implies integrability) and decays at infinity. There exists a constant $C_n = \sup_{\xi} \int_{S^{n-1}} \| \xi \|^{1-n} d\theta < \infty$ , depending on $n$ , such that: + +$$ +\int_ {\mathbb {R} ^ {n}} | \hat {f} (\xi) | ^ {2} \| \xi \| ^ {1 - n} d \xi \leq C _ {n} \int_ {\mathbb {R} ^ {n}} | \hat {f} (\xi) | ^ {2} d \xi = C _ {n} \| f \| _ {L ^ {2} (\mathbb {R} ^ {n})} ^ {2}, +$$ + +using Plancherel's theorem again: $\| \hat{f}\|_{L^2 (\mathbb{R}^n)} = \| f\|_{L^2 (\mathbb{R}^n)}$ + +In conclusion: + +$$ +\| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {n - 1})} ^ {2} \leq 2 C _ {n} \| f \| _ {L ^ {2} (\mathbb {R} ^ {n})} ^ {2} \Rightarrow \| R f \| _ {L ^ {2} (\mathbb {R} \times S ^ {n - 1})} \leq \sqrt {2 C _ {n}} \| f \| _ {L ^ {2} (\mathbb {R} ^ {n})}. +$$ + +Set $C = \sqrt{2C_n}$ , proving the upper bound. + +# Lower Bound Proof + +We prove there exists a constant $c > 0$ such that for all $f\in L^{2}(\mathbb{R}^{n})$ + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {n - 1}\right)} \geq c \| f \| _ {L ^ {2} \left(\mathbb {R} ^ {n}\right)}. +$$ + +Proof: + +Firstly, we introduce the injectivity of $R$ : + +The Radon transform is injective on $L^2(\mathbb{R}^n)$ : if $Rf = 0$ , then $\hat{f}(\sigma\theta) = 0$ for all $\sigma \in \mathbb{R}$ , $\theta \in S^{n-1}$ , implying $\hat{f} = 0$ almost everywhere, so $f = 0$ . + +Then, we talk about boundedness of the Inverse: + +The inverse Radon transform $R^{-1} : \operatorname{Im}(R) \to L^2(\mathbb{R}^n)$ exists and is bounded in $L^2$ under certain conditions: Odd $n$ : For $n$ odd, $R^{-1}$ is given by the filtered backprojection formula, involving derivatives of order $n - 1$ , and is bounded in $L^2$ . There exists $K > 0$ such that: + +$$ +\| R ^ {- 1} g \| _ {L ^ {2} (\mathbb {R} ^ {n})} \leq K \| g \| _ {L ^ {2} (\mathbb {R} \times S ^ {n - 1})}, +$$ + +for all $g\in \operatorname {Im}(R)$ .- Since $f = R^{-1}(Rf)$ + +$$ +\| f \| _ {L ^ {2} \left(\mathbb {R} ^ {n}\right)} = \| R ^ {- 1} (R f) \| _ {L ^ {2} \left(\mathbb {R} ^ {n}\right)} \leq K \| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {n - 1}\right)}. +$$ + +Rearranging, + +$$ +\| R f \| _ {L ^ {2} \left(\mathbb {R} \times S ^ {n - 1}\right)} \geq \frac {1}{K} \| f \| _ {L ^ {2} \left(\mathbb {R} ^ {n}\right)}. +$$ + +Set $c = 1 / K$ + +For $n$ even, $R^{-1}$ involves fractional derivatives (e.g., Hilbert transform for $n = 2$ ), but remains bounded in $L^2$ with an appropriate constant $K$ , ensuring the same inequality holds. + +In conclusion, We have shown: + +- Upper bound $\| Rf\|_{L^2 (\mathbb{R}\times S^{n - 1})}\leq C\| f\|_{L^2 (\mathbb{R}^n)}$ +- Lower bound $\|Rf\|_{L^2(\mathbb{R} \times S^{n-1})} \geq c \|f\|_{L^2(\mathbb{R}^n)}$ + +Thus, $R$ is bilipschitz in $L^2$ , being both bounded and non-degenerate. + +![](images/ce398c258e072bef05cb8e08cdc43674ac1d55c380c884e149b05556bcebd60e.jpg) + +# D Supplement to Sinogram-Domain Convolution + +In the sinogram-domain space obtained after the Radon transform, the sinogram-domain convolution is discretely invariant. Specifically, for data of different resolutions, the siongram-domain convolution of their Radon transforms yields consistent results in the angular direction, independent of the spatial resolution of the input images. This implies that a convolution kernel trained on one resolution can be applied to another without retraining, enabling super-resolution capabilities. + +Consider two resolutions: + +- Low-resolution image: Resolution $M_1 \times N_1$ , with position sampling $s_i = i\Delta s_1, i = 0, 1, \ldots, P_1 - 1$ , where $P_1$ depends on $M_1$ and $N_1$ ; +- High-resolution image: Resolution $M_2 \times N_2$ (where $M_2 > M_1, N_2 > N_1$ ), with position sampling $s_j = j\Delta s_2, j = 0, 1, \ldots, P_2 - 1$ , where $P_2 > P_1$ and $\Delta s_2 < \Delta s_1$ . + +Assumption: The angular sampling $\theta_{k} = \frac{2\pi k}{N_{\theta}}$ remains consistent across both resolutions, i.e., $N_{\theta}$ is fixed. + +# D.1 Definition of Sinogram-Domain Convolution + +In Equation (9) of the main text, we have presented the sinogram domain convolution formula. To maintain notational consistency, we will employ the same symbols as introduced previously while providing a more detailed explanation. + +$$ +S _ {\text {c o n v}} [ k, t ] = \sum_ {m = 0} ^ {N _ {\theta} - 1} S [ m, t ] K [ k - m ] \tag {10} +$$ + +where: $S[m, t] = S(\theta_m, t); K[k - m] = K(\theta_{k - m})$ ; $k - m$ is computed modulo $N_{\theta}$ to account for the periodicity of $\theta$ . + +Here, $K(\theta)$ is the convolution kernel, assumed to depend only on the angular difference $\theta - \theta'$ , and is discretized as $K[k], k = 0, 1, \dots, N_{\theta} - 1$ . + +# D.2 Sinogram-Domain Convolution at Different Resolutions + +Let the sinograms for the low-resolution and high-resolution images be: + +- Low-resolution: $S_{\mathrm{low}}(\theta_k, t_i)$ , where $t_i = i \Delta t_1$ ; +- High-resolution: $S_{\mathrm{high}}(\theta_k, t_j)$ , where $t_j = j \Delta t_2$ . + +The discrete convolutions are: + +- Low-resolution: + +$$ +S _ {\text {c o n v , l o w}} [ k, t _ {i} ] = \sum_ {m = 0} ^ {N _ {\theta} - 1} S _ {\text {l o w}} [ m, t _ {i} ] K [ k - m ] \tag {11} +$$ + +High-resolution: + +$$ +S _ {\text {c o n v , h i g h}} [ k, t _ {j} ] = \sum_ {m = 0} ^ {N _ {\theta} - 1} S _ {\text {h i g h}} [ m, t _ {j} ] K [ k - m ] \tag {12} +$$ + +**Observation:** The convolution operation depends only on the angular index $k$ , the kernel $K[k]$ , and the sinogram values $S[m, t]$ at the respective $t$ -values. The structure of the convolution is identical across resolutions, with differences only in the $t$ -sampling. + +# D.3 Proof of Consistency + +To prove discrete invariance, we need to show that the convolution results in the $\theta$ -direction are consistent across resolutions, independent of the $t$ -sampling. + +For a fixed $t$ , the discrete convolution $S_{\mathrm{conv}}[k,t] = \sum_{m=0}^{N_\theta - 1} S[m,t]K[k - m]$ depends on: + +- The angular sampling $\theta_{k}$ , which is fixed at $\theta_{k} = \frac{2\pi k}{N_{\theta}}$ ; +- The kernel $K[k]$ , which is identical for both resolutions; +- The sinogram values $S[m, t]$ , which vary depending on the specific $t$ -value. + +In the low-resolution case, $t = t_{i} = i\Delta t_{1}$ , so the convolution uses $S_{\mathrm{low}}[m,t_i]$ . In the high-resolution case, $t = t_{j} = j\Delta t_{2}$ , so it uses $S_{\mathrm{high}}[m,t_j]$ . While $t_{i}$ and $t_j$ represent different sampling densities, the convolution operation itself is defined solely over the $\theta$ -direction. The summation over $m$ (i.e., the angular indices) is identical in both cases: + +- Same number of terms $(N_{\theta})$ +- Same kernel values $K[k - m]$ +- Same angular indices $m$ . + +The difference in $t$ -sampling (i.e., $\Delta t_1$ vs. $\Delta t_2$ ) affects the number of convolution outputs along the $t$ -axis ( $P_1$ outputs for low-resolution, $P_2$ for high-resolution), but for any specific $t$ -value, the convolution process in the $\theta$ -direction remains unchanged. If $t_i \approx t_j$ (i.e., they correspond to nearly the same physical detector position, adjusted for sampling), then $S_{\mathrm{low}}[m, t_i] \approx S_{\mathrm{high}}[m, t_j]$ , and the convolution results $S_{\mathrm{conv, low}}[k, t_i] \approx S_{\mathrm{conv, high}}[k, t_j]$ . The operation's form ensures consistency in the $\theta$ -direction across resolutions. + +# E Time Complexity Analysis + +Given a discrete counterpart $u \in \mathbb{R}^{H \times W \times C}$ of a 2D continuous function, we can reform $u$ to get $u \in \mathbb{R}^{N \times C}$ , with $N = H \times W$ . As aforementioned in the main texts, the variable $A$ represents the number of projection angles. + +Table 6: Comparison of runtime complexity + +
ModelsComplexity
GNO (kernel)O(N(N-1))
TransformerO(N2)
FNO (FFT)O(N log N)
CNOO(N)
RNOO(AN)
+ +Graph neural operator (GNO) The integral formulation of GNO is as follows: + +$$ +\begin{array}{l} (\mathcal {K} (v)) (x) = \int_ {U (x)} K (x, y) \cdot v (y) d y \\ \approx \sum_ {y \in U (x)} K (x, y) \cdot v (y) q _ {y}, \\ \end{array} +$$ + +where $K$ is a kernel (typically parametrized by a deep network) and $q_{y} \in \mathbb{R}$ are suitable quadrature weights. Yet GNO is capable of expressing local integral operators by choosing an appropriately small neighborhood $U(x) \subseteq D$ , computing the kernel and performing aggregation within each neighborhood $U(x)$ is computationally expensive and memory-demanding for general applications $k: D \times D \to \mathbb{R}^n$ . For each point, it is required to access all of the neighboring elements. As reported in [24], the runtime complexity of GNO is then $O(N(N - 1))$ . + +Transformer The integration of Transformer-based NO is given as: + +$$ +(\mathcal {K} (v)) (x) = \int_ {\Omega} K (v (x), v (y)) v (y) d y +$$ + +$$ +(\mathcal {K} (v)) (x) \approx \sum_ {i = 1} ^ {n _ {v}} K (v (x), v (y _ {i})) v (y _ {i}) +$$ + +$$ +(\mathcal {K} (v)) (x) \approx \sum_ {i = 1} ^ {n _ {v}} \frac {\exp \left(\frac {\left(W _ {q} v (x) , W _ {k} v \left(y _ {i}\right)\right)}{\sqrt {d _ {v}}}\right)}{\sum_ {j = 1} ^ {n _ {v}} \exp \left(\frac {\left(W _ {q} v (x) , W _ {k} v \left(y _ {i}\right)\right)}{\sqrt {d _ {v}}}\right)} W _ {v} v \left(y _ {i}\right) \tag {13} +$$ + +in which $K$ is perceived as employing a softmax function onto three transformed vectors with length $n_v$ . Evidently, Eq. (13) closely aligns with the commonly applied self-attention mechanism in vanilla transformers, where the matrices $W_{q}, W_{k}, W_{v} \in \mathbb{R}^{d_{v} \times d_{v}}$ concern the learned transformations of queries, keys, and values, respectively. From Eq. (13), it is obvious that two nested loops of length $n_v = N$ must be traversed. On that basis, the runtime complexity of transformer-based NOs is $O(N^2)$ . + +FNO FNO substitutes the global convolution in the time domain with multiplication operations in the frequency counterpart; hence, its kernel integral is written as: + +$$ +\begin{array}{l} (\mathcal {K} (v)) (x) = \int_ {\Omega} K (x, y) v (y) \mathrm {d} y \\ = \int_ {\Omega} K (x - y) v (y) \mathrm {d} y \\ = \mathcal {F} ^ {- 1} \left(R _ {\phi} \cdot (\mathcal {F} v _ {t})\right) (x) \\ \end{array} +$$ + +in which $R_{\phi}$ designs $N$ multiplications, parameterized by $\phi$ , therefore its runtime complexity is $O(N)$ . Moreover, the time complexity of the Fast Fourier Transform is $O(N \log N)$ , leading to an overall complexity of $O(N \log N)$ . + +CNO Typically, the local convolution operation is elaborated by + +$$ +\begin{array}{l} (\mathcal {K} (v)) (x) = \int_ {\Omega} \kappa (x - y) v (y) d y \\ = \sum_ {i, j = 1} ^ {k} \kappa_ {i j} v (x - z _ {i j}) \\ \end{array} +$$ + +where $f \in \mathcal{B}_w$ , $\kappa$ is a discrete kernel with size $k \in \mathbb{N}$ , $z_{ij}$ are the resultant grid points. Evaluating each point involves $k^2$ multiplications. So the total time complexity is $O(Nk^2) \Rightarrow O(N)$ . + +# F PDE Benchmarks + +We tested our data on 5 benchmarks, including three categories: + +Solid material [6]:Plasticity +- Navier-Stokes equations for fluid [28]:Airfoil, Pipe, Navier-Stokes +Darcy's law [14]:Darcy +- Reaction-Diffusion [38]:Allen-Cahn + +Table 7: Summary of experiment benchmarks, where the first six datasets are from FNO and geo-FNO. Mesh records the size of discretized meshes. Dataset is organized as the number of samples in training and test sets. + +
GeometryBenchmarksDimMeshInputOutputDataset
Structured MeshPlasticity2D+Time3,131External ForceMesh Displacement(900, 80)
Airfoil2D11,271StructureMach Number(1000, 200)
Pipe2D16,641StructureFluid Velocity(1000, 200)
Regular GridNavier-Stokes2D+Time4,096Past VelocityFuture Velocity(1000, 200)
Darcy2D7,225Porous MediumFluid Pressure(1000, 200)
Allen-Cahn2D129×129Initial Phase FieldEvolved Phase Field(1000, 200)
+ +Here are the details of each benchmark. + +# F.1 Solid Material + +The fundamental equation governing the behavior of solid materials is expressed as: + +$$ +\rho^ {s} \frac {\partial^ {2} u}{\partial t ^ {2}} + \nabla \cdot \sigma = 0, \tag {14} +$$ + +where $\rho^s\in \mathbb{R}$ represents the density of the solid, and $\nabla$ signifies the nabla operator. The variable $u$ denotes the displacement vector of the material as a function of time $t$ , while $\sigma$ corresponds to the stress tensor. The model Plasticity [22] is governed by the equation, as presented in (14). + +Plasticity. This benchmark addresses the plastic forging process, where a plastic material undergoes impact from an arbitrarily shaped die applied from above. The input consists of the die's geometry, represented on a structured mesh. The output comprises the deformation of each mesh point over the subsequent 20 time steps. The structured mesh has a resolution of $101 \times 31$ . + +# F.2 Navier-Stokes Equation + +The differential form of the fluid dynamics equations is given by: + +$$ +\frac {\partial \rho}{\partial t} + \nabla \cdot (\rho U) = 0, \tag {15} +$$ + +$$ +\frac {\partial U}{\partial t} + U \cdot \nabla U = f + \frac {1}{\rho} \nabla \cdot \left(T _ {i j} e _ {i} e _ {j}\right), \tag {16} +$$ + +$$ +\frac {\partial \left(e + \frac {1}{2} U ^ {2}\right)}{\partial t} + U \cdot \nabla \left(e + \frac {1}{2} U ^ {2}\right) = f \cdot U + \frac {1}{\rho} \nabla \cdot \left(U \cdot T _ {i j} e _ {i} e _ {j}\right) + \frac {\lambda}{\rho} \Delta T, \tag {17} +$$ + +where Equations (15), (16), and (17) represent the conservation of mass, momentum, and energy, respectively. Here, $\rho$ denotes the fluid density, $U$ is the velocity vector, $f$ represents the external force, and $e$ is the internal energy. The stress tensor in the fluid is denoted by $T$ , with $e_i$ as the basis vector, and $T_{ij}e_i e_j$ adheres to the Einstein summation convention. All variables are functions of space and time, and the term $\frac{\lambda}{\rho}\Delta T$ accounts for heat conduction. For a Newtonian fluid, the stress tensor $T$ depends on the pressure $p$ , viscosity coefficient $\nu$ , and velocity vector $U$ . Consequently, Equation (16) for a Newtonian fluid can be reformulated as: + +$$ +\frac {\partial U}{\partial t} + U \cdot \nabla U = f - \frac {1}{\rho} \nabla p + \nu \nabla^ {2} U. \tag {18} +$$ + +Additionally, Equation (17) can be derived similarly, but its complexity precludes inclusion here; see [28] for details. These equations for Newtonian fluids are commonly referred to as the Navier-Stokes equations. Below, we elaborate on the partial differential equations (PDEs) underlying our fluid benchmarks. + +Navier-Stokes. The Navier-Stokes dataset, sourced from [23], models incompressible, viscous flow on a unit torus, where the fluid density $\rho$ in Equation (15) remains constant. In this context, the energy conservation in Equation (17) is decoupled from mass and momentum conservation. Thus, the fluid dynamics are governed by Equations (15) and (18): + +$$ +\nabla \cdot U = 0, +$$ + +$$ +\frac {\partial w}{\partial t} + U \cdot \nabla w = \nu \nabla^ {2} w + f, \tag {19} +$$ + +$$ +w | _ {t = 0} = w _ {0}, +$$ + +where $U = (u, v)$ is the 2D velocity vector, $w = |\nabla \times U| = \frac{\partial u}{\partial y} - \frac{\partial v}{\partial x}$ represents the vorticity, and $w_0 \in \mathbb{R}$ is the initial vorticity at $t = 0$ . The dataset uses a viscosity $\nu = 10^{-5}$ and a 2D field resolution of $64 \times 64$ . Each sample includes 20 consecutive frames, with the task of predicting the next 10 frames based on the previous 10. + +Pipe. The Pipe dataset, from [22], examines incompressible flow within a pipe. The governing equations are derived from Equations (15) and (18): + +$$ +\nabla \cdot U = 0 +$$ + +$$ +\frac {\partial U}{\partial t} + U \cdot \nabla U = f - \frac {1}{\rho} \nabla p + \nu \nabla^ {2} U. \tag {20} +$$ + +The dataset is generated on a structured mesh with a resolution of $129 \times 129$ . For experiments, the mesh structure serves as the input, and the output is the horizontal fluid velocity within the pipe. + +Airfoil. The Airfoil dataset, also from [22], investigates transonic flow over an airfoil. Given the negligible viscosity of air, the viscous term $\nabla^2 U$ is omitted from the Navier-Stokes equations. The governing equations are thus: + +$$ +\frac {\partial \rho^ {f}}{\partial t} + \nabla \cdot (\rho^ {f} U) = 0 +$$ + +$$ +\frac {\partial \rho^ {f} U}{\partial t} + \nabla \cdot \left(\rho^ {f} U U + p \mathbb {I}\right) = 0 \tag {21} +$$ + +$$ +\frac {\partial E}{\partial t} + \nabla \cdot ((E + p) U) = 0, +$$ + +where $\rho^f$ denotes the fluid density, and $E$ represents the total energy. The data is generated on a structured mesh with a resolution of $200\times 50$ . The mesh point locations are used as inputs, and the Mach number at each mesh point is the output. + +# F.3 Darcy Flow + +Darcy. Darcy's law governs the flow of fluids through porous media, such as water permeating sand. We utilize the Darcy dataset introduced by [23], which models 2D Darcy flow equations within a unit + +square, expressed as: + +$$ +- \nabla \cdot (a \nabla u) = f, \quad u | _ {x \in \partial (0, 1) ^ {2}} = 0, \tag {22} +$$ + +where $a \in \mathbb{R}^+$ represents the diffusion coefficient, and $f$ denotes the external force, fixed at 1 in this dataset. The input for this dataset is the diffusion coefficient $a$ , with the output being the solution $u$ . Data samples are structured on a regular grid with a resolution of $85 \times 85$ . + +# F.4 Allen-Cahn equation + +Allen-Cahn Equation. The Allen-Cahn equation, a reaction-diffusion model, is widely applied to study chemical reactions and phase separation in multi-component alloys. We used the dataset proposed by [37]. In two-dimensional space, the Allen-Cahn equation is formulated as: + +$$ +\partial_ {t} u (x, y, t) = \epsilon \Delta u (x, y, t) + u (x, y, t) - u (x, y, t) ^ {3}, \quad x, y \in (0, 3), t \in [ 0, 2 0 ], +$$ + +$$ +u (x = 0, y, t) = u (x = 3, y, t), \quad y \in (0, 3), t \in [ 0, 2 0 ], \tag {23} +$$ + +$$ +u (x, y = 0, t) = u (x, y = 3, t), \quad x \in (0, 3), t \in [ 0, 2 0 ], +$$ + +$$ +u (x, y, 0) = u _ {0} (x, y), \quad x, y \in (0, 3), +$$ + +where $\epsilon \in \mathbb{R}^{+*}$ is a positive constant controlling the diffusion magnitude. The problem is defined with periodic boundary conditions, setting $\epsilon = 1\times 10^{-3}$ . The initial condition is generated from a Gaussian Random Field using the kernel: + +$$ +\mathcal {K} (x, y) = \tau^ {(\alpha - 1)} \left(\pi^ {2} \left(x ^ {2} + y ^ {2}\right) + \tau^ {2}\right) ^ {\frac {\alpha}{2}}, \tag {24} +$$ + +with parameters $\tau = 15$ and $\alpha = 1$ . The objective is to learn the operator $\mathcal{D}:u_0(x,y)\mapsto u(x,y,t)$ . Here, the solution is computed at $t = 20$ s on a grid with a resolution of $129\times 129$ . + +# G More Experiments and Analysis + +# G.1 Implementation Details + +As shown in Table 8, all the baselines are trained and tested under the same training strategy. For the Allen-Cahn equation, plasticity, and airfoil problems, we set the number of Radon transform angles to 64, while for pipe flow, the Navier-Stokes equations, and Darcy flow, we use 32 angles. Given the physical field $u$ and the model-predicted field $\hat{u}$ , the relative L2 norm of the model prediction can be calculated as follows: + +$$ +\text {R e l a t i v e} \mathrm {L} 2 = \frac {\| u - \hat {u} \|}{\| u \|}. \tag {25} +$$ + +Table 8: Training and model configurations of RNO. Here $\mathcal{L}_{\mathrm{v}}$ and $\mathcal{L}_{\mathrm{s}}$ represent the loss on volume and surface fields respectively. As for Darcy, we adopt an additional spatial gradient regularization term $\mathcal{L}_{\mathrm{g}}$ following ONO [44]. + +
BenchmarksTraining Configuration (Shared in all baselines)Model Configuration
LossEpochsInitial LROptimizerLayers LHeadsChannels CAngles A Blocks
Allen-cahn12864 1
Plasticity12864 1
AirfoilRelative L2500\( {10}^{-3} \)AdamW [26]8812864 4
Pipe12832 4
Navier-Stokes25632 2
Darcy\( {\mathcal{L}}_{\mathrm{{rL}}2} + {0.1}{\mathcal{L}}_{\mathrm{g}} \)12832 4
+ +# G.2 Experiment Visualization + +We have already given a relatively complete description of Radon block in Section 3.4 of the main text. We then show the self-learning adjustment of the sinogram domain weights in Figure 9, in which the weight change of the first Radon block is provided. As seen, while the training steps proceed deeper and deeper, the resultant weights would adaptively gather into some specific angles with enriched features. + +![](images/ba01c8afc93b9fd459af2d5c2a44a1fcfde265b760462b012f45238236ac073d.jpg) +Figure 9: Self-learning adjustment of the sinogram domain weight. From top to bottom, from left to right, it represents the adaptive change of the angle domain weight as the training time increases. + +# G.3 Generalization + +The discussion of generalization is an important issue for neural operators, which reflects the learning of the mapping relationship between neural operators and infinite-dimensional function spaces. We have introduced this in Section 4.2 of the main text. The comparison in Table 9 reveals RNO's superior generalization performance over Transolver and FNO when handling larger resolutions. We further show the results of the generalization experiment on the Darcy equation dataset in Figure 10. + +Table 9: Comparative display of generalization performance. Relative L2 is recorded. A smaller value indicates better performance. (Bold: Best performance) + +
Model61 × 6185 × 85141 × 141211 × 211421 × 421
FNO1.16 × 10-11.80 × 10-12.68 × 10-13.16 × 10-13.63 × 10-1
Transolver3.28 × 10-25.11 × 10-26.78 × 10-27.46 × 10-28.64 × 10-2
RNO (Ours)3.21 × 10-25.04 × 10-26.69 × 10-27.09 × 10-27.62 × 10-2
+ +![](images/7ce096a3f1243192b13bf34d0d465de2aa9e3d20f6e5379acb3770b73343d4f1.jpg) +Figure 10: The effect of model generalization is shown. From top to bottom, they are the prediction map, the real map, and the error map, showing the generalization of RNO on the Darcy dataset. The model was trained on a low resolution of $43 \times 43$ and tested on five other high resolutions. + +# G.4 Parameter Setting of Radon Transform + +It is generally considered that an increase in the number of angles in the Radon transform enhances the accuracy of data prediction and reconstruction; however, the opposite is sometimes observed. As demonstrated in Figures 11 and 12, it has been found through experiments that, at ultra-low resolution, an increased number of angles does not improve results; conversely, greater errors are produced. This finding further informs our choice of the number of angles and, as discussed in Section 3.2, a lower angle count reduces computational complexity. This finding provides valuable insights for future applications of the Radon transform. + +# G.5 Supplementary Ablation Experiments + +Note that the positive roles of replacing other transformations with RT have been evidenced in previous discussions. Hereafter, we further evaluate the behaviors of Radon block when imposed on the other architectures. Practically, we performed experiments by integrating Radon block with several global methods, including self-attention (S.A.), and orthogonal attention (O.A.). Similar to our method, these attempts all take advantage of global feature extraction and local feature learning. Similar to our method, these attempts all take advantage of global fea + +Table 10: Ablation studies by integrating Radon block with other off-the-shelf modules All experiments were based on the $85 \times 85$ Darcy flow dataset and tested when the number of angles was set to 32. + +
AblationsMemory (GB)Time (s/epoch)Param (B)Relative L2 Darcy
S.A.+Radon Block19.564508.838.48 × 1066.12 × 10-3
O.A.+Radon Block15.038110.362.03 × 1067.80 × 10-3
ours3.19143.212.83 × 1065.34 × 10-3
+ +ture extraction and local feature learning. We observe that previous works have addressed similar local problems [25]. As demonstrated in our main results table, the proposed FNO+Radon block achieves superior performance compared to approaches using either Differential Kernel or Local Integral Kernel operators. + +As shown in Table 11, we conduct systematic experiments by varying the number of model layers. Taking the Darcy dataset as an example, our results demonstrate that increasing the layer depth does not yield performance improvements, but rather leads to a significant growth in both parameter count and computational time. Similar observations hold for other datasets such as Navier-Stokes, where deeper architectures tend to suffer from overfitting. Based on these empirical findings, we deliberately avoid using larger layer counts in our main experiments. + +We conduct extensive experiments to evaluate the impact of angle parameter settings, with results summarized in Table 12. Our findings reveal that while higher angular resolution leads to improved accuracy, this comes at the cost of increased computational time - a trade-off that aligns with our theoretical complexity analysis. Notably, the angle parameter selection has no measurable effect on the total number of model parameters. + +Table 11: Ablation studies by varying the number of model layers from 16 to 40. + +
SettingsMemory (GB)Time (s/epoch)Param (B)Relative L2 Darcy
layer=4011.136110.031.39 × 1074.89 × 10-3
layer=329.07091.451.12 × 1076.78 × 10-3
layer=247.00473.258.38 × 1065.73 × 10-3
layer=164.94055.645.60 × 1065.20 × 10-3
+ +# G.6 Showcases + +To facilitate improved visualization of the experimental results, visual representations of all datasets, along with prediction images and error maps, are presented. As follows, Figs. 14, 13 show the 6 benchmarks. We provide comprehensive visualization of our experimental results, including input data, ground truth, predicted outputs, and error maps. + +Table 12: Ablation studies by varying the number of angles for Radon transform from 64 to 512. + +
SettingsMemory (GB)Time (s/epoch)Param (B)Relative L2 Darcy
Angles=5124.400283.882.83 × 1064.31 × 10-3
Angles=2563.562145.632.83 × 1064.65 × 10-3
Angles=1283.15285.112.83 × 1064.87 × 10-3
Angles=642.96254.192.83 × 1064.90 × 10-3
+ +# H Limitations + +Our work currently faces two main limitations. Unlike the Fourier transform, which benefits from well-optimized algorithms and seamless PyTorch integration, the Radon transform exhibits potential for further improvement in parallel computing and GPU utilization. To substantiate this claim, we have conducted an ablation study in Section 4.3. Additionally, whereas our model is designed for 2D or time-dependent 3D problems, its application to general 3D PDEs, particularly those involving point cloud data, remains unexplored. Regarding this issue, we provide a rigorous proof of the billipschitz condition for arbitrary dimensions in Appendix C.1.2, which theoretically guarantees the feasibility of our approach. This theoretical foundation ensures the soundness of our proposed method across spaces of different dimensionalities. The dimensionality reduction property of the Radon transform is particularly advantageous for 3D PDEs, as it alleviates the computational burden compared to methods like FNO, which may face scalability challenges in higher dimensions. We emphasize that the GNO module integration strategy proposed in [24] for solving 3D problems can be directly adapted to our framework. Our method inherently supports such an extension, and we are well-positioned to address 3D PDE cases through similar architectural modifications. + +# I Broader Impacts + +Partial differential equations represent the most fundamental mathematical tool in modern physics and engineering, making their numerical simulation a research area of both theoretical and practical significance. While mathematically inspired, our work innovatively incorporates operator learning architectures, achieving superior performance in terms of solution accuracy and generalization capability. This advancement demonstrates substantial potential for both industrial applications and theoretical research in PDE solving. Meanwhile, neural operators have proven to be remarkably versatile. They integrate seamlessly with various algorithms like transformers[43], Mamba[47], and diffusion models[48, 13], and demonstrate broad applicability in domains such as remote sensing[45, 31], medical images[15, 19] and super-resolution[41]. We hope our work offers fresh perspectives for this dynamic field. + +![](images/199aac36ef655a9f31bc9a998ed324e817d52c0c223c85deadfa140b55422a89.jpg) +Figure 11: setting of the number of Radon transform angles at ultra-low resolution. From left to right, the true image, the angle domain image, and the transformed image and error are shown. The image above shows the number of angles 512. + +![](images/28e51aea6728a034b8a109f66f3ede4a8d8b315293d34d1b6e97547977c30674.jpg) +Figure 12: The setting of the number of Radon transform angles at ultra-low resolution. From left to right, the true image, the angle domain image, and the transformed image and error are shown. The image above shows the number of angles 64. + +![](images/585a1a63259c2cc9b9079a538a24c2f60449e75015daf95706811d92342ed53a.jpg) +Figure 13: Experimental visualization of Plasticity, Navier-Stokes equations, including the ground truth plots, predicted plots, and error plots. + +![](images/2870e5ddb74848bfaf06b8f80edec3ab64ab3f40e281428e277339515f2298d9.jpg) +Figure 14: Experimental visualization of Darcy, Allen-cahn, Pipe, Airfoil equations, including the input plots, ground truth plots, predicted plots, and error plots. \ No newline at end of file diff --git a/NeurIPS/2025/Solving Partial Differential Equations via Radon Neural Operator/images.zip b/NeurIPS/2025/Solving Partial Differential Equations via Radon Neural Operator/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..e0337e563086c17cd4465bf3352101909a02e1ac --- /dev/null +++ b/NeurIPS/2025/Solving Partial Differential Equations via Radon Neural Operator/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b1131aafe1f7a1527854124bf5a31986aafced80c3346508da3d1217bcc33a94 +size 1737222 diff --git a/NeurIPS/2025/Solving Partial Differential Equations via Radon Neural Operator/layout.json b/NeurIPS/2025/Solving Partial Differential Equations via Radon Neural Operator/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..b52f3662abdfff07cc020b04d0597d4416c2a070 --- /dev/null +++ b/NeurIPS/2025/Solving Partial Differential Equations via Radon Neural Operator/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e45ea5a3679658f8f611efe008dec93206cbb56d77b48ebdad769f7e2f0d530f +size 1705197 diff --git a/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_content_list.json b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..057606a1a30a2af10bc3ff2a5eebf4e3ffe0120a --- /dev/null +++ b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:176b2650c6b0744ab38a09d216006f832e2f3eb0b1ab07860c3147a102d2e649 +size 220513 diff --git a/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_model.json b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_model.json new file mode 100644 index 0000000000000000000000000000000000000000..50dd710c219bea57634cc02c02dbd4783177d7bb --- /dev/null +++ b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:954bb94c880de198d62f1e7ac9c24eb64ef583a13d60edbb201652f00b472731 +size 263664 diff --git a/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_origin.pdf b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b6034488f3bac4bd33100f2d0dd1b0f21fa0e32b --- /dev/null +++ b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/aefeac69-ce8f-4686-b615-f7e0df3f83e5_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5f9ff9f1ef0127dee1343f49906d513c690906adc3e012a812a5543e34ead53c +size 3535116 diff --git a/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/full.md b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/full.md new file mode 100644 index 0000000000000000000000000000000000000000..515720d3fe32522efe14dd2f8841c2c272316389 --- /dev/null +++ b/NeurIPS/2025/Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes/full.md @@ -0,0 +1,1091 @@ +# Solving and Learning Partial Differential Equations with Variational Q-Exponential Processes + +Guangting Yu Shiwei Lan\* + +School of Mathematical & Statistical Sciences +Arizona State University, Tempe, AZ 85287 + +# Abstract + +Solving and learning partial differential equations (PDEs) lies at the core of physics-informed machine learning. Traditional numerical methods, such as finite difference and finite element approaches, are rooted in domain-specific techniques and often lack scalability. Recent advances have introduced neural networks and Gaussian processes (GPs) as flexible tools for automating PDE solving and incorporating physical knowledge into learning frameworks. While GPs offer tractable predictive distributions and a principled probabilistic foundation, they may be suboptimal in capturing complex behaviors such as sharp transitions or non-smooth dynamics. To address this limitation, we propose the use of the $q$ -exponential process (Q-EP), a recently developed generalization of GPs designed to better handle data with abrupt changes and to more accurately model derivative information. We advocate for Q-EP as a superior alternative to GPs in solving PDEs and associated inverse problems. Leveraging sparse variational inference, our method enables principled uncertainty quantification - a capability not naturally afforded by neural network-based approaches. Through a series of experiments, including the Eikonal equation, Burgers' equation, and an inverse Darcy flow problem, we demonstrate that the variational Q-EP method consistently yields more accurate solutions while providing meaningful uncertainty estimates. + +Keywords: Probabilistic PDE Solvers, Bayesian Inverse Problems, Data Inhomogeneity, Modeling Derivatives, Uncertainty Quantification + +# 1 Introduction + +It is of fundamental importance in science and technology to solve mathematical models represented as a system of differential equations and to learn such a complex system by identifying crucial physical quantities with estimated uncertainty (a.k.a. inverse problems). Over centuries, theoretic foundations and computational methods have been developed for solving and learning partial differential equations (PDEs), which is facilitated by the development of modern computers. Traditional numerical algorithms such as the finite element method remain demanding for both domain knowledge and computing resources. Recently, there has been increasing interest and effort to efficiently automate this process using machine learning techniques. + +The surge of physics-informed machine learning is driven by two main thrusts: neural network-based algorithms and Gaussian process (GP)-based probabilistic solvers. The former works are represented by physics-informed neural networks [PINN 37, 49], the deep Ritz method [9], the deep Galerkin method [44], and operator learning [20] methods including the Fourier neural operator [FNO 25], deep operator networks [DeepONet 26] and the neural inverse operator [NIO 28]. See [19] for a + +review of recent advances. The basic idea is to parametrize the solution with a neural network and to minimize certain loss with respect to network parameters to obtain the solution. Despite empirical successes, these neural network-type approaches typically require large samples but lack convergence guarantee or uncertainty quantification (UQ). On the other hand, GP has been introduced to solve and learn ordinary differential equations (ODEs) [45, 39, 4, 43, 15] and PDEs [32, 33, 35, 38] with theoretic guarantee [34] and UQ [16]. More recent development on GP for solving and learning PDEs includes [6, 27, 14, 13, 29, 3]. Different from neural network approaches, these probabilistic solvers model the solution as a GP conditioned on PDE constraints and identify the solution as the maximum a posteriori (MAP). + +Due to the tractability of conditional and predictive densities, GP has been widely adopted in machine learning and scientific computing [40]. However, as an $L_{2}$ regularization, GP tends to be over-smooth for modeling certain objects with abrupt changes or sharp contrast. For example, it is known in imaging analysis that GP may not detect or preserve edges very well in an image [23, 7]. On the other hand, researchers [42, 48] notice that total variation regularization penalizes the $L_{1}$ norm of derivatives and yields edge-preserving reconstructions. However, the total variation prior degenerates to GP prior with increasingly finer discretization mesh [23] and hence loses its edge-preserving feature. Therefore, [22] propose the Besov prior as an $L_{q}$ regularization and prove its discretization-invariant property. [24] further develop the $q$ -exponential process (Q-EP) as a probabilistic definition of the Besov process with tractable posterior prediction and demonstrate it as a superior generalization of GP (with $q = 2$ ) in modeling inhomogeneous data with sharp transitions. + +In this paper, we discover that Q-EP (with $q = 1$ ) is better in modeling derivative information than GP, and hence presents as a preferable candidate for solving PDEs. Heuristically, this is attributed to Q-EP's enhanced ability to model inhomogeneous objects with sharp variations, resulting in better regularization of large derivatives. Theoretically, this can be explained by a faster posterior convergence rate in Bayesian modeling with Q-EP priors. Unlike optimization-based approaches [6, 27], we adopt sparse variational inference [46, 47] for Q-EP [30, 5] to solve and learn PDEs, allowing natural UQ. An emerging challenge is that in addition to mapping the Q-EP mean function by the nonlinear PDE dynamics, one also needs to propagate the whole variational distribution through, which no longer renders a Q-EP. We solve this difficulty by linearizing the complicated PDE mapping. We also extend the resulting variational Q-EP solver for inverse problems. + +**Connection to the literature** Our work is motivated by [6] which optimizes the log-posterior for MAP as the PDE solution. Our proposed method replaces GP with a more general Q-EP and adopts variational Bayes for UQ. We investigate Q-EPs in solving various forward and inverse PDEs for a spectrum of $q$ 's with $q = 2$ corresponding to GP. As a probabilistic solver, Q-EP may not be best compared with neural network-based approaches. However, we still include PINN [37] and Bayesian PINN [B-PINN 49] as baselines in our comparison. We emphasize that our algorithms rely only on limited data, e.g. boundary values or interior observations, while providing meaningful UQ. Our work is also related to the recently proposed physics-informed state-space GP [13], which however focuses on time-dependent PDEs. It adopts a variational spatiotemporal state-space GP and can be regarded as a related special case of ours for $q = 2$ . Our work on solving and learning PDEs makes multiple contributions to the field of physics-informed machine learning: + +1. It is a novel probabilistic PDE solver based on Q-EP with superior capability of modeling data inhomogeneity and derivative information. +2. It theoretically justifies the preference of Q-EP over GP in solving and learning PDEs. +3. It provides efficient UQ for solving forward and inverse PDE problems. + +The remainder of the paper is organized as follows. Section 2 reviews Q-EP as a flexible prior in Bayesian models for inhomogeneous data and introduces an extension to incorporate derivative information. Section 3 explains the details of applying Q-EP to solve the forward and inverse problems of PDEs. We follow [6] to model the solution as MAP of Q-EP but highlight the challenges of variational inference including distribution propagation and variational lower bound. In Section 4, we justify the preference of Q-EP for $q = 1$ over GP ( $q = 2$ ) in solving PDEs. In Section 5, we demonstrate the numerical advantages, particularly faster convergence, of Q-EP compared with alternatives using forward problems involving Eikonal equation and Burgers' equation and inverse problems of identifying permeability in the Darcy flow. Section 6 concludes with a discussion of limitations and future improvements. + +# 2 Bayesian Modeling with Q-Exponential Process + +# 2.1 $Q$ -Exponential Process + +The univariate $q$ -exponential distribution [7] has density $\pi_q(u) \propto \exp(-\frac{1}{2}|u|^q)$ , whose logarithm yields an $L_{q}$ regularization term. [24] generalize the univariate $q$ -exponential random variable to a multivariate random vector, based on which a stochastic process can be defined. Suppose a function $u(\mathbf{x})$ is observed at $N$ locations, $\mathbf{x}_1, \dots, \mathbf{x}_N \in \Omega \subset \mathbb{R}^d$ . [24] define the multivariate $q$ -exponential distribution for $\mathbf{u} := (u(\mathbf{x}_1), \dots, u(\mathbf{x}_N))$ , as a member of the family of elliptic distributions [18]. + +Definition 1. A multivariate random vector $\mathbf{u} \in \mathbb{R}^N$ follows the $q$ -exponential distribution, denoted as $\mathbf{u} \sim q\text{-ED}_N(\boldsymbol{\mu}, \mathbf{C})$ , if it has the following density: + +$$ +p (\mathbf {u} | \boldsymbol {\mu}, \mathbf {C}, q) = \frac {q}{2} (2 \pi) ^ {- \frac {N}{2}} | \mathbf {C} | ^ {- \frac {1}{2}} r (\mathbf {u}) ^ {\left(\frac {q}{2} - 1\right) \frac {N}{2}} \exp \left\{- \frac {r ^ {\frac {q}{2}}}{2} \right\}, \quad r = (\mathbf {u} - \boldsymbol {\mu}) ^ {\top} \mathbf {C} ^ {- 1} (\mathbf {u} - \boldsymbol {\mu}). \tag {1} +$$ + +Remark 1. The negative log density of q-ED in (1) yields a quantity dominated by some weighted $L_{q}$ norm of $\mathbf{u} - \pmb{\mu}$ , i.e. $\frac{1}{2} r^{\frac{q}{2}} = \frac{1}{2}\| \mathbf{u} - \pmb{\mu}\|_{\mathbf{C}}^{q}$ . From the optimization perspective, q-ED, when used as a prior, imposes $L_{q}$ regularization in obtaining the maximum a posteriori (MAP). + +Li et al. [24] prove that the above multivariate $q$ -exponential distribution satisfies the conditions of Kolmogorov's extension theorem [31] and thus can be generalized to a stochastic process. For this purpose, we scale $\mathbf{u} \sim \mathrm{q-ED}_N(\mathbf{0}, \mathbf{C})$ by a factor $N^{\frac{1}{2}-\frac{1}{q}}$ so that the scaled $q$ -exponential random variable $\mathbf{u}^{*} := N^{\frac{1}{2}-\frac{1}{q}}\mathbf{u} \sim \mathrm{q-ED}_{N}^{*}(\mathbf{0}, \mathbf{C})$ has covariance asymptotically finite [Proposition 3.1 of 24]. With a covariance (symmetric and positive-definite) kernel $\mathcal{C}: \Omega \times \Omega \to \mathbb{R}$ , we define the following $q$ -exponential process ( $Q$ -EP) based on the scaled $q$ -exponential distribution. + +Definition 2. A (centered) $q$ -exponential process $u(\mathbf{x})$ with a kernel $\mathcal{C}$ , q- $\mathcal{EP}(0,\mathcal{C})$ , is a collection of random variables such that any finite set, $\mathbf{u} = (u(\mathbf{x}_1),\dots u(\mathbf{x}_N))$ , follows a scaled multivariate $q$ -exponential distribution $\mathrm{q - ED}^* (\mathbf{0},\mathbf{C})$ , where $\mathbf{C} = [\mathcal{C}(\mathbf{x}_i,\mathbf{x}_j)]_{N\times N}$ . + +Remark 2. When $q = 2$ , q-ED $_N(\pmb{\mu}, \mathbf{C})$ reduces to $\mathcal{N}_N(\pmb{\mu}, \mathbf{C})$ and q-EP $(0, \mathcal{C})$ becomes $\mathcal{GP}(0, \mathcal{C})$ . When $q \in (0, 2)$ , q-EP $(0, \mathcal{C})$ lends flexibility to modeling functional data with more regularization than GP. In practice, $q = 1$ is often adopted for faster posterior convergence [1, 21] and the capability of preserving inhomogeneous features (rough functional data, edges in image, etc). + +The covariance kernel $\mathcal{C}$ is associated with a Hilbert-Schmidt (HS) integral operator $T_{\mathcal{C}}:L^{2}(\Omega)\to$ $L^2 (\Omega),u(\cdot)\mapsto \int_{\Omega}\mathcal{C}(\cdot ,\mathbf{x}^{\prime})u(\mathbf{x}^{\prime})\mu (d\mathbf{x}^{\prime})$ which has eigen-pairs $\{\lambda_{\ell},\phi_{\ell}(\cdot)\}_{\ell = 1}^{\infty}$ such that for $\forall \ell \in \mathbb{N}$ $T_{\mathcal{C}}\phi_{\ell}(\mathbf{x}) = \phi_{\ell}(\mathbf{x})\lambda_{\ell}$ and $\| \phi_{\ell}\|_{2} = 1$ . Assume $T_{\mathcal{C}}$ is trace-class, i.e. $\mathrm{tr}(T_{\mathcal{C}}):= \sum_{\ell = 1}^{\infty}\lambda_{\ell} < \infty$ Theorem 3.4 of [24] presents a series representation of Q-EP similar to GP and the Besov process [8]. + +Theorem 2.1 (Karhunen-Loéve). If $u(\cdot) \sim \mathrm{q - }\mathcal{EP}(0,\mathcal{C})$ with a trace-class HS operator $T_{\mathcal{C}}$ having eigen-pairs $\{\lambda_{\ell},\phi_{\ell}(\cdot)\}_{\ell = 1}^{\infty}$ , then we have the following series representation for $u(\mathbf{x})$ : + +$$ +u (\mathbf {x}) = \sum_ {\ell = 1} ^ {\infty} u _ {\ell} \phi_ {\ell} (\mathbf {x}), \quad u _ {\ell} := \int_ {\Omega} u (\mathbf {x}) \phi_ {\ell} (\mathbf {x}) \stackrel {{i n d}} {{\sim}} \mathrm {q - E D} ^ {*} (0, \lambda_ {\ell}), \tag {2} +$$ + +where $\operatorname{E}[u_{\ell}] = 0$ and $\operatorname{Cov}(u_{\ell}, u_{\ell'}) = \lambda_{\ell} \delta_{\ell \ell'}$ with Dirac function $\delta_{\ell \ell'} = 1$ if $\ell = \ell'$ and $0$ otherwise. Moreover, we have $\operatorname{E}[\|u(\cdot)\|_2^2] = \sum_{\ell=1}^{\infty} \operatorname{E}[u_{\ell}^2] = \operatorname{tr}(T_C) < \infty$ . + +# 2.2 Bayesian Regression with Q-EP Priors + +Given $\mathbf{X}_{N\times d} = \{\mathbf{x}_n\}_{n = 1}^N$ and $\mathbf{y}_{N\times 1} = \{y_n\}_{n = 1}^N$ , we consider the Bayesian regression model: + +$$ +\mathbf {y} = u (\mathbf {X}) + \varepsilon , \quad \varepsilon \sim \mathrm {q - E D} _ {N} (0, \Gamma), +$$ + +$$ +u \sim \mathrm {q} - \mathcal {E P} (0, \mathcal {C}). \tag {3} +$$ + +Li et al. [24, Theorem 3.5] show that the posterior (predictive) distribution is analytically tractable when both the prior and the likelihood are $q$ -exponential. + +Theorem 2.2. For the regression model (3), the posterior distribution of $u(\mathbf{x}_*)$ at $\mathbf{x}_*$ is + +$$ +u (\mathbf {x} _ {*}) | \mathbf {y}, \mathbf {X}, \mathbf {x} _ {*} \sim \mathrm {q - E D} (\boldsymbol {\mu} ^ {*}, \mathbf {C} ^ {*}), +$$ + +$$ +\boldsymbol {\mu} ^ {*} = \mathbf {C} _ {*} ^ {\mathsf {T}} (\mathbf {C} + \Gamma) ^ {- 1} \mathbf {y}, \mathbf {C} ^ {*} = \mathbf {C} _ {* *} - \mathbf {C} _ {*} ^ {\mathsf {T}} (\mathbf {C} + \Gamma) ^ {- 1} \mathbf {C} _ {*}, +$$ + +where $\mathbf{C} = \mathcal{C}(\mathbf{X},\mathbf{X})$ $\mathbf{C}_{*} = \mathcal{C}(\mathbf{X},\mathbf{x}_{*})$ , and $\mathbf{C}_{**} = \mathcal{C}(\mathbf{x}_{*},\mathbf{x}_{*})$ + +![](images/3fe3c5b4eea8e6a17ef75fe1c84fa9a389fbf1fb6c0d2d8da3a75e3f4d778bad.jpg) +Figure 1: Contrasting Q-EP ( $q = 1.0$ , middle row) with GP ( $q = 2.0$ , bottom row) against the truth (top row) for modeling function values and derivatives of Rosenbrock (left) and Rastrigin (right). + +# 2.3 Modeling with Derivative Information + +Let $u \sim \mathrm{q - }\mathcal{EP}(0,\mathcal{C})$ . Denote the function and its derivatives by $\tilde{u} = (u,\frac{\partial}{\partial\mathbf{x}} u,\dots ,\frac{\partial^k}{\partial\mathbf{x}^k} u)$ up to order $k$ . Because linear operation preserves elliptic distributions [18, 10], $\tilde{u} \sim \mathrm{q - }\mathcal{EP}(0,\tilde{\mathcal{C}})$ is also a Q-EP if $\mathcal{C}$ in Definition 2 is differentiable up to order $k$ , where the augmented kernel, $\tilde{\mathcal{C}}$ , has a structure illustrated in Table A.1. For example, the (1,2) block of $\tilde{\mathcal{C}}$ is interpreted as $\operatorname {Cov}(u(\mathbf{x}),\frac{\partial}{\partial\mathbf{x}'} u(\mathbf{x}')) = \frac{\partial}{\partial\mathbf{x}'}\mathcal{C}(\mathbf{x},\mathbf{x}')$ . For solving the second-order PDEs in Section 5, we could adopt Matérn kernel (matern52 for $\nu = 5 / 2$ ), $\mathcal{C}(\mathbf{x},\mathbf{x}') = \sigma^2 (1 + \sqrt{5} r + \frac{5}{3} r^2)\exp (-\sqrt{5} r)$ , $r = \sqrt{\sum_{i = 1}^{d}(x_i - x_i')^2 / \rho_i^2}$ , for which the process is twice differentiable in the mean-square sense. + +To prepare for solving PDEs, we define $\| \cdot \|_{s,q}$ for $u$ based on (2) with a smoothness parameter $s > 0$ and an integrability parameter $q\geq 1$ [22, 7]: $\| u(\cdot)\|_{s,q} = \left(\sum_{\ell = 1}^{\infty}\ell^{\tau_q(s)q}|u_\ell |^q\right)^{\frac{1}{q}}$ , $\tau_q(s) = \frac{s}{d} +\frac{1}{2} -\frac{1}{q}$ . Consider the Banach space $B^{s,q}(\Omega)\coloneqq \{u:\Omega \to \mathbb{R}\mid \| u(\cdot)\|_{s,q} < \infty \}$ . If $q = 2$ and $\{\phi_{\ell}\}_{\ell = 1}^{\infty}$ form the Fourier basis, then $B^{s,2}(\Omega)$ reduces to the Sobolev space $H^{s}(\Omega)$ . For $u$ to be regular enough, we make the assumption on $\tilde{C}$ so that $\tilde{u}\in L^{q}(\Omega)$ almost surely by the following proposition [21]. + +Assumption 1. Suppose $\lambda = \{\lambda_{\ell}\}_{\ell = 1}^{\infty}$ are eigenvalues of HS operator $T_{\tilde{C}}$ for the kernel $\tilde{C}$ . We assume $\lambda \in \ell^{\frac{q}{2}}$ , i.e. $\| \lambda \|_{\frac{q}{2}}^{\frac{q}{2}} = \sum_{\ell = 1}^{\infty} \lambda_{\ell}^{\frac{q}{2}} < \infty$ . + +Proposition 2.1. If $\tilde{u} (\cdot)\sim \mathrm{q - }\mathcal{EP}(0,\tilde{\mathcal{C}})$ with a trace-class HS operator $T_{\tilde{C}}$ satisfying Assumption 1, then $\tilde{u} (\cdot)\in L_{\mathbb{P}}^{q}(\mathbb{R}^{\infty},L^{q}(\Omega))\coloneqq \{\tilde{u}:\Omega \times \mathbb{R}^{\infty}\to \mathbb{R}|\mathbb{E}(\| \tilde{u}\|_{q}^{q}) < \infty \}$ and $\mathbb{E}[\| \tilde{u} (\cdot)\| _q^q ] = \| \lambda \|_{\frac{q}{2}}^{\frac{q}{2}} < \infty$ + +Proof. See Appendix B. + +![](images/86e41d95e57f067b789c8d8a6bdf0a24b305ecca7f394a8c12ca642d0fcc5e34.jpg) + +Let $\tilde{\mathbf{U}}\coloneqq \tilde{u} (\mathbf{X})_{N\times (1 + kd)} = [u(\mathbf{X}),\frac{\partial}{\partial\mathbf{x}} u(\mathbf{X}),\dots ,\frac{\partial^k}{\partial\mathbf{x}^k} u(\mathbf{X})]$ . For brevity, we denote $D = 1 + kd$ . Then $\mathbf{Y}_{N\times D}$ and $\mathbf{E}_{N\times D}$ are the corresponding observations and errors respectively in the model (3). We apply this model to Rosenbrock $(f(\mathbf{x}) = \sum_{i = 1}^{d}[100(x_{i + 1} - x_i^2)^2 +(1 - x_i)^2])$ and Rastrigin $(f(\mathbf{x}) = 10d + \sum_{i = 1}^{d}[x_i^2 -10\cos (2\pi x_i)])$ test functions with function values and their derivatives observed on a $20\times 20$ grid, i.e. $N = 400$ , $d = 2$ , $k = 1$ . Figure 1 contrasts Q-EP $(q = 1.0)$ and GP $(q = 2.0)$ in predicting function and derivative values on the $50\times 50$ grid. Q-EP outperforms GP in yielding a more accurate recovery, especially in the more challenging example of Rastrigin function. + +Heuristically, the superiority of Q-EP in modeling derivatives over GP comes from its improved ability to handle inhomogeneous data with sharp variation. From the perspective of the $L_{q}$ norm of the gradient, the $L_{1}$ norm imposes stronger regularization than the $L_{2}$ norm on large values in $\nabla_{\mathbf{x}}f$ + +resulting in a better model for prediction. In Section 4 we will give a more rigorous justification. In the following, we take advantage of Q-EP's capability of modeling derivative information and apply it to solve PDEs. + +# 3 Solving Partial Differential Equations with Q-EP + +# 3.1 Bayesian Solver + +Consider the following general PDE defined on a bounded domain $\Omega \subset \mathbb{R}^d$ : + +$$ +\begin{array}{l} \mathcal {D} (u) (\mathbf {x}) = f (\mathbf {x}), \quad \mathbf {x} \in \Omega , \\ \mathcal {B} (u) (\mathbf {x}) = g (\mathbf {x}), \quad \mathbf {x} \in \partial \Omega . \tag {4} \\ \end{array} +$$ + +where $\mathcal{D}: B^{s,q}(\Omega) \to L^q(\Omega)$ is a differential operator and $\mathcal{B}: B^{s,q}(\partial \Omega) \to L^q(\partial \Omega)$ is a boundary operator with data $f \in L^q(\Omega)$ and $g \in L^q(\partial \Omega)$ . Here we assume a sufficiently large smoothness index $s > 0$ such that the PDE (4) is well-defined pointwise and has a unique strong solution [6]. Let $\overline{\Omega} = \Omega \cup \partial \Omega$ . For convenience of exposition, we denote the joint operator as $\mathcal{P} = (\mathcal{D}, \mathcal{B}): B^{s,q}(\overline{\Omega}) \to L^q(\overline{\Omega})$ , and the right-hand side function as $h = (f, g) \in L^q(\overline{\Omega})$ . + +A set of collocation points $\mathbf{X} = \{\mathbf{x}_n\}_{n=1}^N$ consists of $N_d$ interior points $\mathbf{X}_d = \{\mathbf{x}_1, \dots, \mathbf{x}_{N_d} \in \Omega\}$ and $N_b$ boundary points $\mathbf{X}_b = \{\mathbf{x}_{N_d+1}, \dots, \mathbf{x}_N \in \partial \Omega\}$ , i.e. $\mathbf{X} = \mathbf{X}_d \cup \mathbf{X}_b$ , and $N = N_d + N_b$ . Regarding the evaluation of $\mathcal{P}(u)$ on $\mathbf{X}$ , we make the following assumption so that we can properly define the likelihood model. + +Assumption 2. There exists a differentiable function $P: \mathbb{R}^D \to \mathbb{R}$ such that $\mathcal{P}(u)(\mathbf{x}) = P(\tilde{u}(\mathbf{x}))$ . And further there is a constant $C > 0$ such that $\|\nabla P\| \leq C$ . + +Then $\mathcal{P}(u)(\mathbf{X})$ becomes a nonlinear function of $\tilde{u} (\mathbf{X})$ , denoted as $P(\tilde{u} (\mathbf{X})) = \mathcal{P}(u)(\mathbf{X})$ . Let $\mathbf{h} = h(\mathbf{X})$ . The probabilistic solver seeks to obtain $\tilde{\mathbf{U}} = \tilde{u} (\mathbf{X})$ based on observations $(P(\tilde{u} (\mathbf{X})),\mathbf{h})$ . + +Even if we model $\tilde{u} (\mathbf{X})\sim \mathrm{q - ED}(\tilde{\mathbf{u}},\mathbf{S})$ with $\tilde{\mathbf{u}},\mathbf{S}$ to be specified in (8) in Section 3.2, the nonlinear mapping $P$ would not render $P(\tilde{u} (\mathbf{X}))$ another q-ED random variable. Therefore, to properly define the likelihood model, we propose the following distribution propagation by linearizing $P$ : + +$$ +\begin{array}{l} P (\tilde {u} (\mathbf {X})) \approx P (\tilde {\mathbf {u}} _ {0}) + \nabla P (\tilde {\mathbf {u}} _ {0}) (\tilde {u} (\mathbf {X}) - \tilde {\mathbf {u}} _ {0}) \sim \mathrm {q - E D} (\mathbf {m}, \boldsymbol {\Gamma}), \\ \mathbf {m} = P \left(\tilde {\mathbf {u}} _ {0}\right) + \nabla P \left(\tilde {\mathbf {u}} _ {0}\right) \left(\tilde {\mathbf {u}} - \tilde {\mathbf {u}} _ {0}\right), \quad \boldsymbol {\Gamma} = \nabla P \left(\tilde {\mathbf {u}} _ {0}\right) \mathbf {S} \nabla P \left(\tilde {\mathbf {u}} _ {0}\right) ^ {\top} + \delta \mathbf {I} _ {N}. \tag {5} \\ \end{array} +$$ + +where the Taylor expansion of $P$ is about $\tilde{\mathbf{u}}_0$ , which can be chosen as $\tilde{\mathbf{u}}_{n - 1}$ from the previous training epoch or simply $\tilde{\mathbf{u}}$ , and $\delta >0$ is a small nugget to ensure positive definiteness of $\Gamma$ . + +Let $\mathbf{Y}_{N\times 1}\coloneqq P(\tilde{u} (\mathbf{X}))$ . The Q-EP solver aims to minimize the discrepancy $\mathbf{E} = \mathbf{Y} - \mathbf{h}$ . The potential (negative log-likelihood) function, $\Phi :L^q (\Omega)\times \mathbb{R}^N\to \mathbb{R}$ , can then be defined: + +$$ +\Phi (\tilde {u}; \mathbf {Y}) = - \varphi (r; \boldsymbol {\Gamma}, N), \quad r = (\mathbf {m} - \mathbf {h}) ^ {\top} \boldsymbol {\Gamma} ^ {- 1} (\mathbf {m} - \mathbf {h}). \tag {6} +$$ + +where $\varphi (r;\Gamma ,N)\coloneqq -\frac{1}{2}\log |\Gamma | + \frac{N}{2}\left(\frac{q}{2} -1\right)\log r - \frac{1}{2} r^{\frac{q}{2}}$ . Under Assumption 2, $\Phi$ is Lipschitz continuous in $\tilde{u}$ , which is used in the convergence theorem in Section 4. + +Proposition 3.1. Suppose that the PDE mapping $P$ satisfies Assumption 2. Let $q \in (0,2]$ . Then for every $r > 0$ , there exists $L = L(r) > 0$ such that for every $\mathbf{Y} \in \mathbb{R}^N$ and for all $\tilde{u}_1, \tilde{u}_2 \in L^q(\Omega)$ with $\max \{\|\tilde{u}_1\|_q, \|\tilde{u}_2\|_q\} < r$ , $|\Phi(\tilde{u}_1; \mathbf{Y}) - \Phi(\tilde{u}_2; \mathbf{Y})| \leq L\|\tilde{u}_1 - \tilde{u}_2\|_q$ . + +Proof. See Appendix B. + +![](images/572f242126160d328c243cb8cedf2d28d4e17cdc2370bac4bb086f5e673bf7f2.jpg) + +Therefore, the Bayesian model for the solution $u$ to (4) can be summarized as + +$$ +\begin{array}{l} \mathbf {Y} | \tilde {u} (\mathbf {X}), \mathbf {h} \sim \mathrm {q - E D} _ {N} (\mathbf {h}, \boldsymbol {\Gamma}), \\ \tilde {u} \sim \mathrm {q -} \mathcal {E P} (0, \tilde {\mathcal {C}}). \tag {7} \\ \end{array} +$$ + +Our goal is to infer the posterior $p(\tilde{\mathbf{U}}|\mathbf{Y}) \propto p(\mathbf{Y}|\tilde{\mathbf{U}})p(\tilde{\mathbf{U}})$ . Note that because the extended function $\tilde{u}(\mathbf{X})$ enters the likelihood model in a nonlinear way, Theorem 2.2 does not apply. In the following, we solve the inference problem using variational Bayes. + +# 3.2 Variational Inference + +We approximate the posterior $p(\tilde{\mathbf{U}}|\mathbf{Y})$ with some variational distribution $q(\tilde{\mathbf{U}})$ using the variational Bayes method, which aims to minimize the Kullback-Leibler divergence $\mathrm{KL}(q(\tilde{\mathbf{U}})\| p(\tilde{\mathbf{U}}|\mathbf{Y}))$ . Because $\log p(\mathbf{Y}) = \mathrm{KL}(q(\tilde{\mathbf{U}})\| p(\tilde{\mathbf{U}}|\mathbf{Y})) + \mathcal{L}(q(\tilde{\mathbf{U}}))$ , it reduces to maximizing the lower bound $\mathcal{L}(q(\tilde{\mathbf{U}}))$ . The sparse variational approximation [46, 47] is adopted by introducing inducing points $\tilde{\mathbf{X}} \in \mathbb{R}^{M \times d}$ with their function values $\tilde{\mathbf{V}} = \tilde{u}(\tilde{\mathbf{X}}) \in \mathbb{R}^{M \times D}$ . + +With the variational distribution for inducing values $q(\tilde{\mathbf{V}}) \sim \mathrm{q - ED}_{MD}(\pmb {\mu},\pmb {\Sigma})$ , the marginal variational distribution $q(\tilde{\mathbf{U}})$ can be obtained as [17, 30] + +$$ +q (\tilde {\mathbf {U}}) = \int q (\tilde {\mathbf {U}}, \tilde {\mathbf {V}}) d \tilde {\mathbf {V}} = \int p (\tilde {\mathbf {U}} | \tilde {\mathbf {V}}) q (\tilde {\mathbf {V}}) d \tilde {\mathbf {V}} \sim \mathrm {q} - \mathrm {E D} (\tilde {\mathbf {u}}, \mathbf {S}), \tag {8} +$$ + +$$ +\tilde {\mathbf {u}} = \tilde {\mathbf {C}} _ {N M} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \mathrm {v e c} (\boldsymbol {\mu}), \quad \mathbf {S} = \tilde {\mathbf {C}} _ {N N} + \tilde {\mathbf {C}} _ {N M} \tilde {\mathbf {C}} _ {M M} ^ {- 1} (\boldsymbol {\Sigma} - \tilde {\mathbf {C}} _ {M M}) \tilde {\mathbf {C}} _ {M M} ^ {- 1} \tilde {\mathbf {C}} _ {M N}. +$$ + +The final evidence lower bound (ELBO) $\mathcal{L}^{*}(q(\tilde{\mathbf{U}}))$ is (Refer to Section A.1 for more details.) + +$$ +\begin{array}{l} \mathcal {L} ^ {*} (q (\tilde {\mathbf {U}})) = \varphi (\tilde {\mathbf {u}} ^ {\top} \boldsymbol {\Gamma} ^ {- 1} \tilde {\mathbf {u}} + \operatorname {t r} (\boldsymbol {\Gamma} ^ {- 1} \mathbf {S}); \boldsymbol {\Gamma}, N) \\ + \frac {1}{2} \log | \boldsymbol {\Sigma} | + \varphi (\left\langle \operatorname {v e c} (\boldsymbol {\mu}) ^ {\top} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \operatorname {v e c} (\boldsymbol {\mu}) \right\rangle + \operatorname {t r} (\boldsymbol {\Sigma} \tilde {\mathbf {C}} _ {M M} ^ {- 1}); \tilde {\mathbf {C}} _ {M M}, M D). \tag {9} \\ \end{array} +$$ + +The variational solution $q(\tilde{\mathbf{U}})$ can be obtained by maximizing the ELBO (9) with respect to the variational parameters $(\pmb{\mu},\pmb{\Sigma},\tilde{\mathbf{X}})$ and hyper-parameters in the kernel $\mathcal{C}$ . By introducing the $M$ inducing points, the overall computational complexity is reduced from $\mathcal{O}(N^3)$ to $\mathcal{O}(NM^2)$ [47]. + +# 3.3 Bayesian Inverse Problems + +The above Bayesian framework can be readily extended to solve inverse problems. The following adaptation enables us to obtain both forward and inverse PDE solutions simultaneously. + +Suppose that the PDE (4) contains a quantity of interest, $a(\mathbf{x})$ , which could appear in the differential equation $\mathcal{D}$ or as part of the boundary condition $\mathcal{B}$ . The task of Bayesian inverse problems is to find a true solution, $a^{\dagger}$ , with proper UQ based on observations. Suppose $a$ is differentiable enough and we denote $\tilde{a} = (a,\frac{\partial}{\partial\mathbf{x}} a,\dots ,\frac{\partial^{k^{\prime}}}{\partial\mathbf{x}^{k^{\prime}}} a)$ to the order $k^{\prime}\leq k$ . Now the joint operator $\mathcal{P}$ applies to both $u$ and $a$ , which produces a nonlinear function of $\tilde{u} (\mathbf{X}),\tilde{a} (\mathbf{X})$ when evaluated on $\mathbf{X}$ , denoted as $P(\tilde{u} (\mathbf{X}),\tilde{a} (\mathbf{X})) = \mathcal{P}(u,a)(\mathbf{X})$ . + +In addition, there is an observation operator $\mathcal{O}$ such that observations, $\mathcal{O}(u)(\mathbf{X}) = u(\mathbf{X}_o)$ , are obtained on some set of $N_{o}$ observation points, $\mathbf{X}_o\subset \overline{\Omega}$ with $|\mathbf{X}_o| = N_o$ . This can be achieved by solving (4) with true $a^\dagger$ in simulations or simply modeling measurement data as noisy realization of (4) in real-world applications. Let $\tilde{N} = N + N_{o}$ . We supplement the joint equation operator $\mathcal{P}$ with the observation operator $\mathcal{O}$ to form an augmented operator $\tilde{\mathcal{P}} = (\mathcal{P},\mathcal{O})$ . Therefore, we have $\tilde{\mathbf{Y}}_{\tilde{N}\times 1} = \tilde{\mathcal{P}} (u,a)(\mathbf{X}) = \tilde{P} (\tilde{u} (\mathbf{X}),\tilde{a} (\mathbf{X})) = [P(\tilde{u} (\mathbf{X}),\tilde{a} (\mathbf{X}))^{\top},O(\tilde{u} (\mathbf{X}))^{\top}]^{\top}$ . Similarly, we augment the right-hand side data $\mathbf{h}$ with observed $u(\mathbf{X}_o)$ to make $\tilde{\mathbf{h}}_{\tilde{N}\times 1} = [\mathbf{h}^{\top},u(\mathbf{X}_o)^{\top}]^{\top}$ . + +If we model $\tilde{a}(\mathbf{X})$ using $q(\tilde{a}(\mathbf{X})) \sim \mathrm{q-ED}(\tilde{a}, \mathbf{S}_a)$ with variational mean and covariance $\tilde{a}, \mathbf{S}_a$ respectively, then $q(\tilde{u}(\mathbf{X})) q(\tilde{a}(\mathbf{X}))$ propagates through the PDE dynamics similarly as in (5). Finally, we summarize the Bayesian inverse model for $a(\mathbf{x})$ as follows: + +$$ +\begin{array}{l} \tilde {\mathbf {Y}} | \tilde {u} (\mathbf {X}), \tilde {a} (\mathbf {X}), \tilde {\mathbf {h}} \sim \mathrm {q - E D} _ {\tilde {N}} (\tilde {\mathbf {h}}, \tilde {\boldsymbol {\Gamma}}), \quad \tilde {\boldsymbol {\Gamma}} = \nabla \tilde {P} (\tilde {\mathbf {u}} _ {0}, \tilde {\mathbf {a}} _ {0}) \left[ \begin{array}{l l} \mathbf {S} _ {u} & \mathbf {0} \\ \mathbf {0} & \mathbf {S} _ {a} \end{array} \right] \nabla \tilde {P} (\tilde {\mathbf {u}} _ {0}, \tilde {\mathbf {a}} _ {0}) ^ {\top} + \delta \mathbf {I} _ {\tilde {N}}, \tag {10} \\ \tilde {u} \sim \mathrm {q -} \mathcal {E P} (0, \tilde {\mathcal {C}} _ {u}), \quad \tilde {a} \sim \mathrm {q -} \mathcal {E P} (0, \tilde {\mathcal {C}} _ {a}). \\ \end{array} +$$ + +where $\tilde{\mathbf{a}}_0$ can be similarly chosen as $\tilde{\mathbf{a}}_{n - 1}$ from the previous training epoch or simply $\tilde{\mathbf{a}}$ . The variational Bayes procedure in Section 3.2 can be modified accordingly to obtain the variational solution of $\tilde{a} (\mathbf{X})|\tilde{\mathbf{Y}}$ . Meanwhile, we obtain the variational solution of $\tilde{u} (\mathbf{X})|\tilde{\mathbf{Y}}$ as a byproduct. + +# 4 Convergence Theorem + +In this section, we study the posterior contraction of the Bayesian model (7) in the infinite data limit. Similar theory can be developed for the model (10). We focus on $q \in [1, 2]$ and leave the technically more challenging case $q \in (0, 1)$ to future study. For brevity, we denote $u = \tilde{u}$ , $\mathcal{C} = \tilde{\mathcal{C}}$ , and $n = N$ . + +Consider the separable Banach space $\mathbb{X} = (L^q(\Omega), \| \cdot \|_q) \supset (B^{s,q}(\Omega), \| \cdot \|_{s,q})$ for $s > d\left(\frac{2}{q} - \frac{1}{2}\right)$ and $\mathbb{Y} = \mathbb{R}^n$ . Define the concentration function of Q-EP measure $\Pi$ at $u = u^\dagger$ as + +$$ +\varphi_ {u ^ {\dagger}} (\varepsilon) = \inf _ {h \in B ^ {s, q} (\Omega): \| h - u ^ {\dagger} \| _ {q} \leq \varepsilon} \frac {1}{2} \| h \| _ {s, q} ^ {q} - \log \Pi (\| u \| _ {q} \leq \varepsilon). \tag {11} +$$ + +Let $\mathbb{P}_u^{(n)}$ be the measure of the observations $\mathbf{Y}^{(n)}$ on $(\mathbb{Y},\mathcal{B},\mu_0)$ having density $p_u$ and corresponding potential function $\Phi (u;\cdot)$ with respect to the Lebesgue measure $\mu_0$ , i.e. $\frac{d\mathbb{P}_u^{(n)}}{d\mu_0} (\mathbf{Y}) = p_u(\mathbf{Y})\propto \exp (-\Phi (u;\mathbf{Y}))$ . Define the Hellinger distance as $d_{n,H}^2 (u,u') = \int (\sqrt{p_u} -\sqrt{p_{u'}})^2 d\mu_0$ . We have the following posterior contraction theorem. + +Theorem 4.1 (Posterior Contraction). Let $u \sim \mathrm{q - }\mathcal{EP}(0,\mathcal{C})$ with $\mathcal{C}$ satisfying Assumption 1 in $\Theta \coloneqq L^{q}(\Omega)$ and $\mathbb{P}_u^{(n)}$ is the measure of $\mathbf{Y}^{(n)}$ parameterized by $u$ with PDE (4) satisfying Assumption 2. If the true value $u^{\dagger} \in \Theta$ is in the support of $u$ , and $\varepsilon_{n}$ satisfies the rate equation $\varphi_{u^{\dagger}}(\varepsilon_n) \leq n\varepsilon_n^2$ with $\varepsilon_{n} \geq n^{-\frac{1}{2}}$ , then there exists $\Theta_{n} \subset \Theta$ such that $\Pi_{n}(u \in \Theta_{n}: d_{n,H}(u, u^{\dagger}) \geq M_{n}\varepsilon_{n}|\mathbf{Y}^{(n)}) \to 0$ in $P_{u^{\dagger}}^{(n)}$ -probability for every $M_{n} \to \infty$ . + +Proof. See Appendix B. + +![](images/b4bb6adf5b36d6e94b0df38ecee59be67c9507051aef415eb9370726f8a3b1c9.jpg) + +Denote $a \wedge b = \min\{a, b\}$ , $a \vee b = \max\{a, b\}$ , and $x_+ = x \vee 0$ . By solving the inequality $\varphi_{u^\dagger}(\varepsilon_n) \leq n\varepsilon_n^2$ for the minimal $\varepsilon_n$ , we obtain the posterior contraction rate as follows. + +Theorem 4.2 (Contraction Rate). Let $u \sim \mathrm{q - }\mathcal{EP}(0,\mathcal{C})$ with $\mathcal{C}$ satisfying Assumption 1 in $\Theta \coloneqq L^q (\Omega)$ . The rest of the settings are the same as in Theorem 4.1. If the true value $u^{\dagger} \in B^{s^{\dagger},q^{\dagger}}(\Omega)$ with $s^{\dagger} > s' + \left(\frac{d}{q^{\dagger}} -\frac{d}{q}\right)_+$ , $s' = \frac{d}{q} -\frac{d}{2}$ , and $q^{\dagger},q \in [1,2]$ , then we have the rate of the posterior contraction as $\varepsilon_{n} = n^{-\frac{\sigma(s,q,s^{\dagger},q^{\dagger}) - s^{\prime}}{2(\sigma(s,q,s^{\dagger},q^{\dagger}) - s^{\prime}) + q(s - \sigma(s,q,s^{\dagger},q^{\dagger}))}}$ , where $\sigma (s,q,s^{\dagger},q^{\dagger}) = \left(s - \frac{d}{q}\right)\Lambda \left(s^{\dagger} - \left(\frac{d}{q^{\dagger}} -\frac{d}{q}\right)_+\right)$ . + +Proof. See Appendix B. + +![](images/ad19c56ef6cbd67ed2c27155affac6d28f96f9962b7f7ceea790f435aeed7db8.jpg) + +Remark 3. If we set the smoothness parameter $s = s^{\dagger} + \frac{d}{q} - \left(\frac{d}{q^{\dagger}} - \frac{d}{q}\right)_+$ , and allow the integrability parameter $q \leq q^{\dagger}$ , then the contraction rate $\varepsilon_{n}$ is maximized as $\varepsilon_{n}^{\dagger} = n^{-\frac{1}{2 + \frac{d}{s^{\dagger} - s^{\prime}}} > n^{-\frac{1}{2}}}$ . Note that this optimal rate is achieved regardless of the value of modeling regularization parameter $q$ as long as $q \leq q^{\dagger}$ . This implies that when modeling inhomogeneous data and derivative information, under-smoothing (with smaller regularization parameter $q$ ) is preferred to over-smoothing. When the true integrability $q^{\dagger}$ is at least $L_{1}$ , setting $q = 1$ guarantees the fastest convergence of the posterior. In the following section, we will demonstrate such optimal choice with various numerical examples. + +# 5 Numerical Experiments + +In this section, we test variational Q-EP for solving and learning PDEs. Using the Eikonal equation, Burgers' equation, a nonlinear elliptic equation (Section C.3), and an inverse problem involving the Darcy flow, we investigate the proposed method for a variety of different $q_{s}$ . We include PINN [37] and B-PINN [49] as baselines and the variational GP as a special case with $q = 2$ and mainly compare them using the relative error of the estimated solution $\hat{u}$ with reference to the true solution $u^{\dagger}$ : RLE- $p = \frac{\|u^{\dagger} - \hat{u}\|_p}{\|u^{\dagger}\|_p}$ for $p = 1,2,\infty$ . Q-EP with $q = 1$ outperforms PINN and variational GP in all cases and attains the best results in most comparisons. The numerical evidence also supports that Q-EP converges the fastest with the optimal choice of $q = 1$ . + +![](images/f14d916eaab9a65aa5388568a96d781e2937375c787c526fb0d3940665ff1de9.jpg) +Figure 2: Solving Eikonal equation (12) using high-resolution finite difference method (upper left), PINN (lower left), Q-EPs (middle two in upper row: $q = 0.8$ ; right two in upper row: $q = 1.0$ ; middle two in lower row: $q = 1.5$ ), and GP (right two in lower row: $q = 2.0$ ) respectively. Blue crosses are learned inducing points. + +In most experiments, we choose the interior collocation points on a $24 \times 24$ mesh grid and the corresponding 100 boundary collocation points unless otherwise stated. For the sparse variational inference, $M = 256$ inducing points are randomly initialized and learned by optimizing the ELBO (9). The kernel $\mathcal{C}$ of Q-EP/GP is chosen to be matern52 with the hyperparameters, e.g. the correlation strength, automatically tuned in the python package GPyTorch [11] implemented based on PyTorch. PINN is configured to have neural network parameters (weights and biases) of similar size to the variational Q-EP model. The computer codes are publicly available at https://github.com/lanzithinking/Diff_QEP. + +# 5.1 Eikonal Equation + +First, we consider the following regularized Eikonal equation on $\Omega = [0,1]^2$ also considered in [6]: + +$$ +\left| \nabla u (\mathbf {x}) \right| ^ {2} - \varepsilon \Delta u (\mathbf {x}) = f (\mathbf {x}) ^ {2}, \quad \mathbf {x} \in \Omega , \tag {12} +$$ + +$$ +u (\mathbf {x}) = 0, \quad \mathbf {x} \in \partial \Omega . +$$ + +where $f \equiv 1$ and $\varepsilon = 0.1$ . Based on the set-up in Section 3.1, we define the nonlinear function $D(u,d_1u,d_2u,d_1^2 u,d_2^2 u) = (d_1u)^2 + (d_2u)^2 - \varepsilon (d_1^2 u + d_2^2 u)$ . Then the observations are $\mathbf{Y} = P(\tilde{u} (\mathbf{X})) = \left[ \begin{array}{c}D(u(\mathbf{X}_d),\frac{\partial}{\partial x_1} u(\mathbf{X}_d),\frac{\partial}{\partial x_2} u(\mathbf{X}_d),\frac{\partial^2}{\partial x_1^2} u(\mathbf{X}_d),\frac{\partial^2}{\partial x_2^2} u(\mathbf{X}_d))\\ u(\mathbf{X}_b) \end{array} \right]$ , and $\mathbf{h} = \left[ \begin{array}{c}f(\mathbf{X}_d)^2\\ \mathbf{0}_{N_b} \end{array} \right]$ . We obtain $N_{d} = 24^{2}$ interior and $N_{b} = 100$ boundary collocation points. Then we apply the variational Bayes in Section 3.2 to solve (12) with $M = 256$ inducing points. Though not required for convergence, we train each algorithm for 5000 iterations for fair comparison (PINN and B-PINN need much more training epochs than Q-EP solvers to converge). + +Figure 2 compares solutions and uncertainty estimates generated by a variety of Q-EP solvers for $q = 0.8, 1.0, 1.5$ and $2.0$ (GP) respectively. We solve the equation using a highly-resolved finite difference method with the Cole-Hopf transformation [6] and use it as the true solution for comparison (upper left). We notice that the solution by PINN (lower left) is much worse and UQ is not available. All Q-EP solvers ( $q < 2$ ) produce better solutions than GP ( $q = 2$ ) which does not precisely characterize the pyramid feature of the true solution. Only two solvers with $q = 0.8$ and $1.0$ yield solutions that correctly match the range of true solution. GP also manifests higher uncertainty in its generated solution. Table C.1 further verifies the superior accuracy of Q-EP solvers compared to GP and PINN (B-PINN) in terms of multiple error metrics including MAE, MSE and RLE by repeating the experiments for 10 times with different random seeds. In this example, Q-EP solver with $q = 1.0$ attains the result comparable to the most accurate solution. Note that with similar size of collocation points, our best results (4.43e-2 in $L_{2}$ error and $1.03\mathrm{e - }2$ in $L_{\infty}$ error) are comparable to those reported (1.64e-2 in $L_{2}$ error and $7.76\mathrm{e - }2$ in $L_{\infty}$ error) in [6] from which UQ is absent. + +# 5.2 Burgers' Equation + +Next, we test our Q-EP solvers on the Burgers' equation [6] with $\nu = 0.1$ : + +![](images/481390cb4bb8f15e0e7ea77dbdd8a9182a9b951233ae8624439c92c68ae0a6d0.jpg) + +![](images/a620a33d28540b99449a14af7f8ef701b9880cd33c21e1954085aaf69472caf5.jpg) + +![](images/87235b1dc5967398b9dbba9fbdbc6bbefb67980e3c05b5c7c7120cda7cc54f54.jpg) + +![](images/1bfb33ecc6c032b31a0949947f7f84f6315acbe6c4ed6a7a22b842b15860a52a.jpg) + +![](images/a97787e8dfe3364a5c195f52e54f435c408918dfc6aaae12ac80a4e94e5cf36b.jpg) + +![](images/2f24b545f86ac43f280f31f02a0247803a2c43f78b87ebcb454fcab761bd4e1c.jpg) + +![](images/655730c81d95203302848fb5f17eb2d4b7ecde5d94863e0219b5f8da96ba0ff4.jpg) + +![](images/55f2fde234488ad9623190b6cb1c571bd17d272ef119ff33fa9f1f67a7fd44ce.jpg) + +![](images/cc80e20c78eabd8c8d2d4295b1aff6603d0eab25a6cd30e348bf09af270a3ad7.jpg) +Figure 3: Comparing convergence of Q-EP (left three in top row: $q = 1.0$ ), GP (left three in middle row: $q = 2.0$ ) and PINN (left three in bottom row) in solving Burgers' equation (13), with the right column illustrating the error reducing in $L_{1}$ norm (top), $L_{2}$ norm (middle), and $L_{\infty}$ norm (bottom) respectively. Blue crosses are learned inducing points. Shaded regions are standard errors based on 10 repeated experiments. + +![](images/892d63d7dfeeffa03d026bd1d6d4493d31450a9ca649c1a3b6763377308fb644.jpg) + +![](images/1b5e2db910899415c900fc38843987d2b19e5146c0da0f02e5d52a4d25511db0.jpg) + +![](images/039fc0de957f363a13da737acd95f1683318e9345a8a6265031b9309e21fd105.jpg) + +$$ +\frac {\partial}{\partial t} u + u \frac {\partial}{\partial x} u - \nu \frac {\partial^ {2}}{\partial x ^ {2}} u = 0, \quad (x, t) \in (- 1, 1) \times (0, 1 ], +$$ + +$$ +u (x, 0) = - \sin (\pi x), \quad x \in (- 1, 1), \tag {13} +$$ + +$$ +u (- 1, t) = u (1, t) = 0, \quad t \in (0, 1 ]. +$$ + +We use the same experiment setup as above. Figure C.1 compares the solutions by a highly-resolved finite difference method (upper left, treated as true solution for comparison purpose), PINN (lower left), Q-EP (right three in upper row: $q = 1.0$ ), and GP (right three in lower row: $q = 2.0$ ). Q-EP is about one order of magnitude more accurate than PINN and GP, which can be verified from the pointwise error plots and Table 1. The plots of posterior standard deviation on the right column also indicate meaningful uncertainty in the middle area around $x = 0$ where the shock is difficult to resolve. Q-EP still achieves one order of magnitude lower uncertainty compared with GP. + +Table 1: Comparing accuracy of various solvers for Burgers' equation (13) in terms of mean absolute error (MAE), mean squared error (MSE), and relative errors in $L_{1}$ norm (RLE-1), $L_{2}$ norm (RLE-2), and $L_{\infty}$ norm (RLE- $\infty$ ) respectively. Result in each cell are averaged over 10 experiments with different random seeds; values after $\pm$ are standard deviations of these repeated experiments. + +
Model (q)MAEMSERLE-1RLE-2RL-∞
PINN5.81e-2 ± 1.03e-47.12e-3 ± 3.96e-40.1508 ± 0.00170.1896 ± 0.00520.2842 ± 0.0071
B-PINN2.94e-2 ± 1.57e-21.67e-3 ± 1.58e-30.0785 ± 0.04200.0833 ± 0.04090.1327 ± 0.0446
0.57.77e-2 ± 7.09e-23.50e-2 ± 7.15e-20.2018 ± 0.37510.2056 ± 0.37640.2177 ± 0.3838
0.81.56e-2 ± 4.96e-31.59e-3 ± 5.25e-30.0405 ± 0.07800.0434 ± 0.08040.0529 ± 0.0924
1.09.33e-3 ± 1.76e-41.77e-4 ± 2.16e-40.0242 ± 0.01280.0266 ± 0.01400.0324 ± 0.0145
1.21.87e-2 ± 3.23e-45.92e-4 ± 4.06e-40.0485 ± 0.01510.0522 ± 0.01680.0560 ± 0.0181
1.52.64e-2 ± 8.58e-41.31e-3 ± 1.23e-30.0684 ± 0.02620.0754 ± 0.03160.0854 ± 0.0430
2.0(Gaussian)7.13e-2 ± 8.68e-31.08e-2 ± 1.28e-20.1848 ± 0.09080.2068 ± 0.11180.2376 ± 0.1456
2.52.26e-1 ± 3.20e-12.49e-1 ± 6.04e-10.5879 ± 0.82960.6669 ± 0.95760.7558 ± 1.0521
+ +Table 1 compares the accuracy of solutions in terms of MAE, MSE and RLE, for which Q-EP with $q = 1.0$ attains the most accurate one, verifying its best performance. The error for Q-EP models does not decrease monotonically with $q > 0$ , but reaches the lowest at $q = 1.0$ . To further investigate the convergence, we plot solutions by Q-EP (top row: $q = 1.0$ ), GP (middle row: $q = 2.0$ ), and PINN (bottom row) in Figure 3 at 10 (first column), 100 (second column), and 1000 (third column) iterations respectively. We can tell that Q-EP with $q = 1.0$ converges the fastest to the best estimate, which is confirmed by the error-reducing plots on the right column. Note that in this example, our algorithm does not need random perturbation of mesh grid as required by [6]. + +![](images/8cf1f78585d0743eb4a6e2bef210b3ddb103409fa9f4d73e214e50898f054cae.jpg) +Figure 4: Solving inverse Darcy flow (14) using PINN (second column), Q-EP (right three in upper row: $q = 1.0$ ), and GP (right three in lower row: $q = 2.0$ ) respectively. Upper left: true inverse solution $a^\dagger$ ; lower left: fine-resolution finite element solution $u^\dagger$ to (14) with $a^\dagger$ . Blue crosses are learned inducing points, and red dots indicate locations of observations. + +![](images/73018cb3800f66098c79afbd0bef4258798547f6151570ac3153301d03920003.jpg) + +![](images/415ca3178f0d18f39ab03d60cfe8d9e9c613a6cc988bf9f522b444a5cf9099e3.jpg) + +![](images/601df453e935fd45c90307446a02dbead999aec5f9b7773b1512c5b5c5be1d03.jpg) + +![](images/e47620b62583e2c293c4432e1329f047afe2b8969217d37367b4b5b5f3b512ca.jpg) + +# 5.3 Inverse Darcy Flow + +Now we consider an inverse problem that involves the following Darcy flow: + +$$ +- \operatorname {d i v} (\exp (a) \nabla u) (\mathbf {x}) = f (\mathbf {x}), \quad \mathbf {x} \in \Omega := [ 0, 1 ] ^ {2}, \tag {14} +$$ + +$$ +u (\mathbf {x}) = 0, \quad \mathbf {x} \in \partial \Omega . +$$ + +where the true coefficient such that $\exp(a^\dagger(\mathbf{x})) = \exp(\sin(2\pi x_1) + \sin(2\pi x_2)) + \exp(-\sin(2\pi x_1) - \sin(2\pi x_2))$ [6] is plotted in the upper left panel of Figure 4. We generate data by solving (14) with $a^\dagger$ on a mesh $80 \times 80$ to obtain $u^\dagger$ using the finite element method, illustrated in the lower left panel in Figure 4. On a (coarser) mesh $N_d = 20 \times 20$ used for inference, we randomly select $N_o = 100$ points $\mathbf{X}_o$ in $\Omega$ and obtain observations as $u^\dagger(\mathbf{X}_o) + \varepsilon$ with noise $\varepsilon \sim N(0, \gamma^2 \mathbf{I}_{N_o})$ for $\gamma = 10^{-3}$ . + +We solve the inverse problem of finding $a$ in (14) given these observations. With $N_{d} = 400$ interior and $N_{b} = 84$ boundary collocation points and $M = 256$ inducing points, we train Q-EP solvers for 2000 iterations. Figure 4 illustrates the forward PDE solution $u$ (third column), the inverse solution $a$ (forth column), and the uncertainty of $a$ (rightmost column). Compared with the true $a^{\dagger}$ , Q-EP $(q = 1.0)$ recovers $a$ more faithfully than GP $(q = 2.0)$ and PINN (upper in the second column). Meanwhile, Q-EP also generates the solution $u$ much closer to that by the finite element method, as shown in the lower left panel. Higher uncertainty is observed by Q-EP $(q = 1.0)$ around the corners with less data (Figure C.3), reflecting the configuration of observations. If run for longer (e.g. 5000) iterations, PINN may improve its forward solution, but still gets a poor inverse solution (Figure C.4). + +Table C.5 compares the relative errors (RLEs) of forward and inverse solutions. Q-EP with $q = 1.0$ also achieves the best or comparable solutions. Here we emphasize that all the results are based on only $N_{o} = 100$ observations from one solution, as opposed to the thousands of PDE solutions typically used in operator learning algorithms. Figure C.5 compares the inverse solutions of Q-EP with those of GP in increasingly finer meshes. If we view training in finer mesh as a process to see more data, Q-EP ( $q = 1.0$ ) has already converged fast to better estimates in coarser mesh, leaving less room to improve compared to GP ( $q = 2.0$ ). + +# 6 Conclusion + +In this paper, we propose variational Q-EP to solve and learn PDEs. We advocate Q-EP with $q = 1.0$ over GP ( $q = 2.0$ ) for modeling derivative information and hence a better probabilistic solver for PDEs. The fastest convergence at $q = 1.0$ is theoretically justified and empirically verified using two nonlinear forward PDE problems and an inverse Darcy flow problem. + +One of the limitations might be the variational inference adopted in this paper. The highly nonlinear nature of some PDEs imposes challenges on the quality of variational approximation to the resulting posterior, which in turn may undermine both the solution accuracy and the associated UQ. 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Cov(·,·)u(x')∂/∂x' u(x')∂2/∂(x')2 u(x')
u(x)C(x, x')∂/∂x' C(x, x')∂2/∂(x')2 C(x, x')
∂/∂x u(x)∂/∂x C(x, x')∂2/∂x ∂x' C(x, x')∂3/∂x ∂(x')2 C(x, x')
∂2/∂x2 u(x)∂2/∂x2 C(x, x')∂3/∂x2 ∂x' C(x, x')∂4/∂x2 ∂(x')2 C(x, x')
+ +# A.1 Variational Lower Bound + +Because $\log p(\mathbf{Y}) = \mathrm{KL}(q(\tilde{\mathbf{U}})\| p(\tilde{\mathbf{U}}|\mathbf{Y})) + \mathcal{L}(q(\tilde{\mathbf{U}}))$ , it reduces to maximizing the lower bound $\mathcal{L}(q(\tilde{\mathbf{U}}))$ . The sparse variational approximation [46, 47] is adopted by introducing inducing points $\tilde{\mathbf{X}} \in \mathbb{R}^{M \times d}$ with their function values $\tilde{\mathbf{V}} = \tilde{u}(\tilde{\mathbf{X}}) \in \mathbb{R}^{M \times D}$ . Hence the joint distribution of $\mathbf{Y}$ and $\tilde{\mathbf{U}}$ can be augmented by including $\tilde{\mathbf{V}}$ : + +$$ +p (\mathbf {Y}, \tilde {\mathbf {U}}) \propto p (\mathbf {Y} | \tilde {\mathbf {U}}) p (\tilde {\mathbf {U}} | \tilde {\mathbf {V}}, \mathbf {X}, \tilde {\mathbf {X}}) p (\tilde {\mathbf {V}} | \tilde {\mathbf {X}}), +$$ + +where we have $\mathrm{vec}(\tilde{\mathbf{V}}) |\tilde{\mathbf{X}} \sim \mathrm{q-ED}_{MD}(\mathbf{0}, \tilde{\mathbf{C}}_{MM})$ and the conditional distribution + +$$ +\mathrm {v e c} (\tilde {\mathbf {U}}) | \tilde {\mathbf {V}}, \mathbf {X}, \tilde {\mathbf {X}} \sim \mathrm {q - E D} _ {N D} (\tilde {\mathbf {C}} _ {N M} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \mathrm {v e c} (\tilde {\mathbf {V}}), \tilde {\mathbf {C}} _ {N N} - \tilde {\mathbf {C}} _ {N M} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \tilde {\mathbf {C}} _ {M N}). +$$ + +Now we approximate the joint posterior $p(\tilde{\mathbf{U}},\tilde{\mathbf{V}} |\mathbf{Y})$ with the following variational distribution + +$$ +q (\tilde {\mathbf {U}}, \tilde {\mathbf {V}}) = p (\tilde {\mathbf {U}} | \tilde {\mathbf {V}}) q (\tilde {\mathbf {V}}), \quad q (\tilde {\mathbf {V}}) \sim \mathrm {q - E D} _ {M D} (\boldsymbol {\mu}, \boldsymbol {\Sigma}), +$$ + +where the covariance $\pmb{\Sigma}$ is of size $MD\times MD$ and can be chosen as a (block)-diagonal matrix for convenience. A standard variational Bayes procedure yields the variational bound: + +$$ +\begin{array}{l} \log p (\mathbf {Y}) \geq \int q (\tilde {\mathbf {U}}, \tilde {\mathbf {V}}) \log \frac {p (\mathbf {Y} | \tilde {\mathbf {U}}) p (\tilde {\mathbf {U}} | \tilde {\mathbf {V}} , \mathbf {X}) p (\tilde {\mathbf {V}})}{q (\tilde {\mathbf {U}} , \tilde {\mathbf {V}})} d \tilde {\mathbf {U}} d \tilde {\mathbf {V}} \\ = \int p (\tilde {\mathbf {U}} | \tilde {\mathbf {V}}) q (\tilde {\mathbf {V}}) d \tilde {\mathbf {V}} \log p (\mathbf {Y} | \tilde {\mathbf {U}}) d \tilde {\mathbf {U}} + \int q (\tilde {\mathbf {V}}) \log \frac {p (\tilde {\mathbf {V}})}{q (\tilde {\mathbf {V}})} d \tilde {\mathbf {V}} \\ = \mathbb {E} _ {q (\tilde {\mathbf {U}})} \log p (\mathbf {Y} | \tilde {\mathbf {U}}) - \mathrm {K L} (q (\tilde {\mathbf {V}}) \| p (\tilde {\mathbf {V}})). \\ \end{array} +$$ + +where the marginal variational distribution $q(\mathbf{\tilde{U}})$ can be obtained as [17, 30] + +$$ +\begin{array}{l} q (\tilde {\mathbf {U}}) = \int q (\tilde {\mathbf {U}}, \tilde {\mathbf {V}}) d \tilde {\mathbf {V}} = \int p (\tilde {\mathbf {U}} | \tilde {\mathbf {V}}) q (\tilde {\mathbf {V}}) d \tilde {\mathbf {V}} \sim \mathrm {q - E D} (\tilde {\mathbf {u}}, \mathbf {S}) \\ \tilde {\mathbf {u}} = \tilde {\mathbf {C}} _ {N M} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \mathrm {v e c} (\pmb {\mu}), \quad \mathbf {S} = \tilde {\mathbf {C}} _ {N N} + \tilde {\mathbf {C}} _ {N M} \tilde {\mathbf {C}} _ {M M} ^ {- 1} (\pmb {\Sigma} - \tilde {\mathbf {C}} _ {M M}) \tilde {\mathbf {C}} _ {M M} ^ {- 1} \tilde {\mathbf {C}} _ {M N}. \\ \end{array} +$$ + +Denote by $\varphi(r; \mathbf{C}, N) := -\frac{1}{2} \log |\mathbf{C}| + \frac{N}{2} \left( \frac{q}{2} - 1 \right) \log r - \frac{1}{2} r^{\frac{q}{2}}$ which is convex for $q \in (0, 2]$ . Let $\varphi_0(r) = \varphi(r; \Gamma, N)$ and $r(\mathbf{Y}) = (\mathbf{Y} - \mathbf{h})^\top \boldsymbol{\Gamma}^{-1}(\mathbf{Y} - \mathbf{h})$ be a quadratic form of random variable $\mathbf{Y}$ . Then $\log p(\mathbf{Y}|\tilde{\mathbf{U}}) = \varphi_0(r(\mathbf{Y}))$ . Therefore, by Jensen's inequality, we can bound from below as + +$$ +\langle \log p (\mathbf {Y} | \tilde {\mathbf {U}}) \rangle_ {q (\tilde {\mathbf {U}})} = \langle \varphi_ {0} (r (\mathbf {Y})) \rangle_ {q (\tilde {\mathbf {U}})} \geq \varphi_ {0} (\langle r (\mathbf {Y}) \rangle_ {q (\tilde {\mathbf {U}})}), \quad \langle r (\mathbf {Y}) \rangle_ {q (\tilde {\mathbf {U}})} = \tilde {\mathbf {u}} ^ {\intercal} \boldsymbol {\Gamma} ^ {- 1} \tilde {\mathbf {u}} + \mathrm {t r} (\boldsymbol {\Gamma} ^ {- 1} \mathbf {S}). +$$ + +Now we compute the K-L divergence $\mathrm{KL}_{\tilde{\mathbf{V}}} \coloneqq \mathrm{KL}(q(\tilde{\mathbf{V}}) \| p(\tilde{\mathbf{V}}))$ : + +$$ +\mathrm {K L} _ {\tilde {\mathbf {V}}} = \int q (\tilde {\mathbf {V}}) \log q (\tilde {\mathbf {V}}) d \tilde {\mathbf {V}} - \int q (\tilde {\mathbf {V}}) \log p (\tilde {\mathbf {V}}) d \tilde {\mathbf {V}} = - \mathcal {H} _ {q} (\tilde {\mathbf {V}}) - \langle \log p (\tilde {\mathbf {V}}) \rangle_ {q (\tilde {\mathbf {V}})}. +$$ + +Denote by $r(\tilde{\mathbf{V}}) = \mathrm{vec}(\tilde{\mathbf{V}} -\boldsymbol {\mu})^{\top}\boldsymbol{\Sigma}^{-1}\mathrm{vec}(\tilde{\mathbf{V}} -\boldsymbol {\mu})$ . Then $\log q(\tilde{\mathbf{V}}) = \varphi (r(\tilde{\mathbf{V}});\boldsymbol {\Sigma},MD)$ . From [Proposition A.1. of 24] we know that $r^{\frac{q}{2}}\sim \chi^2 (MD)$ . Therefore + +$$ +\begin{array}{l} \mathcal {H} _ {q} (\tilde {\mathbf {V}}) = \frac {1}{2} \log | \boldsymbol {\Sigma} | + \frac {M D}{2} \left(\frac {q}{2} - 1\right) \frac {2}{q} \mathcal {H} \left(\chi^ {2} (M D)\right) + \frac {M D}{2} \\ = \frac {1}{2} \log | \boldsymbol {\Sigma} | + \frac {M D}{2} \left(1 - \frac {2}{q}\right) \left[ \frac {M D}{2} + \log \left(2 \Gamma \left(\frac {M D}{2}\right)\right) + \left(1 - \frac {M D}{2}\right) \psi \left(\frac {M D}{2}\right) \right] + \frac {M D}{2}. \\ \end{array} +$$ + +Let $\varphi_{1}(r) \coloneqq \varphi(r; \tilde{\mathbf{C}}_{MM}, MD)$ . Then by Jensen's inequality + +$$ +\langle \log p (\tilde {\mathbf {V}}) \rangle_ {q (\tilde {\mathbf {V}})} = \langle \varphi_ {1} (\operatorname {v e c} (\tilde {\mathbf {V}}) ^ {\top} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \operatorname {v e c} (\tilde {\mathbf {V}})) \rangle_ {q (\tilde {\mathbf {V}})} \geq \varphi_ {1} (\langle \operatorname {v e c} (\tilde {\mathbf {V}}) ^ {\top} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \operatorname {v e c} (\tilde {\mathbf {V}}) \rangle_ {q (\tilde {\mathbf {V}})}), +$$ + +$$ +\left\langle \operatorname {v e c} (\tilde {\mathbf {V}}) ^ {\top} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \operatorname {v e c} (\tilde {\mathbf {V}}) \right\rangle_ {q (\tilde {\mathbf {V}})} = \left\langle \operatorname {v e c} (\boldsymbol {\mu}) ^ {\top} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \operatorname {v e c} (\boldsymbol {\mu}) \right\rangle + \operatorname {t r} (\boldsymbol {\Sigma} \tilde {\mathbf {C}} _ {M M} ^ {- 1}). +$$ + +Therefore, the final evidence lower bound (ELBO) $\mathcal{L}^{*}(q(\hat{\mathbf{U}}))$ is + +$$ +\begin{array}{l} \log p (\mathbf {E}) \geq \mathcal {L} ^ {*} (q (\tilde {\mathbf {U}})) = \varphi (\tilde {\mathbf {u}} ^ {\top} \boldsymbol {\Gamma} ^ {- 1} \tilde {\mathbf {u}} + \operatorname {t r} (\boldsymbol {\Gamma} ^ {- 1} \mathbf {S}); \boldsymbol {\Gamma}, N) \\ + \frac {1}{2} \log | \boldsymbol {\Sigma} | + \varphi (\langle \operatorname {v e c} (\boldsymbol {\mu}) ^ {\top} \tilde {\mathbf {C}} _ {M M} ^ {- 1} \operatorname {v e c} (\boldsymbol {\mu}) \rangle + \operatorname {t r} (\boldsymbol {\Sigma} \tilde {\mathbf {C}} _ {M M} ^ {- 1}); \tilde {\mathbf {C}} _ {M M}, M D). \\ \end{array} +$$ + +# B Proofs + +Notations: $\lesssim$ means "less than or approximately equal to"; $a_{n} \lesssim b_{n}$ implies $a_{n} \leq C b_{n}$ for some constant $C > 0$ . $\asymp$ means "asymptotically equal to"; $a_{n} \asymp b_{n}$ implies $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = c$ for some constant $c$ . + +Proof of Proposition 2.1. Note $r(\tilde{u}_\ell)^{\frac{q}{2}} = \lambda_\ell^{-\frac{q}{2}}|\tilde{u}_\ell|^q \sim \chi^2(1)$ for all $\ell \in \mathbb{N}$ by Proposition A.1. of [24]. Denote $\chi_\ell^2 \coloneqq \lambda_\ell^{-\frac{q}{2}}|\tilde{u}_\ell|^q \stackrel{iid}{\sim} \chi^2(1)$ . Hence $\| \tilde{u} \|_q^q = \sum_{\ell=1}^\infty \lambda_\ell^{\frac{q}{2}} \chi_\ell^2$ becomes an infinite mixture of chi-squared random variables whose density is analytically intractable. Yet we have + +$$ +\mathbb {E} [ \| \tilde {u} (\cdot) \| _ {q} ^ {q} ] = \sum_ {\ell = 1} ^ {\infty} \lambda_ {\ell} ^ {\frac {q}{2}} \mathbb {E} [ \chi_ {\ell} ^ {2} ] = \sum_ {\ell = 1} ^ {\infty} \lambda_ {\ell} ^ {\frac {q}{2}} < \infty , +$$ + +if Assumption 1 holds. Thus it completes the proof. + +Proof of Proposition 3.1. Based on (5) and (6), the potential $\Phi (\tilde{u};\mathbf{Y}) = -\varphi (r;\mathbf{\Gamma},N)$ where $\varphi (r;\mathbf{\Gamma},N)\coloneqq -\frac{1}{2}\log |\mathbf{\Gamma}| + \frac{N}{2}\left(\frac{q}{2} -1\right)\log r - \frac{1}{2} r^{\frac{q}{2}}$ is convex in $r$ if $q\in (0,2]$ , and $r = r(\tilde{\mathbf{u}})$ with + +$$ +r (\tilde {\mathbf {u}}) = \left(\mathbf {A} \tilde {\mathbf {u}} - \mathbf {b}\right) ^ {\top} \boldsymbol {\Gamma} ^ {- 1} \left(\mathbf {A} \tilde {\mathbf {u}} - \mathbf {b}\right), \mathbf {A} = \nabla P \left(\tilde {\mathbf {u}} _ {0}\right), \mathbf {b} = - P \left(\tilde {\mathbf {u}} _ {0}\right) + \nabla P \left(\tilde {\mathbf {u}} _ {0}\right) \tilde {\mathbf {u}} _ {0} + \mathbf {h}, \boldsymbol {\Gamma} = \mathbf {A} \mathbf {S} \mathbf {A} ^ {\top} + \delta \mathbf {I}. +$$ + +Since convex function is Lipschitz continuous over compact domain, it suffices to prove that $r(\tilde{\mathbf{u}})$ is bounded (both bounds achievable) and Lipschitz. + +Note that $r(\tilde{\mathbf{u}})\leq \delta^{-1}\| \mathbf{A}\tilde{\mathbf{u}} -\mathbf{b}\| _2^2\leq \delta^{-1}(\| \mathbf{A}\| \| \tilde{\mathbf{u}}\| +\| \mathbf{b}\|)^2$ where $\tilde{\mathbf{u}}$ represents a PDE solution in the compact domain $\overline{\Omega}\subset \mathbb{R}^d$ and is therefore bounded by some $M > 0$ . By Assumption 2, $\| \mathbf{A}\| \leq C$ . Therefore, $0\leq r(\tilde{\mathbf{u}})\leq \delta^{-1}(CM + \| \mathbf{b}\|)^2$ . + +On the other hand, we have the gradient of the quadratic form $r(\tilde{\mathbf{u}})$ bounded as + +$$ +\| \nabla r (\tilde {\mathbf {u}}) \| = 2 \| \mathbf {A} ^ {\mathsf {T}} \mathbf {\Gamma} ^ {- 1} (\mathbf {A} \tilde {\mathbf {u}} - \mathbf {b}) \| \leq 2 \delta^ {- 1} \| \mathbf {A} \| \| \mathbf {A} \tilde {\mathbf {u}} - \mathbf {b} \| \leq 2 \delta^ {- 1} C (C M + \| \mathbf {b} \Vert). +$$ + +Hence $r(\tilde{\mathbf{u}})$ is Lipschitz. + +Lastly, there exists $L_{1}, L_{2} > 0$ such that + +$$ +\begin{array}{l} \left| \Phi (\tilde {u} _ {1}; \mathbf {Y}) - \Phi (\tilde {u} _ {2}; \mathbf {Y}) \right| = \left| \varphi \left(r \left(\tilde {\mathbf {u}} _ {1}\right); \boldsymbol {\Gamma}, N\right) - \varphi \left(r \left(\tilde {\mathbf {u}} _ {2}\right); \boldsymbol {\Gamma}, N\right) \right| \\ \leq L _ {1} \left| r \left(\tilde {\mathbf {u}} _ {1}\right) - r \left(\tilde {\mathbf {u}} _ {2}\right) \right| \leq L _ {1} L _ {2} \| \tilde {\mathbf {u}} _ {1} - \tilde {\mathbf {u}} _ {2} \| _ {q} \\ \leq 2 L _ {1} L _ {2} \left(\frac {N}{| \Omega |}\right) ^ {\frac {1}{q}} \| \tilde {u} _ {1} - \tilde {u} _ {2} \| _ {q} \\ \end{array} +$$ + +assuming the collocation points are uniformly sampled. Therefore, it completes the proof. + +![](images/3244e208e240993378a6538313e6aa469a3c8a63e1c0d531bc66a390809fac19.jpg) + +According to Theorem 3.1 and Lemma 5.14 of [1], the following general contraction conditions hold for Q-EP and will be used in the proof of posterior contraction Theorem 4.1. + +Theorem B.1. Let $\mu$ be a q- $\mathcal{EP}(0,\mathcal{C})$ measure satisfying Assumption 1 in the separable Banach space $(L^q (\Omega),\| \cdot \| _q)$ , where $q\in [1,2]$ . Let $u\sim \mu$ and the true parameter $u^{\dagger}\in L^{q}(\Omega)$ . Assume $\varepsilon_{n} > 0$ such that $\varphi_{u^{\dagger}}(\varepsilon_n)\leq n\varepsilon_n^2$ , where $n\varepsilon_n^2\geq 1$ . Then for any $C > 1$ , there exists a measurable set $B_{n}\subset L^{q}(\Omega)$ and a constant $R > 0$ depending on $C$ and $q$ , such that + +$$ +\log N \left(4 \varepsilon_ {n}, B _ {n}, \| \cdot \| _ {q}\right) \leq R n \varepsilon_ {n} ^ {2}, \tag {15} +$$ + +$$ +\mu (u \notin B _ {n}) \leq \exp \left(- C n \varepsilon_ {n} ^ {2}\right), \tag {16} +$$ + +$$ +\mu \left(\left\| u - u ^ {\dagger} \right\| _ {s ^ {\prime}, q} < 2 \varepsilon_ {n}\right) \geq \exp \left(- n \varepsilon_ {n} ^ {2}\right), \tag {17} +$$ + +where $N(4\varepsilon_n,B_n,\| \cdot \| _q)$ is the minimal number of $\| \cdot \| _q$ -balls of radius $4\varepsilon_{n}$ to cover $B_{n}$ + +We need the following lemma to bound the Hellinger distance, Kullback-Leibler (K-L) divergence, and K-L variation to complete the proof of Theorem 4.1 [21]. + +Lemma B.1. Suppose the potential function $\Phi$ (6) satisfies Lipschitz continuity in $u$ as in Proposition 3.1. Then we have + +$d_H(p_u,p_{u'})\lesssim \| u - u'\| _q$ +- $K(p_{u}, p_{u'}) \lesssim \| u - u' \|_{q}$ . +$V(p_u,p_{u'})\lesssim \| u - u'\| _q^2$ + +Proof. First, we consider K-L divergence: + +$$ +K \left(p _ {u}, p _ {u ^ {\prime}}\right) = \int \log \frac {p _ {u}}{p _ {u ^ {\prime}}} p _ {u} d \mu = \int \left(\Phi \left(u ^ {\prime}; y\right) - \Phi (u; y)\right) p _ {u} d \mu (y) \leq L \| u - u ^ {\prime} \| _ {q} +$$ + +by Proposition 3.1. Similarly, we have for K-L variation: + +$$ +V (p _ {u}, p _ {u ^ {\prime}}) = \int \left(\log \frac {p _ {u}}{p _ {u ^ {\prime}}}\right) ^ {2} p _ {u} d \mu = \int | \Phi (u ^ {\prime}; y) - \Phi (u; y) | ^ {2} p _ {u} d \mu (y) \leq L ^ {2} \| u - u ^ {\prime} \| _ {q} ^ {2}. +$$ + +Lastly, we bound the Hellinger distance: + +$$ +\begin{array}{l} 2 d _ {H} ^ {2} (p _ {u}, p _ {u ^ {\prime}}) = \int (\sqrt {p _ {u}} - \sqrt {p _ {u ^ {\prime}}}) ^ {2} d \mu = \int \left[ 1 - \exp \left(\frac {1}{2} \Phi (u; y) - \frac {1}{2} \Phi (u ^ {\prime}; y)\right) \right] ^ {2} p _ {u} d \mu (y) \\ \leq \int \frac {C}{4} | \Phi (u ^ {\prime}; y) - \Phi (u; y) | ^ {2} p _ {u} d \mu (y) \leq \frac {C L ^ {2}}{4} \| u - u ^ {\prime} \| _ {q} ^ {2}. \\ \end{array} +$$ + +where the inequality holds for $\| u - u' \|_q^2$ small enough. + +![](images/5a5afbb374788c3d951151a482a56c51b347693602bf4928eafcb04f527d0b75.jpg) + +Proof of Theorem 4.1. Based on [Theorem 1 of 12], it suffices to verify the following two conditions (the entropy condition (2.4), and the prior mass condition (2.5)) for some universal constants $\eta, K > 0$ and sufficiently large $k \in \mathbb{N}$ + +$$ +\sup _ {\varepsilon > \varepsilon_ {n}} \log N \left(\eta \varepsilon / 2, \{u \in \Theta_ {n}: d _ {n, H} (u, u ^ {\dagger}) < \varepsilon \}, d _ {n, H}\right) \leq n \varepsilon_ {n} ^ {2}, \tag {18} +$$ + +$$ +\frac {\Pi_ {n} \left(u \in \Theta_ {n} : k \varepsilon_ {n} < d _ {n , H} \left(u , u ^ {\dagger}\right) \leq 2 k \varepsilon_ {n}\right)}{\Pi_ {n} \left(B _ {n} \left(u ^ {\dagger} , \varepsilon_ {n}\right)\right)} \leq e ^ {K n \varepsilon_ {n} ^ {2} k ^ {2} / 2}, \tag {19} +$$ + +where the left side of (18) is logarithm of the minimal number of $d_{n,H}$ -balls of radius $\xi \varepsilon /2$ needed to cover a ball of radius $\varepsilon$ around the true value $u^{\dagger}$ ; $B_{n}(u^{\dagger},\varepsilon_{n}) = \{u\in \Theta :\frac{1}{n} K(u^{\dagger},u)\leq \varepsilon^{2},\frac{1}{n} V(u^{\dagger},u)\leq \varepsilon^{2}\}$ with $K(u^{\dagger},u) = K(p_{u^{\dagger}},p_u)$ and $V(u^{\dagger},u) = V(p_{u^{\dagger}},p_u)$ . + +Since $u(\cdot) \in L^{q}(\Omega)$ satisfy conditions for Theorem B.1, there exists $B_{n} \subset L^{q}(\Omega)$ such that (15)-(17) holds. Now we set $\Theta_{n} = B_{n}$ . For $\forall u, u' \in \Theta_{n}$ such that $\| u(\cdot) - u'(\cdot)\|_{q} \leq \varepsilon_{n}$ , we have $d_{n,H}(u,u') \lesssim \| u - u'\|_{q} \leq \varepsilon_{n}$ by Lemma B.1. Therefore by (15) we have the following global entropy bound holds + +$$ +\log N \left(\varepsilon_ {n}, \Theta_ {n}, d _ {n, H}\right) \leq R n \varepsilon_ {n} ^ {2}, +$$ + +which is stronger than the local entropy condition (18). + +Now by Lemma B.1 and (17), we have + +$$ +\Pi_ {n} (B _ {n} (u ^ {\dagger}, \varepsilon_ {n})) \geq \Pi_ {n} (\| u ^ {\dagger} - u \| _ {q} \leq \varepsilon_ {n} ^ {2}, \| u ^ {\dagger} - u \| _ {q} ^ {2} \leq \varepsilon_ {n} ^ {2}) = \Pi_ {n} (\| u ^ {\dagger} - u \| _ {q} ^ {q} \leq \varepsilon_ {n} ^ {2 q}) \geq e ^ {- n \varepsilon_ {n} ^ {2}}. +$$ + +Then the prior mass condition (19) is satisfied because the numerator is bounded by 1. The proof is hence completed. + +The following lemma studies the small ball probability in the concentration function (11) [21]. + +Lemma B.2 (Small ball probability). Let $\Pi$ be a q- $\mathcal{EP}(0,\mathcal{C})$ prior on $B^{s',q}(\Omega)$ with $s' < s - \frac{d}{q}$ . Then as $\varepsilon \to 0$ , we have + +$$ +- \log \Pi (\| u \| _ {s ^ {\prime}, q} \leq \varepsilon) \asymp \varepsilon^ {- \frac {1}{\frac {s - s ^ {\prime}}{d} - \frac {1}{q}}}. +$$ + +Proof. We can compute + +$$ +\Pi (\| u \| _ {s ^ {\prime}, q} \leq \varepsilon) = \mathbb {P} \left[ \sum_ {\ell = 1} ^ {\infty} \left(\ell^ {\tau_ {q} (s ^ {\prime}) - \tau_ {q} (s)} | u _ {\ell} |\right) ^ {q} \leq \varepsilon^ {q} \right], +$$ + +where $\mathbb{P}$ is the probability measure on the infinite product space $(L^q (\Omega))^{\infty}$ . From the proof of Proposition 2.1 we know $\| u\| _q^q = \sum_{\ell = 1}^{\infty}\lambda_\ell^{\frac{q}{2}}\chi_\ell^2$ is an infinite mixture of $\chi^2 (1)$ random variables, so the condition of [Theorem 4.2 of 2] is trivially met and we have + +$$ +\log \mathbb {P} \left[ \sum_ {\ell = 1} ^ {\infty} \left(\ell^ {\tau_ {q} (s ^ {\prime}) - \tau_ {q} (s)} | u _ {\ell} |\right) ^ {q} \leq \varepsilon^ {q} \right] \asymp \varepsilon^ {- \frac {1}{\tau_ {q} (s) - \tau_ {q} (s ^ {\prime}) - \frac {1}{q}}}. +$$ + +![](images/9c3d3e21fa6516c769c1a3853edbacb187043c224743e0c3dc68210dea2f4f53.jpg) + +The second lemma gives an upper bound of the first term of the concentration function (11) [21]. + +Lemma B.3 (Decentering function). Assume $u^{\dagger} \in B^{s^{\dagger},q^{\dagger}}(\Omega)$ for some $s^{\dagger} > s'$ and $q^{\dagger} \in [1,2]$ . Then as $\varepsilon \to 0$ , we have the following bounds + +(i) If $q^{\dagger} \geq q$ , we require $s^{\dagger} > s'$ : + +$$ +\inf_{h\in B^{s,q}(\Omega):\| h - u^{\dagger}\|_{s^{\prime},q}\leq \varepsilon}\| h\|_{s,q}^{q}\lesssim \left\{ \begin{array}{ll}1, & if s < s^{\dagger}\\ (-\log \varepsilon)^{1 - \frac{q}{q^{\dagger}}}, & if s = s^{\dagger};\\ \varepsilon^{-\frac{s - s^{\dagger}}{s^{\dagger} - s^{\prime}} (q\wedge q^{\dagger})}, & if s > s^{\dagger} \end{array} \right.; +$$ + +(ii) If $q^{\dagger} < q$ , we require $s^{\dagger} > s' - \frac{d}{q} + \frac{d}{q^{\dagger}}$ : + +$$ +\inf_{h\in B^{s,q}(\Omega):\| h - u^{\dagger}\|_{s^{\prime},q}\leq \varepsilon}\| h\|_{s,q}^{q}\lesssim \left\{ \begin{array}{ll}1, & if s\leq s^{\dagger} + \frac{d}{q} -\frac{d}{q^{\dagger}}\\ -\frac{\frac{s - s^{\dagger}}{d} - \frac{1}{q} + \frac{1}{q^{\dagger}}}{d} q\\ \varepsilon \end{array} \right.. +$$ + +Proof. We identify $u^{\dagger} \in B^{s^{\dagger},q^{\dagger}}$ with $\{u_{\ell}^{\dagger}\}_{\ell = 1}^{\infty} \in \ell^{q^{\dagger},\tau_{q}{}^{\dagger}(s^{\dagger})}$ . Then we follow [1] to approximate $u^{\dagger}$ with $h_{1:L} = \{u_{\ell}^{\dagger}\}_{\ell = 1}^{\infty}$ where $u_{\ell}^{\dagger} \equiv 0$ for all $\ell > L$ . Note $h_{1:L} \in \ell^{q,\tau_q(s)}$ for any finite $L \in \mathbb{N}$ . Identifying $h_{1:L}$ with $h \in B^{s,q}(\Omega)$ , we could get + +$$ +\| h - u ^ {\dagger} \| _ {s ^ {\prime}, q} ^ {q} = \sum_ {\ell = L + 1} ^ {\infty} \ell^ {\tau_ {q} (s ^ {\prime}) q} | u _ {\ell} ^ {\dagger} | ^ {q} \leq \left\{ \begin{array}{l l} \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}} ^ {q ^ {\dagger}} L ^ {\frac {s ^ {\prime} - s ^ {\dagger}}{d} q ^ {\dagger}}, & i f q ^ {\dagger} = q \\ \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}} ^ {q} L ^ {\frac {s ^ {\prime} - s ^ {\dagger}}{d} q}, & i f q ^ {\dagger} > q \\ \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}, q} ^ {q} L ^ {\left(\frac {s ^ {\prime} - s ^ {\dagger}}{d} - \frac {1}{q} + \frac {1}{q ^ {\dagger}}\right) q}, & i f q ^ {\dagger} < q \end{array} \right.. +$$ + +Therefore, to have $\| h - u^{\dagger}\|_{s',q}\leq \varepsilon$ we let + +$$ +L \gtrsim \left\{ \begin{array}{l l} \varepsilon^ {- \frac {d}{s ^ {\dagger} - s ^ {\prime}}}, & i f q ^ {\dagger} \geq q \\ \varepsilon^ {- \frac {1}{\frac {s ^ {\dagger} - s ^ {\prime}}{d} + \frac {1}{q} - \frac {1}{q ^ {\dagger}}}}, & i f q ^ {\dagger} < q \end{array} . \right. \tag {20} +$$ + +On the other hand, the infimum is less than $\| h\|_{s,q}^{q}$ with above $h$ , which can be bounded as follows. If $q^{\dagger} = q$ , + +$$ +\| h \| _ {s, q} ^ {q} = \sum_ {\ell = 1} ^ {L} \ell^ {\tau_ {q ^ {\dagger}} (s) q ^ {\dagger}} | u _ {\ell} ^ {\dagger} | ^ {q ^ {\dagger}} \leq \left\{ \begin{array}{l l} \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}} ^ {q ^ {\dagger}}, & i f s \leq s ^ {\dagger} \\ \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}} ^ {q ^ {\dagger}} L ^ {\frac {s - s ^ {\dagger}}{d} q ^ {\dagger}}, & i f s > s ^ {\dagger} \end{array} \right.. +$$ + +If $q^{\dagger} > q$ , by similar argument using Hölder inequality, + +$$ +\| h \| _ {s, q} ^ {q} = \sum_ {\ell = 1} ^ {L} \ell^ {\tau_ {q} (s) q} | u _ {\ell} ^ {\dagger} | ^ {q} \leq \left\{ \begin{array}{l l} C \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}} ^ {q}, & i f s < s ^ {\dagger} \\ \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}} ^ {q} (\log L) ^ {1 - \frac {q}{q ^ {\dagger}}}, & i f s = s ^ {\dagger} \\ \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}} ^ {q} L ^ {\frac {s - s}{d} ^ {\dagger} q}, & i f s > s ^ {\dagger} \end{array} \right.. +$$ + +If $q^{\dagger} < q$ , by similar argument, + +$$ +\| h \| _ {s, q} ^ {q} = \sum_ {\ell = 1} ^ {L} \ell^ {\tau_ {q} (s) q} | u _ {\ell} ^ {\dagger} | ^ {q} \leq \left\{ \begin{array}{l l} \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}, q} ^ {q}, & i f s \leq s ^ {\dagger} + \frac {d}{q} - \frac {d}{q ^ {\dagger}} \\ \| u ^ {\dagger} \| _ {s ^ {\dagger}, q ^ {\dagger}, q} ^ {q} L ^ {\left(\frac {s - s ^ {\dagger}}{d} - \frac {1}{q} + \frac {1}{q ^ {\dagger}}\right) q}, & i f s > s ^ {\dagger} + \frac {d}{q} - \frac {d}{q ^ {\dagger}} \end{array} \right.. +$$ + +Substituting $L$ in (20) to the above equations yields the conclusion. + +![](images/fe0ce37db181e3320cae0c2d3fb79a582b0166a540f0e6cebf86eae39effdc5c.jpg) + +Proof of Theorem 4.2. By Lemmas B.2 and B.3, we have the following bounds for the concentration function (11) as $\varepsilon \to 0$ , if $q^{\dagger} \geq q$ , + +$$ +\varphi_ {u ^ {\dagger}} (\varepsilon) \lesssim \left\{ \begin{array}{l l} 1 + \varepsilon^ {- \frac {1}{\frac {s - s ^ {\prime}}{d} - \frac {1}{q}}}, & i f s < s ^ {\dagger} \\ (- \log \varepsilon) ^ {1 - \frac {q}{q ^ {\dagger}}} + \varepsilon^ {- \frac {1}{\frac {s - s ^ {\prime}}{d} - \frac {1}{q}}}, & i f s = s ^ {\dagger} \\ \varepsilon^ {- \frac {s - s ^ {\dagger}}{s ^ {\dagger} - s ^ {\prime}} (q \wedge q ^ {\dagger})} + \varepsilon^ {- \frac {1}{\frac {s - s ^ {\prime}}{d} - \frac {1}{q}}}, & i f s > s ^ {\dagger} \end{array} \right.. +$$ + +For $s \leq s^{\dagger}$ , the bound is dominated by $\varepsilon^{-\frac{1}{s - s' - \frac{1}{q}}}$ . For the last case, we need to determine a balancing point of $s$ for the two terms by setting their powers equal. The calculation shows that if $s \leq s^{\dagger} + \frac{d}{q}$ , the bound is still dominated by $\varepsilon^{-\frac{1}{s - s' - \frac{1}{q}}}$ , but otherwise is dominated by $\varepsilon^{-\frac{s - s^{\dagger}}{s^{\dagger} - s'^{t}} q}$ . Therefore, we have + +$$ +\varphi_ {u ^ {\dagger}} (\varepsilon) \lesssim \left\{ \begin{array}{l l} \varepsilon^ {- \frac {1}{\frac {s - s ^ {\prime}}{d} - \frac {1}{q}}}, & i f s \leq s ^ {\dagger} + \frac {d}{q} \\ \varepsilon^ {- \frac {s - s ^ {\dagger}}{s ^ {\dagger} - s ^ {\prime}} q}, & i f s > s ^ {\dagger} + \frac {d}{q} \end{array} \right.. +$$ + +We need to determine minimal $\varepsilon_{n}$ such that $\varphi_{u^{\dagger}}(\varepsilon_n)\leq n\varepsilon_n^2$ . Hence for $q^{\dagger}\geq q$ + +$$ +\varepsilon_ {n} \asymp \left\{ \begin{array}{l l} n ^ {- \frac {q (s - s ^ {\prime}) - d}{2 q (s - s ^ {\prime}) + (q - 2) d}}, & i f s \leq s ^ {\dagger} + \frac {d}{q} \\ n ^ {- \frac {s ^ {\dagger} - s ^ {\prime}}{2 (s ^ {\dagger} - s ^ {\prime}) + q (s - s ^ {\dagger})}}, & i f s > s ^ {\dagger} + \frac {d}{q} \end{array} \right.. +$$ + +Now if $q^{\dagger} < q$ , by similar argument we have the concentration function (11) as $\varepsilon \rightarrow 0$ + +$$ +\varphi_ {u ^ {\dagger}} (\varepsilon) \lesssim \left\{ \begin{array}{l l} 1 + \varepsilon^ {- \frac {1}{\frac {s - s ^ {\prime}}{d} - \frac {1}{q}}}, & i f s \leq s ^ {\dagger} + \frac {d}{q} - \frac {d}{q ^ {\dagger}} \\ \varepsilon^ {- \frac {\frac {s - s ^ {\dagger}}{d} - \frac {1}{q} + \frac {1}{q ^ {\dagger}}}{\frac {s ^ {\dagger} - s ^ {\prime}}{d} + \frac {1}{q} - \frac {1}{q ^ {\dagger}}} q} + \varepsilon^ {- \frac {1}{\frac {s - s ^ {\prime}}{d} - \frac {1}{q}}}, & i f s > s ^ {\dagger} + \frac {d}{q} - \frac {d}{q ^ {\dagger}} \end{array} \right.. +$$ + +Thus the contraction rate for $q^{\dagger} < q$ becomes + +$$ +\varepsilon_ {n} \asymp \left\{ \begin{array}{l l} n ^ {- \frac {q (s - s ^ {\prime}) - d}{2 q (s - s ^ {\prime}) + (q - 2) d}}, & i f s \leq s ^ {\dagger} + \frac {2 d}{q} - \frac {d}{q ^ {\dagger}} \\ n ^ {- \frac {s ^ {\dagger} - s ^ {\prime} + \frac {d}{q} - \frac {d}{q ^ {\dagger}}}{2 (s ^ {\dagger} - s ^ {\prime}) + q (s - s ^ {\dagger}) - (q - 2) (\frac {d}{q} - \frac {d}{q ^ {\dagger}})}}}, & i f s > s ^ {\dagger} + \frac {2 d}{q} - \frac {d}{q ^ {\dagger}} \end{array} \right.. +$$ + +Rewriting the equations into one yields the conclusion. + +![](images/5308558b251c572c6aa6a4deb93d0996b69bebefde2f7d593f33a888f8cde243.jpg) + +# C More Numerical Results + +# C.1 Eikonal Equation + +Table C.1: Comparing accuracy of various solvers for Eikonal equation (12) in terms of mean absolute error (MAE), mean squared error (MSE), and relative errors in $L_{1}$ norm (RLE-1), $L_{2}$ norm (RLE-2), and $L_{\infty}$ norm (RLE- $\infty$ ) respectively. Result in each cell are averaged over 10 experiments with different random seeds; values after $\pm$ are standard deviations of these repeated experiments. + +
Model (q)MAEMSERLE-1RLE-2RL-∞
PINN1.77e-2 ± 9.03e-44.65e-4 ± 4.08e-50.1120 ± 0.00570.1167 ± 0.00510.1354 ± 0.0061
B-PINN1.81e-2 ± 2.59e-21.24e-3 ± 3.52e-30.1253 ± 0.17920.1162 ± 0.17000.2143 ± 0.2193
0.51.96e-3 ± 5.09e-46.20e-6 ± 2.87e-60.0124 ± 0.00320.0132 ± 0.00290.0387 ± 0.0067
0.81.44e-3 ± 3.44e-43.50e-6 ± 1.28e-60.0091 ± 0.00220.0100 ± 0.00180.0339 ± 0.0074
1.01.68e-3 ± 5.41e-44.42e-6 ± 2.26e-60.0106 ± 0.00340.0110 ± 0.00300.0321 ± 0.0071
1.21.69e-3 ± 3.85e-44.11e-6 ± 1.52e-60.0107 ± 0.00240.0108 ± 0.00200.0284 ± 0.0057
1.51.68e-3 ± 3.31e-44.65e-6 ± 1.35e-60.0106 ± 0.00210.0116 ± 0.00180.0398 ± 0.0080
2.0 (Gaussian)9.64e-3 ± 1.33e-31.30e-4 ± 3.61e-50.0610 ± 0.00840.0612 ± 0.00870.1009 ± 0.0124
2.57.71e-2 ± 3.18e-28.36e-3 ± 5.29e-30.4881 ± 0.20110.4668 ± 0.17430.4442 ± 0.1390
+ +To verify the systematic superiority of Q-EP with $q = 1.0$ than GP ( $q = 2.0$ ), we extend our comparison to more kernels including the radius basis function (rbf), $\mathcal{C}(\mathbf{x},\mathbf{x}^{\prime}) = \sigma^{2}\exp (-0.5r^{2})$ and rational quadratic (rq), $\mathcal{C}(\mathbf{x},\mathbf{x}^{\prime}) = \sigma^{2}(1 + \frac{1}{2\alpha} r^{2})^{-\alpha}$ , $\alpha > 0$ , for $r = \sqrt{\sum_{i=1}^{d}(x_i - x_i')^2 / \rho_i^2}$ in Table C.2. Within each type of kernel, Q-EP with $q = 1.0$ consistently outperforms GP ( $q = 2.0$ ). This indicates that the advantage of Q-EP over GP is independent of the choice of kernels (as long as they are compared with the same kernel). + +Table C.2: Comparing accuracy of Q-EP ( $q = 1$ ) against GP ( $q = 2$ ) solvers with various kernels for Eikonal equation (12) in terms of mean absolute error (MAE), mean squared error (MSE), and relative errors in $L_{1}$ norm (RLE-1), $L_{2}$ norm (RLE-2), and $L_{\infty}$ norm (RLE- $\infty$ ) respectively. Result in each cell are averaged over 10 experiments with different random seeds; values after $\pm$ are standard deviations of these repeated experiments. + +
Model (q)kernelMAEMSERLE-1RLE-2RL-∞
1.0Matern1.68e-3 ± 5.41e-44.42e-6 ± 2.26e-60.0106 ± 0.00340.0110 ± 0.00300.0321 ± 0.0071
2.0 (Gaussian)Matern9.64e-3 ± 1.33e-31.30e-4 ± 3.61e-50.0610 ± 0.00840.0612 ± 0.00870.1009 ± 0.0124
1.0rbf4.39e-3 ± 1.05e-33.91e-5 ± 2.52e-50.0278 ± 0.00660.0327 ± 0.01070.0648 ± 0.0324
2.0 (Gaussian)rbf1.55e-2 ± 4.97e-43.07e-4 ± 2.87e-50.0982 ± 0.00310.0949 ± 0.00440.1236 ± 0.0048
1.0rq1.86e-3 ± 5.40e-45.51e-6 ± 3.44e-60.0118 ± 0.00340.0122 ± 0.00380.0168 ± 0.0064
2.0 (Gaussian)rq3.07-3 ± 1.70e-31.89e-5 ± 2.83e-50.0194 ± 0.01070.0200 ± 0.01340.0325 ± 0.0242
+ +# C.2 Burgers' Equation + +Define the nonlinear function $D(u, d_1u, d_2u, d_1^2u) = d_2u + ud_1u - \nu d_1^2u$ . The observations $\mathbf{Y}$ can then be expressed as $P(\tilde{u}(\mathbf{X})) = \begin{bmatrix} D(u(\mathbf{X}_d), \frac{\partial}{\partial x}u(\mathbf{X}_d), \frac{\partial}{\partial t}u(\mathbf{X}_d), \frac{\partial^2}{\partial x^2}u(\mathbf{X}_d)) \\ u(\mathbf{X}_b) \end{bmatrix}$ , and $\mathbf{h} = \begin{bmatrix} \mathbf{0}_{N_d} \\ \mathbf{g} \end{bmatrix}$ , where $\mathbf{g}$ is a vector of size $N_b$ whose elements are $-\sin(\pi x)$ or 0 depending on the order of the corresponding elements in $\mathbf{X}_b$ . We use the same experiment setup as above. + +In Table C.3, we also observe a consistently better performance of Q-EP ( $q = 1.0$ ) compared to GP ( $q = 2.0$ ) in solving Burgers' equation (13) using various kernels. + +One can possibly find a GP with fine-tuned kernel, e.g. rational quadratic (rq), to have better (Q-EP with rbf in Table C.2) or matching (Q-EP with Matern52 in Table C.3) results. However, as a + +![](images/5023c9909fa59f6d20e206fb6b78a8e24f8a575fe8493542c9cd520470020414.jpg) +Figure C.1: Solving Burgers' equation (13) using high-resolution finite difference method (upper left), PINN (lower left), Q-EP (right three in upper row: $q = 1.0$ ), and GP (right three in lower row: $q = 2.0$ ) respectively. Blue crosses are learned inducing points. + +![](images/4498e6026b1fa8e3fc19f1fcce88e6e59cceb13863aee471654cafe5e880d55c.jpg) + +![](images/a150166e1bb4f5fae088690b1ccce34cb40c52ec79e5040556447d1aaefd7816.jpg) + +![](images/5d42ac37ffe0ef418e4a62e7ad6bc297c8de94af7ed00ff2cf75740b23b2f198.jpg) + +Table C.3: Comparing accuracy of Q-EP ( $q = 1$ ) against GP ( $q = 2$ ) solvers with various kernels for Burgers' equation (13) in terms of mean absolute error (MAE), mean squared error (MSE), and relative errors in $L_{1}$ norm (RLE-1), $L_{2}$ norm (RLE-2), and $L_{\infty}$ norm (RLE- $\infty$ ) respectively. Result in each cell are averaged over 10 experiments with different random seeds; values after $\pm$ are standard deviations of these repeated experiments. + +
Model (q)kernelMAEMSERLE-1RLE-2RL-∞
1.0Matern9.33e-3 ± 1.76e-41.77e-4 ± 2.16e-40.0242 ± 0.01280.0266 ± 0.01400.0324 ± 0.0145
2.0 (Gaussian)Matern7.13e-2 ± 8.68e-31.08e-2 ± 1.28e-20.1848 ± 0.09080.2068 ± 0.11180.2376 ± 0.1456
1.0rbf2.51e-3 ± 1.30e-31.37e-5 ± 1.59e-50.0065 ± 0.00330.0074 ± 0.00400.0150 ± 0.0088
2.0 (Gaussian)rbf2.49e-2 ± 3.72e-31.06e-3 ± 2.63e-40.0646 ± 0.00970.0726 ± 0.00940.1144 ± 0.0073
1.0rq2.57e-3 ± 5.52e-41.39e-5 ± 3.15e-60.0067 ± 0.00140.0083 ± 0.00100.0199 ± 0.0049
2.0 (Gaussian)rq1.09e-2 ± 1.97e-31.94e-4 ± 7.02e-50.0283 ± 0.00510.0309 ± 0.00570.0551 ± 0.0096
+ +principled regularization over function spaces, Q-EP facilitates effective modeling of derivatives and differential equations, thereby alleviating the struggle of GP on meticulous kernel engineering. + +# C.3 Nonlinear Elliptic Equation + +Now we consider a nonlinear elliptic equation with Dirichlet boundary condition on $\Omega = [0,1]^2$ : + +$$ +- \Delta u (\mathbf {x}) + \tau (u (\mathbf {x})) = f (\mathbf {x}), \quad \mathbf {x} \in \Omega , +$$ + +$$ +u (\mathbf {x}) = g (\mathbf {x}), \quad \mathbf {x} \in \partial \Omega . \tag {21} +$$ + +where we choose $\tau(u) = u^3$ , $g(\mathbf{x}) \equiv 0$ , and $f$ such that the following true solution $u^\dagger$ satisfies equation (21) [6] + +$$ +u ^ {\dagger} (\mathbf {x}) = \sin (\pi x _ {1}) \sin (\pi x _ {2}) + 4 \sin (4 \pi x _ {1}) \sin (4 \pi x _ {2}). +$$ + +We follow the variational procedure in Section 3.1 and use the same experiment setup in Section 5. Figure C.2 compares the solutions by a highly-resolved finite difference method (upper left, treated as true solution), PINN (lower left), Q-EP (right three in upper row: $q = 1.0$ ), and GP (right three in lower row: $q = 2.0$ ). Compared to the true solution, Q-EP with $q = 1.0$ yields an estimate more accurate than PINN and GP. Based on Table C.4, the best solution is obtained by Q-EP with $q = 0.5$ . Since the true solution is highly fluctuating over the domain $\Omega$ , imposing more challenges on the boundary, where the largest pointwise errors occur. Adding more boundary points might help with Q-EP solvers. + +# C.4 Inverse Darcy Flow + +Finally, we consider a more realistic Darcy flow data used by Fourier Neural Operator (FNO) and Physics-Informed Neural Operator (PINO) available in NVIDIA PhysicsNeMo. This dataset contains thousands of permeability-solution pairs that reflect realistic porous media. Since our method is + +![](images/06b311c54c4fc79cef0b78d96562ae53d647360c3487bcbefc53ab52898cb7c1.jpg) + +![](images/7ad8d1beee3bd02bb1dd40dc46255c3cd4ebc8c5c5823196e2b3f69099f968ba.jpg) + +![](images/0588d4a16011626bf2dfbcba72301547dad8a6658c896bb190529e4a6b3b7eb3.jpg) + +![](images/62dfc60dfd67f8c69e65c63931630e0da550b820008ecf62a213c3d086b8a9b1.jpg) + +![](images/1b9b618abdfca367d54253a39d35ad5c12a2ebb47517133d206a2e3e5b5cab46.jpg) +Figure C.2: Solving nonlinear elliptic equation (21) using high-resolution finite difference method (upper left), PINN (lower left), Q-EP (right three in upper row: $q = 1.0$ ), and GP (right three in lower row: $q = 2.0$ ) respectively. Blue crosses are learned inducing points. + +![](images/875e3155271da78e9176d26f64b94b78f178967c08493c30ec80ac4c98f1a504.jpg) + +![](images/210937beec8862c28d32f56b6d0beacd0efec55fe05074106fc8386959d8e7be.jpg) + +![](images/fb4d3addd852022ff8b50ea29713f5d404903de85e29129b7d772e29474b8064.jpg) + +Table C.4: Comparing accuracy of various solvers for nonlinear elliptic equation (21) in terms of relative error in $L_{1}$ norm (RLE-1), $L_{2}$ norm (RLE-2), $L_{\infty}$ norm (RLE- $\infty$ ), and time per iteration. Result in each cell are averaged over 10 experiments with different random seeds; values after $\pm$ are standard deviations of these repeated experiments. + +
Model (q)RLE-1RLE-2RLE-∞time/iteration
PINN1.6428 ± 0.71331.5979 ± 0.65361.5595 ± 0.44620.0052 ± 0.0023
B-PINN0.7026 ± 0.28790.7868 ± 0.28721.2395 ± 0.34800.0108 ± 0.0013
0.50.0341 ± 0.02030.0514 ± 0.03220.1617 ± 0.08800.2530 ± 0.0008
0.80.3020 ± 0.09150.3159 ± 0.07490.5549 ± 0.10920.2604 ± 0.0016
1.00.2835 ± 0.05320.2958 ± 0.04340.5122 ± 0.08700.2614 ± 0.0005
1.20.3379 ± 0.08450.3342 ± 0.06890.5229 ± 0.10940.2714 ± 0.0004
1.50.3953 ± 0.10580.4429 ± 0.15910.8938 ± 0.57140.2645 ± 0.0003
2.0(Gaussian)0.9876 ± 0.27771.1028 ± 0.32542.0255 ± 1.12550.2562 ± 0.0004
+ +not to train a surrogate model, we take only one pair and impose the data on $200 \times 200$ mesh to obtain 2000 randomly sampled observations. We then train Q-EP solvers on $60 \times 60$ mesh (with collocation points taken from the grid) and with 512 inducing points. This problem has much larger scale ( $\sim 20$ times larger) than all the previous examples. As illustrated in Table C.6, Q-EP with $q = 1$ still achieves remarkable advantage compared with GP ( $q = 2$ ) and PINN. Note, Q-EP with $q = 1$ has comparable accuracy with $q = 0.5$ in forward solution. With only one training pair in the NIVIDA FNO-Darcy data, it is understandably more challenging to obtain the inverse solution, yet for which Q-EP with $q = 1$ attains the best accuracy. In both solutions, GP ( $q = 2$ ) is much worse by most metrics. + +Table C.5: Comparing accuracy of various solvers for inverse Darcy flow (14) in terms of relative error in $L_{1}$ norm (RLE-1), $L_{2}$ norm (RLE-2), and $L_{\infty}$ norm (RLE- $\infty$ ) respectively. Result in each cell are averaged over 10 experiments with different random seeds; values after $\pm$ are standard deviations of these repeated experiments. + +
Forward PDE SolutionInverse Solution
Model (q)RLE-1RLE-2RLE-∞RLE-1RLE-2RLE-∞
PINN0.3145 ± 0.01630.3331 ± 0.01370.4225 ± 0.02506.6922 ± 0.58006.2856 ± 0.54324.1070 ± 0.3189
0.50.2397 ± 0.09750.2689 ± 0.11370.3445 ± 0.14900.4827 ± 0.20110.5217 ± 0.19160.8711 ± 0.1414
0.80.1669 ± 0.02420.1784 ± 0.02480.2155 ± 0.03210.3577 ± 0.07580.4154 ± 0.06750.8110 ± 0.1239
1.00.1354 ± 0.02860.1432 ± 0.02530.1810 ± 0.01930.4340 ± 0.03760.4942 ± 0.03760.8494 ± 0.2132
1.20.1523 ± 0.09670.1549 ± 0.08980.2066 ± 0.10350.5468 ± 0.10780.5921 ± 0.08900.7801 ± 0.1301
1.50.3270 ± 0.32570.4723 ± 0.50481.2103 ± 1.09813151.6317 ± 0.81621.5914 ± 0.78101.7561 ± 0.8064
2.0(Gaussian)0.6328 ± 0.59541.0658 ± 0.91863.1391 ± 2.24922.2096 ± 0.91002.1675 ± 0.88232.8022 ± 1.6839
+ +![](images/b6052113e7e020277b441d4988ccf6b57c5afa31287109f1a1bc9df0bab7532c.jpg) + +![](images/610087773abec79175fd789eb1a79951a66985fff447c0ecba4188da1ada6195.jpg) + +![](images/dc768f0c582292ed83dadf695447c6f3f0677e19461306af6a4aa4404d1a83b9.jpg) + +![](images/9747d77645061b08a724c652439d0c8f7627797b28fa57466ddc10c320f89885.jpg) + +![](images/c7e282d3ae09e99ce4f5afb17b5bda579b7309a82703a26ed30241f01dd6df9b.jpg) +Figure C.3: Solving inverse Darcy flow (14) with sparse data using Q-EP (right three in upper row: $q = 1.0$ ) and GP (right three in lower row: $q = 2.0$ ) respectively. Upper left: true inverse solution $a^\dagger$ ; lower left: fine-resolution finite element solution $u^\dagger$ to (14) with $a^\dagger$ . Blue crosses are learned inducing points, and red dots indicate locations of observations. + +![](images/5174bd694c558b7e204eb0c5c8fda01a71048052dce793b1e64f393bb1f4f828.jpg) + +![](images/9003d5e88ce0e6d57d9f6264f1e732003e59c07077afb0dbe03f71c7fbe9ad25.jpg) + +![](images/39cc2045186d64089e3263d59d13a6dbaf5ee801ec9ea9edf814b0c21cc0be74.jpg) + +![](images/7b4ee0f8eae508e7b571bf09ce8ed29da6ac2c3ef3b14cc69dcff3754c652747.jpg) +Figure C.4: Solving inverse Darcy flow (14) with PINN running for longer (5000) iterations. + +![](images/cce59ba69354c2d9429fe3125f05345533c1ae2b15edb33df850d47653b1827d.jpg) + +![](images/b97ca7cfc6a30654ec6381e7f73b1ac309d9e7fbfd1a18e91a104142af6b9c36.jpg) + +![](images/eeef07a300339b8de01c63c39c18df00f2646baedeb9ba3974651ea14c8f1dcf.jpg) + +![](images/7990e151a697331752f0a1916a31f558a5f0e14458dcb4a6c99dbb17b3a70d72.jpg) + +Table C.6: Comparing accuracy of various solvers for the inverse problem with FNO-Darcy data in terms of relative error in $L_{1}$ norm (RLE-1), $L_{2}$ norm (RLE-2), and $L_{\infty}$ norm (RLE- $\infty$ ) respectively. Result in each cell are averaged over 10 experiments with different random seeds; values after $\pm$ are standard deviations of these repeated experiments. + +
Forward PDE SolutionInverse Solution
Model (q)RLE-1RLE-2RLE-∞RLE-1RLE-2RLE-∞
PINN0.9168 ± 0.90190.8170 ± 0.72720.8190 ± 0.38591.6550 ± 0.18761.5170 ± 0.15471.4303 ± 0.1190
0.50.3316 ± 0.06770.3191 ± 0.04020.5978 ± 0.05350.7847 ± 0.13940.8548 ± 0.16831.2072 ± 0.2975
1.00.3239 ± 0.12970.3195 ± 0.09850.6041 ± 0.08300.7724 ± 0.17980.8349 ± 0.18411.2895 ± 0.2849
1.50.2338 ± 0.01970.2490 ± 0.01050.6149 ± 0.10560.8466 ± 0.16170.8744 ± 0.12131.2070 ± 0.3958
2.0(Gaussian)0.5341 ± 0.52960.4904 ± 0.42920.6881 ± 0.23560.9124 ± 0.17910.9251 ± 0.15021.1018 ± 0.1219
+ +![](images/03867eae208ddcb5987eaabe1aaf4ca30f25687c1bd6a5db9c4abf104fc7d57a.jpg) + +![](images/69ff4c833f27543154ff48c44eb9c84509107d89dcbfa3669bd4a696de8f17e1.jpg) + +![](images/46d79500bdadea26276cdfab7bc5a6f264efd6f8c6bde7ef8ea140ab2d738b99.jpg) + +![](images/8372dabc35bfe4b8898f33d9ff73be18740f7d0c473fedde4e5dd850b095a0ca.jpg) + +![](images/d0fd9ec0ed87da57734d90997e1f7a2696900036ce061662c42c6478b587d01b.jpg) + +![](images/c2289bd4368c1059ea4a901b3f3bb7ee176f89a588c169992b8c86649f9f563d.jpg) +Figure C.5: Comparing solutions to inverse Darcy flow (14) at refined mesh sizes using Q-EP (left four in upper row: $q = 1.0$ ) and GP (left four in lower row: $q = 2.0$ ) respectively, with relative errors in $L_{1}$ norm (upper right) and $L_{2}$ norm (lower right). Blue crosses are learned inducing points. Error bars indicate standard errors based on 10 repeated experiments with different random seeds. + +![](images/a3a6632d6b6120fa4c20585dae3524c3d863501feb93ab269527b658e1fbb3f7.jpg) + +![](images/6bc3ac0b41cee5306d198142a7743daa58c7571e695a672b72cda4a758425783.jpg) + +![](images/6a9a5cbfe283d0a34c00994834ab035bfcfa299cff15037b4c66ed489bf9a7d7.jpg) + +![](images/129d5bdb9ed4ec69a118737be79c3a517429ff62ca2b7f4e8b50a6bf20830950.jpg) + +# NeurIPS Paper Checklist + +# 1. 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Trace-Guided Cost Augmentation/dcb5d3d1-71cb-4a16-80fc-53d931e4c4f4_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dd8b99e364766bfd2756a955493bab97dce11e16518a2ee20b466e5b951ed9c0 +size 338362 diff --git a/NeurIPS/2025/Solving the Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation/full.md b/NeurIPS/2025/Solving the Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation/full.md new file mode 100644 index 0000000000000000000000000000000000000000..d1303b7499dab79c94acdf8130ce4a7f16b39181 --- /dev/null +++ b/NeurIPS/2025/Solving the Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation/full.md @@ -0,0 +1,611 @@ +# Solving Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation + +# Zhen Zhang + +$^{1}$ Responsible AI Research Centre Australian Institute for Machine Learning + +$^{2}$ The University of Adelaide & $^{3}$ CSRIO + +zhen@zzhang.org + +# Javen Qingfeng Shi + +$^{1}$ Responsible AI Research Centre Australian Institute for Machine Learning + +2The University of Adelaide + +javen.shi@adelaide.edu.au + +# Wee Sun Lee + +School of Computing + +National University of Singapore + +leews@comp.nus.edu.sg + +# Abstract + +The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. We propose a novel continuous relaxation framework for ATSP that leverages differentiable constraints to encourage acyclic structures and valid permutations. Our approach integrates a differentiable trace-based Directed Acyclic Graph (DAG) constraint with a doubly stochastic matrix relaxation of the assignment problem, enabling gradient-based optimization over soft permutations. We further develop a projected exponentiated-gradient method with adaptive step size to minimize tour cost while satisfying the relaxed constraints. To recover high-quality discrete tours, we introduce a greedy post-processing procedure that iteratively eliminates subtours through cost-aware cycle merging. Empirically, our method achieves state-of-the-art performance on standard asymmetric TSP benchmarks and demonstrates strong scalability and accuracy, particularly on large or highly asymmetric instances where heuristic solvers such as LKH-3 often struggle. + +# 1 Introduction + +The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. Many important combinatorial optimization (CO) problems are theoretically reducible to ATSP. Recent work by Pan et al. [22] has shown that such reductions are not only of theoretical interest but can also be practically feasible for certain small-scale CO tasks, suggesting that advances in ATSP solvers could yield broad benefits across the CO landscape. + +Although the symmetric TSP is a special case of ATSP, solvers designed specifically for the asymmetric case often perform poorly on symmetric instances. Conversely, any ATSP instance can be transformed into a symmetric TSP by doubling the number of nodes [9], enabling symmetric solvers to be applied to asymmetric problems. Consequently, much of the CO community has focused on the symmetric TSP [1, 8, 20, 26], leading to the development of increasingly effective handcrafted and learning-based heuristics. Recently, methods such as those proposed by Drakulic et al. [4, 5], Pan et al. [22] have employed reinforcement learning to directly search for optimal ATSP tours. These approaches provide faster inference compared to state-of-the-art heuristics such as LKH-3 [8], but typ + +ically yield lower-quality solutions—even though LKH-3 was originally developed for the symmetric TSP. + +Assignment-problem (AP) relaxations have long been used to provide strong lower bounds within branch-and-bound and branch-and-cut TSP solvers [17, 19], and they remain effective for large-scale asymmetric instances. Several constructive heuristics, including the Karp-Steele Patching (KSP) algorithm [12, 13] and its variants, attempt to lift assignment solutions into valid tours. However, based on existing open-source implementations, these heuristics often underperform relative to LKH-3. Interestingly, AP relaxations tend to be tighter for asymmetric TSPs than for symmetric ones, motivating the development of stronger heuristics grounded in AP formulations. + +A general TSP can be formulated as a linear assignment problem with an additional constraint ensuring that the solution forms exactly one cycle [9]. We observe that this constraint is equivalent to requiring all proper subgraphs of the assignment to be Directed Acyclic Graphs (DAGs). Inspired by recent progress in differentiable DAG learning [25, 28, 29], we propose a novel continuous relaxation of the TSP that incorporates DAG constraints into the assignment formulation. We show that the global minimum of this relaxed problem exactly recovers the true TSP solution. + +Although the proposed relaxation is non-convex, we find empirically that local minima obtained via approximate projected gradient descent often yield high-quality solutions. In many cases, our method recovers valid tours with shorter lengths than those produced by LKH-3. For instances where the relaxed optimization does not directly yield a valid tour, we introduce a simple greedy post-processing heuristic to eliminate subtours and construct feasible solutions. Finally, we apply a 2-opt local refinement to further improve tour quality. Together, these components constitute a new heuristic for solving general (asymmetric, non-metric) TSP instances that consistently outperforms LKH-3, particularly on large-scale problems. When applied to small-scale symmetric TSPs, our method remains competitive—achieving slightly lower solution quality than LKH-3 but offering significantly faster runtime performance. + +In summary, our main contributions are: + +- Novel TSP formulation: A novel exact continuous formulation that suppresses subtours via a trace-based acyclicity loss. +- Soft assignment optimization: A doubly stochastic matrix relaxation of ATSP solved via projected exponentiated gradient descent. +- Efficient greedy decoding: A post-processing algorithm that reconstructs valid tours from fractional assignments with minimal additional cost. +- Strong empirical performance: Extensive experiments show our method outperforms LKH-3 on ATSP benchmarks and scales well to large problem sizes. + +# 2 Related Works + +Learning-based TSP Solvers Recently, learning-based approaches have been increasingly applied to combinatorial optimization (CO) problems, including the Traveling Salesman Problem (TSP). While many existing works focus on the Euclidean TSP, where inputs are given as point coordinates [11, 14, 15, 18, 20, 23, 24, 31], more recent studies [4, 5, 16, 22] have extended these methods to handle general asymmetric TSPs. Kwon et al. [16] treat distance matrices directly as transformer input tokens, which is computationally inefficient and unsuitable for large graphs. Subsequent methods [4, 5, 22] address this limitation by encoding distance matrices more efficiently, enabling scalability to graphs with thousands of nodes. These approaches [4, 5, 16, 22] generate tours in an auto-regressive manner, formulating the tour-construction process as a Markov Decision Process (MDP). Reinforcement learning is then applied to optimize over the MDP, thereby avoiding reliance on expensive ground-truth optimal solutions. + +In contrast to these auto-regressive frameworks, Min et al. [20] proposed a surrogate loss based on assignment-problem relaxations of the TSP. However, their approach is limited to the Euclidean setting and only enforces the absence of length-2 loops in the solution, necessitating a post-processing step to construct valid tours. + +Learning-based solvers generally outperform classical heuristics such as LKH-3 in terms of computational efficiency, although they may produce slightly less accurate tours. In reinforcement learning + +settings, high-quality solutions from fast heuristics (even if suboptimal) can serve as valuable offline data for off-policy algorithms, potentially accelerating training and improving solution quality. + +Beyond methods that directly generate tours from node coordinates or distance matrices, another line of research [27, 30] focuses on learning improved search strategies—such as branch-and-bound or local search policies. These approaches replace handcrafted heuristics in traditional solvers with learned decision mechanisms, leading to more adaptive and data-driven optimization behavior. + +Assignment Problem Relaxations for TSP Relaxations of the Traveling Salesman Problem (TSP) based on the classical linear Assignment Problem (AP) have been extensively studied due to their computational efficiency and strong theoretical foundations [9, 17]. These relaxations typically involve solving a linear program in which the tour constraints are relaxed, allowing the solution to contain multiple disconnected subtours. To recover a valid TSP tour, additional strategies such as iterative subtour elimination or branch-and-bound frameworks are employed [19]. Although AP relaxations do not always yield feasible tours directly, they provide tight lower bounds [6] and serve as key components in exact solvers and heuristic algorithms. Several tour-construction heuristics have also been developed based on AP relaxations [12, 13]; however, existing open-source implementations often perform worse than modern LKH-3 solvers. For a more comprehensive discussion, we refer readers to Cirasella et al. [2], Glover et al. [7], and Öncan et al. [21]. + +# 3 Exact Continuous Relaxation of TSP + +Let $\mathbf{D} \in \mathbb{R}_+^{n \times n}$ be the non-negative distance (or cost) matrix, where its entry $d_{ij}$ denotes the distance or cost of traveling from node $i$ to node $j$ . The general TSP seeks a tour (i.e., a permutation of the nodes forming a single cycle that visits each node exactly once) that minimizes the total travel cost. This can be formulated as: + +$$ +\begin{array}{l} \min _ {\mathbf {X} \in \{0, 1 \} ^ {n \times n}} \quad \langle \mathbf {D}, \mathbf {X} \rangle = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} x _ {i j} \\ \text {s u b j e c t} \quad \sum_ {j = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} i \in \{1, \dots , n \} \quad (\text {e a c h n o d e e x i t s e x a c t l y o n c e}), \tag {1} \\ \sum_ {i = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} j \in \{1, \dots , n \} \quad (\text {e a c h n o d e i s e n t e r e d e x a c t l y o n c e}), \\ \mathbf {X} \text {f o r m s a s i n g l e H a m i t o n i a n c y c l e (n o s u b t o u r s)}. \\ \end{array} +$$ + +Here, $\mathbf{X}$ is a permutation matrix corresponding to a single cycle over the nodes. The final constraint prohibits subtours—i.e., disjoint cycles that do not span all nodes—ensuring that the solution corresponds to a valid TSP tour. + +Differentiable DAG Constraints Recently, a family of differentiable constraints for enforcing acyclicity in graphs was proposed in the literature [25, 28, 29]. For any positive adjacency matrix $\mathbf{X}$ , acyclicity can be enforced by the following constraint: + +$$ +h (\mathbf {X}) = \sum_ {i = 1} ^ {n} \operatorname {t r} \left(\mathbf {X} ^ {i}\right) = 0. \tag {2} +$$ + +In this constraint, each term $\mathrm{tr}(\mathbf{X}^i)$ is strictly positive when $\mathbf{X}$ contains cycles of length $i$ , since all entries of $\mathbf{X}$ are positive. Therefore, setting the total sum to zero guarantees that there are no cycles of any length, ensuring that $\mathbf{X}$ represents a Directed Acyclic Graph (DAG). + +Based on these differentiable DAG constraints, we propose a novel continuous relaxation of the TSP by enforcing that any proper subgraph of the assignment matrix $\mathbf{X}$ forms a DAG. This leads to the following formulation: + +$$ +\min _ {\mathbf {X} \in \mathcal {D} ^ {n}} \left\langle \mathbf {D}, \mathbf {X} \right\rangle = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} x _ {i j} \quad \text {s u b j e c t t o} \quad \sum_ {i = 1} ^ {n - 1} \operatorname {t r} \left(\mathbf {X} ^ {i}\right) = 0, \tag {3} +$$ + +where $\mathcal{D}^n = \left\{\mathbf{X}\in [0,1]^{n\times n}\mid \mathbf{X}\mathbf{1} = \mathbf{1},\mathbf{1}^\top \mathbf{X} = \mathbf{1}^\top \right\}$ is the set of doubly stochastic matrices of size $n\times n$ + +One appealing property of this relaxation is its tightness, as stated below: + +Proposition 1 (Exactness of the Continuous Relaxation). The global minimum of the relaxed problem in (3) coincides with the optimal solution of the original TSP. + +Proof. To prove the result, it suffices to show that any fractional matrix $\mathbf{X} \in \mathcal{D}^n$ cannot satisfy the constraint $\sum_{i=1}^{n-1} \operatorname{tr}(\mathbf{X}^i) = 0$ . In other words, all feasible solutions must be permutation matrices representing Hamiltonian cycles. + +The proof follows from the Birkhoff-von Neumann theorem [10], which states that any doubly stochastic matrix $\mathbf{X} \in \mathcal{D}^n$ can be expressed as a convex combination of permutation matrices: + +$$ +\mathbf {X} = \sum_ {i = 1} ^ {k} \theta_ {i} \mathbf {P} _ {i}, \quad \theta_ {i} \geq 0, \sum_ {i = 1} ^ {k} \theta_ {i} = 1, \tag {4} +$$ + +where $\mathbf{P}_i$ are permutation matrices. + +Assume for contradiction that a fractional $\mathbf{X} \in \mathcal{D}^n$ satisfies the constraint in (3). Then its Birkhoff-von Neumann decomposition contains at least two distinct permutation matrices $\mathbf{P}_i$ . If any $\mathbf{P}_i$ corresponds to a subtour (i.e., not a single Hamiltonian cycle), then $\mathrm{tr}(\mathbf{P}_i^j) > 0$ for some $j < n$ , and hence $\mathrm{tr}(\mathbf{X}^j) > 0$ for some $j < n$ due to linearity and positivity. This violates the constraint $\sum_{j=1}^{n-1} \mathrm{tr}(\mathbf{X}^j) = 0$ , making $\mathbf{X}$ infeasible. + +Even if all $\mathbf{P}_i$ represent valid Hamiltonian cycles, a convex combination of two or more distinct tours will produce fractional entries in $\mathbf{X}$ . These fractional values imply the presence of edges from multiple tours, which results the presence of subtours and nonzero traces in $\mathbf{X}^j$ for some $j < n$ . Thus, $\mathbf{X}$ would again be infeasible. + +Therefore, the only feasible solutions are permutation matrices representing single Hamiltonian cycles. Among them, the one minimizing $\langle \mathbf{D},\mathbf{X}\rangle$ is exactly the TSP solution. + +![](images/20d57ad8c85ab32957a217096c85f86252b68a329fe174be939cd0a772443bb4.jpg) + +# 4 Trace Guided Tour Search + +To efficiently optimize the relaxation in Equation (3), we introduce an auxiliary variable $\mathbf{Y} \in \mathbb{R}^{n \times n}$ with elementwise constraints $\mathbf{Y} \leqslant 0$ , and define the assignment matrix as $\mathbf{X} = \exp(\mathbf{Y})$ , where the exponential is applied elementwise. This reparameterization ensures that $\mathbf{X}$ is strictly positive and facilitates optimization over the interior of the doubly stochastic polytope $\mathcal{D}^n$ . Compared to optimizing directly over $\mathbf{X}$ , working in the log-domain (i.e., optimizing $\mathbf{Y}$ ) avoids boundary issues, improves numerical stability, and enables smooth gradients that are more amenable to gradient-based learning methods. The relaxed optimization problem is then formulated as: + +$$ +\min _ {\exp (Y) \in \mathcal {D} ^ {n}} \langle \mathbf {D}, \exp (\mathbf {Y}) \rangle = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} \exp \left(y _ {i j}\right) \text {s u b j e c t t o} \operatorname {t r} \sum_ {i = 1} ^ {n - 1} \left(\exp (\mathbf {Y}) ^ {i}\right) = 0, \tag {5} +$$ + +where $\mathbf{D}$ is the cost matrix, and the trace constraint eliminates self-loops and short cycles. By introducing a Lagrange multiplier $\lambda$ , the constrained problem can be rewritten as the following unconstrained objective: + +$$ +\min _ {\exp (\mathbf {Y}) \in \mathcal {D} ^ {n}} f (\mathbf {Y}) = \langle \mathbf {D}, \exp (\mathbf {Y}) \rangle + \lambda \operatorname {t r} \sum_ {i = 1} ^ {n - 1} \left(\exp (\mathbf {Y}) ^ {i}\right). \tag {6} +$$ + +The gradient of this objective with respect to $\mathbf{Y}$ is given by: + +$$ +\nabla_ {\mathbf {Y}} f (\mathbf {Y}) = \mathbf {D} \odot \exp (\mathbf {Y}) + \lambda \left[ \sum_ {i = 0} ^ {n - 2} (i + 1) \exp (\mathbf {Y}) ^ {i} \right] ^ {\top} \odot \exp (\mathbf {Y}), \tag {7} +$$ + +where $\odot$ denotes elementwise multiplication. This formulation enables efficient and scalable optimization while preserving the structural constraints required by the TSP. + +Algorithm 1 Approximate Projected Gradient Descent for Exponentiated TSP Relaxation +1: Input: Cost matrix $\mathbf{D} \in \mathbb{R}^{n \times n}$ , initial step size $\alpha > 0$ , scaling factor $\gamma > 0$ , max iterations $T$ +2: Initialize $\mathbf{Y} \gets -\mathbf{D}$ +3: Compute initial assignment $\pi_{\mathrm{prev}} \gets \mathrm{LinearAssignment}(-\mathbf{Y})$ +4: for $t = 1$ to $T$ do +5: Compute gradient: + $\nabla_{\mathbf{Y}} f(\mathbf{Y}) = \mathbf{D} \odot \exp(\mathbf{Y}) + \lambda \left[ \sum_{i=0}^{n-2} (i+1) \exp(\mathbf{Y})^i \right]^\top \odot \exp(\mathbf{Y})$ +6: Update $\mathbf{Y}_{\mathrm{new}} \gets \mathbf{Y} - \alpha \nabla_{\mathbf{Y}} f(\mathbf{Y})$ +7: Project: $\mathbf{X} = \mathrm{Sinkhorn}(\exp(\mathbf{Y}_{\mathrm{new}}))$ , $\mathbf{Y}_{\mathrm{new}} \gets \log(\mathbf{X}) \triangleright$ elementwise projection onto $\mathbf{D}^n$ +8: Compute new assignment $\pi_{\mathrm{new}} \gets \mathrm{LinearAssignment}(-\mathbf{Y}_{\mathrm{new}})$ +9: if $\pi_{\mathrm{new}} \neq \pi_{\mathrm{prev}}$ then +10: $\alpha \gets \alpha / 2$ +11: else +12: $\alpha \gets 2\alpha$ +13: end if +14: $\mathbf{Y} \gets \mathbf{Y}_{\mathrm{new}}, \pi_{\mathrm{prev}} \gets \pi_{\mathrm{new}}$ +15: if $\pi_{\mathrm{new}}$ has only one loop then +16: break +17: end if +18: end for +19: Output: $\mathbf{Y}$ , permutation $\pi_{\mathrm{new}}$ + +Approximate Projected Gradient Descent for Exponentiated TSP Relaxation: To optimize the exponentiated relaxation of the TSP problem, we propose an approximate projected gradient descent algorithm over the log-domain variable $\mathbf{Y}$ . The algorithm initializes $\mathbf{Y}$ as the negative of the cost matrix, $\mathbf{Y} \gets -\mathbf{D}$ , and iteratively updates it using the gradient of the relaxed objective. At each step, the gradient is computed and used to update $\mathbf{Y}$ with a step size $\alpha$ . To maintain feasibility within the doubly stochastic set, we apply Sinkhorn [3] normalization to the exponentiated matrix $\exp(\mathbf{Y})$ and then project back to the log-domain via $\log(\cdot)$ , ensuring that $\exp(\mathbf{Y}) \in \mathcal{D}^n$ . The step size $\alpha$ is dynamically adjusted based on whether the solution to the linear assignment problem (using $-\mathbf{Y}$ as the cost) changes after the update: if it does, the step size is reduced; otherwise, it is increased to accelerate convergence. This adaptive strategy helps balance exploration and stability. The process terminates either after a fixed number of iterations or once a valid TSP tour (a single loop) is found in the assignment. The algorithm returns the optimized $\mathbf{Y}$ and the corresponding assignment matrix $\exp(\mathbf{Y})$ . + +Approximate Gradient Computation: In our gradient expression, the term $\left[\sum_{i=0}^{n-2}(i+1)\exp(\mathbf{Y})^i\right]^\top$ involves computing a weighted sum of matrix powers, which can be computationally expensive. Although efficient approximations such as the truncated Neumann series proposed in Zhang et al. [28] can reduce the number of operations to $O(\log n)$ matrix multiplications, the cost remains significant for large instances due to the $O(n^3)$ complexity of matrix multiplication. To address this, we adopt a lightweight approximation by substituting $\exp(\mathbf{Y})$ with the hard assignment matrix corresponding to the current solution $\pi_{\mathrm{prev}}$ . Since permutation matrices are sparse and binary, their powers are simply cyclic shifts, and the sum can be computed in $O(n^2)$ time. This approximation significantly accelerates training while preserving useful structural signals from the current assignment during optimization. + +Greedy Tour Decoding from Invalid Assignments: Although the proposed optimization approach typically converges rapidly for asymmetric TSP instances, there exist hard cases where convergence to a valid permutation (i.e., a single-loop tour) does not occur. To address such situations, we apply a greedy decoding strategy to extract a valid TSP tour from the relaxed assignment matrix. Specifically, we begin by selecting a random starting node and iteratively follow the highest-assigned edge (i.e., the node with the largest assignment probability) to determine the next node in the tour. If the selected + +
MethodATSP20ATSP50
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%1.680.00%0.00%5.18
LKH-3 (1, 500)0.02%0.00%0.021.00%0.40%0.060
LKH-3 (1, 10000)0.0005%0.00%0.180.28%0.00%0.54
MatPOENet27.36%27.80%0.0067688.67%675.04%0.016
GOAL36.50%33.59%0.002108.29%109.38%0.0095
Ours1.9%0.08%0.0052.6%0.8%0.012
+ +Table 1: Comparison of methods on small-scale pure random ATSP (TSP20 and TSP50). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP. + +
MethodATSP20ATSP50
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%2.540.00%0.00%4.51
LKH-3 (1, 500)0.00%0.00%0.030.06%0.00%0.05
LKH-3 (1, 10000)0.00%0.00%0.270.004%0.00%0.48
MatPOENet7.4%5.3%0.00631.4%1.1%0.016
GOAL9.26%10.15%0.0025.87%5.83%0.0095
Ours0.97%0.00%0.0060.6%0.00%0.009
+ +Table 2: Comparison of methods on small-scale random ATSP with triangle inequality problems (TSP20 and TSP50). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. + +edge would result in the formation of a premature loop or revisiting a node, we instead choose the closest unvisited node according to the original cost matrix. This process continues until all nodes have been visited exactly once, ensuring the construction of a valid Hamiltonian cycle. This decoding heuristic is simple, efficient, and empirically effective in recovering feasible tours from imperfect or fractional assignments. + +Time Complexity. The dominant computational cost in each iteration of the proposed algorithm lies in solving the linear assignment problem for approximating $\exp (\mathbf{Y})$ , which has a worst-case time complexity of $O(n^{3})$ when using the Hungarian algorithm. However, in practice, the Hungarian algorithm is typically much faster due to early termination and efficient implementations. Additionally, viewing the assignment step through the lens of primal-dual linear programming reveals an opportunity to reuse the dual variables as warm starts for subsequent iterations, potentially accelerating convergence. Alternatively, the assignment problem can also be solved using the Auction algorithm, which offers better parallelism and is more amenable to GPU implementation. This opens up promising directions for future work, where the entire optimization process could be ported to GPUs to achieve significant speed-ups and scalability for large-scale TSP instances. + +Local Search Refinement. To further improve the quality of the obtained tour, our implementation applies a simple 2-opt heuristic as a post-processing step. We observe that for asymmetric TSP instances, the 2-opt heuristic often brings negligible improvements due to the directional nature of the cost structure. In contrast, for symmetric TSPs, the 2-opt refinement can lead to noticeable performance gains by eliminating crossing edges and reducing the overall tour length. While 2-opt is lightweight and easy to implement, it may be suboptimal for larger or more complex instances. As a future direction, one could consider replacing the 2-opt heuristic with more powerful local search methods such as the Lin-Kernighan-Helsgaun (LKH) algorithm [8]. In such a setup, our learned solution could serve as a high-quality initialization for LKH, potentially reducing the number of local search iterations required and improving overall efficiency. + +# 5 Experimental Results + +Hardware All experiments were conducted on a workstation equipped with an AMD Ryzen 9 3900X CPU, 128GB DDR4 memory, and an NVIDIA RTX 3090 GPU. The LKH-3 heuristic was implemented in $\mathrm{C + + }$ and executed in a single-threaded mode on the CPU. Our proposed method was implemented using Python and SciPy, and it was also executed on a single CPU process without utilizing the GPU. In contrast, all learning-based approaches (e.g., GOAL and MatPOENet) were executed on the GPU to leverage hardware acceleration during inference. + +Dataset We evaluate our method on randomly generated Asymmetric TSP (ATSP) instances of sizes 20, 50, 100, 200, 500, and 1000. Following prior works [16], the cost matrices for these instances are generated by sampling each entry independently from a uniform distribution over the interval $[0,1]$ . Kwon et al. [16] add an post-processing procedure to ensure the triangle inequality in enforced in the generated ATSP, while we also consider the cases where the triangle inequality is not satisfied. For solvers that require integer-valued cost matrices, we scale the costs by $10^{6}$ and round them to the nearest integer to preserve precision. Additionally, we evaluate our approach on symmetric TSP instances of sizes 20 and 50. In line with previous settings, we first sample $n$ random points in 2D space from a uniform distribution over $[0,1]^{2}$ , and use the Euclidean distances between these points to construct the symmetric cost matrices. + +Competitors We compare our method against several strong baselines, including both classical heuristics and recent learning-based approaches. Specifically, we consider LKH-3, a state-of-the-art heuristic algorithm for solving TSP, GOAL[5], a reinforcement learning-based approach using graph neural networks, and MatPOENet[22], a transformer-based model for solving combinatorial optimization problems. For GOAL and MatPOENet, we use their publicly released pretrained weights to ensure a fair comparison. For LKH-3, we evaluate three configurations: (i) one run with 500 search trials, (ii) one run with 10,000 search trials, and (iii) ten runs with 10,000 search trials each. These configurations allow us to assess both the performance and robustness of LKH-3 under varying computational budgets. Exact solvers such as Gurobi and Concorde [1] are not compared as they are often too slow for large scale problems. + +
MethodATSP100ATSP200
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%11.400.00%0.00%18.67
LKH-3 (1, 500)5.38%5.06%0.1312.59%12.72%0.27
LKH-3 (1, 10000)1.06%0.91%1.192.02%1.90%2.03
MatPOENet295.18%295.62%0.047747.56%750.27%0.165
GOAL181.92%177.47%0.058265.03%261.43%0.16
Ours3.39%1.77%0.0430.32%-1.01%0.13
+ +Table 3: Comparison of methods on medium-scale pure random ATSP problems (TSP100 and TSP200). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP. + +# 5.1 Asymmetric Problems + +Small Scale Problems The comparison of running time and solution quality for small-scale problems is presented in Table 1 and Table 2. Our method, while yielding slightly worse average performance than LKH-3, demonstrates significantly faster running times. For the learning-based approaches, which are trained on ATSP with triangle inequality instances, their performance degrades substantially on purely random ATSP problems—underperforming all other competitors. This indicates that current neural combinatorial solvers may struggle to generalize across different problem distributions. Additionally, MatPOENet performs considerably worse on our test set than reported in its original paper, suggesting a potential distribution shift between the two test scenarios. + +Surprisingly, GOAL achieves the best performance on asymmetric TSP20 with triangle inequality, outperforming the state-of-the-art LKH-3 heuristic by a substantial margin. Given that GOAL was + +
MethodATSP100ATSP200
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%7.310.00%0.00%11.69
LKH-3 (1, 500)0.25%0.11%0.110.64%0.06%0.23
LKH-3 (1, 10000)0.09%0.00%0.7870.18%0.01%1.34
MatPOENet14.91%13.84%0.04735.58%35.72%0.165
GOAL3.83%4.04%0.0580.77%0.74%0.44
Ours0.52%0.04%0.0270.16%0.00%0.10
+ +Table 4: Comparison of methods on medium-scale random ATSP with triangle inequality problems (TSP100 and TSP200). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. There may potentially be a distribution shift between the training set of MatPOENet and our test set. + +
MethodATSP500ATSP1000
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%29.990.00%0.00%46.64
LKH-3 (1, 500)24.47%24.86%1.0851.23%51.48%3.81
LKH-3 (1, 10000)2.50%2.45%3.853.25%2.94%7.91
MatPOENet6400.51%6400.50%1.5115017.16%15016.70%10.24
GOAL369.69%369.40%6.84305.84%304.62%58.53
Ours-10.77 %-11.59%0.87-21.40%-21.86%5.22
+ +Table 5: Comparison of methods on large-scale pure random ATSP problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP. + +trained on 100-node triangle asymmetric instances, this result suggests that training neural solvers on large, structured, and challenging problems may enhance their generalization to smaller problem instances. + +Medium and Large Scale Problems The comparison of running time and solution quality for medium- and large-scale problems is presented in Table 3, Table 4, Table 5, and Table 6. Consistent with the small-scale setting, learning-based approaches perform significantly worse than others on purely random asymmetric TSPs. In these settings, our algorithm outperforms LKH-3(1,500) for graphs with more than 100 nodes, and even surpasses LKH-3(1,10000) for graphs with more than 200 nodes. As the graph size increases, our method increasingly outperforms all LKH-3 heuristics, with the performance gap becoming more pronounced for larger instances. On 1000-node problems, our algorithm achieves over $20\%$ improvement compared to LKH-3(10,10000) and over $70\%$ compared to LKH-3(1,500). Meanwhile, our runtime on large graphs is comparable to that of LKH-3(1,500) and remains faster than the learning-based approaches. On asymmetric TSPs with triangle inequality, similar trends are observed. Our method continues to outperform LKH-3 on large graphs, although the performance margin is smaller compared to the purely random setting. + +In terms of running time, it is notable that, despite our implementation not being carefully optimized, its execution speed is often faster than that of LKH-3. On very large graphs, our method can outperform learning-based approaches, even when those methods are accelerated by GPUs. Furthermore, most operations in our algorithm are amenable to parallel execution and could potentially benefit significantly from GPU acceleration. + +# 5.2 Symmetric Problems + +Although our algorithm is not specifically designed for symmetric TSPs and does not exploit any symmetry properties, we evaluated its performance on small-scale symmetric instances. The results are presented in Table 7. The performance trends closely resemble those observed on triangle asymmetric TSPs. + +
MethodATSP500ATSP1000
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%20.880.00%0.00%36.72
LKH-3 (1, 500)1.08%1.00%1.0251.23%51.48%3.93
LKH-3 (1, 10000)0.25%0.24%2.933.25%2.94%7.05
MatPOENet112.48%112.19%1.50156.48%156.24%10.21
GOAL1.07%1.01%6.851.26%1.22%58.48
Ours-0.06 %-0.11%0.74-0.35%-0.36%5.10
+ +Table 6: Comparison of methods on large-scale random ATSP with triangle inequality problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality. + +
MethodTSP20TSP50
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%3.210.00%0.00%4.51
LKH-3 (1, 500)0.00%0.00%0.030.06%0.00%0.05
LKH-3 (1, 10000)0.00%0.00%0.330.004%0.00%0.48
MatPOENet1.50%1.25%0.00640.63%0.58%0.015
GOAL14.06%15.14%0.00214.34%15.18%0.0095
Ours0.04%0.00%0.0130.86%0.66%0.094
+ +Table 7: Comparison of methods on small-scale random symmetric TSP problems (TSP20 and TSP50). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. + +# 6 Conclusions and Future Works + +In this work, we introduced a novel continuous relaxation framework for the Asymmetric Traveling Salesman Problem (ATSP) that combines differentiable acyclicity constraints with doubly stochastic matrix relaxations. Our key innovation lies in the formulation of ATSP as a constrained optimization problem where trace-based DAG constraints naturally eliminate subtours while maintaining differentiability for gradient-based optimization. The proposed projected exponentiated gradient method with adaptive step sizes demonstrates efficient convergence properties, and our greedy post-processing procedure effectively recovers valid tours from fractional assignments. + +Through extensive experiments on standard benchmarks, we have shown that our method achieves state-of-the-art performance on asymmetric TSP instances, particularly excelling on large-scale problems where traditional heuristics like LKH-3 struggle. The results reveal several important insights: (1) our continuous relaxation provides a tight formulation that frequently converges to valid tours without post-processing, (2) the approach scales favorably with problem size, maintaining both computational efficiency and solution quality, and (3) the method remains competitive even on symmetric TSP instances despite not being specifically designed for them. + +The success of our framework suggests several promising directions for future research. First, the differentiable nature of our formulation makes it potentially amenable to integration with learning-based approaches, where the solver could be trained to predict good initializations or adaptive penalty parameters. Second, the trace-based constraints may generalize to other combinatorial optimization problems that require acyclic substructures. Finally, investigating more sophisticated optimization techniques within our framework could lead to further improvements in both solution quality and computational efficiency. + +This work bridges an important gap between continuous relaxations and discrete combinatorial optimization, offering a new perspective on solving challenging problems like ATSP through differentiable programming. The demonstrated effectiveness of our approach on large-scale instances suggests that similar techniques could prove valuable for other NP-hard optimization problems where traditional methods face scalability limitations. + +# Acknowledgments and Disclosure of Funding + +This research is supported by Responsible AI Research Center (RAIR) and the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (A-8001814-00-00). + +# References + +[1] David L Applegate, Robert E Bixby, Vasek Chvátal, William Cook, Daniel G Espinoza, Marcos Guycoolea, and Keld Helsgaun. Certification of an optimal tsp tour through 85,900 cities. Operations Research Letters, 37(1):11-15, 2009. +[2] Jill Cirasella, David S Johnson, Lyle A McGeoch, and Weixiong Zhang. The asymmetric traveling salesman problem: Algorithms, instance generators, and tests. In Workshop on algorithm engineering and experimentation, pages 32-59. Springer, 2001. +[3] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, 26, 2013. +[4] Darko Drakulic, Sofia Michel, Florian Mai, Arnaud Sors, and Jean-Marc Andreoli. Bq-nc: Bisimulation quotienting for efficient neural combinatorial optimization. Advances in Neural Information Processing Systems, 36:77416-77429, 2023. +[5] Darko Drakulic, Sofia Michel, and Jean-Marc Andreoli. Goal: A generalist combinatorial optimization agent learner. In The Thirteenth International Conference on Learning Representations, 2024. +[6] Alan Frieze and Gregory B Sorkin. The probabilistic relationship between the assignment and asymmetric traveling salesman problems. SIAM Journal on Computing, 36(5):1435-1452, 2007. +[7] Fred Glover, Gregory Gutin, Anders Yeo, and Alexey Zverovich. Construction heuristics for the asymmetric tsp. European Journal of Operational Research, 129(3):555-568, 2001. +[8] Keld Helsgaun. An extension of the lin-kernighan-helsgaun tsp solver for constrained traveling salesman and vehicle routing problems. Roskilde: Roskilde University, 12:966-980, 2017. +[9] Karla L Hoffman, Manfred Padberg, Giovanni Rinaldi, et al. Traveling salesman problem. Encyclopedia of operations research and management science, 1:1573-1578, 2013. +[10] GLENN Hurlbert. A short proof of the birkhoff-von neumann theorem. preprint (unpublished), 2008. +[11] Chaitanya K Joshi, Thomas Laurent, and Xavier Bresson. An efficient graph convolutional network technique for the travelling salesman problem. arXiv preprint arXiv:1906.01227, 2019. +[12] Richard M Karp. A patching algorithm for the nonsymmetric traveling-salesman problem. SIAM Journal on Computing, 8(4):561-573, 1979. +[13] Richard M Karp and J Michael Steele. Probabilistic analysis of heuristics. The traveling salesman problem, pages 181-205, 1985. +[14] Wouter Kool, Herke van Hoof, and Max Welling. Attention, learn to solve routing problems! In International Conference on Learning Representations, 2019. +[15] Yeong-Dae Kwon, Jinho Choo, Byoungjip Kim, Iljoo Yoon, Youngjune Gwon, and Seungjai Min. Pomo: Policy optimization with multiple optima for reinforcement learning. Advances in Neural Information Processing Systems, 33:21188-21198, 2020. +[16] Yeong-Dae Kwon, Jinho Choo, Iljoo Yoon, Minah Park, Duwon Park, and Youngjune Gwon. Matrix encoding networks for neural combinatorial optimization. Advances in Neural Information Processing Systems, 34:5138-5149, 2021. +[17] Gilbert Laporte. The traveling salesman problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(2):231-247, 1992. +[18] Yang Li, Jinpei Guo, Runzhong Wang, and Junchi Yan. T2t: From distribution learning in training to gradient search in testing for combinatorial optimization. Advances in Neural Information Processing Systems, 36:50020-50040, 2023. +[19] Donald L Miller and Joseph F Pekny. Exact solution of large asymmetric traveling salesman problems. Science, 251(4995):754-761, 1991. +[20] Yimeng Min, Yiwei Bai, and Carla P Gomes. Unsupervised learning for solving the travelling salesman problem. Advances in Neural Information Processing Systems, 36:47264-47278, 2023. +[21] Temel Öncan, I Kuban Altinel, and Gilbert Laporte. A comparative analysis of several asymmetric traveling salesman problem formulations. Computers & Operations Research, 36(3):637-654, 2009. +[22] Wenzheng Pan, Hao Xiong, Jiale Ma, Wentao Zhao, Yang Li, and Junchi Yan. UniCO: On unified combinatorial optimization via problem reduction to matrix-encoded general tsp. In The Thirteenth International Conference on Learning Representations, 2025. +[23] Ruizhong Qiu, Zhiqing Sun, and Yiming Yang. Dimes: A differentiable meta solver for combinatorial optimization problems. Advances in Neural Information Processing Systems, 35:25531-25546, 2022. +[24] Zhiqing Sun and Yiming Yang. Difusco: Graph-based diffusion solvers for combinatorial optimization. Advances in neural information processing systems, 36:3706-3731, 2023. + +[25] Dennis Wei, Tian Gao, and Yue Yu. Dags with No Fears: A closer look at continuous optimization for learning bayesian networks. Advances in Neural Information Processing Systems, 33:3895-3906, 2020. +[26] Yifan Xia, Xianliang Yang, Zichuan Liu, Zhihao Liu, Lei Song, and Jiang Bian. Position: Rethinking post-hoc search-based neural approaches for solving large-scale traveling salesman problems. arXiv preprint arXiv:2406.03503, 2024. +[27] Haoran Ye, Jiarui Wang, Helan Liang, Zhiguang Cao, Yong Li, and Fanzhang Li. Glop: Learning global partition and local construction for solving large-scale routing problems in real-time. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 20284-20292, 2024. +[28] Zhen Zhang, Ignavier Ng, Dong Gong, Yuhang Liu, Ehsan Abbasnejad, Mingming Gong, Kun Zhang, and Javen Qinfeng Shi. Truncated matrix power iteration for differentiable dag learning. Advances in Neural Information Processing Systems, 35:18390-18402, 2022. +[29] Zhen Zhang, Ignavier Ng, Dong Gong, Yuhang Liu, Mingming Gong, Biwei Huang, Kun Zhang, Anton van den Hengel, and Javen Qinfeng Shi. Analytic dag constraints for differentiable dag learning. arXiv preprint arXiv:2503.19218, 2025. +[30] Jiongzhi Zheng, Kun He, Jianrong Zhou, Yan Jin, and Chu-Min Li. Combining reinforcement learning with lin-kernighan-helsgaun algorithm for the traveling salesman problem. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pages 12445-12452, 2021. +[31] Zhi Zheng, Changliang Zhou, Tong Xialiang, Mingxuan Yuan, and Zhenkun Wang. 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Let $\mathbf{X} \in \mathbb{R}^{n \times n}$ be the assignment matrix, where $x_{ij} = 1$ indicates that row $i$ is assigned to column $j$ , and zero otherwise. The problem can be formulated as the following linear program: + +$$ +\text {m i n i m i z e} \quad \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} x _ {i j} +$$ + +$$ +\text {s u b j e c t} \quad \sum_ {j = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} i = 1, \dots , n, \tag {8} +$$ + +$$ +\sum_ {i = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} j = 1, \dots , n, +$$ + +$$ +x _ {i j} \geq 0 \quad \text {f o r a l l} i, j. +$$ + +The dual problem introduces variables $u_{i}$ and $v_{j}$ corresponding to the row and column constraints, respectively, and is given by: + +$$ +\text {m a x i m i z e} \quad \sum_ {i = 1} ^ {n} u _ {i} + \sum_ {j = 1} ^ {n} v _ {j} \tag {9} +$$ + +$$ +\text {s u b j e c t} \quad u _ {i} + v _ {j} \leq d _ {i j} \quad \text {f o r a l l} i, j. +$$ + +Strong duality holds for this problem, guaranteeing that the optimal values of the primal and dual problems are equal when a feasible solution exists. + +The Hungarian Algorithm is, in fact, a primal-dual method for solving the Linear Assignment Problem (LAP). This insight allows us to exploit the structure of Algorithm 1, where multiple LAPs are solved iteratively. Specifically, we can retain the previously optimal dual variables $\mathbf{u}$ and $\mathbf{v}$ and use them as the initialization for subsequent runs of the Hungarian Algorithm. This warm-start strategy significantly accelerates convergence, as the optimal dual variables often remain close across successive LAP instances. + +# B Further Refining Performance by Branch-and-Bound + +We further refined our implementation by incorporating two key strategies: (1) reusing the dual variables $\mathbf{u}$ and $\mathbf{v}$ , as described in the previous section, and (2) introducing a simple branch-and-bound heuristic. In our branch-and-bound strategy, we first solve the Linear Assignment Problem (LAP) to obtain a lower bound for the ATSP instance. We then identify the edge with the highest cost in the LAP solution and branch on it. For each branch, we apply Algorithm 1 to compute a potentially global upper bound, while the LAP provides a local lower bound. Our heuristic allows up to five branch-and-bound iterations. + +With these enhancements, the results on large-scale ATsP instances satisfying the triangle inequality are presented in Table 8. As shown, reusing the dual variables accelerates our algorithm by up to a factor of five compared to the original implementation. Additionally, the branch-and-bound strategy offers further performance improvements. Similar gains are observed on purely random ATsP instances, as shown in Table 9. + +
MethodTSP500TSP1000
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%20.880.00%0.00%36.72
LKH-3 (1, 500)1.08%1.00%1.0251.23%51.48%3.93
LKH-3 (1, 10000)0.25%0.24%2.933.25%2.94%7.05
MatPOENet112.48%112.19%1.50156.48%156.24%10.21
GOAL1.07%1.01%6.851.26%1.22%58.48
Ours-0.06 %-0.11%0.74-0.35%-0.36%5.10
Ours-reuse[u, v]-0.04%-0.09%0.19-0.35%-0.36%0.97
Ours-reuse[u, v]-BnB-0.11%-0.12%0.70-0.38%-0.38%2.91
+ +Table 8: Comparison of methods on large-scale random ATSP with triangle inequality problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality. + +
MethodATSP500ATSP1000
Avg. GAPMedian GAPRuntime (s)Avg. GAPMedian GAPRuntime (s)
LKH-3 (10, 10000)0.00%0.00%29.990.00%0.00%46.64
LKH-3 (1, 500)24.47%24.86%1.0851.23%51.48%3.81
LKH-3 (1, 10000)2.50%2.45%3.853.25%2.94%7.91
MatPOENet6400.51%6400.50%1.5115017.16%15016.70%10.24
GOAL369.69%369.40%6.84305.84%304.62%58.53
Ours-10.77 %-11.59%0.87-21.40%-21.86%5.22
Ours-reuse[u,v]-10.89%-11.67%0.33-21.49%-21.98%1.54
Ours-reuse[u,v]-BnB-11.88%-12.02%0.8422.31%-22.39%3.63
+ +Table 9: Comparison of methods on large-scale pure random ATSP problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP. + +# C Experimental Results on TSPLIB instances + +We evaluated our algorithm on large-scale ATSP benchmark instances from TSPLIB, particularly those derived from stacker crane problems. Our method consistently outperforms LKH-3 (1-100) in terms of solution quality while also requiring less computational time. These results demonstrate the practical effectiveness of our approach on real-world structured benchmarks, further validating its scalability and competitiveness beyond synthetic instances. + +# D Comparison with ILP Solver + +We conducted a preliminary comparison between our method and an exact ILP solver on both symmetric and asymmetric TSP instances. Specifically, we tested the solver on 100-node symmetric problems and 500-node asymmetric problems and found that the ILP solver was able to return optimal solutions within a reasonable time budget. + +Due to time constraints, we evaluated both methods on 10 randomly sampled instances for each case. The results are summarized below. While the ILP solver guarantees optimality, our method achieves competitive solution quality with significantly faster runtimes, particularly for larger asymmetric instances—highlighting its scalability and practical utility for large-scale problems. + +# D.1 Performance comparison on 100-node symmetric TSP problems + +# D.2 Performance comparison on 500-node asymmetric TSP problems + +Table 10: Comparison of algorithms on asymmetric TSP benchmark instances. + +
InstanceAlgorithmBest Known ResultRunning Time (s)Result
rbg323LKH(1-100)13261.461388
LKH(1-10000)3.381346
LKH(10-10000)20.321346
Ours0.421360
Ours (no 2-opt)0.201365
rbg358LKH(1-100)11631.671294
LKH(1-10000)4.031175
LKH(10-10000)25.161175
Ours0.091180
Ours (no 2-opt)0.041180
rbg403LKH(1-100)24656.272536
LKH(1-10000)26.992498
LKH(10-10000)9.062498
Ours1.532473
Ours (no 2-opt)1.282473
rbg443LKH(1-100)27205.442813
LKH(1-10000)7.872762
LKH(10-10000)30.062756
Ours0.522760
Ours (no 2-opt)0.272760
+ +Table 11: Performance comparison on 100-node symmetric TSP problems. + +
AlgorithmRunning Time (s)Ratio to Optimal (Mean)Ratio to Optimal (Median)
LKH(1–100)0.070.61%0.17%
LKH(1–10000)1.240.12%0.00%
LKH(10–10000)12.140.00%0.00%
Gurobi2.060.00%0.00%
Ours (no local search)0.0315.2%14.4%
Ours (local search)0.051.47%1.35%
+ +Table 12: Performance comparison on 500-node asymmetric TSP problems. + +
AlgorithmRunning Time (s)Ratio to Optimal (Mean)Ratio to Optimal (Median)
LKH(1-100)0.872.22%0.17%
LKH(1-10000)2.820.45%0.00%
LKH(10-10000)20.230.18%0.00%
Gurobi6.360.00%0.00%
Ours (no local search)0.170.005%0.001%
Ours (local search)0.300.005%0.001%
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Through stochastic differential equation analysis of optimization dynamics in an analytically tractable setup, we demonstrate that the choice of optimization algorithm indeed influences fairness outcomes, particularly under severe imbalance. Furthermore, we show that when comparing two categories of optimizers, adaptive methods and stochastic methods, RMSProp (from the adaptive category) has a higher likelihood of converging to fairer minima than SGD (from the stochastic category). Building on this insight, we derive two new theoretical guarantees showing that, under appropriate conditions, RMSProp exhibits fairer parameter updates and improved fairness in a single optimization step compared to SGD. We then validate these findings through extensive experiments on three publicly available datasets, namely CelebA, FairFace, and MS-COCO, across different tasks as facial expression recognition, gender classification, and multi-label classification, using various backbones. Considering multiple fairness definitions including equalized odds, equal opportunity, and demographic parity, adaptive optimizers like RMSProp and Adam consistently outperform SGD in terms of group fairness, while maintaining comparable predictive accuracy. Our results highlight the role of adaptive updates as a crucial yet overlooked mechanism for promoting fair outcomes. We release the source code at: https://github.com/Mkolahdoozi/Some-Optimizers-Are-More-Equal. + +# 1 Introduction + +Machine learning is increasingly applied to a wide variety of domains, many of which have significant social implications [5, 43]. These include, but are not limited to, decision-making systems, risk assessment models, recommendation engines, and personalized health interventions [12, 46]. These systems influence people's lives and can lead to unintended biases or disparities [38], for instance by treating different groups of people differently. To address this, researchers have developed various techniques to ensure group fairness in machine learning algorithms [20, 13]. + +Prior work has demonstrated that a variety of factors such as training data imbalance [45] and model hyper-parameters [10] can impact the performance of deep neural networks in terms of group fairness and bias. In this paper, we pose the following question for the first time: + +Does the choice of optimization algorithm impact group fairness in deep neural networks? and how? + +The limited theoretical and empirical analysis in this area leaves a major gap in our understanding of how optimizers impact the inherent performance of models in terms of bias. Consequently, we believe that this question can have significant implications in developing models that perform consistently across various groups, as optimization is a fundamental component of any deep learning model. + +In this paper, we bridge this gap by first analyzing the impact of optimization in group fairness in an analytically tractable setup. This initial study confirms our hypothesis: that indeed the choice of optimizer impacts group fairness in downstream applications, particularly in the presence of severe bias in the training set. Next, we delve deep into the problem by proposing and proving two new theorems that demonstrate that among the two primary families of optimization algorithms, adaptive gradient methods such as Adam and RMSProp, and stochastic gradient methods such as SGD, the former is more likely to exhibit better group fairness. Next, we validate our theoretical findings by conducting extensive experiments on three publicly available datasets-MS-COCO [27], CelebA [31], and FairFace [16] across three different tasks: multi-label classification, facial expression recognition, and gender classification. We carefully evaluate fairness using three widely adopted metrics: Equalized Odds, Equal Opportunity, and Demographic Parity. The results are strongly aligned with our theoretical analysis, consistently demonstrating that adaptive gradient methods achieve better fairness scores compared to stochastic gradient methods. + +Our contributions in this paper are summarized as follows: (1) First, we provide a detailed mathematical analysis to study the impact of optimization algorithms on fairness in an analytically tractable setting. Here, we demonstrate that adaptive gradient algorithms like RMSProp have a higher probability of converging to fairer minima. (2) Next, we establish and prove two theorems showing that under appropriate conditions: (a) RMSProp provides fairer updates than SGD, and (b) the worst-case bias introduced by RMSProp in a single optimization step has an upper bound equal to that of SGD. These theoretical insights highlight key properties of optimization algorithms that contribute to their advantages in terms of fairness. (3) Through extensive experiments on multiple publicly available datasets and tasks, we empirically validate our theoretical findings. We systematically evaluate the fairness of models optimized using adaptive gradient methods and compare the outcomes to those obtained from stochastic gradient methods. Our results consistently show that adaptive gradient methods lead to fairer minima. (4) To enable rapid reproducibility and contribute to the area, we make our code publicly available at: https://github.com/Mkolahdoozi/Some-Optimizers-Are-More-Equal. + +# 2 Related work + +Group fairness. Bias in machine learning models leads to discriminatory outcomes, and as these models are increasingly being deployed in real-life applications, such biases can impose risks to society [38]. This risk is particularly important in high-stakes applications, such as hiring [17], healthcare [22], etc. When models yield biased outcomes, they favor certain demographic groups, which reinforces social inequalities. As an example, facial expression recognition models show higher error rates for darker skin tones [55]. Such biases undermine trust in machine learning models and pose ethical and legal challenges. To analyze fairness in machine learning, researchers in this area often distinguish the notion of "group fairness" and "individual fairness" [38]. Group fairness focuses on the notion that machine learning models treat different demographic groups, such as those defined by race, the same. However, individual fairness dictates that similar individuals should be treated similarly. In this paper, we focus on group fairness, which is more widely recognized in practical applications [7]. + +One of the main sources of bias in machine learning models is data imbalance [9]. When certain demographic subgroups are underrepresented, models prioritize patterns from the majority group, leading to biased decisions. To tackle this, a variety of methods have been proposed that can be categorized into 3 main groups [6]: pre-processing, in-processing, and post-processing. Pre-processing methods often apply some type of transformation to data to make the demographic information less recognizable to the model. As an example, [48] augments the training set with synthetically generated samples, thereby ensuring an equal distribution for demographic subgroups. In-processing methods directly modify the learning algorithm of the model. As an example, [20] proposes a new loss function based on maximum mean discrepancy statistical distance to enhance fairness. Post-processing methods intervene after training is completed. For instance, [44] introduces a post-processing approach using graph Laplacian regularization to enforce individual fairness constraints while preserving accuracy. Nevertheless, in practice, these fairness-enhancing methods are rarely used in real-world pipelines [39]. One major limitation is their computational overhead or interference with the training phase. As an example, pre-processing methods require extensive data augmentation or transformations. Considering these challenges, an alternative approach to enhance fairness is to use components that are inherently present in every training pipeline. One of + +the fundamental building blocks of modern deep learning is optimization [52]. Understanding the role of different optimization algorithms in fairness is crucial for practitioners seeking to develop fairer models without additional fairness-specific modifications. The work in [29] provides a key theoretical analysis, showing that standard and unconstrained machine learning implicitly favors one fairness criterion over others. The authors prove that a model's deviation from calibration is bounded by its excess risk. This is related to our work as it also investigates the fairness properties that emerge from a standard training process. Furthermore, [56] focuses on in-processing fairness methods that use surrogate functions to ensure fairness. Their theory also highlights that balanced datasets are beneficial for achieving tighter fairness and stability guarantees. This is relevant to our findings, as we also identify data imbalance as a critical factor through stochastic differential equation analysis. + +Optimization algorithms. Optimization algorithms in deep learning generally fall into two main categories: SGD and adaptive gradient methods (e.g., Adam [18] and RMSProp [34]). SGD updates model parameters using a constant learning rate [15], while adaptive methods dynamically adjust per-parameter learning rates based on past gradients, often leading to faster convergence [35]. The impact of these two types of optimization algorithms on group fairness remains not well understood. Nonetheless, several studies have explored their effects in other areas. For instance, [34] empirically demonstrates that models trained with SGD are more robust to input perturbations than those trained with adaptive methods. Fairness and robustness are often competing objectives in deep learning [36]. This hints that adaptive optimizers may yield better fairness compared to SGD. However, this relationship remains largely unexplored. + +# 3 Method + +# 3.1 Preliminaries + +**Optimization algorithms.** Stochastic gradient methods like SGD and adaptive gradient methods such as RMSProp and Adam are widely used for optimizing deep neural networks [58]. Given a loss function $\mathcal{L}(w)$ , SGD updates the model parameters $w$ at each iteration $k$ as: + +$$ +w _ {k + 1} = w _ {k} - \eta \nabla \mathcal {L} (w _ {k}), \tag {1} +$$ + +where $\eta$ denotes the learning rate. Adaptive gradient methods improve the speed of convergence by applying coordinate-wise scaling to the gradient [41]. Specifically, RMSProp normalizes the gradient with an exponentially decaying average of its squared values: + +$$ +v _ {k} = \gamma v _ {k - 1} + (1 - \gamma) \nabla \mathcal {L} \left(w _ {k}\right) ^ {2}, \quad w _ {k + 1} = w _ {k} - \frac {\eta}{\sqrt {v _ {k} + \epsilon}} \nabla \mathcal {L} \left(w _ {k}\right), \tag {2} +$$ + +where $v_{k}$ is the moving average of squared gradients, $\gamma$ is the decay factor, and $\epsilon$ is a small constant added for numerical stability. On the other hand, Adam works by considering both the first and the second moments of gradients: + +$$ +m _ {k} = \beta_ {1} m _ {k - 1} + (1 - \beta_ {1}) \nabla \mathcal {L} \left(w _ {k}\right), \tag {3} +$$ + +$$ +v _ {k} = \beta_ {2} v _ {k - 1} + (1 - \beta_ {2}) \nabla \mathcal {L} \left(w _ {k}\right) ^ {2}, \tag {4} +$$ + +$$ +\hat {m} _ {k} = \frac {m _ {k}}{1 - \beta_ {1} ^ {k}}, \quad \hat {v} _ {k} = \frac {v _ {k}}{1 - \beta_ {2} ^ {k}}, \tag {5} +$$ + +$$ +w _ {k + 1} = w _ {k} - \frac {\eta}{\sqrt {\hat {v} _ {k}} + \epsilon} \hat {m} _ {k}, \tag {6} +$$ + +where $\beta_{1}$ and $\beta_{2}$ are the first and second moment decay factors, respectively. + +Stochastic Differential Equations (SDE). Given the interval $[0, T]$ where $T > 0$ , we have [51]: + +$$ +d X _ {t} = a (X (t), t) d t + b (X (t), t) d B (t), \tag {7} +$$ + +where $X_{t} \in R^{d}$ is a stochastic process (the solution of the SDE), $a: R^d \times [0, T] \to R^d$ is the drift term, $b: R^d \times [0, T] \to R^{d \times s}$ is the diffusion term, and $B(t) \in R^s$ is $s$ -dimensional Brownian motion. Intuitively, SDEs extend ordinary differential equations (ODEs) by incorporating stochastic noise. Analyzing functions involving stochastic processes requires tools beyond classical calculus. A fundamental theorem in stochastic analysis is Itô's lemma, extending the chain rule to the functions + +of stochastic processes [1]. Please see the Appendix A for Itô's lemma. Unlike ODEs, the solution of an SDE is a stochastic process $X_{t}$ . Formally, the solution can be represented as: + +$$ +X _ {t} = X _ {0} + \int_ {0} ^ {t} a (X _ {\tau}, \tau) d \tau + \int_ {0} ^ {t} b (X _ {\tau}, \tau) d B _ {\tau}. \tag {8} +$$ + +The second integral, also known as the Itô's integral, captures the stochastic contribution driven by Brownian motion. It is defined as: + +$$ +\int_ {0} ^ {t} b \left(X _ {\tau}, \tau\right) d B _ {\tau} = \lim _ {n \rightarrow \infty} \sum_ {\tau_ {k}, \tau_ {k - 1} \in \pi (n)} b \left(X _ {\tau_ {k - 1}}, \tau_ {k - 1}\right)\left(B _ {\tau_ {k}} - B _ {\tau_ {k - 1}}\right), \tag {9} +$$ + +where $\pi (n)$ denotes a sequence of $n$ partitions in $[0,t]$ , and lim shows convergence in probability. In many cases, explicitly computing $X_{t}$ is intractable. Instead, it is often more practical to analyze the probability distribution of $X_{t}$ . The Fokker-Planck equation, also known as the forward Kolmogorov equation, describes the time evolution of the probability density function $p(x,t)$ of $X_{t}$ [21] as: + +$$ +\frac {\partial p (x , t)}{\partial t} = - \nabla \cdot (a (x, t) p (x, t)) + \frac {1}{2} \nabla \cdot (\nabla \cdot (b (x, t) b (x, t) ^ {t} p (x, t))), \tag {10} +$$ + +where $\nabla$ represents the divergence operator. For completeness, we provide the derivation of the Fokker-Planck equation from Eq. 7 in Appendix B. + +SDE approximations for optimization algorithms. SDEs provide a powerful framework for analyzing the dynamics of discrete optimization algorithms such as SGD and RMSProp in a continuous setup. While direct analysis of discrete updates is often challenging [26], SDE approximations enable the study of optimization trajectories. In the context of optimization, the sources of stochasticity include mini-batch gradient noise which are variations due to data heterogeneity (e.g., differences in mini-batches drawn from different subpopulations), among others. To formalize this, we adopt the Noisy Gradient Oracle with Scale Parameter (NGOS) [37], which models stochastic gradients as $g(w) = \nabla \mathcal{L}(w) + \Theta z$ , where $\Theta$ is a noise scale parameter, and $z$ is a random variable that follows a distribution with covariance matrix $\Sigma(w)$ and mean of zero. NGOS formally models the gradients used in mini-batch training. It describes the gradient from a small data batch as the "true" gradient (which would come from the full dataset) plus random noise introduced by the sampling process. A well-behaved NGOS is defined as $\nabla \mathcal{L}(w)$ being Lipschitz and $\sqrt{\Sigma(w)}$ being bounded. See Appendix C for a detailed description of a well-behaved NGOS. Note that these assumptions are common and standard in prior works that study optimization using SDEs [25]. + +Under well-behaved NGOS, SGD can be approximated by the following SDE [25]: + +$$ +d W _ {t} = - \nabla \mathcal {L} \left(W _ {t}\right) d t + \left(\eta \Sigma \left(W _ {t}\right)\right) ^ {1 / 2} d B (t), \tag {11} +$$ + +where $W_{t}$ represents an approximation of $w_{k}$ at discrete time steps $k\eta$ . More recently, SDE approximation for RMSProp has been proposed as a coupled system of equations [37]: + +$$ +d W _ {t} = - P _ {t} ^ {- 1} \left(\nabla \mathcal {L} \left(W _ {t}\right) d t + \Theta \eta \Sigma^ {1 / 2} \left(W _ {t}\right) d B (t)\right), \quad d u _ {t} = \frac {1 - \gamma}{\eta^ {2}} \left(d i a g \left(\Sigma \left(W _ {t}\right)\right) - u _ {t}\right) d t, \tag {12} +$$ + +where $P_{t} = \Theta \eta \, diag(u_{t})^{\frac{1}{2}} + \epsilon \eta I$ and $u_{t}$ is defined as $\frac{v_k}{\Theta^2}$ . Note that these SDE approximations converge to $w_{k}$ in the weak sense, meaning that while individual trajectories of the discrete and continuous processes may differ, their probability distributions remain close. Please see Appendix D for a formal formulation of the convergence. + +# 3.2 SDE analysis of optimizers + +To illustrate the different behaviors of SGD and RMSProp in terms of group fairness, let's consider a warm-up example. In this example, assume a training set consisting of two equally-sized subgroups, denoted as subgroups 0 and 1. That is, each group constitutes $50\%$ of the total population. Please note this example can be generalized over more than 2 subgroups. For simplicity, assume each subgroup has its own loss function, denoted as: + +$$ +\mathcal {L} _ {0} (w) = \frac {1}{2} (w - 1) ^ {2}, \quad \mathcal {L} _ {1} (w) = \frac {1}{2} (w + 1) ^ {2}. \tag {13} +$$ + +The objective is to minimize a population-level loss function defined as the weighted sum of the subgroup losses as: + +$$ +\mathcal {L} _ {\text {p o p}} (w) = 0. 5 \mathcal {L} _ {0} (w) + 0. 5 \mathcal {L} _ {1} (w), \tag {14} +$$ + +where the weights correspond to the proportion of each subgroup in the population. In this setup, the optimal solution for subgroup 0 is $w_0^* = 1$ , while the optimal solution for subgroup 1 is $w_1^* = -1$ . However, The population's optimal solution is obtained by minimizing $\mathcal{L}_{\mathrm{pop}}(w)$ , leading to $w_{\mathrm{pop}}^* = 0$ . The following lemma shows that under the demographic parity definition of group fairness [38], $w_{\mathrm{pop}}^* = 0$ is the fairest minima. + +Lemma 1. Let $\mathcal{L}_0(w) = \frac{1}{2} (w - 1)^2$ and $\mathcal{L}_1(w) = \frac{1}{2} (w + 1)^2$ be the loss functions for subgroups $0$ and $1$ , respectively. Define the population loss as $\mathcal{L}_{pop}(w) = 0.5\mathcal{L}_0(w) + 0.5\mathcal{L}_1(w)$ . Under the demographic parity definition of fairness, the fairest minimizer of $\mathcal{L}_{pop}(w)$ is $w_{pop}^{*} = 0$ . + +Proof. See Appendix E. + +![](images/acb52e36319b33337c8f8bf89ff56f17b2f60d081e58fdce2e5f15e346f1c9cf.jpg) + +In practice, we do not have access to the full population above, rather we have access to the set $\Omega$ of $N$ individual samples, sampled uniformly from the population. Additionally, set $\Omega$ may not contain an equal number of samples from subgroups 0 and 1, thus introducing bias. We assume that the probability of subgroup 0 in $\Omega$ is $p_0$ while the probability of subgroup 1 is $p_1 = 1 - p_0$ . For a given set $\Omega$ , the empirical minimization problem used in practice is defined as: $\mathcal{L}_{emp}(w) = \frac{1}{N}\sum_{r\in \Omega}\mathcal{L}_{q_r}(w)$ , where $q_{r}\in \{0,1\}$ corresponds to samples from subgroups 0 and 1. Note that as $N\to \infty$ , $\mathcal{L}_{emp}\rightarrow \mathcal{L}_{pop}$ . In min-batch training with batch size of 1, at each iteration, the optimizer processes a single individual from subgroup 0 with a probability of $p_0$ and from subgroup 1 with a probability $p_1$ . In other words, with both SGD and RMSProp, the model receives $\nabla \mathcal{L}_0(w)$ with a probability of $p_0$ and $\nabla \mathcal{L}_1(w)$ with a probability of $p_1$ , updating $w$ according to Eqs. 1 and 2, respectively. If $p_0 > p_1$ , the optimization trajectory is biased towards the group-specific minimum at $w = 1$ , and $w = -1$ otherwise. This demonstrates that disproportionate sampling of subgroups leads to unfair minima, which is particularly relevant in practical settings. The following theorem establishes that when the sampling bias $|p_0 - p_1|$ exceeds a certain threshold, RMSProp has a higher probability of converging to $w_{pop}^{*}$ compared to SGD. + +Theorem 1. Let $p_0, p_1 \in (0,1)$ with $p_0 + p_1 = 1$ be the subgroup sampling probabilities for the loss functions $\mathcal{L}_0(w) = \frac{1}{2}(w - 1)^2$ and $\mathcal{L}_1(w) = \frac{1}{2}(w + 1)^2$ . Suppose we optimize the empirical objective $\mathcal{L}_{\mathrm{emp}}(w) = \frac{1}{N} \sum_{r \in \Omega} \mathcal{L}_{qr}(w)$ , where each sample $q_r \in \{0,1\}$ is drawn i.i.d. with probability $p_0$ for subgroup 0 or $p_1$ for subgroup 1. Consider mini-batch gradient updates of size 1 (i.e., one sample per iteration) using SGD and RMSProp optimization algorithms. Then there exists a constant $\Delta(p_1p_2,\eta) > 0$ such that, whenever $|p_0 - p_1| > \Delta(p_1p_2,\eta)$ , we have: $\frac{p_{rms}(w_{pop}^*)}{p_{sgd}(w_{pop}^*)} > 1$ , where $p_{rms}(w_{pop}^*)$ and $p_{sgd}(w_{pop}^*)$ are the probabilities of RMSProp and SGD converging to fair minima $w_{pop}^* = 0$ , respectively. + +Proof. As mentioned earlier, the gradient of $\mathcal{L}_{emp}$ is $\nabla \mathcal{L}_0(w)$ with a probability $p_0$ and $\nabla \mathcal{L}_1(w)$ with a probability of $p_1$ . Thus, the covariance is $\Sigma(W_t) = 4p_0p_1$ . This means $\Sigma(W_t)$ is independent of $W_t$ . When convergence occurs for RMSProp, $u_t$ defined in Eq. 12 converges to $\Sigma(W_t)$ . Thus, $u_t = 4p_0p_1$ . With this in mind, let's derive the SDE for RMSProp: + +$$ +d W _ {t} = \frac {- 1}{\eta \left(2 \Theta \sqrt {p _ {0} p _ {1}} + \epsilon\right)} \left(\nabla \mathcal {L} _ {p o p} \left(W _ {t}\right) d t + 2 \Theta \eta \sqrt {p _ {0} p _ {1}} d B _ {t}\right). \tag {15} +$$ + +Here, $\epsilon$ is added for numerical stability, but it can be ignored as it is commonly done so in prior work [37]. Next, let's derive the SDE for SGD: + +$$ +d W _ {t} = - \nabla \mathcal {L} _ {p o p} (W _ {t}) d t + 2 \sqrt {\eta} \sqrt {p _ {0} p _ {1}} d B (t). \tag {16} +$$ + +For analyzing the two SDEs above, we use the Fokker-Planck equations, presented in Eq. 10. When convergence occurs, $\frac{\partial p(w,t)}{\partial t} = 0$ . Thus, Fokker-Planck for RMSProp can be written as: + +$$ +\frac {\partial}{\partial w} (p (w) \times \frac {\nabla \mathcal {L} _ {p o p} (w)}{2 \eta \Theta \sqrt {p _ {0} p _ {1}}}) + \frac {1}{2} \frac {\partial^ {2} p (w)}{\partial w ^ {2}} = 0. \tag {17} +$$ + +Substituting $\nabla \mathcal{L}_{pop} = \frac{p_0}{2} (w - 1) + \frac{p_1}{2} (w + 1)$ in the equation above, we obtain: + +$$ +\frac {\partial}{\partial w} (p (w) \times \frac {p _ {0} (w - 1) + p _ {1} (w + 1)}{4 \eta \Theta \sqrt {p _ {0} p _ {1}}}) + \frac {1}{2} \frac {\partial^ {2} p (w)}{\partial w ^ {2}} = 0. \tag {18} +$$ + +Similarly, the Fokker-Planck equation for SGD can be written as: + +$$ +\frac {\partial}{\partial w} (p (w) \times \frac {p _ {0} (w - 1) + p _ {1} (w + 1)}{2}) + 2 \eta p _ {0} p _ {1} \frac {\partial^ {2} p (w)}{\partial w ^ {2}} = 0. \tag {19} +$$ + +Eq. 18 is an analytically solvable ordinary differential equation, which can be written as: + +$$ +p _ {r m s} (w) = \sqrt {\frac {\kappa}{\pi}} \exp \left(- \kappa \left(w - \left(p _ {0} - p _ {1}\right)\right) ^ {2}\right), \tag {20} +$$ + +where $\kappa = \frac{1}{4\eta\Theta\sqrt{p_0p_1}}$ . As a sanity check, we can see that the expectation of the distribution in Eq. 20 is equal to $p_0 - p_1$ . This means in the case of $p_0 = p_1$ , i.e. no sampling bias, the weight converges to $w^{*} = 0$ , the unbiased minimum. Similarly, we can write the analytical solution of Eq. 19 as: + +$$ +p _ {s g d} (w) = \sqrt {\frac {\vartheta}{\pi}} e x p \left(- \vartheta \left(w - \left(p _ {0} - p _ {1}\right)\right) ^ {2}\right), \tag {21} +$$ + +where $\vartheta = \frac{1}{8\eta p_0p_1}$ . As we use a batch size of 1, we can set $\Theta = 1$ [37]. Then, $1 > = 2\sqrt{p_0p_1}$ inequality holds. With this in mind, by substituting $w_{pop}^{*} = 0$ in Eqs. 20 and 21, we can conclude $\frac{p_{rms}(w_{pop}^{*})}{p_{sgd}(w_{pop}^{*})} > 1$ holds if and only if $|p_0 - p_1| > \Delta(p_1p_2, \eta)$ , where $\Delta(p_1p_2, \eta) = \sqrt{\frac{1}{2}\ln\frac{\vartheta}{\vartheta - \kappa}}$ . + +Now, let's perform a quick simulation to verify our findings. We use the loss functions defined in Eq. 13 and set $p_0 = 0.1$ and $p_1 = 0.9$ to induce a severe sampling bias. We fix the learning rate to $\eta = 0.1$ and apply SGD and RMSProp for 100 epochs. Each experiment is repeated 1000 times, and at every epoch we record the fraction of runs converging to the neighborhood of the fair optimum $w_{pop}^*$ , defined by $|w - w_{pop}^*| < 0.2$ . The result is shown in Fig. 1 (left). As it is evident from this figure, approximately $10\%$ of the runs under RMSProp lie in the neighborhood of $w_{pop}^*$ , whereas SGD + +![](images/5f461a8734b484505f300802872459edd01a401ce20848a368d25a3840cd6eb1.jpg) +Figure 1: Percentage of 1000 runs converging within the fair neighborhood for SGD and RM-SProp, under severe bias $(p_0 = 0.1)$ and mild bias $(p_0 = 0.3)$ . + +rarely converges to this fair region. Now, let's induce a milder bias by setting $p_0 = 0.3$ and $p_1 = 0.7$ and run the same experiment. The results are illustrated in Fig. 1 (right). As shown in this figure, in the milder bias, the two optimizers behave more similarly, yet RMSProp attains a slightly higher fraction of solutions near $w_{pop}^*$ . These observations align with our theoretical findings in Theorem 1. To show that our findings are true for a wider range of $\eta$ and alternative definitions of the fair neighborhood, we refer the reader to Appendix F for additional experiments. + +# 3.3 Group fairness analysis + +The detailed SDE analysis carried out in Subsection 3.2 highlights the distinct behaviors of SGD and RMSProp concerning group fairness. However, extending this analysis to higher dimensions, particularly through the Fokker-Planck Eq. 10, introduces significant complexity. To address this challenge, we present two theorems that leverage alternative mathematical tools beyond SDE, to study the impact of SGD and RMSProp on group fairness. First, we illustrate how adaptive optimizers like RMSProp suppress subgroup disparities at the level of parameter updates. The second theorem quantifies the consequence, showing that this mechanism translates into provably smaller fairness violations in terms of demographic parity. + +Theorem 2. Consider a population that consists of two subgroups with subgroup-specific loss functions $\mathcal{L}_0(w)$ and $\mathcal{L}_1(w)$ , sampled with probabilities $p_0$ and $p_1$ , respectively. Suppose a stochastic (online) training regime, in which each parameter update is computed from a sample drawn from + +one of the two subgroups. Suppose the gradients $\nabla \mathcal{L}_0(w_k)$ and $\nabla \mathcal{L}_1(w_k)$ are well-behaved isotropic NGOs, namely $\mathcal{N}(\mu_0,\Theta_0^2 I)$ and $\mathcal{N}(\mu_1,\Theta_1^2 I)$ , respectively. Then, the difference in parameter updates between subgroups 0 and 1 under RMSProp has an upper bound given by the corresponding difference under SGD. Consequently, RMSProp offers fairer updates across subgroups. + +Proof. For SGD, the difference between the parameter updates across subgroups can be quantified as $||\nabla \mathcal{L}_0 - \nabla \mathcal{L}_1||$ . However, as the RMSProp scales the updates using exponentially decaying average, computing the difference is more complex. Let's compute the expected value of $v_{k}$ defined in Eq. 2: + +$$ +\mathbb {E} \left[ v _ {k} \right] = \gamma^ {k} v _ {0} + (1 - \gamma) \Sigma_ {i = 0} ^ {k - 1} \gamma^ {i} \mathbb {E} \left[ \nabla \mathcal {L} _ {i} ^ {2} \right]. \tag {22} +$$ + +Given that $\mathcal{L}$ (loss function) is a discrete mixture of Gaussians, $\mathbb{E}[\nabla \mathcal{L}_i^2 ]$ can be calculated as: + +$$ +\mathbb {E} \left[ \nabla \mathcal {L} _ {i} ^ {2} \right] = p _ {0} \left(\mu_ {0} ^ {2} + \Theta_ {0} ^ {2} \mathbf {1}\right) + p _ {1} \left(\mu_ {1} ^ {2} + \Theta_ {1} ^ {2} \mathbf {1}\right), \tag {23} +$$ + +where $\mathbf{1}$ denotes a vector of all 1s. Substituting the equation above in Eq. 22 yields: + +$$ +\mathbb {E} \left[ v _ {k} \right] = \gamma^ {k} v _ {0} + \left(1 - \gamma^ {k}\right) \left[ p _ {0} \left(\mu_ {0} ^ {2} + \Theta_ {0} ^ {2} \mathbf {1}\right) + p _ {1} \left(\mu_ {1} ^ {2} + \Theta_ {1} ^ {2} \mathbf {1}\right) \right]. \tag {24} +$$ + +The equation above demonstrates that after enough steps, given that $\gamma < 1$ , $\mathbb{E}[v_k]$ converges to Eq. 23. Next, let's analyze the variance of the $j$ th coordinate of $v_{k}$ , $v_{k_j}$ , to see its spread. To this end, we compute the variance of $\nabla \mathcal{L}_{ij}^2$ using the law of total variance as: + +$$ +\operatorname {v a r} \left(\nabla \mathcal {L} _ {i _ {j}} ^ {2}\right) = \mathbb {E} \left[ \operatorname {v a r} \left(\nabla \mathcal {L} _ {i _ {j}} ^ {2} | Z\right) \right] + \operatorname {v a r} \left(\mathbb {E} \left[ \nabla \mathcal {L} _ {i _ {j}} ^ {2} | Z \right]\right), \tag {25} +$$ + +where $Z$ is a random variable denoting the subgroup. Finally, assuming that (a) $\Theta_0^2$ and $\Theta_1^2$ are larger than $\mu_0^2$ and $\mu_1^2$ , as shown experimentally in prior work [37], and (b) there is bias in the training set (i.e., $p_0p_1$ is small), the variance $\mathrm{var}(v_{k_j})$ can be computed as $\mathcal{O}((1 - \gamma)(\Theta_{0_j}^4 +\Theta_{1_j}^4))$ . For a large $\gamma$ , typical in practical training, $var(v_{k_j})$ becomes negligible, and thus $v_{k}\rightarrow E[v_{k}]$ . Using this, we can approximate the difference between the parameter updates across subgroups for RMSProp as $||D(\nabla \mathcal{L}_0 - \nabla \mathcal{L}_1)||_1$ , where $D$ is a diagonal matrix, whose $jj$ th element can be computed as: $D_{jj} = \frac{1}{\sqrt{p_0(\mu_{0_j}^2 + \Theta_{0_j}^2) + p_1(\mu_{1_j}^2 + \Theta_{1_j}^2) + \epsilon}}$ . As $\Theta^2 >\mu^2$ and in stochastic (online) training setup, we + +can set $\Theta^2\to 1$ , then $D_{jj} < 1$ . In other words, the eigenvalues of matrix $D$ is less than 1, thus proving the theorem. + +Theorem 2 shows that RMSProp, unlike SGD, shrinks disparities between subgroups in the updates. In other words, RMSProp prevents large gradient disparities from dominating the training dynamics. Consequently, the final model is less likely to be pulled toward the subgroup with the larger gradients. Over many iterations, this can help prevent large performance gaps between subgroups. While this does not establish a universal fairness guarantee, it outlines why RMSProp's dynamics yield fairer minima across subgroups. We relax the isotropic noise assumption in Theorem 2 and generalize it to the anisotropic case in Appendix G. The following theorem analyzes fairness through the lens of demographic parity and establishes that, under appropriate conditions in a single iteration, RMSProp's worst-case increase in the gap of demographic parity has an upper-bound no greater than that of SGD. + +Theorem 3. Consider a population that consists of two subgroups with subgroup-specific loss functions $\mathcal{L}_0(w)$ and $\mathcal{L}_1(w)$ , sampled with probabilities $p_0$ and $p_1$ , respectively. Suppose a stochastic (online) training regime in which each parameter update is computed from a sample drawn from one of the two subgroups. Suppose that the gradients $\nabla \mathcal{L}_0(w_k)$ and $\nabla \mathcal{L}_1(w_k)$ are well-behaved isotropic NGOs, namely $\mathcal{N}(\mu_0, \Theta_0^2 I)$ and $\mathcal{N}(\mu_1, \Theta_1^2 I)$ , respectively, with $\mu_1 > \mu_0$ . Then, in expectation, the worst-case increase in the demographic parity gap after one iteration of RMSProp has an upper-bound no greater than the corresponding increase under SGD. + +Proof. See Appendix H. + +![](images/69bc78358361cb1f21ef55f5b4c6d9eae163f653bf9e7babeb6363e0d62481f3.jpg) + +Theorem 3 establishes that the adaptive learning rate of RMSProp, which scales updates based on past squared gradients, helps mitigate demographic parity gaps that can occur during training, whereas SGD does not offer such a mechanism. We relax the isotropic noise assumption in Theorem 3 and generalize the result to the anisotropic case in Appendix I. + +# 4 Experiments + +Setup. We use three public datasets that are widely used in this area: CelebA [31], FairFace [16], and MS-COCO [27]. CelebA and FairFace are datasets commonly used for group fairness analysis, while we also use a general-purpose dataset, MS-COCO, for further completeness. The details about these datasets are presented in Appendix J. The training protocol and backbones used in our experiments are described in Appendix K. We report the performance across different tasks using accuracy and F1. We evaluate group fairness using three widely recognized fairness criteria [38]: equalized odds, equal opportunity, and demographic parity. A predictor is fair under equalized odds definition if the true positive and false positive rates are equal across demographic subgroups. We measure the violation of this criterion as: + +$$ +F _ {E O D} = \min _ {i, j, q} [ \min (\frac {p (\hat {y} = c _ {i} | y = c _ {i} , z _ {j})}{p (\hat {y} = c _ {i} | y = c _ {i} , z _ {q})}, \frac {p (\hat {y} = c _ {i} | y \neq c _ {i} , z _ {j})}{p (\hat {y} = c _ {i} | y \neq c _ {i} , z _ {q})}) ]. \tag {26} +$$ + +In the equation above, $c_{i}$ demonstrates class $i$ . A classifier satisfies equal opportunity if the true positive rate is equal across demographic subgroups. The related fairness measure is defined as: + +$$ +F _ {E O P} = \min _ {i, j} \left(\frac {\sum_ {c = 1} ^ {C} T P R _ {i , c}}{\sum_ {c = 1} ^ {C} T P R _ {j , c}}\right), \tag {27} +$$ + +where $TPR_{i,c}$ denotes the true positive rate for the $i^{\mathrm{th}}$ subgroup on class $c$ . Finally, a predictor is said to be fair from a demographic parity point of view if the predicted label distribution is independent of sensitive attributes. The corresponding measure is defined as: + +$$ +F _ {D P A} = \min _ {i, j, q} \frac {p (\hat {y} = c _ {i} | z _ {j})}{p (\hat {y} = c _ {i} | z _ {q})}. \tag {28} +$$ + +For all the above fairness metrics, higher values indicate better (fairer) outcomes. + +Implementation details. We use PyTorch [42] and up to four Nvidia A100 GPUs. The hyperparameter values are provided in Appendix L, which have been optimized using Weights & Biases [4]. + +Results and analysis. We first calculate the fairness metrics for the ViT backbone across different datasets and different sensitive attributes (gender: G, race: R, and age: A). Fig. 2 presents the results for SGD, RMSProp, and Adam. Considering $F_{EOD}$ , Adam consistently outperforms SGD across all scenarios, mostly by substantial margins. For instance, on CelebA dataset with gender as the sensitive attribute, Adam achieves an $8\%$ improvement over SGD. Furthermore, RMSProp outperforms SGD in 4 out of 5 scenarios, and they both achieve zero $F_{EOD}$ for MS-COCO dataset. Given that MS-COCO contains some extremely underrepresented object categories, this result suggests that the disparity captured by $F_{EOD}$ is largely driven by rare categories receiving no correct classifications. The largest gap between RMSProp and SGD is observed on the FairFace dataset with race as the sensitive attribute, where RMSProp outperforms SGD by $9\%$ . Another point to mention is that Adam and RMSProp yield comparable $F_{EOD}$ values, as expected. Considering $F_{EOP}$ , both Adam and RMSProp outperform SGD in all 5 scenarios. To shed more light, the largest difference between the two is $6\%$ in the case of FairFace dataset with age as the sensitive attribute. Additionally, unlike SGD, both RMSProp and Adam achieve a near-perfect $F_{EOD}$ in the case of CelebA dataset with gender as the sensitive attribute, indicating their effectiveness in reaching fairer minima. + +![](images/b78615035f4727660ccd2dcfe245dd37d53cd29b975e7cf0ef88243c8093be0c.jpg) +Figure 2: Fairness for ViT across different datasets, attributes (G: gender, A: age, R: race), and metrics. + +![](images/6d80d498153e252f74ed67017412c936f441f0a5931e8b6f6470938bd9aa0df9.jpg) + +![](images/db4dd66289edb2e34770bef36f06062de81195bc2a5f0e1a60ffacaaecbf3182.jpg) + +Regarding $F_{DPA}$ , Adam consistently outperforms SGD, while RMSProp surpasses SGD in 4 out of 5 scenarios. Both optimizers achieve zero $F_{DPA}$ on MS-COCO with gender as the sensitive + +attribute. Interestingly, the largest discrepancy between RMSProp and SGD occurs on the FairFace dataset with age as the sensitive attribute. This aligns with our theoretical insights from Theorem 1. Specifically, FairFace exhibits extreme imbalance, with a minority-to-all ratio of approximately $0.9\%$ , whereas CelebA with gender as the sensitive attribute has a substantially higher ratio of $42\%$ . These findings highlight that as dataset imbalance intensifies, the gap between adaptive optimizers and SGD becomes more evident. We refer the reader to Appendix M for the results on fairness metrics for other backbones across CelebA, FairFace, and MS-COCO datasets with different sensitive attributes. The results are well-aligned with the results above. To further validate our findings across different variants of optimizers, we conduct additional experiments with SGD w/ momentum [30], AdamW [32], and AdaBound [33] optimizers. The results are provided in Appendix N. + +Beyond tuning the learning rate, we further investigate the influence of additional hyperparameters on fairness outcomes. Specifically, we extend the hyperparameter tuning using Bayesian Optimization over the learning rate, decay rate, and batch size. This comprehensive search is performed using the ViT backbone on the CelebA dataset, considering gender and age as sensitive attributes. The results are reported in Table 1. As shown, adaptive optimizers like Adam still achieve better fairness compared to SGD. This confirm that the + +fairness trends observed in our main study remain consistent even when multiple hyperparameters are jointly tuned. + +Table 1: Fairness comparison across optimizers with tuning learning rate, decay rate, and batch size. + +
GenderAge
AdamRMSPropSGDAdamRMSPropSGD
FEOD65.2165.1862.6672.3471.9968.40
FEOP99.9099.9196.6099.1099.1097.77
FDPA73.5073.6860.8098.2098.2095.40
+ +To demonstrate that our training is successful and the models are well-behaved, we evaluate the classification performance of ViT on the three datasets for facial expression recognition, multi-label classification, and gender classification, respectively. The results are presented in Table 2. The performance of the other models are presented in Appendix O. Here, we can confirm that + +Table 2: Comparison of accuracy and F1 across different optimizers and datasets. + +
DatasetAccuracyF1 score
SGDRMSPropAdamSGDRMSPropAdam
CelebA91.2391.5492.0892.1291.1792.09
MS-COCO89.6289.7190.0368.3571.0374.10
FairFace89.4191.3792.2091.1392.0792.17
+ +in most cases, all three optimizers exhibit comparable accuracy and F1 scores across all tasks. For instance, on the CelebA dataset, SGD and RMSProp achieve an accuracy of $91\%$ , while Adam attains $92\%$ . This trend is consistent with previous findings in the prior work [34]. As evident, the improved fairness observed with adaptive gradient algorithms cannot be solely attributed to their performance. For instance, on the CelebA dataset, the F1 score for SGD is higher than that of RMSProp, whereas on FairFace, RMSProp achieves a higher F1 score than SGD. However, in both cases, RMSProp demonstrates better fairness performance, as shown in Fig. 2. This further supports our theoretical conclusion that RMSProp is fairer than SGD, regardless of overall performance. + +To rigorously assess the significance of fairness improvement of Adam and RMSProp over SGD, we conduct the following experiment. We train ViT on CelebA dataset 10 times using SGD, RMSProp, and Adam, while recording fairness metrics for both gender and age attributes in each run. To evaluate statistical significance, we perform a paired Wilcoxon signed-rank test [3] on the fairness metrics and report the corresponding p-values in Table 3. As shown in the table, most scenarios yield $p$ -values in the order of 0.001, indicating a statistically significant improvement in + +fairness for adaptive gradient optimizers compared to SGD. The worst $p$ -value in the table corresponds to SGD vs. RMSProp on CelebA with age as the sensitive attribute. However, even this value remains statistically significant under the conventional threshold of 0.05 [14, 40]. + +Table 3: $p$ -values from the Wilcoxon test comparing SGD vs. RMSProp and SGD vs. Adam optimizers on the CelebA dataset for gender and age attributes. + +
MetricGenderAge
SGD-RMSPropSGD-AdamSGD-RMSPropSGD-Adam
FEOD1 × 10-31 × 10-31 × 10-31 × 10-3
FEOP1 × 10-31 × 10-31 × 10-35 × 10-3
FDPA2 × 10-31 × 10-37 × 10-33 × 10-3
+ +Theorem 1 provides theoretical insight that under the demographic parity fairness criterion, RMSProp is more likely to converge to a fairer minimum than SGD in highly imbalanced datasets. Moreover, it predicts that as the dataset becomes more balanced, the fairness advantage of RMSProp over SGD + +Table 4: Comparison of fairness metrics across optimizers, with and without fairness-enhancing methods. Lower values indicate better fairness. + +
DatasetGap in Equal OpportunityGap in Equalized OddsGap in Demographic Parity
AdamRMSPropSGDAdamRMSPropSGDAdamRMSPropSGD
With fairness-enhancing
ProPublica COMPAS0.450.480.712.792.782.900.860.862.60
AdultCensus3.153.163.382.392.414.287.006.8011.79
Without fairness-enhancing
ProPublica COMPAS13.9913.9015.1913.9913.9514.9811.4911.4511.80
AdultCensus21.0420.9121.2920.9020.9121.1412.1912.2312.42
+ +gradually diminishes. To empirically validate this behavior, we conduct an experiment where we randomly downsample the 'male' subgroup in the CelebA dataset from its native male-to-all ratio of $42\%$ down to $22\%$ and $2\%$ . We then train a model on each downsampled version using both RMSProp and SGD. We repeat this process five times while recording the values of $F_{DPA}$ and $F_{EOD}$ . Fig. 3 shows the results, illustrating the absolute difference in $F_{DPA}$ and $F_{EOD}$ between SGD and RMSProp across different male-to-all ratios. As expected, the gap in demographic parity is maximum at the most severe imbalance level ( $2\%$ male-to-all ratio). As the male-to-all ratio increases (indicating less imbalance), the absolute difference between the two optimizers gradually decreases, a trend that also holds for equalized odds. This empirical finding strongly aligns with the theoretical predictions in Theorem 1. + +Theorems 2 and 3 are general with respect to the loss function, which suggests their fairness benefits should be complementary to existing fairness-enhancing methods that add regularization terms to the primary loss. To validate this hypothesis, we conduct an experiment where we test the impact of optimizers in addition to the fairness-enhancing method [47] on two tabular datasets (ProPublica COMPAS [2] and AdultCensus [19]) with gender as the sensitive attribute. The results are presented in Table 4 for SGD, Adam, and RMSProp. Following [47], we report the average gap in equal opportunity, gap in equalized odds, and gap in demographic parity (lower is better) over 10 runs. Regarding the fairness enhancing method, across all metrics on both datasets, Adam and RMSProp achieve substantially lower (better) fairness gaps than SGD. This + +experiment demonstrates that the fairness benefits of adaptive optimizers are not limited to standard training but are complementary to and can amplify the effectiveness of existing fairness interventions. Additionally, Table 4 shows that in the absence of any fairness-enhancing method, adaptive optimizers yield better fairness outcomes than SGD. This finding further indicates that the fairness advantage of adaptive optimizers extends beyond computer vision applications. + +# 5 Conclusion + +This paper provided both theoretical and empirical evidence that the choice of optimizer can significantly affect the group fairness of resulting deep learning models. Through establishing an analytically tractable setup and analyzing it via stochastic differential equations, we showed that SGD and RMSProp can converge to different minima with regards to fairness, with RMSProp more frequently reaching fairer optima. Next, we provided two theorems that showed 1) adaptive gradient methods (e.g., RMSProp) shrink subgroup-specific gradients more effectively, thereby reducing unfair updates compared to their SGD counterparts; 2) In a single optimization step, the worst case demographic disparity of RMSProp has the upper bound given by that of SGD. We validated these theoretical findings on three diverse datasets, namely CelebA, FairFace, and MS-COCO, and on multiple tasks. Across a range of group fairness criteria, i.e., equalized odds, equal opportunity, and demographic parity, adaptive optimizers consistently led to fairer solutions. We hope our work will encourage broader adoption of adaptive optimizers in fairness-critical domains, and we envision this research as a step toward developing more equitable deep learning models. + +![](images/0a68b1acb52db3c4bc96c08b3afe5ecce8134459642383d46f1d9281e3b7426d.jpg) +Figure 3: Difference of RMSProp and SGD's fairness in different male-to-all ratios for CelebA dataset with gender as sensitive attribute. + +# Acknowledgment + +This work was partially funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), and was enabled in part by the support provided by the Digital Research Alliance of Canada. + +# References + +[1] Jing An, Lexing Ying, and Yuhua Zhu. Why resampling outperforms reweighting for correcting sampling bias with stochastic gradients. In International Conference on Learning Representations, 2020. +[2] Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner. 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Suppose that the following stochastic differential equation (SDE) is given: + +$$ +d X (t) = M (t, X (t)) d t + N (t, X (t)) d B (t). \tag {29} +$$ + +In the equation above, $\mathrm{X(t)}$ is a stochastic process, $\mathbf{M}()$ and $\mathbf{N}()$ are the functions of $X(t)$ and $t$ . Let's define stochastic process $Y(t)$ by: + +$$ +Y (t) = Q (t, X (t)), \tag {30} +$$ + +where, $Q()$ is a function of stochastic process $X(t)$ and $t$ . Itô's lemma states that we can compute $dY(t)$ by using the following: + +$$ +\begin{array}{l} d Y (t) = \left(\frac {d}{d t} Q (t, X) + \frac {d}{d x} Q (t, X) \cdot M (t, X (t)) + 0. 5 \frac {d ^ {2}}{d x ^ {2}} Q (t, X) N ^ {2} (t, X (t))\right) d t + \tag {31} \\ \left(\frac {d}{d x} Q (t, X) \cdot N (t, X (t))\right) d B (t). \\ \end{array} +$$ + +In the above equation, as the term $0.5\frac{d^2}{dx^2} Q(t,X)N^2 (t,X(t))$ does not appear in classic calculus, it is often referred to as Ito's correction term. + +# B Fokker-Planck equation + +In this section, for completeness, we provide the derivation of fokker-planck equation from a given SDE. This is a well-known equation in the analysis of SDEs. Suppose the following SDE is given: + +$$ +d X (t) = M (t, X (t)) d t + N (t, X (t)) d B (t). \tag {32} +$$ + +Also, suppose that $Q(x)$ is a function which is 0 outside of a bounded interval. For simplicity, we use $X$ instead of $X(t)$ . By Ito's lemma, we have: + +$$ +d Q (X) = \left(M (t, X) Q ^ {\prime} (X) + 0. 5 N ^ {2} (t, X) Q ^ {\prime \prime} (X)\right) d t + N (t, X) Q ^ {\prime} (X) d B (t). \tag {33} +$$ + +Let's integrate the equation above from 0 to $t$ and then take the expected value from both sides: + +$$ +\begin{array}{l} \mathbb {E} [ Q (X) - Q (0) ] = \int_ {0} ^ {t} \mathbb {E} \left[ \left(M (\tau , X) Q ^ {\prime} (X) + 0. 5 N ^ {2} (\tau , X) Q ^ {\prime \prime} (X)\right) \right] d \tau + \tag {34} \\ \int_ {0} ^ {t} \mathbb {E} [ N (\tau , X) Q ^ {\prime} (X) d B (\tau) ]. \\ \end{array} +$$ + +Suppose $X$ is independent of $B$ , then as $\mathbb{E}[dB(\tau)] = 0$ , the second integral above is zero. Next, let's take the derivative with regards to $t$ from both sides: + +$$ +\frac {d}{d t} \mathbb {E} [ Q (X) ] = \mathbb {E} \left[ \left(M (t, X) Q ^ {\prime} (X) + 0. 5 N ^ {2} (t, X) Q ^ {\prime \prime} (X)\right) \right] \tag {35} +$$ + +To make the distribution of $X$ appear in the equations, let's use the definition of $\mathbb{E}[.]$ in the equation above: + +$$ +\int_ {- \infty} ^ {+ \infty} \frac {\partial}{\partial t} p (t, x) Q (x) = \int_ {- \infty} ^ {+ \infty} p (t, x) M (t, x) Q ^ {\prime} (x) d x + \int_ {- \infty} ^ {+ \infty} p (t, x) 0. 5 N ^ {2} (t, x) Q ^ {\prime \prime} (x) d x \tag {36} +$$ + +In the equation above, $p(t,x)$ is the distribution of stochastic process $X$ . By applying integration by part to the integrals in the Eq. 36, and re-arranging the terms, we get: + +$$ +\int_ {- \infty} ^ {+ \infty} \left(\frac {\partial}{\partial t} p (t, x) + \frac {\partial}{\partial x} (p (t, x) M (t, x)) - 0. 5 \frac {\partial^ {2}}{\partial x ^ {2}} (p (t, x) N ^ {2} (t, x))\right) Q (x) d x = 0 \tag {37} +$$ + +As the equation above holds for all $Q(x)$ , then the other factor must equal to zero. Thus, we have: + +$$ +\frac {\partial}{\partial t} p (t, x) + \frac {\partial}{\partial x} (p (t, x) M (t, x)) - 0. 5 \frac {\partial^ {2}}{\partial x ^ {2}} (p (t, x) N ^ {2} (t, x)) = 0 \tag {38} +$$ + +The equation above is known as Fokker-Planck equation. + +# C Well-behaved NGOS + +Noisy Gradient Oracle with Scale Parameter (NGOS) models stochastic gradients as $g(w) = \nabla \mathcal{L}(w) + \Theta z$ , where $\Theta$ is a noise scale parameter, and $z$ is a random variable that follows a distribution with covariance matrix $\Sigma(w)$ and a mean of zero. An NGOS is well-behaved if the conditions below are satisfied [37]: + +1. $\nabla \mathcal{L}(w)$ is Lipschitz. +2. Square root of covariance matrix is Lipschitz and bounded. +3. Both $\nabla \mathcal{L}(w)$ and $\Sigma (w)$ are differentiable and their partial derivatives up to 4-th order have polynomial growth. +4. Noise distribution in the NGOS must have low skewness. In other words, there must exist a function of polynomial growth such that $|\mathbb{E}[z^3]|$ is equal or less than that function divided by $\Theta$ . +5. Noise distribution must have bounded moments. In other words, there must exist a constant $\kappa$ such that $\mathbb{E}[||z||^{2k}]^{\frac{1}{2k}}\leq \kappa (1 + ||w||)$ + +Gaussian noise is a well-behaved NGOS. However, the noise in NGOS is not restricted to Gaussian. Particularly, if the noise distribution is not heavy-tailed, then it satisfies requirements 4 and 5. Whether or not the noise distribution in real-world applications is heavy-tailed is still an open problem. Nevertheless, several experimental results denote that the conditions described above are not too strong [37]. + +# D Approximation Quality of SDE for SGD + +The dynamics of SGD can be approximated by the following SDE [25]: + +$$ +d W _ {t} = - \nabla \mathcal {L} \left(W _ {t}\right) d t + \left(\eta \Sigma \left(W _ {t}\right)\right) ^ {1 / 2} d B (t), \tag {39} +$$ + +where $W_{t}$ represents an approximation of $w_{k}$ at discrete time steps $k \eta$ and $\Sigma$ is the covariance matrix of the underlying NGOS. The noises that make the individual paths for the above SDE and the SGD are independent processes. Hence, SGD and its SDE approximation do not share the same sample paths; rather, they are close to each other in "distribution" or "weak sense" [25]. To formalize this mathematically, suppose $\Xi$ be the set of functions, $f(x)$ , of polynomial growth, i.e. $\exists a, b \in \mathbb{R}^{+}$ such that $|f(x)| < a(1 + |x|^{b})$ . Then, SDE defined in Eq. 39 is the order $r$ weak approximation of SGD if for each function $f(x)$ in $\Xi$ , there exists a constant $C$ such that for all $k = 1 \dots N$ , $|\mathbb{E}[f(W_{k\eta})] - \mathbb{E}[f(w_k)]| < C\eta^r$ . This is a well-known result in analyzing optimization algorithms using SDEs, first time proposed in [25]. Similar theorems for RMSProp and Adam are available in [37]. + +# E Proof of Lemma 1 + +The demographic parity criterion for fairness between two subgroups is defined as: + +$$ +P (\hat {y} | z = 0) = P (\hat {y} | z = 1), \tag {40} +$$ + +where $z$ denotes the subgroup, and $\hat{y}$ represents the output of a model. The gap in demographic parity serves as a fairness measure [38]. A common approach to quantifying this gap is through the zero-one loss difference [24]: $|[ \ell_{0/1}(w) | z = 0 ] - [ \ell_{0/1}(w) | z = 1 ]|$ . As established in prior work [24], the zero-one loss can be approximated by the training loss, leading to the following fairness measure $\mathcal{F}(w) = |[\mathcal{L}(w) | z = 0 ] - [\mathcal{L}(w) | z = 1 ]|$ . Substituting the given loss functions $\mathcal{L}_0(w)$ and $\mathcal{L}_1(w)$ , we obtain $\mathcal{F}(w) = 2 | w|$ . Since $\mathcal{F}(w)$ achieves its minimum at $w = 0$ , this proves that $w = 0$ is the fairest minimizer. + +# F Additional Results for Theorem 1 + +In this section, we extend our empirical analysis of Theorem 1 by evaluating our findings with broader choices of learning rates and alternative definitions of the fair neighborhood. + +Table 5: Hyperparameter configurations for the experimental scenarios. + +
ConfigurationηRMSPropηSGDFairness Threshold
10.010.10.2
20.10.20.2
30.010.10.1
40.010.10.4
50.10.20.1
60.10.20.4
+ +![](images/9826663baba023c98a462cbc3b84e3c59de243b9de92ce8478982d1294682d4e.jpg) +Figure 4: Convergence rates of RMSProp ( $\eta = 0.01$ ) and SGD ( $\eta = 0.1$ ) to the fair neighbourhood, defined by threshold of 0.2. + +![](images/bad5ec4b972ae7cef846fab301fbd770d23d096d8f104f860dfa5115aa13a5b7.jpg) + +Theorem 1 establishes that for subgroup sampling probabilities $p_0, p_1 \in (0,1)$ with $p_0 + p_1 = 1$ , and loss functions $\mathcal{L}_0(w) = \frac{1}{2} (w - 1)^2$ and $\mathcal{L}_1(w) = \frac{1}{2} (w + 1)^2$ , the optimization of the empirical loss function: + +$$ +\mathcal {L} _ {e m p} (w) = \frac {1}{N} \sum_ {r \in \Omega} \mathcal {L} _ {q _ {r}} (w), \tag {41} +$$ + +where each sample $q_{r} \in 0,1$ is drawn i.i.d. with probability $p_0$ for subgroup 0 or $p_1$ for subgroup 1, leads to an increased likelihood of RMSProp converging to the fair optimum $w_{pop}^{*} = 0$ as the imbalance between $p_0$ and $p_1$ grows. To empirically validate these theoretical insights, we conducted simulations using a fixed learning rate of $\eta = 0.1$ , training for 1000 independent runs per experiment. The fraction of runs converging to the neighborhood of the fair optimum, defined as $|w - w_{pop}^{*}| < 0.2$ , was recorded at each epoch. We call 0.2 the fairness threshold. To assess the robustness of our findings, we extend our analysis by examining a broader range of $\eta$ values and explore more restrictive and relaxed fairness thresholds. Table 5 summarizes the hyperparameter configurations used in our experiments. Fig.s 4 to 9 illustrate the convergence behavior of RMSProp and SGD under the specified configurations in the table. + +The figures reveal a consistent trend in the convergence behavior of RMSProp and SGD. In the severely biased setting $(p_0 = 0.1)$ , RMSProp exhibits a higher likelihood of converging to the fair neighborhood, regardless of the learning rate. As the bias decreases $(p_0 = 0.3)$ , the behavior of RMSProp and SGD becomes more similar. These experimental findings are in line with theoretical findings in Theorem 1. Additionally, a more restrictive fairness threshold reduces the probability of convergence to the fair neighborhood, whereas a relaxed threshold increases it. This is expected, as stricter definitions impose tighter constraints on acceptable fair solutions, while relaxed criteria naturally allow a higher fraction of runs to qualify as fair. + +# G Extension of Theorem 2 to Anisotropic Noise + +Theorem 4. Consider a population that consists of two subgroups with subgroup-specific loss functions $\mathcal{L}_0(w)$ and $\mathcal{L}_1(w)$ , sampled with probabilities $p_0$ and $p_1$ , respectively. Suppose a stochastic (online) training regime, in which each parameter update is computed from a sample drawn from one of the two subgroups. Suppose the gradients $\nabla \mathcal{L}_0(w_k)$ and $\nabla \mathcal{L}_1(w_k)$ are well-behaved anisotropic NGOs, namely $\mathcal{N}(\mu_0, \Sigma_0)$ and $\mathcal{N}(\mu_1, \Sigma_1)$ , respectively. Then, the difference in parameter updates + +![](images/e1dba32fff40160028282d755afe9e8e7a5f454edc7942415573fd7f253221fb.jpg) +Figure 5: Convergence rates of RMSProp ( $\eta = 0.1$ ) and SGD ( $\eta = 0.2$ ) to the fair neighbourhood, defined by threshold of 0.2. + +![](images/0b3baf5be95d4a86b5017973f1a33fe4c7e6f512dab0e4afcfdc9d2f2ff77759.jpg) + +![](images/4c78b12eadbd4065005f458d078ad1120616f48bea1a35f4cf27136e519d60f4.jpg) +Figure 6: Convergence rates of RMSProp ( $\eta = 0.01$ ) and SGD ( $\eta = 0.1$ ) to the fair neighbourhood, defined by threshold of 0.1. + +![](images/550790bfa85f6281971808b809c9f123b40119254f23a15e95797e158c81cbfe.jpg) + +between subgroups 0 and 1 under RMSProp has an upper bound given by the corresponding difference under SGD. Consequently, RMSProp offers fairer updates across subgroups. + +Proof. The proof is similar to Theorem 2. For SGD, the difference between the parameter updates across subgroups is $||\nabla \mathcal{L}_0 - \nabla \mathcal{L}_1||$ . Regarding RMSProp, let's compute the expected value and the variance of $v_{k}$ : + +$$ +\mathbb {E} \left[ v _ {k} \right] = \gamma^ {k} v _ {0} + \left(1 - \gamma^ {k}\right) \left[ p _ {0} \left(\mu_ {0} ^ {2} + \operatorname {d i a g} \left(\Sigma_ {0}\right)\right) + p _ {1} \left(\mu_ {1} ^ {2} + \operatorname {d i a g} \left(\Sigma_ {1}\right)\right) \right]. \tag {42} +$$ + +Where $\text{diag}(\Sigma_0)$ and $\text{diag}(\Sigma_1)$ are the vectors with the diagonal elements of $\Sigma_0$ and $\Sigma_1$ , respectively. Similar to the isotropic case, we can show that the variance of $v_k$ is negligible and thus $v_k \to E[v_k]$ . Hence, the difference between parameter updates are $||D(\nabla \mathcal{L}_0 - \nabla \mathcal{L}_1)||$ , where $D$ is a diagonal matrix, whose $jj$ th element can be computed as: + +$$ +D _ {j j} = \frac {1}{\sqrt {p _ {0} \left(\mu_ {0 _ {j}} ^ {2} + \sigma_ {0 _ {j}} ^ {2}\right) + p _ {1} \left(\mu_ {1 _ {j}} ^ {2} + \sigma_ {1 _ {j}} ^ {2}\right) + \epsilon}}. \tag {43} +$$ + +Prior work [37] has experimentally shown that $\sigma^2 >\mu^2$ . Additionally, in stochastic (online) training setup: $\Theta^2\to 1$ . Thus, $D_{jj} < 1$ . This completes the proof. + +# H Proof of Theorem 3 + +Proof. As described in the proof of Lemma 1, the following can be used to approximate the gap in demographic parity: $\mathcal{F}(w) = |[\mathcal{L}(w)|z = 0] - [\mathcal{L}(w)|z = 1]$ . Let's define the function $\Psi(w_k) = \mathcal{L}_0(w_k) - \mathcal{L}_1(w_k)$ . Also, define $\varphi(w_k) = \frac{L_0(w_k) - L_1(w_k)}{|L_0(w_k) - L_1(w_k)|}$ . Let's expand $\mathcal{F}(w_k)$ around $w_k$ using Taylor series: + +$$ +\mathcal {F} \left(w _ {k} + \delta\right) - \mathcal {F} \left(w _ {k}\right) = \nabla \mathcal {F} \left(w _ {k}\right) ^ {t} \delta . \tag {44} +$$ + +We can compute $\nabla \mathcal{F}(w_k)$ as: + +$$ +\nabla \mathcal {F} \left(w _ {k}\right) = \varphi \left(w _ {k}\right) \cdot \left(\nabla \Psi \left(w _ {k}\right)\right). \tag {45} +$$ + +![](images/e4d50ce6152c0e6881dee9cbdf2ea4375d668b2e2e0f3e0bcaa1c6fc49d8489b.jpg) +Figure 7: Convergence rates of RMSProp ( $\eta = 0.01$ ) and SGD ( $\eta = 0.1$ ) to the fair neighbourhood, defined by threshold of 0.4. + +![](images/5fab86c7afff257af2a4666d2f0950c1e2a8a37d2606c833e6cdf8d851cd4257.jpg) + +![](images/37dfea4fb319842d0e24a7d75f1eb97fa56cc6eaf05ee713ed4ecb59d7d2eed5.jpg) +Figure 8: Convergence rates of RMSProp ( $\eta = 0.1$ ) and SGD ( $\eta = 0.2$ ) to the fair neighbourhood, defined by threshold of 0.1. + +![](images/9e65744401ca4d94a9c241d01e01bfb0fef0026bba858ec835948981e335a71f.jpg) + +If in Eq. 44 we set $\delta - \eta \nabla \mathcal{L}(w_k)$ , given the updates of SGD in Eq. 1, we can re-write Eq. 44 as: + +$$ +\mathcal {F} \left(w _ {k + 1 _ {s g d}}\right) - \mathcal {F} \left(w _ {k}\right) = - \eta \varphi \left(w _ {k}\right) \nabla \Psi \left(w _ {k}\right) ^ {t} \nabla \mathcal {L}, \tag {46} +$$ + +where $\mathcal{F}(w_{k + 1_{sgd}})$ is the value of function $\mathcal{F}()$ after one iteration using SGD. Using the proof of Theorem 2, we can perform a similar operation for RMSProp: + +$$ +\mathcal {F} \left(w _ {k + 1 _ {r m s}}\right) - \mathcal {F} \left(w _ {k}\right) = - \eta \varphi \left(w _ {k}\right) \nabla \Psi \left(w _ {k}\right) ^ {t} D \nabla \mathcal {L}. \tag {47} +$$ + +In the equation above, $\mathcal{F}(w_{k + 1_{r_m s}})$ is the value of $\mathcal{F}()$ after one iteration using RMSProp, and matrix $D$ is a diagonal matrix defined in the proof of Theorem 2. As $\mu_0 < \mu_1$ , then the worst case increase in the gap of demographic parity for SGD occurs when $\nabla \Psi (w_k)$ aligns with $\nabla \mathcal{L}$ . Mathematically, the increase can be modeled as: + +$$ +\left| \mathcal {F} \left(w _ {k + 1 _ {s g d}}\right) - \mathcal {F} \left(w _ {k}\right) \right| = \eta \left| \varphi \left(w _ {k}\right) \nabla \Psi \left(w _ {k}\right) ^ {t} \nabla \mathcal {L} \right| \leq \eta \| \nabla \Psi \left(w _ {k}\right) \| \cdot \| \nabla \mathcal {L} \|. \tag {48} +$$ + +Similarity, for RMSProp, we have: + +$$ +\begin{array}{l} \left| \mathcal {F} \left(w _ {k + 1 _ {r m s}}\right) - \mathcal {F} \left(w _ {k}\right) \right| = \eta \left| \varphi \left(w _ {k}\right) \nabla \Psi \left(w _ {k}\right) ^ {t} D \nabla \mathcal {L} \right| \leq \eta \| D \nabla \Psi \left(w _ {k}\right) \| \cdot \| \nabla \mathcal {L} \| \tag {49} \\ < \eta \| \nabla \Psi (w _ {k}) \| \cdot \| \nabla \mathcal {L} \|. \\ \end{array} +$$ + +In the equation above, as diagonal elements of $D$ are strictly less than 1, we have a strict inequality. Comparing Eqs. 48 and 49, we conclude the proof. + +# I Extension of Theorem 3 to Anisotropic Noise + +Theorem 5. Consider a population that consists of two subgroups with subgroup-specific loss functions $\mathcal{L}_0(w)$ and $\mathcal{L}_1(w)$ , sampled with probabilities $p_0$ and $p_1$ , respectively. Suppose a stochastic (online) training regime in which each parameter update is computed from a sample drawn from one of the two subgroups. Suppose that the gradients $\nabla \mathcal{L}_0(w_k)$ and $\nabla \mathcal{L}_1(w_k)$ are well-behaved anisotropic NGOs, namely $\mathcal{N}(\mu_0, \Sigma_0)$ and $\mathcal{N}(\mu_1, \Sigma_1)$ , respectively, with $\mu_1 > \mu_0$ . Then, in expectation, the worst-case increase in the demographic parity gap after one iteration of RMSProp has an upper-bound no greater than the corresponding increase under SGD. + +![](images/5458d45fc31af637d91886b64b24726a2efbfee4cffb714339f3da3542bc6d9d.jpg) +Figure 9: Convergence rates of RMSProp ( $\eta = 0.1$ ) and SGD ( $\eta = 0.2$ ) to the fair neighbourhood, defined by threshold of 0.4. + +![](images/1d1f750b6b982b960d5d791a2d2778e66714e2eaa7d8dc3107ea076d778c9650.jpg) + +Proof. The proof follows directly as the proof of this Theorem relies on the conclusion of the extension of the Theorem 2, which is already proved in Theorem 4. $\square$ + +# J Datasets + +CelebA [31] is a large-scale and real-world dataset, which include 202,599 facial images from over 10,000 individuals. Each image is annotated with 40 distinct attributes, and the dataset is officially partitioned into training, validation, and test sets. In this study, we use gender (male vs. female) and age (young vs. old) as sensitive attributes, while performing facial expression recognition based on the "happy" attribute. Notably, the dataset exhibits an imbalance in demographic distribution, with the probability of an image depicting a male being 0.42 and the probability of depicting a young individual being 0.77. This skewed distribution indicates the presence of bias in the training set. + +FairFace [16] is a large-scale face attribute dataset, mainly designed to mitigate racial bias in facial analysis tasks. It includes 108,501 facial images sourced primarily from the YFCC-100M Flickr dataset [53]. Each image is annotated for gender, age (9 sub-groups), and race (7 sub-groups). In this study, we leverage FairFace for gender classification, treating age and race as sensitive attributes. Given its broad demographic representation, FairFace serves as a robust benchmark in fairness related tasks. + +MS-COCO [27] is a large-scale, real-world benchmark dataset widely used for multi-label classification, object detection, and image captioning. It consists of over 330,000 images, including more than 200,000 labeled images spanning 80 object categories. In this study, we leverage MS-COCO for multi-label classification. Notably, the dataset does not provide explicit gender annotations. To infer gender labels, we follow a protocol established in previous works [57], utilizing the five associated captions per image. Specifically, we retain images where at least one caption contains either the term "man" or "woman". Additionally, we remove object categories that are not strongly associated with humans. Human-associated objects are defined as those appearing at least 100 times alongside either "man" or "woman" in the captions, resulting in a reduced set of 75 object categories. Fig. 10 provides sample captions and their corresponding inferred gender labels. It is worth mentioning that the probability of an image depicting a male is 0.71. + +# K Training Protocol + +For facial expression recognition, we employ RetinaFace [8] to detect and crop faces from images. The images in the FairFace dataset are already cropped by default. As part of the preprocessing pipeline for all tasks, we normalize the images and resize them to $224 \times 224$ . Data augmentation consists of random horizontal flipping and random rotation. Facial expression recognition and gender classification tasks are trained using the cross-entropy loss $\mathcal{L}_{ce}$ , defined as: + +$$ +\mathcal {L} _ {c e} = - \frac {1}{N} \sum_ {j = 1} ^ {N} \left[ y _ {j} \log \left(\hat {y} _ {j}\right) + \left(1 - y _ {j}\right) \log \left(1 - \hat {y} _ {j}\right) \right]. \tag {50} +$$ + +![](images/fcb097195d9eb89027e525077bb2ed38d6b1074f93e49f1c87afa45915270a5f.jpg) +Figure 10: Two example images from MS-COCO dataset along with their captions. The inferred gender labels from the captions for the left and right images are "woman" and "man", respectively. + +(a) 1. a woman prepares several sub sandwiches at a deli counter. + +2. a person behind a display glass preparing food +3. a lady behind a sneeze guard making sub sandwiches. +4. a woman behind a deli counter making sub sandwiches. +5. there is a woman that is making sandwiches at a deli + +![](images/a6e89a1a3aeac796690e42e442b838700a37d330500e3e7b8e42e37d2858c0ce.jpg) + +(b) 1. a man in a yellow shirt holding a white dog and making a face +2. a man holding a hot dog bun beside his puppy. +3. a man putting a hot dog bun on a puppy and pretending to eat +4. a man holding a white puppy and a red leash. +5. a man who is pretending to bite a puppy. + +For the multi-label classification task, given the class imbalance in MS-COCO, we employ the focal loss [28] to mitigate overfitting to majority classes. It is formulated as: + +$$ +\mathcal {L} _ {\text {f o c a l}} = - \frac {1}{N} \sum_ {j = 1} ^ {N} \left[ \xi y _ {j} \left(1 - \hat {y} _ {j}\right) ^ {v} \log \left(\hat {y} _ {j}\right) + (1 - \xi) \left(1 - y _ {j}\right) \hat {y} _ {j} ^ {v} \log \left(1 - \hat {y} _ {j}\right) \right], \tag {51} +$$ + +where $\xi$ is the balancing factor and $v$ is the focusing parameter. To optimize the learning rate during training, we leverage Bayesian optimization [50] combined with the Hyperband algorithm [23]. The best-performing models are trained twice, and we report the average performance across these runs. + +We experiment with multiple backbone architectures, including ResNet-18 and ResNet-50 [11], VGG-16 [49], and a recently proposed vision transformer (tiny ViT)[54]. These architectures have been widely used in studies analyzing the impact of optimization methods on different learning paradigms[34], as well as in fairness-focused works [55]. ResNet models leverage residual connections to facilitate deep representation learning, while VGG-16 provides a structured CNN design with uniform filter sizes. The tiny ViT, a transformer-based model, processes images as patch embeddings, and showed state-of-the-art performance [54]. This diverse selection allows us to investigate fairness effects across both convolutional and transformer-based architectures. + +# L Hyperparameters + +We employ Bayesian Optimization [50] with Hyperband [23] algorithm to systematically tune the learning rate across different optimizers. Hyperband efficiently allocates training resources to more promising setups and progressively eliminate underperforming ones. It evaluates training at specific checkpoints, called "brackets". We set Hyperband brackets to [10, 20, 40], allowing enough time for each run to demonstrate its potential. Additionally, the learning rate is sampled using a log-uniform distribution within the following search spaces: + +- SGD: $[10^{-3}, 10^{-1}]$ +- RMSProp: $[10^{-5}, 10^{-2}]$ +Adam: $[10^{-5}, 10^{-2}]$ + +All models are trained for maximum of 60 epochs with a batch size of 64 and weight decay of $10^{-4}$ . We use Pytorch default values for optimizer specific hyperparameters: + +![](images/ad02140f2ffdebf5e3b5b532b3c7a74d28b0da95bcc3093c567e93d4327c8d32.jpg) +Figure 11: Fairness metrics for ResNet-18 across different datasets and sensitive attributes. + +![](images/4ecd499b691fae5b2d074816b6b9935f4725f4c3c845d338c560283b76e855a8.jpg) + +![](images/5b25f1f7ed5bcb218a09fec7dccf74adf89087bfb2ac7d625ff158db3e6b3460.jpg) + +- RMSProp: $\gamma = 0.9$ +- Adam: $\beta_{1} = 0.9$ , $\beta_{2} = 0.999$ + +Additionally, we use "Reduce-on-Plateau" scheduler of Pytorch to decrease the learning rate during the training. if no improvement is observed for 5 epochs, then the learning rate is multiplied by 0.5. + +# M Additional Fairness Results across Different Backbones and Datasets + +In this section, we present fairness metrics for SGD, RMSProp, and Adam across ResNet-18, ResNet-50, and VGG-16 backbones, evaluated on multiple datasets with different sensitive attributes. + +ResNet-18: Fig. 11 illustrates fairness metrics for the ResNet-18 backbone. Considering $F_{EOD}$ , both Adam and RMSProp consistently outperform SGD across all datasets and sensitive attributes. The only exception is the MS-COCO dataset, where both RMSProp and SGD yield an $F_{EOD}$ of zero. This is due to MS-COCO's 75 object classes, some of which are highly underrepresented, leading to an effective fairness score of zero. The largest difference in $F_{EOD}$ is observed on the CelebA dataset with age as the sensitive attribute, where RMSProp significantly surpasses SGD. Adam and RMSProp exhibit comparable performance, as expected. + +For the $F_{EOP}$ metric, both RMSProp and Adam outperform SGD across all datasets. For example, in the FairFace dataset with age as the sensitive attribute, Adam achieves an $F_{EOP}$ of $80\%$ , compared to $72\%$ with SGD. Regarding $F_{DPA}$ , Adam surpasses SGD in four out of five cases. The only exception is CelebA with age as the sensitive attribute, where both optimizers yield similar $F_{DPA}$ . This aligns with Theorem 1, as CelebA is not sufficiently biased for adaptive gradient methods to demonstrate a significant fairness advantage. + +ResNet-50: Fig. 12 presents results for the ResNet-50 backbone. For $F_{EOD}$ , Adam consistently outperforms SGD, while RMSProp surpasses SGD in four out of five cases, with MS-COCO again being the exception. The difference between Adam and SGD reaches approximately $15\%$ in CelebA when using age as the sensitive attribute. Similar trends emerge for $F_{EOP}$ , where both Adam and RMSProp outperform SGD across all datasets. Finally, for $F_{DPA}$ , more biased datasets, such as FairFace with age as the sensitive attribute, show a higher gap between adaptive gradient methods and SGD. + +VGG-16: Fig. 13 shows fairness metrics for the VGG-16 backbone. As with the previous architectures, Adam outperforms SGD in all cases for $F_{EOD}$ . For $F_{EOP}$ , both Adam and RMSProp demonstrate superior fairness compared to SGD. In FairFace with age as the sensitive attribute, the difference between Adam and SGD reaches approximately $10\%$ . For $F_{DPA}$ , the most significant difference is again observed in FairFace with age as the sensitive attribute, which reinforces our theoretical findings. + +Across all three fairness metrics, we observe consistent trends across multiple architectures, datasets, and problems, including gender classification, facial expression recognition, and multi-label classification. These extensive experiments provide strong empirical evidence that adaptive gradient methods, such as Adam and RMSProp, converge to fairer minima with higher probability compared to SGD. + +![](images/dc6d15626e71277632024a60a1c47161537f861c4cbf1bec5ef4dd8938a45c70.jpg) +Figure 12: Fairness metrics for ResNet-50 across different datasets and sensitive attributes. + +![](images/2993f70a2db67a0e98c371cf061d8705e66c7bdfac5e8644c9ab82fa8dc8480a.jpg) + +![](images/b639b71c933b8f53d3b506c435eef5924eaf7d02e5b21c328026edad9ef668cc.jpg) + +![](images/6dfb68fa4ee52593d99451a165ae6c3dd97c79ea1e491e0e00c628dff26c7820.jpg) +Figure 13: Fairness metrics for VGG-16 across different datasets and sensitive attributes. + +![](images/a26d6cb0b0a20963b6c034af5cbf01cb3cdc5493ab9888ab2d4ff194e52280bc.jpg) + +![](images/a3fc8c079fddef6489410997e0e8fc5f2db768bbddb7af1075902a5af271a925.jpg) + +# N Additional Fairness Results across Different Variants of Optimizers + +Our theoretical analysis identifies gradient normalization via second moment as the key mechanism for improving group fairness. This provides a clear framework for predicting the behavior of other optimizers with regards to fairness. To validate these predictions, we have conducted a new set of experiments with SGD w/ momentum [30], AdamW [32], and AdaBound [33] variants. Following the protocol from our main experiments, we use the CelebA dataset with a ViT backbone, treating gender and age as sensitive attributes. We tuned the learning rate for each optimizer using Bayesian optimization with Hyperband and report the average fairness scores in Table 6. As shown in the table, AdamW consistently achieves the best fairness scores. This is expected, as AdamW retains the normalization mechanism of Adam and RMSProp via the second moment. In contrast, SGD w/momentum, lacking the normalization via the second moment, performs more biased. The performance of AdaBound is comparable to SGD w/momentum. The reason is that the AdaBound algorithm deliberately clips the second moments to make it act like SGD, which in turn limits the fairness advantage. + +# O Additional Classification Results across Different Backbones and Datasets + +Table 7 reports the accuracy and F1 scores of SGD, RMSProp, and Adam optimizers across ResNet-18, ResNet-50, and ViT backbones. We evaluate these optimizers on CelebA (facial expression recognition), MS-COCO (multi-label classification), and FairFace (gender classification) datasets. For CelebA, the performance of SGD, RMSProp, and Adam is comparable, with only minor variations. For instance, using the VGG-16 backbone, SGD achieves an accuracy of $92.22\%$ , while RMSProp and Adam yield $92.37\%$ and $91.42\%$ , respectively. Similarly, for MS-COCO and FairFace, the performance differences across the optimizers remain limited. The largest accuracy gap between SGD + +Table 6: Fairness comparison of optimizer variants. + +
GenderAge
AdamWAdaBoundSGD w/ mom.AdamWAdaBoundSGD w/ mom.
FEOD64.9061.1061.1271.1167.2067.25
FEOP99.9096.1095.0798.9596.6096.55
FDPA73.1170.0169.5098.1195.3995.39
+ +Table 7: Performance of different backbones across datasets. Each cell reports three numbers corresponding to SGD, RMSProp, and Adam, respectively. + +
DatasetBackbones
ResNet18ResNet50VGG16
Acc.F1Acc.F1Acc.F1
CelebA91.12, 92.21, 92.1991.41, 92.76, 92.8091.19, 92.60, 92.3991.33, 92.09, 92.2592.22, 92.37, 91.4292.20, 92.28, 92.50
MS-COCO88.31, 91.10, 90.4266.96, 71.26, 69.3789.15, 91.89, 91.7067.68, 70.48, 70.1689.48, 90.11, 91.5766.47, 72.04, 73.20
FairFace86.16, 90.36, 90.3187.15, 88.71, 90.3088.48, 90.58, 92.4189.67, 90.77, 91.5791.70, 91.81, 92.0990.59, 91.66, 92.69
+ +and adaptive optimizers occurs on MS-COCO with the VGG-16 backbone, reaching approximately $5.5\%$ . However, the highest disparity in fairness metrics does not occur on MS-COCO but rather on FairFace dataset. These findings indicate that the differences between SGD and adaptive optimization algorithms cannot be solely attributed to their performance. Instead, as demonstrated in Theorems 1-3, the normalization behavior of these optimizers plays a crucial role in shaping their fairness properties. + +# NeurIPS Paper Checklist + +# 1. 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$^{3,4,*}$ + +Hangting Chen + +Wei Tan² + +Jianwei Yu + +Haizhou Li $^{1,4,*}$ + +1The Chinese University of Hong Kong, Shenzhen 2Tencent AI Lab + +$^{3}$ Nanjing University $^{4}$ Shenzhen Research Institution of Big Data + +# Abstract + +Generating music with coherent structure, harmonious instrumental and vocal elements remains a significant challenge in song generation. Existing language models and diffusion-based methods often struggle to balance global coherence with local fidelity, resulting in outputs that lack musicality or suffer from incoherent progression and mismatched lyrics. This paper introduces SongBloom1, a novel framework for full-length song generation that leverages an interleaved paradigm of autoregressive sketching and diffusion-based refinement. SongBloom employs an autoregressive diffusion model that combines the high fidelity of diffusion models with the scalability of language models. Specifically, it gradually extends a musical sketch from short to long and refines the details from coarse to fine-grained. The interleaved generation paradigm effectively integrates prior semantic and acoustic context to guide the generation process. Experimental results demonstrate that SongBloom outperforms existing methods across both subjective and objective metrics and achieves performance comparable to the state-of-the-art commercial music generation platforms. Audio samples are available on our demo page: https://cypress-yang.github.io/SongBloom_demo. + +# 1 Introduction + +Music, as one of the most expressive forms of human art, serves as a universal language that transcends cultural and linguistic boundaries. Lyric-to-song generation, which involves producing both vocals and accompaniment from given lyrics, requires comprehensive modeling of diverse musical elements, including instrumentation, vocal expressiveness, structural arrangement, and emotional dynamics. These complexities pose significant challenges for existing generative approaches [1, 2]. In particular, end-to-end models that directly synthesize music face considerable challenges in maintaining sound quality, as music's wide frequency range and complex temporal dynamics impose stringent demands on generative fidelity and model capacity. + +Advanced approaches to long-form generation typically adopt either a unified non-autoregressive (NAR) architecture [3] that models the entire process with a diffusion transformer, or an autoregressive (AR) framework [4, 5] that employs language models (LMs) and separates the generation into semantic and acoustic stages. However, NAR models often struggle to capture precise alignments between phonemes and audio frames, whereas AR approaches typically predict quantized tokens that suffer from low bitrates, resulting in weak musicality and degraded audio quality. + +
SystemArchitectureSizeMax LengthSample Rate (kHz)Structure
SongGen [9]LM1.1B30s16×
SongEditor [5]LM + Diff.0.7B + 1B120s44.1✓
DiffRhythm-base [3]Diff.1.1B95s44.1×
DiffRhythm-full [3]Diff.1.1B285s44.1×
YuE [4]LM + LM7B + 1B300s44.1✓
SongBloom-tinyLM & Diff.1.3B60s48✓
SongBloom-fullLM & Diff.2B150s48✓
+ +Table 1: Comparison of existing song generation models. “+” indicates two cascaded models, while “&” indicates one unified model with jointly optimized objectives. + +To address these challenges, we propose SongBloom, a novel lyric-to-song generation approach that combines the strengths of both AR and NAR approaches. SongBloom is built upon an autoregressive diffusion architecture [6], which is well-suited for continuous-valued modalities such as music and speech, and has shown strong scalability and robust long-range generation capabilities [7] comparable to that of audio language models. Given structured lyrics and a 10-second reference audio clip, SongBloom can generate full-length songs with diverse sections up to 150 seconds long. + +To further improve the semantic alignment in long-form song generation, we decouple high-level sketch planning from low-level acoustic synthesis, and integrate both within the unified autoregressive framework that supports joint modeling in the style of Chain-of-Thought (CoT) prompting [8]. Notably, we observe that the entire sketch sequence is not always necessary for predicting each acoustic frame. Instead, the acoustic context itself can provide valuable guidance for shaping subsequent sketch planning. Based on this insight, we partition both semantic and acoustic sequences into patches and generate them in an interleaved manner. This paradigm not only facilitates bidirectional contextual exchange between sketching and refinement stages but also significantly reduces the sequence length needed during acoustic synthesis. + +Both subjective and objective evaluations are conducted in our experiments. The assessment employs a comprehensive suite of metrics covering multiple dimensions, including musicality, audio quality, structural coherence, and overall musical aesthetics. The results show that SongBloom consistently outperforms all open-source baselines and surpasses most commercial systems, achieving state-of-the-art performance across several key metrics. + +The main contributions of this paper can be concluded as: + +- We propose SongBloom, the first autoregressive diffusion-based model for full-length song generation, which is capable of producing high-quality and expressive songs with coherent structure and rich acoustic details. +- We design a unified framework that combines coarse and fine stages into a single, jointly optimized model, and treats sketch tokens as CoT-like prompts for acoustic generation directly. Additionally, we introduce a novel interleaved generation paradigm that alternates between sketch and acoustic patches, effectively utilizing both semantic and acoustic contexts. +- Experiments demonstrate that SongBloom surpasses all non-commercial baselines and achieves competitive performance with the state-of-the-art Suno-v4.5 in terms of both subjective and objective metrics. Additionally, our approach achieves a relatively low real-time factor (RTF) compared to other LM-based methods, indicating a favorable trade-off between generation quality and computational efficiency. + +# 2 Related work + +# 2.1 Song Generation + +Song generation models aim to produce both singing voice and accompanying music. Previous studies, such as Melodist [10], SongGen [9], and MelodyLM [11], have shown promising results in generating sentence-level singing and accompaniment. However, generating long-form song pieces that encompass diverse musical structures, such as verses, choruses, and instrumental-only sections, + +remains a significant challenge. JukeBox [12] was one of the earliest attempts at generating long-form songs, though it suffers from limited genre and timbre control. SongComposer [13] supports long-form lyric-to-melody generation but struggles to preserve certain aspects such as timbre. + +Recent research has increasingly focused on long-form and structure-guided song generation. DiffRhythm [3] introduces a single diffusion model capable of generating songs up to 285 seconds in length while maintaining a low real-time factor. YuE [4] employs a coarse-to-fine generation strategy, leveraging two large-scale language models to produce acoustic tokens with improved fidelity and structure. SongEditor [5] proposes a two-stage framework that supports both full song generation and infilling-based editing tailored to specific lyrics, offering greater flexibility for user-driven customization. In addition, commercial platforms such as $\mathrm{Sun}^{2}$ and $\mathrm{Ud}^{3}$ have demonstrated promising capabilities in complete song generation. However, due to the lack of publicly available technical details, these systems cannot be thoroughly evaluated or compared. + +# 2.2 Autoregressive Diffusion Models + +While language models predominantly operate in discrete-valued space, this paradigm might be suboptimal for inherently continuous audio signals. A common workaround involves transforming waveforms into discrete token sequences through vector quantization (VQ)[14, 15] or residual vector quantization (RVQ)[16]. Recent work, however, challenges the necessity of discretization. For instance, Li et al. [6] demonstrates that discrete tokens may not be essential for autoregressive language models in image generation. Some research has also explored autoregressive diffusion techniques for text [17] and speech [18] generation. ARDiT [19] introduces a decoder-only diffusion transformer for zero-shot text-to-speech generation, achieving impressive results without relying on discrete tokens. DiTAR [7] further simplifies the architecture through a divide-and-conquer strategy, making the model more adaptable for large-scale deployment. + +However, the approaches mentioned above remain restricted to the speech domain, and direct migration faces some limitations. First, songs are typically longer than spoken utterances, imposing stricter requirements on contextual consistency across extended durations. Second, the presence of musical accompaniment leads to a higher semantic density of each frame, making generation more difficult. Recent works [20, 21] introduce a shared transformer architecture capable of handling both discrete and continuous tokens across different modalities. Inspired by this design, we propose a unified autoregressive framework that integrates both sketch prediction and audio synthesis with distinct training objectives. + +# 3 Preliminary + +Next-Token Prediction Given a sequence of tokens $\mathbf{x} = (x_{1}, x_{2}, \ldots, x_{t - 1})$ , the objective is to model the conditional probability of the next token $P(x_{t} \mid x_{1}, x_{2}, \ldots, x_{t - 1})$ . For an entire sequence $(x_{1}, x_{2}, \ldots, x_{T})$ , the joint probability is factorized as: + +$$ +P \left(x _ {1}, x _ {2}, \dots , x _ {T} \mid C\right) = \prod_ {t = 1} ^ {T} P \left(x _ {t} \mid x _ {1}, x _ {2}, \dots , x _ {t - 1}; C\right), \tag {1} +$$ + +where $C$ represents the conditions. + +Rectified Flow Matching Rectified flow-matching [22] defines a linear interpolation between a source point $z_0$ and a target point $z_1$ over time $t \in [0,1]$ : + +$$ +z _ {t} = (1 - t) z _ {0} + t z _ {1} \tag {2} +$$ + +$$ +\mathrm {d} z _ {t} = \left(z _ {1} - z _ {0}\right) \mathrm {d} t = v \left(z _ {t}, t\right) \mathrm {d} t \tag {3} +$$ + +where $z_0$ denotes the original data and $z_1 \sim \mathcal{N}(0,1)$ is a random noise. $v(z_t,t)$ represents the velocity field guiding the transformation from $z_0$ to $z_1$ . The reverse process enables the generation by inverting this transformation: + +$$ +z _ {t - \Delta t} = z _ {t} - v \left(z _ {t}, t\right) \cdot \Delta t \tag {4} +$$ + +This formulation allows the model to sample from the data distribution by iteratively denoising from the noise distribution. + +![](images/49bb68850263bb26feca8cce5095d61e45404723588cb4d6b2e1abe07487c2ec.jpg) +Figure 1: Overall architecture of SongBloom. + +# 4 SongBloom + +# 4.1 Task Formulation + +MusicLM [23] proposes a two-stage framework that leverages semantic tokens, referred to as sketch, extracted from self-supervised learning (SSL) models such as MERT [24] as intermediate representations. In the first stage, an LM is employed to generate the sketch sequence from scratch, as its variable length and discrete nature make it well-suited for language modeling approaches, which have demonstrated strong scalability. In the second stage, either diffusion or LM subsequently generates acoustic tokens in a coarse-to-fine manner. The overall generation process can be formulated as: + +$$ +p \left(a _ {(0: T ]}, s _ {(0: T ]} \mid C\right) = p _ {\theta} \left(a _ {(0: T ]} \mid s _ {(0: T ]}, C\right) \cdot p _ {\phi} \left(s _ {(0: T ]} \mid C\right), \tag {5} +$$ + +where $s_i$ denotes the sketch token at frame $i$ , and $a_i$ denotes the acoustic token. $T$ is the total length of the frame sequence. The parameters $\phi$ and $\theta$ represent the models responsible for generating semantic and acoustic tokens, respectively. This two-stage architecture has been widely adopted in previous long-form music generation approaches such as [5, 4, 25]. + +However, the aforementioned formulation faces several limitations: 1) Future sketch tokens contribute little to the prediction of current acoustic tokens due to the strict token-level alignment between the two representations. As noted in Yang et al. [5], Xu et al. [26], using chunk-level semantic-to-acoustic reconstruction introduces minimal degradation in fluency, suggesting that generating the complete semantic sequence in advance may be unnecessary. 2) Given the rich and expressive nature of music, prior acoustic latents can provide valuable contextual cues for sketch token generation. However, existing approaches typically condition only on the semantic history, thereby overlooking potentially useful acoustic information. + +To address the aforementioned challenges, we propose a novel generation paradigm that interleaves the generation of sketch and acoustic sequences. Both types of tokens are first segmented into fixed-size patches. The modified generation process is formulated as: + +$$ +p \left(a _ {(0: T ]}, s _ {(0: T ]} \mid C\right) = \prod_ {i = 0} ^ {N} p _ {\theta} \left(s _ {(i P: (i + 1) P ]} \mid s _ {(0: i P ]}, a _ {(0: i P ]}, C\right) \cdot p _ {\phi} \left(a _ {(i P: (i + 1) P ]} \mid s _ {(0: (i + 1) P ]}, a _ {(0: i P ]}, C\right), \tag {6} +$$ + +where $P$ denotes the patch size, $N = \lceil T / P\rceil -1$ denotes the number of patches. $\theta$ and $\phi$ correspond to the parameters responsible for generating sketch tokens and acoustic features, respectively. Although a sequential dependency between the two stages still exists, the interleaved generation paradigm enables bidirectional information exchange between the semantic and acoustic representations. The two stages share a subset of model parameters and are jointly optimized within a unified framework, facilitating coherent and high-fidelity song generation. + +# 4.2 Data Representation + +Lyric preprocessing To incorporate structural information, we introduce two categories of structure flags into the lyric sequence. Vocal-based flags, which indicate the structure of vocal sections (e.g., verse, chorus), are prepended to the beginning of each paragraph. Accompaniment-based flags, which represent non-vocal regions such as intros and outros, are inserted between paragraphs and expanded proportionally according to their exact durations. All lyric text is normalized and then transformed into a phoneme sequence to serve as input for the subsequent sketch generation stage. + +Sketch tokens In this paper, we adopt the embeddings extracted from MuQ [27] as our sketch representation. A single vector quantization layer is attached to discretize the sketch embeddings into flattened tokens. Additionally, we have explored various sketch alternatives including musical signals, which are further discussed in Appendix A. + +Acoustic latents Due to the broader frequency spectrum of music compared to speech, modeling and reconstruction become significantly more challenging. Codec-based discretization methods [28, 29] inherently suffer from a trade-off between reconstruction fidelity and the depth of codebooks, often increasing generation complexity. To tackle this issue, we substitute the discrete acoustic tokens with continuous latents derived from an autoencoder, which compresses 2-channel $48\mathrm{kHz}$ music into continuous-valued acoustic latent sequences at a reduced frame rate. This continuous representation preserves high-frequency detail while simplifying the generation process, making it more suitable for high-fidelity music synthesis. + +# 4.3 Architecture + +# 4.3.1 Overview + +The overall model architecture is illustrated in Figure 1, which consists of three main modules: an autoregressive transformer decoder, a non-autoregressive diffusion transformer, and an acoustic encoder. Both semantic and acoustic sequences are first segmented into fixed-length patches and then generated in an interleaved manner, enabling the model to maintain consistency between high-level structure and fine-grained acoustic detail. + +# 4.3.2 Autoregressive Sketch Generation + +The autoregressive sketch generation stage is designed to generate both sketch tokens $p_{\theta}(s_{(iP:(i + 1)P]}\mid \cdot)$ and a corresponding hidden vector $p_{\theta}(h_i|\cdot)$ per patch, which serve as conditioning inputs for the downstream diffusion stage. The architecture is composed of a stack of transformer decoders with causal masks, enabling left-to-right autoregressive prediction over the sketch tokens. Conditions, including lyric text and style prompt, are prepended to the semantic stream. Each semantic patch is generated based on all previously generated sketch tokens, as well as the acoustic latent features from preceding patches. Specifically, the acoustic features of the current patch are compressed via an acoustic encoder and then inserted as the next token at the position of the hidden vector. This design eliminates the need for custom attention masks, thereby facilitating compatibility with acceleration techniques such as FlashAttention2 [30]. + +The training objective for the autoregressive sketch generation is formulated as a cross-entropy loss: + +$$ +\mathcal {L} _ {\mathrm {L M}} = - \frac {1}{N P} \sum_ {i = 0} ^ {N - 1} \sum_ {j = 1} ^ {P} \log p _ {\theta} \left(s _ {i P + j} \mid s _ {< i P + j}, a _ {< i P}, C\right), \tag {7} +$$ + +where $s_t$ denotes the sketch token at timestep $t$ . + +# 4.3.3 Non-Autoregressive Latent Diffusion + +The non-autoregressive diffusion module predicts acoustic latents within each patch in parallel using a full-attention DiT [31] architecture. The model is trained with the Rectified Flow-Matching (RFM) objective [22], which aims to predict the velocity field $v_{\phi}(\cdot)$ governing the dynamics of the latent trajectories: + +$$ +\frac {\mathrm {d} \mathbf {z} _ {t} ^ {i}}{\mathrm {d} t} = v _ {\phi} \left(t, \mathbf {z} _ {t} ^ {i} \mid h _ {i}, s _ {(i P: (i + 1) P ]}, \mathbf {z} _ {0} ^ {i - 1}\right). \tag {8} +$$ + +Here, $t$ denotes the diffusion time step. In our experiments, we sample $t$ from a logit-normal distribution $\pi_t$ following Esser et al. [32]. $\mathbf{z}_t^i$ represents the latent variables at time $t$ for the $i$ -th patch, where $\mathbf{z}_0^i = a_{(iP:(i + 1)P]}$ and $\mathbf{z}_1^i\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ . The variable $h_i$ is the hidden vector inherited from the first stage. $s_{(iP:(i + 1)P]}$ refers to the sketch tokens for the current patch, and $\mathbf{z}_0^{i - 1}$ refers to the acoustic latents from the preceding patch. + +The loss function for training is defined as: + +$$ +\mathcal {L} _ {\text {f l o w}} = \mathbb {E} _ {t \sim \pi_ {t}, \mathbf {z} ^ {i}} \left[ | | v _ {\phi} (t, \mathbf {z} _ {t} ^ {i} \mid \cdot) - (\mathbf {z} _ {1} ^ {i} - \mathbf {z} _ {0} ^ {i}) | | ^ {2} \right], \tag {9} +$$ + +where $\pi_t$ denotes the distribution over diffusion steps. The total training objective combines the sketch generation loss and the diffusion loss: + +$$ +\mathcal {L} = \mathcal {L} _ {\mathrm {L M}} + \lambda \cdot \mathcal {L} _ {\text {f l o w}}, \tag {10} +$$ + +where $\lambda = 0.1$ is an empirically chosen weighting factor. + +Gradients are backpropagated from the diffusion stage to the sketch generation stage via the hidden vector $h_i$ . Additionally, the sketch tokens share the same embedding layer across both stages, which improves training efficiency and accelerates convergence. + +Jia et al. [7] introduced a classifier-free guidance (CFG) mechanism for autoregressive diffusion models, which is applied only to the hidden vector. In our work, we treat the hidden vector and sketch tokens as a unified conditioning signal, and jointly mask them during training. Additional masks are applied solely to the sketch tokens to further encourage the hidden vector to carry more semantic information. + +# 5 Experimental Setup + +# 5.1 Dataset + +We use a large-scale song dataset totaling 100K hours, including both Chinese and English songs, to train our model. The original lyrics are of relatively low quality, prompting the use of a dedicated data cleaning pipeline. First, we separate the vocal and instrumental tracks using the Demucs model [33]. Then we use the WhisperX toolkit [34] to filter out mismatched lyric sentences and recover missing words. The time boundaries of lyrics are also refined based on the results of automatic speech recognition (ASR). Additionally, structural information is extracted from the original tracks using the structure analyzer proposed by Kim and Nam [35]. + +For inference, considering that most current song generation models are trained on large-scale but closed-source datasets, it is crucial to ensure that the test samples are not seen during training. To this end, the lyrics are generated using GPT-4o, and the reference audio is randomly clipped from unseen real songs with diverse genres, followed by random shuffling to prevent potential data leakage. A subset comprising 20 samples is used for subsequent human evaluation. + +# 5.2 Configuration & Training Setup + +In our experiments, the codebook size for semantic tokens is set to 16384, at a frame rate of 25. The stable-audio-vae [36] is adopted as the implementation of our waveform autoencoder. Minor modifications are made to its hyperparameters to ensure they have the same frame rate, thereby facilitating synchronized sketches and latents. + +The core component of SongBloom is based on the LLaMA-2 decoder architecture [37], utilizing causal attention as the backbone of the autoregressive LM. This architecture is further modified to support bidirectional attention, forming our diffusion transformer. Rotary Positional Embeddings (RoPE) [38] are employed in both autoregressive and non-autoregressive transformers to encode positional information. The acoustic encoder is a simple two-layer convolutional network. All conditioning inputs are prepended to the input sequence. Attention modules in each layer use 24 heads with a hidden dimension of 1536, consistent across both autoregressive and non-autoregressive components. The patch size is set to 16, spanning 0.64 seconds. We evaluate two model configurations in our experiments: (1) SongBloom-tiny, comprising 16 layers for the autoregressive LM and 8 layers for the non-autoregressive diffusion transformer, capable of generating songs up to 60 seconds in length; and (2) SongBloom-full, comprising 24 autoregressive layers and 12 non-autoregressive + +Table 2: Objective evaluation results across all models. The upper section reports the performance of commercial platforms (valid until June 23, 2025), while the lower section presents results for open-source baselines as well as our proposed models. + +
ModelsPromptPER(%)↓MCC↑FAD↓SER(%)↓Aesthetic ScoreRTF↓
CE↑CU↑PC↑PQ↑
Suno-v4.5 †text24.670.693.3910.437.777.936.038.40-
Udio-v1.510s wav20.040.794.0417.927.477.636.298.20-
Haimiantext10.030.635.4513.397.557.875.758.28-
Mureka-O110s wav7.790.863.3931.377.697.846.418.45-
ACE-step [46]text54.340.628.0212.277.377.526.267.85-
YuE-7B [4]text + 30s wav27.300.625.9913.937.257.595.968.0313.724
DiffRhythm-full [3]‡10s wav15.770.705.0446.625.817.294.527.730.034
SongEditor [5]10s wav16.200.774.8518.067.447.806.068.271.717
SongBloom-full10s wav6.750.883.4317.677.717.885.868.431.649
SongBloom-full-ft10s wav5.490.863.2014.507.797.965.888.471.649
+ +$\dagger$ DiffRhythm requires additional sentence-level timestamps. We first extract paragraph-level timestamps from ASR results of generated samples, then employ GPT-4o to predict the start time of each sentence. + +layers, enabling song generation up to 150 seconds. The former is used for analysis and ablation studies, while the latter is included to enable fair comparisons with other baselines. The model is trained with 16 A100 GPUs for approximately one week. + +All models are trained using the AdamW optimizer [39] with a learning rate of 1e-4. A cosine learning rate scheduler [40] with 2000 warm-up steps is employed to stabilize early training. Each model is trained for approximately 150K steps with a batch size of 128. The DeepSpeed strategy [41] is adopted to support efficient training. For inference, both stages share a classifier-free guidance coefficient of 1.5. Next-token prediction is performed using top-k sampling with $k = 200$ and a temperature of 0.9. The diffusion process employs the Euler ODE solver with 36 diffusion steps. + +# 5.3 Evaluation Metrics + +We evaluate the proposed models using the following objective metrics: (1) Phoneme Error Rate (PER), computed based on the separated vocal tracks and the corresponding lyrics; (2) MuLan Cycle Consistency (MCC), which measures the cosine similarity of MuLan [42] embeddings between generated samples and reference audio or textual descriptions; (3) Fréchet Audio Distance (FAD)[43], which quantifies the distributional similarity between generated samples and real songs from which the clips are intercepted; and (4) Structural Error Rate (SER), which measures the mismatch between detected structural patterns and the target lyric structure. The Dynamic Time Warping (DTW) algorithm [44] is first employed to obtain the optimal temporal alignment, and then we calculate the proportion of error duration. Additionally, we leverage the Audiobox-Aesthetic [45] to assess musical aesthetics, including content enjoyment (CE), content usefulness (CU), production complexity (PC), and production quality (PQ). + +For subjective evaluation, we conduct a Mean Opinion Score (MOS) listening test. A group of at least 10 participants with musical expertise are invited to rate each sample on a scale from 1 to 5. The evaluation focuses on several aspects: musicality (MUS) and audio quality (QLT) assessed separately for both the vocal and accompaniment components, correctness (CRR) of lyrics, and consistency (CST) between samples and prompts. + +# 6 Results + +# 6.1 Full-Length Lyric-to-Song Generation + +Based on the SongBloom-full model, we fine-tuned it for 1,000 additional steps on synthesized data with a clear alternating structure of verses and choruses, denoted as SongBloom-full-ft. This fine-tuning process enhances the model's ability to emulate synthesized lyric compositions, leading to improved performance on lyrics with similar stylistic patterns. + +Comparison of objective metrics As shown in Table 2, SongBloom demonstrates superior performance across various metrics and is competitive with Suno-v4.5, the state-of-the-art commercial song + +Table 3: Subjective evaluation results. Consistency (CST) is reported only for models that accept a 10-second reference audio as the style prompt. + +
ModelsMUSV↑MUSA↑QLTV↑QLTA↑CRR↑CST↑
Suno-v4.53.87 ± 0.284.04 ± 0.183.83 ± 0.123.96 ± 0.062.95 ± 0.14-
Udio-v1.53.28 ± 0.273.55 ± 0.253.62 ± 0.233.74 ± 0.212.57 ± 0.192.76 ± 0.31
Haimian3.39 ± 0.263.65 ± 0.243.86 ± 0.173.90 ± 0.113.09 ± 0.20-
Mureka-O13.91 ± 0.263.93 ± 0.143.85 ± 0.183.89 ± 0.123.38 ± 0.163.41 ± 0.39
ACE-step [46]2.95 ± 0.292.93 ± 0.322.66 ± 0.272.68 ± 0.252.11 ± 0.16-
YuE-7B [4]2.93 ± 0.273.15 ± 0.243.16 ± 0.273.28 ± 0.232.54 ± 0.19-
DiffRhythm-full [3]2.99 ± 0.273.48 ± 0.272.97 ± 0.213.33 ± 0.262.51 ± 0.232.45 ± 0.28
SongEditor [5]2.90 ± 0.323.11 ± 0.273.03 ± 0.323.24 ± 0.282.89 ± 0.162.44 ± 0.28
SongBloom-full3.59 ± 0.253.60 ± 0.233.83 ± 0.093.81 ± 0.083.27 ± 0.173.62 ± 0.27
SongBloom-full-ft3.91 ± 0.243.92 ± 0.123.95 ± 0.043.93 ± 0.103.42 ± 0.183.45 ± 0.31
+ +generation platform. After fine-tuning on downstream data, SongBloom-full-ft even outperforms Suno-v4.5 in several metrics. + +During our evaluation, we observed that Suno tends to follow rigid structural patterns, such as redundantly repeating the chorus at the end, which leads to structural hallucinations and degraded PER performance. In contrast, SongBloom adheres more faithfully to the structure of the input lyrics, enabling flexible and structurally coherent song generation that significantly reduces the PER. In terms of the MCC metric, waveform-based style prompts generally possess a natural advantage over text descriptions, as they always provide a comprehensive and unbiased representation of all musical components. Among these approaches, SongBloom-full achieves the highest MCC score. In terms of the automated-evaluated aesthetic scores, SongBloom-full-ft outperforms all other baselines in three out of the four metrics, further demonstrating its ability to generate high-quality, musically coherent, and aesthetically pleasing songs. + +Apart from generation performance, we also assessed the inference speed of SongBloom compared to other open-source models. The integrated design of SongBloom enables outstanding inference efficiency, yielding a lower RTF than all other autoregressive baselines. Compared to SongEditor, our model achieves a similar RTF despite having a larger size, since SongEditor takes the entire semantic sequence as input during the diffusion stage, leading to unnecessary computational overhead. Meanwhile, YuE utilizes two LMs and generates multi-layer codec tokens in a flattened pattern, significantly extending the generation sequence. While autoregressive models are naturally slower than their non-autoregressive counterparts, SongBloom achieves an excellent trade-off between efficiency and performance. + +Comparison of subjective metrics Table 3 shows that SongBloom and Suno-v4.5 dominate the human evaluation results, outperforming all other models by a clear margin. Suno-v4.5 demonstrates particular strength in accompaniment generation, whereas SongBloom excels in metrics related to vocal tracks. Notably, SongBloom achieves the highest correctness score, indicating a stronger adherence to the provided lyrics compared to other systems. This suggests that its interleaved generation paradigm effectively preserves semantic intent throughout the song. Furthermore, SongBloom's superior performance in consistency is corroborated by its high MCC score, reflecting its ability to maintain thematic coherence over every section of songs. + +# 6.2 Ablation Study on Diffusion Conditions + +Table 4 compares the performance of SongBloom-tiny under different combinations of diffusion conditions. In the absence of sketch tokens, the LM stage produces only a sequence of hidden vectors, and the diffusion stage reconstructs the audio solely based on them. As shown in the table, the sketch plays a critical role in generation. Without it as a coarse-grained CoT, the model fails to learn proper alignment between phoneme sequences and audio, resulting in extremely high PER. In contrast, when the sketch is provided, even without additional conditions, the diffusion transformer can generate intelligible vocals solely based on the hidden vector from the LM stage. Meanwhile, incorporating both the acoustic context and sketch tokens as input conditions during the diffusion stage further enhances both objective and subjective performance, leading to higher-quality generation results. + +Table 4: Ablation study of SongBloom-tiny under different diffusion conditions. "H" represents the hidden vector, "C" represents the acoustic context from the previous patch, and "S" represents the sketch tokens of the current patch. + +
w/ sketchConditionsObjectiveAesthetic ScoreSubjective
PER(%)↓FAD↓CE↑CU↑PC↑PQ↑MUS↑QLT↑
✓H+C+S9.445.607.577.666.008.253.553.77
H+C10.495.167.437.576.428.153.433.46
H11.357.587.467.505.808.093.143.41
XH+C109.768.867.067.405.927.71--
+ +![](images/11bbfdc094d4c0697eea181011801f4dc9e0b996044e657a6ec95c97a9995318.jpg) +Figure 2: Aesthetic scores and sketch token accuracy of SongBloom-tiny with various patch sizes. + +![](images/bf9840de88e8153ea80be2830af3cefbe4265317d33d40d195086f3870944ade.jpg) +Figure 3: Aesthetic scores of SongBloom-tiny with increasing diffusion steps. + +# 6.3 Effect of Hyper-Parameters + +Figure 2 illustrates the impact of various patch sizes, where $"+"\infty$ denotes separating the generation of sketch and acoustic sequence. For speech-oriented autoregressive diffusion models [7, 19], a small patch size is essential, as larger patch sizes make the model difficult to converge. For SongBloom, since the sketch tokens have served as the CoT of hidden vectors and guide the diffusion directly, a small patch size is no longer the optimal choice. Although smaller patch sizes provide more acoustic information for sketch generation, thereby improving sketch token accuracy, they also hinder the fluency of acoustic latents during the diffusion stage, as the window for preceding contexts becomes smaller. Figure 3 illustrates the relationship between performance and diffusion steps during inference. The RTF increases proportionally with the number of diffusion steps, while near-optimal generation performance is achieved in as few as 10 steps, indicating that the inference process can be further accelerated. + +# 7 Discussions and Limitations + +In this paper, we propose SongBloom, a novel approach for full-song generation that produces expressive, high-quality songs and achieves state-of-the-art performance across multiple metrics. Nevertheless, several open challenges remain. First, the current sketch representation is derived from SSL models, which lack interpretability. Replacing these with some symbolic formats could enable more fine-grained control and user customization, which will be our future work. Second, we believe that some reinforcement learning-based techniques, such as DPO [47] or PPO [48], can be applied to SongBloom, which aligns the generation process with user preferences, thereby enabling outputs that better match human aesthetic judgments. + +We fully acknowledge the potential ethical risks associated with music generation models. We ensure that both our models and training data are strictly used for academic research purposes only. We respect the intellectual property rights of all original artists and content creators. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: LLM is only used in very few text processing steps. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Ablation Study on Different Sketch Choices + +Table 5 demonstrates our early efforts at decomposing the semantic information of sketches. We compare different sketch representations, including pitch, pitch + chromagram, and semantic embeddings. As shown in the table, simple pitch-based sketches offer limited alignment capability, resulting in high PER and suboptimal performance in other metrics. The inclusion of chroma features slightly improves alignment, but still falls short of fully capturing the rich semantics needed for coherent vocal generation. These results validate the importance of incorporating abstract, semantically meaningful information into the sketch stage, and lay the groundwork for future exploration of interpretable yet powerful sketch formats. + +
TypePER(%)↓FAD↓CE↑CU↑PC↑PQ↑
no sketch109.768.867.067.405.927.71
pitch103.225.437.347.586.038.04
pitch + chroma68.475.477.377.575.988.06
SSL embedding9.445.607.557.666.008.25
+ +Table 5: Impact of sketch types on SongBloom's performance. + +# B Time Complexity of SongBloom Inference + +We analyze the inference time complexity of SongBloom compared to a decoupled two-stage model, assuming both have the same number of layers. + +Let $L_{1}$ denote the number of layers in the language model stage, and $L_{2}$ the number of layers in the diffusion stage. Let $T$ be the total number of frames for both semantic and acoustic sequences (eg. $30\mathrm{s}\times 25\mathrm{fps} = 600$ ), $P$ the patch size, $N = T / P$ the number of patches, and $S$ the number of diffusion steps. + +Assuming key-value caching is used during inference, we analyze the leading-order time complexity: + +Decoupled 2-stage model: + +$$ +O _ {0} = L _ {1} \cdot \frac {(1 + T) T}{2} + L _ {2} \cdot (2 T) ^ {2} \cdot S +$$ + +SongBloom: + +$$ +O _ {1} = L _ {1} \cdot \frac {(1 + T + N) (T + N)}{2} + L _ {2} \cdot (2 P) ^ {2} \cdot N \cdot S +$$ + +Substituting $N = T / P$ , we compute the difference: + +$$ +O _ {1} - O _ {0} = \left(\frac {L _ {1}}{2 P ^ {2}} + \frac {L _ {1}}{P} - 4 L _ {2} S\right) T ^ {2} + \left(\frac {L _ {1}}{2 P} + 4 L _ {2} P S\right) T +$$ + +When $T$ is sufficiently large, the $T^2$ term dominates. In most practical cases, where: + +$$ +L _ {1} < 4 S P \cdot \frac {2 P}{2 P + 1} \cdot L _ {2} +$$ + +the coefficient of $T^2$ is negative. Therefore, we conclude that SongBloom is asymptotically more efficient than the decoupled two-stage models, owing to its patch-wise diffusion mechanism and reduced per-step input length during inference. + +# C Criteria of the Subjective Listening Test + +1. Musicality of vocal: (1-5 points) Does the main melody of the generated vocal match the subjective expectation? + +- 5 points: The melody is pleasant and emotionally expressive, with strong musical phrasing. It aligns well with expectations. +- 4 points: The melody generally meets expectations and conveys the song's theme and emotion, but lacks standout features. +- 3 points: The melody mostly aligns with expectations and conveys the theme and emotion, though some notes feel abrupt. +- 2 points: Only parts of the melody are coherent; most notes are scattered, and the theme and emotion are vaguely presented. +- 1 point: The melody significantly deviates from expectations, lacks coherent musical phrasing, and fails to convey the song's theme and emotion. + +2. Musicality of accompaniment: (1-5 points) Does the accompaniment of the generated song sound harmonious? + +- 5 points: The accompaniment is richly colored and features diverse instrumentation. The melody is beautiful and complements the main melody harmoniously. +- 4 points: The accompaniment supports the main melody, but uses limited instrumentation or has a generally average melodic performance. +- 3 points: The accompaniment mostly supports the main melody, with only minor discord. However, it sometimes clashes with the main melody and lacks variety and color in instrumentation. +- 2 points: Some segments show disorganized instrumentation and monotonous melody, barely supporting the main melody. +- 1 point: The instrumentation is chaotic and the melody is discordant. There is a clear conflict with the main melody, failing to provide support. + +3. Quality of vocal: (1-5 points) Is the vocal in the generated music clear and bright, with a full high-frequency range? Are there any noises or distortions present? + +- 5 points: The vocal quality is rich and clear, with no noise, approaching studio-recording quality. +- 4 points: The vocal quality is relatively clear, with slight noise that is either imperceptible or barely noticeable. +- 3 points: The vocal quality contains some noise or distortion, but it does not significantly affect the listening experience. +- 2 points: The vocal quality is unclear and unstable, resulting in a poor listening experience. Noticeable noise or distortion is present. +- 1 point: The vocal quality is extremely poor, with an unpleasant listening experience, and the vocal characteristics are barely recognizable. + +4. Quality of accompaniment: (1-5 points) Is the high-frequency range of the generated music's accompaniment full? Are there any noises or instrumental distortions? + +- 5 points: The accompaniment has a full and clear sound quality with no flaws. The characteristics and melodies of different instruments are clearly distinguishable. +- 4 points: The accompaniment has good sound quality with slight noise. Only a few instruments in certain segments are hard to distinguish or slightly distorted, but this does not affect the overall listening experience. +- 3 points: The accompaniment has average sound quality. Some instruments are unclear or unidentifiable in certain segments. There is noticeable noise, distortion, or a lack of clarity. +- 2 points: The accompaniment has poor sound quality. In most parts of the piece, most instruments are unrecognizable. There is clear noise, distortion, or lack of clarity. +- 1 point: The accompaniment has extremely poor sound quality, with severe distortion, making it nearly impossible to identify any instrumental characteristics. + +5. Correctness of lyrics: (1-4 points) Does the song content match the lyrics? Are there any errors such as extra words, missing words, or mechanical repetition? + +- 4 points: The song content fully matches the lyrics, with no missing or extra words, and no mechanical repetition of musical segments. + +- 3 points: The generated song contains a small number (within 5 words) of unclear, repeated, or missing lyrics. +- 2 points: The generated song contains multiple segments with unclear, repeated, or missing lyrics. +- 1 point: The generated song does not match the lyrics at all. + +6. 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Aggregation Graph Neural Networks/18ea1710-e494-42c0-badc-28eafa7418a4_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c48b3c244d107a5045ba57a3a17e2d238f9851bfbe1fe8fc4716676eb634a6e2 +size 460347 diff --git a/NeurIPS/2025/Sound Logical Explanations for Mean Aggregation Graph Neural Networks/full.md b/NeurIPS/2025/Sound Logical Explanations for Mean Aggregation Graph Neural Networks/full.md new file mode 100644 index 0000000000000000000000000000000000000000..473473aa3eee10f1c6b3b6897666e3be71e1accc --- /dev/null +++ b/NeurIPS/2025/Sound Logical Explanations for Mean Aggregation Graph Neural Networks/full.md @@ -0,0 +1,909 @@ +# Sound Logical Explanations for Mean Aggregation Graph Neural Networks + +Matthew Morris + +Department of Computer Science + +University of Oxford + +matthew.morris@cs.ox.ac.uk + +Ian Horrocks + +Department of Computer Science + +University of Oxford + +ian.horrocks@cs.ox.ac.uk + +# Abstract + +Graph neural networks (GNNs) are frequently used for knowledge graph completion. Their black-box nature has motivated work that uses sound logical rules to explain predictions and characterise their expressivity. However, despite the prevalence of GNNs that use mean as an aggregation function, explainability and expressivity results are lacking for them. We consider GNNs with mean aggregation and non-negative weights (MAGNNs), proving the precise class of monotonic rules that can be sound for them, as well as providing a restricted fragment of first-order logic to explain any MAGNN prediction. Our experiments show that restricting mean-aggregation GNNs to have non-negative weights yields comparable or improved performance on standard inductive benchmarks, that sound rules are obtained in practice, that insightful explanations can be generated in practice, and that the sound rules can expose issues in the trained models. + +# 1 Introduction + +Knowledge graphs (KGs) [17] find use in a number of applications [16, 41, 45]. However, KGs are often incomplete, creating the need for models to predict their missing facts. Neural models such as graph neural networks (GNNs) [13] are frequently used for this task [18, 22, 25], as well as a variety of others, including predicting properties of drug combinations [9] and recommender systems [11]. + +However, the predictions of neural models cannot easily be explained and verified [12]. In contrast, logic-based and neuro-symbolic methods for KG completion often yield logical rules which can be used to explain predictions [24, 29, 31, 46]. To ensure that the rules truly express the reasons why the model makes a particular prediction, it is important to ensure that they are sound, in the sense that applying the rules to an arbitrary dataset produces only facts that are predicted by the model. Thus, there is growing interest in models whose predictions can be characterized by sound rules [36, 42]. + +When it comes to GNNs, Tena Cucala et al. [35, 37] provide sound Datalog rules and equivalent programs for GNNs with non-negative weights and max or sum aggregation, respectively. Morris et al. [25] consider GNNs with sum aggregation and show that, in practice, they often provably have no sound Datalog rules. There is also a plethora of related work considering the logical expressivity of general aggregate-combine GNNs. Barceló et al. [7] show that if a GNN captures a rule, then it can be expressed in the logic ALCQ. Other works provide GNN expressivity results for various extensions of first-order logic, including linear programming [26], Presburger quantifiers [8], and counting terms [14]. Ahvonen et al. [3], Pflueger et al. [27] characterise recurrent GNNs using fixpoint operators. Cuenca Grau et al. [10] consider GNNs with bounded aggregation functions and show their equivalence to fragments of first-order logic. + +Although any function mapping multisets of real numbers to real numbers can be used for the aggregation function in a GNN, standard options include mean, sum, max, and attention [40]. In practice, mean aggregation often emerges as the default choice [16, 19, 30, 32, 44] due to its + +simplicity, stability during training, and good empirical performance. With regard to the expressivity of mean-GNNs, Rosenbluth et al. [30] study mean, sum, and max aggregation in terms of their ability to approximate one another. However, to our knowledge, there are no prior works analysing the logical expressivity of or attempting to extract sound rules from GNNs that use mean aggregation (mean-GNNs). + +Our Contribution To obtain explanatory sound rules for mean-GNNs, we consider mean-GNNs with non-negative weights (MAGNNs). This isolates the source of the non-monotonicity to the aggregation function (since negative weights can also be a source of non-monotonicity), enabling a simpler analysis. This extends the work of Tena Cucala et al. [37], who proved that GNNs with non-negative weights and max or sum aggregation are monotonic under injective homomorphisms, enabling the verification of sound rules, and of Tena Cucala et al. [35] and Morris et al. [25], who found that restricting GNNs to non-negative weights did not greatly impact performance when using max or sum aggregation, respectively. + +We provide two main theoretical results. First, we prove the exact class of "monotonic rules" that can be sound for MAGNNs, which turns out to be very limited. We also provide a means for checking if such rules are sound. Second, in light of the fact that some MAGNNs do not have equivalent first-order logic (FOL) programs, we instead provide a restricted fragment of FOL that contains sound rules that can be used to explain any prediction of a MAGNN. In our experiments, we show that mean-GNNs still perform well on benchmark datasets when restricted to having non-negative weights. We also demonstrate that they recover a variety of sound monotonic rules, as well as providing examples of the generated rules that explain predicted facts of the GNN. Despite good model performance on the test set, we find that MAGNNs still learn some nonsensical sound rules, showing the importance of explainability. + +# 2 Background + +Datasets and Graphs We fix a signature of countably infinite, disjoint sets of unary/binary predicates and constants. We also consider a countably infinite set of variables disjoint with the sets of predicates and constants. A term is a variable or a constant. An atom is an expression of the form $R(t_{1},t_{2})$ or $U(t_{1})$ , where each $t_i$ is a term and $R, U$ are binary and unary predicates, respectively. An atom is ground if it contains no variables. A fact is a ground atom and a dataset $D$ is a finite set of facts. We denote the set of all constants in $D$ with $\operatorname{con}(D)$ . + +For $\mathbf{v}$ a vector and $i > 0$ , $\mathbf{v}[i]$ denotes the $i$ -th element of $\mathbf{v}$ (likewise for matrices). For a finite set $\mathrm{Col}$ of colours and $\delta \in \mathbb{N}$ , a (Col, $\delta$ )-graph $G$ is a tuple $\langle V, \{E^c\}_{c \in \mathrm{Col}}, \lambda \rangle$ where $V$ is a finite vertex set, each $E^c \subseteq V \times V$ is a set of directed edges with colour $c$ , and $\lambda$ assigns to each $v \in V$ a vector of dimension $\delta$ . Each vertex $v^a \in V$ is defined to correspond uniquely to a constant $a$ from the signature. When $\lambda$ is clear from the context, we abbreviate the labelling $\lambda(v)$ as $\mathbf{v}$ . We uniquely associate each $\mathbf{v}[i]$ with a unary predicate $U_i$ and each colour in $c \in \mathrm{Col}$ with a binary predicate $R^c$ . There is thus a one-to-one correspondence between graphs and datasets [37]. We use $N_c(v)$ to denote the $c$ -coloured neighbours of a vertex $v$ . Graph $G$ is undirected if $E^c$ is symmetric for each $c \in \mathrm{Col}$ and is Boolean if $\mathbf{v}[i] \in \{0,1\}$ for each $v \in V$ and $i \in \{1,\dots,\delta\}$ . + +FOL Rules We use standard first-order logic (FOL) syntax, including equality. A variable in a FOL formula is free if it is not quantified. Barceló et al. [7] define first-order logic (FOL) classifiers over coloured graphs, where for graph $G$ , node $v$ , and FOL formula $\varphi$ with a free variable, $(G, v) \models \varphi$ denotes $\varphi$ being satisfied by node $v$ of $G$ . Similarly, given the correspondence between graphs and datasets, for a dataset $D$ , constant $a \in D$ , and FOL formula $\varphi$ with free variable $x$ , we let $(D, a) \models \varphi$ denote $\varphi$ being satisfied by constant $a$ of $D$ . More precisely, $(D, a) \models \varphi$ if for variable substitution $\mu$ mapping $x \mapsto a$ , we have $D \models \varphi \mu$ . As usual in this setting, $D$ is treated as an interpretation and $D \models \varphi \mu$ iff $\varphi \mu$ evaluates to true in $D$ . + +We define a FOL rule to be of the form $B \to A(x)$ , where $B$ is a FOL formula with free variable $x$ and $A$ a unary predicate. A program is a set of rules. Given a dataset $D$ and FOL rule $r: B \to A(x)$ , we define the immediate consequence operator $T_r(D) = \{A(a) \mid a \in \mathrm{con}(D), (D, a) \models B\}$ . For a program $\alpha$ , we likewise define $T_\alpha(D) = \bigcup_{r \in \alpha} T_r(D)$ . This approach is similar to that of Barceló et al. [7], except that instead of binary classification, we are essentially doing multi-class classification. + +It is the same approach of Tena Cucala et al. [37], where they find equivalent programs and sound rules for different GNNs, but use Datalog semantics. + +We also consider fragments of FOL defined by description logics such as ALCQ [6], the concepts of which are defined by: + +$$ +C := \top \mid \bot \mid A \mid \neg C \mid C \cap D \mid C \sqcup D \mid \forall P. C \mid \exists P. C \mid \leq_ {n} P. C \mid \geq_ {n} P. C, +$$ + +for any relation (binary predicate) $P$ , ALCQ concepts $C, D$ , and atomic concept (unary predicate) $A$ . We define an ALCQ rule to be a subsumption of the form $C \sqsubseteq A$ . Immediate consequences are defined in the same way as for FOL, since every ALCQ concept corresponds to a FOL formula with one free variable. + +Graph Neural Networks A function $\sigma : \mathbb{R} \to \mathbb{R}$ is monotonically increasing if $x < y$ implies $\sigma(x) \leq \sigma(y)$ . We apply functions to vectors element-wise. A (Col,δ)- graph neural network $\mathcal{M}$ with $L \geq 1$ layers is a tuple + +$$ +\langle \left\{\mathbf {A} _ {\ell} \right\} _ {1 \leq \ell \leq L}, \left\{\mathbf {B} _ {\ell} ^ {c} \right\} _ {c \in \operatorname {C o l}, 1 \leq \ell \leq L}, \left\{\mathbf {b} _ {\ell} \right\} _ {1 \leq \ell \leq L}, \left\{\sigma_ {\ell} \right\} _ {1 \leq \ell \leq L}, \left\{\operatorname {a g g} _ {\ell} \right\} _ {1 \leq \ell \leq L}, \left. \operatorname {c l s} _ {t} \right\rangle , \tag {1} +$$ + +where, for each $\ell \in \{1,\dots ,L\}$ and $c\in \mathrm{Col}$ , matrices $\mathbf{A}_{\ell}$ and $\mathbf{B}_{\ell}^{c}$ are of dimension $\delta_{\ell}\times \delta_{\ell -1}$ with $\delta_0 = \delta_L = \delta$ , $\mathbf{b}_{\ell}$ is a vector of dimension $\delta_{\ell}$ , $\sigma_{\ell}:\mathbb{R}\to \mathbb{R}^{+}\cup \{0\}$ is a monotonically increasing continuous activation function with non-negative range, $\mathrm{agg}_{\ell}$ is an aggregation function from finite real multisets to real values, and $\operatorname {cl}\mathfrak{s}_t:\mathbb{R}\to \{0,1\}$ for threshold $t_\mathcal{M}\in \mathbb{R}$ is a step classification function such that $\operatorname {cl}\mathfrak{s}_t(x) = 1$ if $x\geq t$ and $\operatorname {cl}\mathfrak{s}_t(x) = 0$ otherwise. + +Applying $\mathcal{M}$ to a $(\mathrm{Col},\delta)$ -graph induces a sequence of labels $\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_L$ for each vertex $v$ in the graph as follows. First, $\mathbf{v}_0$ is the initial labelling of the input graph; then, for each $1\leq \ell \leq L$ , $\mathbf{v}_{\ell}$ is defined by the following expression: + +$$ +\sigma_ {\ell} \left( \right.\mathbf {b} _ {\ell} + \mathbf {A} _ {\ell} \mathbf {v} _ {\ell - 1} + \sum_ {c \in \operatorname {C o l}} \mathbf {B} _ {\ell} ^ {c} \operatorname {a g g} _ {\ell} \left( \right.\left\{ \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left. \mid (v, u) \in E ^ {c} ]\right)\right]\right]\right]\right]\right]\right]\right]\right]\right]\right]\right]\right]\right]\right]\right)\right) +$$ + +The output of $\mathcal{M}$ is a $(\mathrm{Col},\delta)$ -graph with the same vertices and edges as the input graph, but where each vertex is labelled by $\operatorname{c1s}_t(\mathbf{v}_L)$ . + +When each $\mathrm{agg}_{\ell}$ is mean, we call it a mean-GNN (likewise for max and sum). When all values in each $\mathbf{A}_{\ell}$ and $\mathbf{B}_{\ell}^{c}$ are non-negative, we say that the GNN is monotonic. In this paper, we consider in particular the class of monotonic mean GNNs (MAGNNs). Besides the restriction of non-negative weights, this is the same model used in R-GCN [32] (where the normalisation constant is set as described in their paper, to be the number of $c$ -coloured neighbours of the node). + +Dataset Transformations Through GNNs A GNN $\mathcal{M}$ induces a transformation $T_{\mathcal{M}}$ from datasets to datasets over a given finite signature [37]. To this end, the input dataset must be first encoded into a graph that can be directly processed by the GNN, and the graph resulting from the GNN application must be subsequently decoded back into an output dataset. We adopt the so-called canonical scheme, where $\mathrm{enc}(D)$ of a dataset $D$ is the Boolean $(\mathrm{Col},\delta)$ -graph with a vertex $v^a$ for each constant $a$ in $D$ and a $c$ -coloured edge $(v^{a},v^{b})$ for each fact $R^c (a,b)\in D$ . Furthermore, given a vertex $v^{a}$ , vector component $\mathbf{v}^a [p]$ is set to 1 if and only if $U_{p}(a)\in D$ , for $p\in \{1,\ldots ,\delta \}$ . The decoder dec is the inverse of the encoder. The canonical dataset transformation induced by a GNN $\mathcal{M}$ is then defined as: $T_{\mathcal{M}}(D) = \mathrm{dec}(\mathcal{M}(\mathrm{enc}(D)))$ . We abbreviate $\mathcal{M}(\mathrm{enc}(D))$ by $\mathcal{M}(D)$ . + +In this paper, we will only consider $(\mathrm{Col},\delta)$ -graphs, datasets, and GNNs, unless specified otherwise. When performing the task of link prediction, we use the additional dataset transformations of Tena Cucala et al. [37], which enable GNNs to predict binary facts. + +Soundness and Subsumption A FOL (or ALCQ) logic program or rule $\alpha$ is sound for a MAGNN $\mathcal{M}$ if $T_{\alpha}(D) \subseteq T_{\mathcal{M}}(D)$ for each dataset $D$ . Conversely, $\alpha$ is complete for $\mathcal{M}$ if $T_{\mathcal{M}}(D) \subseteq T_{\alpha}(D)$ for each dataset $D$ . We say that $\alpha$ is equivalent to $\mathcal{M}$ if it is both sound and complete for $\mathcal{M}$ . Finally, we say that a rule or program $R$ subsumes a rule $r$ if for any dataset $D$ , $T_{r}(D) \subseteq T_{R}(D)$ . + +# 3 Sound Rules and Explanations + +We prove a series of theoretical results for MAGNNs, to identify which monotonic rules can be sound for them and obtain sound explanatory rules. The restriction to non-negative weights isolates the source of the non-monotonicity to the aggregation function, making analysis easier; however, MAGNNs still possess the defining feature of mean-GNNs in their choice of aggregation function. This restriction is the same approach that was used effectively for GNNs with sum or max aggregation [25, 35, 37]. First, we show that there exist MAGNNs with no equivalent FOL programs. Intuitively, this follows from FOL's inability to express numerical comparisons. + +Proposition 1. There exists a MAGNN $\mathcal{M}$ such that for any dataset $D$ and constant $a\in \mathsf{con}(D)$ , $U(a)\in T_{\mathcal{M}}(D)$ if and only at least half of the neighbours $b$ of $a$ in $D$ are such that $U(b)\in D$ , where $U$ is a unary predicate. This logical function cannot be defined in FOL [21]. + +A full proof is given in Appendix A.1. This means that the task of trying to find equivalent programs for them, such as the approach of Tena Cucala et al. [37], is impossible. Instead, we aim to extract sound rules from MAGNNs, to explain their predictions. Thus, we consider in particular a class of "monotonic" rules encompassing many common inference patterns, and identify the restricted fragment that can be sound for MAGNNs. Finally, we define a rule language using a fragment of FOL that, for any mean-GNN prediction, contains a sound rule that entails the prediction. + +# 3.1 Which Monotonic Rules can be Sound for MAGNNs + +Since equivalent programs cannot be found in some cases, we instead first show which simple monotonic rules can be sound for MAGNNs. Monotonic rules appear commonly in practice [1, 23, 34], are often easily human-readable, and are easier to use for extracting sound rules. + +Definition 2. A rule $r$ is monotonic (under dataset extension) if for all datasets $D, D'$ such that $D \subseteq D'$ , $T_r(D) \subseteq T_r(D')$ . + +Description logics, such as ALCQ [6], are a natural choice for providing a language for sound rules due to their widespread use in data modelling and their theoretical relationship to GNNs [7]. However, some ALCQ concepts are inherently non-monotonic: for example, if $(D,c)\models \forall P.A$ for some atomic concept $A$ , binary predicate $P$ , dataset $D$ , and constant $c\in \mathrm{con}(D)$ , then for $D^{\prime} = D\cup \{P(c,d)\}$ with a fresh constant $d$ , $(D^{\prime},c)\nVdash \forall P.A(c)$ . + +EL [6] is a more restricted description logic than ALCQ, where concepts use only intersection and existential quantification—any EL rule is thus monotonic. We define ELUQ, a language in-between EL and ALCQ, as the language containing concepts defined by + +$$ +C := \top \mid A \mid C \sqcap D \mid C \sqcup D \mid \exists P. C \mid \ge_ {n} P. C, +$$ + +where $C, D$ are ELUQ concepts, $P$ is a binary predicate, $A$ is an atomic concept, and $n$ a positive integer. Note that any concept $\exists P.C$ can be written equivalently as $\geq_1 P.C$ . An ELUQ rule has the form $C \sqsubseteq A$ . This language is similar (but not equivalent) to that of Datalog with inequalities considered by Tena Cucala et al. [37] for GNNs with sum aggregation and non-negative weights, given constraints on the inequalities and the addition of disjunction. All ELUQ rules are monotonic; however, it remains an open question as to whether this is a maximal monotonic fragment of ALCQ. + +We find that for any ELUQ rule $r$ that is sound for a MAGNN $\mathcal{M}$ , there exists a set of rules of a very simple form that (1) are each sound for $\mathcal{M}$ and (2) collectively subsume $r$ . This is formalised in the following theorem. + +Theorem 3. Let $\mathcal{M}$ be a MAGNN and $r$ an ELUQ rule that is sound for $\mathcal{M}$ . Then there exists a finite set of rules $R$ such that: + +1. Each $r' \in R$ has the form $\exists P_1. \top \cap \dots \cap \exists P_j. \top \cap A_1 \cap \dots \cap A_k \cap \top \subseteq A_{k+1}$ , where each $P_i$ is a binary predicate, $A_i$ a unary predicate, and $j, k \in \mathbb{N}_0$ . +2. Each $r' \in R$ is sound for $\mathcal{M}$ . +3. $R$ subsumes $r$ . + +The proof provides a construction of $R$ . For example, given an ELUQ rule $r: \geq_3 P_1.(A_1 \sqcup A_2) \sqcap \exists P_2.(\exists P_1.A_3) \sqcup A_4 \sqsubseteq A_5$ be sound for $\mathcal{M}$ , we can construct rules $r_1: \exists P_1. \top \sqcap \exists P_2. \top \sqsubseteq A_5$ and $r_2: A_4 \sqsubseteq A_5$ such that both $r_1$ and $r_2$ are sound for $\mathcal{M}$ and $\{r_1, r_2\}$ subsumes $r$ . + +As a consequence of this, when checking for sound ELUQ rules it suffices to check for rules of the above restricted form. In addition, whilst the space of all ELUQ rules is infinite, the restricted fragment we provide is finite, meaning that all sound ELUQ rules can be covered by instead iterating over a finite number of restricted rules. Finally, this raises concerns for the logical expressivity of MAGNNs, since any sound ELUQ rule is ultimately subsumed by a set of much simpler sound rules. + +Proof sketch. The full proof of Theorem 3 is given in Appendix A.2. A different $r'$ is defined for each concept in the outer disjunction of the body of $r$ . To see the crucial step of the proof, consider some rule $r$ without disjunction in the outer concept, and let $r_1 \coloneqq r$ . Then $r_1$ is of the form $\geq_n P.C_1 \sqcap C_2 \sqsubseteq A$ , where $P$ is a binary predicate, $C_1$ an ELUQ concept, $C_2$ an ELUQ concept without disjunction, and $n$ a positive integer. We inductively define a sequence $r_2, \dots, r_k$ of ELUQ rules that such for all $i \in \{1, \dots, k-1\}$ , $r_{i+1}$ is sound for $\mathcal{M}$ and $r_{i+1}$ subsumes $r_i$ . Given $r_i$ of the form $\geq_n P.C_1 \sqcap C_2 \sqsubseteq A$ , we define $r_{i+1}$ as the rule $\exists P.\top \sqcap C_2 \sqsubseteq A$ . Trivially, $r_{i+1}$ subsumes $r_i$ . + +Now consider the following datasets $D_{1}$ and $D_{m}$ (parametrised by $m \in \mathbb{N}$ ). In $D_{1}$ , a concept $C_{2}$ annotating a constant $a$ denotes that there exists other facts in the dataset such that $(D_{1}, a) \models C_{2}$ (similarly for $D_{m}$ ). + +Daset $D_{1}$ .. $C_2\bigoplus a\longrightarrow \bigoplus b$ Daset $D_{m}$ .. P C2 P Cn C1 + +Note that $D_{1}$ is an instantiation of the body of $r_{i + 1}$ . The soundness of $r_{i + 1}$ depends on the fact that as $m\to \infty$ , the computation of $\mathcal{M}$ on $D_{m}$ at constant $a$ tends to that of $\mathcal{M}$ on $D_{1}$ (and that $A(a)\in T_{r_i}(D_m)$ ). + +Since $r_1$ contains a finite number of $\exists$ or $\geq_n$ operators and each $r_{i+1}$ has one fewer $\exists$ or $\geq_n$ operator than $r_i$ , this inductive construction is guaranteed to terminate for some $k$ . We include $r_k$ in $R$ . Once this has been done for every $r_1 \in R'$ , we have $R$ subsumes $r$ . + +Checking for Monotonic Rule Soundness It remains to be shown how one can verify the soundness of ELUQ rules. Since we only need to consider rules of the form given in Theorem 3, there are exactly $\delta \cdot 2^{\delta \cdot |\mathrm{Col}|}$ possible rules to check. For rules $r_1, r_2$ , when considering the sets of body concepts $B_{r_1}$ and $B_{r_2}$ , if $B_{r_1} \subseteq B_{r_2}$ and $r_1$ is sound, then $r_2$ is sound: we use this to optimise the procedure. The following proposition allows one to check the soundness of ELUQ rules. + +Proposition 4. Let $\mathcal{M}$ be a MAGNN and $r:\exists P_1.\top \sqcap \ldots \sqcap \exists P_j.\top \sqcap A_1\sqcap \ldots \sqcap A_k\sqsubseteq A_{k + 1}$ be a rule of the form given in Theorem 3. Define $D_{base}:= \{A_1(a),\dots,A_k(a),P_1(a,b),\dots,P_j(a,b)\}$ for constants $a\neq b$ . Then $r$ is sound for $\mathcal{M}$ if and only if $A_{k + 1}(a)\in T_{\mathcal{M}}(D_{base})$ . + +For example, to check if a rule $r: \exists P_1. \top \sqcap \exists P_2. \top \sqcap A_1 \sqcap A_2 \sqsubseteq A_3$ is sound, it suffices to compute the output of $\mathcal{M}$ on the following dataset $D_{\mathrm{base}} = \{A_1(a), A_2(a), P_1(a,b), P_2(a,b)\}$ , which can be represented as follows: + +![](images/4521d5505a875f87f125a62d74c8e939848a465dbf18355653dca96c0186805e.jpg) + +Then $r$ is sound for $\mathcal{M}$ if and only if $A_{3}(a) \in T_{\mathcal{M}}(D_{\mathrm{base}})$ . Note that this method also works if the body of the rule $r$ is empty, by simply checking if $A_{k + 1}(a) \in T_{\mathcal{M}}(D_{\emptyset})$ , where $D_{\emptyset} = \{P(b,a)\}$ . + +Proof sketch. The full proof of Proposition 4 is given in Appendix A.3. It relies on the fact that $(D_{\mathrm{base}}, a) \models B$ , where $B$ is the body of $r$ . Furthermore, no extension to $D_{\mathrm{base}}$ will decrease the output + +at $v^a$ , and any dataset satisfying $B$ is an extension of $D_{\mathrm{base}}$ (up to constant relabelling, un-merging the neighbours of $v^a$ , and letting $b = a$ ). Note that none of these merges / un-merges will decrease the output at $v^a$ and that MAGNNs are agnostic to the particular constants used in the input dataset. + +Consequences for Link Prediction To perform link prediction, we combine the canonical transformation with the dataset encoding-decoding scheme defined by Tena Cucala et al. [37], which has found wide use [22, 25, 35, 37, 43]; it introduces additional nodes in the graph to represent pairs of constants. Node labels then encode binary facts, instead of unary. Graph edges are used only to connect nodes that have constants in common; and so the presence of an edge is independent of whether any specific fact holds in the dataset. + +The encoding and decoding can be described by a set of rules [37, Section 3.2], each expressed in a simple extension of Datalog. Given a sound rule for the canonical transformation, we can combine it (via unfolding) with the rules representing the encoder and decoder to obtain a Datalog rule that is sound for the entire link prediction transformation. This observation, combined with Theorem 3, ensures that each monotonic rule sound for the entire link prediction transformation is subsumed by rules of the form $R_{1}(x,y)\wedge \ldots \wedge R_{m - 1}(x,y)\to R_{m}(x,y)$ , where $R_{1},\dots,R_{m}$ are binary predicates from the signature. This is obtained by unfolding a rule of the form given in Theorem 3, since the unary predicates $A_{i}$ are unfolded into the binary predicates they represent, and the existentials $\exists P_{j}.\top$ can safely be removed since the edge presence is independent of any specific fact. + +# 3.2 Explaining MAGNN Predictions + +In this section, we provide a family of rules $\Omega$ (a fragment of FOL) to explain any prediction of a MAGNN. More precisely, for any MAGNN $\mathcal{M}$ , dataset $D$ , and predicted fact $A(a) \in T_{\mathcal{M}}(D)$ , there exists a rule $r \in \Omega$ such that $A(a) \in T_r(D)$ and $r$ is sound for $\mathcal{M}$ . To achieve this, we define a procedure to derive the rule $r \in \Omega$ , given $\mathcal{M}$ , $D$ , and $A(a)$ . + +Logic Language Needed We first define the family of rules using ALCQ syntax, with the addition of a new operator, rather than defining it directly in FOL. The operator is $\exists_{n}$ ("exists $n$ unique"): for example: $\exists_{3}P.(A_{1},A_{2},\top)$ . Its semantics are defined as follows. For concepts $C_1,\dots,C_n$ , binary predicate $P$ , dataset $D$ , and constant $c\in D$ , we define $(D,c)\models \exists_{n}P.(C_{1},\dots,C_{n})$ if and only if there exist distinct constants $c_{1},\ldots ,c_{n}\in \mathrm{con}(D)$ such that $(D,c_{1})\models C_{1},\dots,(D,c_{n})\models C_{n}$ and $P(c,c_1),\dots,P(c,c_n)\in D$ . Written in first-order logic, it is defined as $(D,c)\models \exists_{n}P.(C_{1},\dots,C_{n})\equiv$ + +$$ +\begin{array}{l} (D, c) \vDash \exists x _ {1} \dots \exists x _ {n} (x _ {i} \neq x _ {j} \text {f o r a l l} i, j \in \{1, \dots , n \} \text {s u c h t h a t} i < j) \\ \wedge C _ {1} \left(x _ {1}\right) \wedge \dots \wedge C _ {n} \left(x _ {n}\right) \wedge P \left(x, x _ {1}\right) \wedge \dots \wedge P \left(x, x _ {n}\right), \\ \end{array} +$$ + +for free variable $x$ . We then define the family of rules $\Omega$ as follows. An $\Omega$ -concept $C$ is defined by $C := \top \mid A \mid C_1 \cap C_2 \mid \exists_n P.(C_1, \dots, C_n) \sqcap \leq_m P. \top$ , where $A$ is any atomic concept, $n$ any positive integer, $m \geq n$ an integer, $C_1, \dots, C_n$ any $\Omega$ -concepts, and $P$ any binary predicate. A rule in $\Omega$ has the form $C \sqsubseteq A$ , where $C$ is any $\Omega$ -concept and $A$ any atomic concept. + +Explanatory Rule Let $\mathcal{M}$ be a MAGNN, $D$ a dataset, and $A(a)\in T_{\mathcal{M}}(D)$ a prediction of the MAGNN. We define a rule $r\in \Omega$ by $C_L^a\subseteq A$ , where the only part of $\mathcal{M}$ that the rule depends on is $L$ , which is what guarantees that the rule will be finite. The body concept $C_L^a$ is defined as follows: + +Definition 5. For a dataset $D$ , $c \in \mathsf{con}(D)$ , $\ell \in \mathbb{N}_0$ , we inductively define a concept $C_{\ell}^{c}$ by $C_{\ell}^{c} \coloneqq \top \sqcap A_{1} \sqcap \ldots \sqcap A_{k}$ , where $A_{1}, \ldots, A_{k}$ are all atomic concepts $A_{i}$ such that $A_{i}(c) \in D$ . Then, if $\ell = 0$ , we are done: $C_{\ell}^{c}$ has been defined. + +Otherwise, for each binary predicate $P$ , let $c_{1},\ldots ,c_{n}$ be all the constants $c_{i}\in \mathsf{con}(D)$ such that $P(c,c_i)\in D$ . Then, if $n > 0$ , extend $C_\ell^c$ as follows: $C_\ell^c\coloneqq C_\ell^c\cap \exists_nP.(C_{\ell -1}^{c_1},\dots,C_{\ell -1}^{c_n})\cap \leq_nP.\top$ . $C_\ell^c$ is extended for each binary predicate $P$ . Note that each $C_{\ell -1}^{c_i}$ is defined inductively. + +The following theorem then shows that the rule both explains the prediction (by producing the same fact on the dataset as the MAGNN) and is sound for the MAGNN. + +Theorem 6. For any MAGNN $\mathcal{M}$ , dataset $D_{a}$ , and fact $A(a) \in T_{\mathcal{M}}(D_a)$ , the rule $r: C_L^a \subseteq A$ (dependent on $D_{a}$ ) is sound for $\mathcal{M}$ , and $A(a) \in T_r(D_a)$ . + +Consider a dataset $D_{a} = \{A_{1}(a),P_{1}(a,b_{1}),P_{1}(a,b_{2}),P_{2}(b_{3},a),A_{2}(b_{2}),P_{2}(b_{2},c),P_{3}(c,d)\}$ and 2-layer GNN $\mathcal{M}$ , for example, such that $A_{3}(a)\in T_{\mathcal{M}}(D)$ . Then, using Theorem 6, we construct the rule $r:C_2^a\sqsubseteq A_3$ , where $C_2^a$ is $\top \sqcap A_1\sqcap \exists_2P_1.$ $(\top ,\top \sqcap A_2\sqcap \exists_1P_2.$ $(\top)\sqcap \leq_1P_2.\top)$ $\sqcap \leq_2P_1.\top$ . Notice that $(D_a,a)\models C_2^a$ , so we have $A_{3}(a)\in T_{r}(D)$ . Furthermore, $A_{3}(a)\in T_{\mathcal{M}}(D)$ is a sufficient witness for the soundness of $r$ , as we will prove below. + +Proof sketch. The full proof of Theorem 6 is given in Appendix A.5. + +$(D_a, a) \models C_L^a$ follows from the inductive construction of $C_L^a$ , from which we obtain $A(a) \in T_r(D_a)$ . It remains to be shown that $r$ is sound for $\mathcal{M}$ . Let $D_b$ be a dataset: to show soundness, we prove that $T_r(D_b) \subseteq T_{\mathcal{M}}(D_b)$ , so let $A(b) \in T_r(D_b)$ . + +First, we define a dataset $D_{L}^{a}$ to be the $L$ -hop neighbourhood of the constant $a$ in $D_{a}$ (analogously to how $L$ -hop neighbourhoods are defined for nodes in graphs). Then we show that $A(a) \in T_{\mathcal{M}}(D_L^a)$ , since $A(a) \in T_{\mathcal{M}}(D_a)$ — this follows from the definition of $D_{L}^{a}$ as the $L$ -hop neighbourhood. Finally, we prove $A(b) \in T_{\mathcal{M}}(D_b)$ , since $A(a) \in T_{\mathcal{M}}(D_L^a)$ — this follows from the fact that any differences between $D_{b}$ and $D_{L}^{a}$ cannot yield a lower MAGNN output for $v^{b}$ in comparison to $v^{a}$ , plus a mapping between constants of $D_{b}$ and $D_{L}^{a}$ . + +# 4 Experiments + +We train GNNs across several benchmark link prediction datasets and a node classification dataset, showing that sound rules and explanations can be found for MAGNNs in practice, and that the restriction to monotonicity does not significantly decrease performance. For the model architecture, we fix a hidden dimension of twice the input dimension, 2 GNN layers, ReLU after the first layer, and sigmoid after the second layer. The GNN definition given in Section 2, which was chosen for ease of presentation, describes GNNs aggregating in the reverse direction of the edges. For our experiments, we follow the standard approach and aggregate in the direction of the edges. Thus, when presenting a rule, we write each binary predicate as its inverse. For example, "advisor" is written as "advisorOf". + +We use GNNs with max, sum, and mean aggregation. We train each model for 8000 epochs, stopping training early if loss does not improve for 50 epochs. For all trained models, we compute standard classification metrics, such as precision, recall, accuracy, and F1 score. For each model, we choose the classification threshold by computing the accuracy on the validation set across a range of 108 thresholds between 0 and 1, selecting the one which maximises accuracy. We train all our models using binary cross entropy loss and the Adam optimiser with a learning rate of 0.001. We train models without restrictions as baselines (denoted by "Standard"), as well as restricting the models to having non-negative weights (denoted by "Non-Neg") by clamping negative weights to 0 after each optimiser step, as in the approaches of Morris et al. [25], Tena Cucala et al. [35]. We run each experiment across 5 different random seeds and present the aggregated metrics. To compute $95\%$ confidence intervals, we assume a normal distribution and compute the interval as $1.97 \times$ SEM (standard error of the mean). Experiments were run using PyTorch Geometric, with 2 CPUs and 16GB of memory on a Linux server, using 34 days of compute time. + +Datasets We use 3 standard benchmarks: WN18RRv1, FB237v1, and NELLv1 [38], each of which provides datasets for training, validation, and testing, as well as negative examples and positive targets. Importantly, these benchmarks are also inductive, meaning that the validation and testing sets contain constants not seen during training. We also use the LUBM dataset [15, LUBM(1,0)], with the train/test split from Liu et al. [23]; this is a node classification dataset, all others are link prediction. + +Finally, we utilise LogInfer [23], a framework which augments a dataset by considering Datalog rules conforming to a particular pattern and adding the consequences of the rules to the dataset. We use the datasets LogInfer-WN-hier (WN-hier) and LogInfer-WN-sym (WN-sym) [23], which are enriched with the hierarchy and symmetry patterns, respectively. We also use LogInfer-WN-hier_nmhier [25], which was created using a mixture of monotonic and non-monotonic rules: rules from the "hierarchy" and "non-monotonic hierarchy" $(R(x,y)\land \neg S(y,z)\to T(x,y))$ patterns, in this case. We use the dataset to test whether monotonic rules can be recovered from the dataset, despite the presence of non-monotonic rules. For the LogInfer datasets, during each training epoch, $10\%$ of the input facts are randomly set aside and used as ground truth positive targets, whilst the rest of the facts are used as input to the model. + +
DatasetAggWeights%Acc%Prec%RecF1
LUBMMeanStandard97.1 ± 0.096.9 ± 0.097.2 ± 0.097.1 ± 0.0
Non-Neg91.5 ± 0.087.8 ± 0.096.4 ± 0.091.9 ± 0.0
WN-hierMeanStandard99.5 ± 0.099.5 ± 0.099.5 ± 0.099.5 ± 0.0
Non-Neg99.8 ± 0.099.7 ± 0.099.9 ± 0.099.8 ± 0.0
WN-symMeanStandard99.4 ± 0.099.5 ± 0.099.3 ± 0.099.4 ± 0.0
Non-Neg100 ± 0.0100 ± 0.0100 ± 0.0100 ± 0.0
WN-hier_nmhierMeanStandard86.2 ± 0.084.7 ± 0.088.4 ± 0.086.5 ± 0.0
Non-Neg71.6 ± 0.080.1 ± 0.158.8 ± 0.167.2 ± 0.0
FB237v1MeanStandard68.7 ± 0.095.4 ± 0.039.3 ± 0.055.7 ± 0.0
Non-Neg71.8 ± 0.075.4 ± 0.064.8 ± 0.069.7 ± 0.0
WN18RRv1MeanStandard93.7 ± 0.098.5 ± 0.088.8 ± 0.093.4 ± 0.0
Non-Neg95.5 ± 0.098.1 ± 0.092.7 ± 0.095.3 ± 0.0
NELLv1MeanStandard75.2 ± 0.193.8 ± 0.053.4 ± 0.265.7 ± 0.2
Non-Neg93.4 ± 0.088.8 ± 0.099.4 ± 0.093.8 ± 0.0
+ +Table 1: Results for mean-GNNs with standard / non-negative weights, across all datasets. Metrics are computed on the test set and shown with a $95\%$ CI. + +
DatasetTotUnBinMix012345678
LUBM11.61.49.80.429.61.40.60
WN-hier22.615.60702.65.46.61.43.82.20.40.2
WN-sym0.4000.400000.40000
WN-hier_nmhier53.64.6049001.816.217.812.64.800.4
FB237v11361360029136
WN18RRv10000000000000
NELLv11100001
+ +Table 2: Counts of monotonic rules of the form given in Theorem 3. Tot, Un, Bin, Mix are the counts of the total number of rules, rules with only unary atoms, rules with only binary atoms, and rules with a mix of atom arities. Each remaining column $i$ counts the number of rules with $i$ body concepts. + +Rule Extraction and Explanations On all datasets, we use Proposition 4 to check for monotonic rules of the form given in Theorem 3. This procedure is described in Algorithm 1 of Appendix B.1. Given the differing number of unary and binary predicates in each dataset and the exponential growth of the rule space, we check all rules up to a differing number of a body concepts: 4 for LUBM, all 15 for WN-based-datasets, 1 for FB237v1, and 2 for NELLv1. This yields a total possible number of sound rules: 383670 for LUBM, all 360448 for WN18RRv1 and the LogInfer datasets, 56169 for FB237v1, and 216600 for NELLv1. When a rule is a subsumed by another with fewer body atoms, it is not counted. + +We also compute sound explanatory rules of the form given in Theorem 6 for all true positive predictions made on the test set of LUBM. Given that the explanatory rules are often large, we improve the process by first checking if a sound rule of the form given in Theorem 3 derives the fact on the input dataset, and return the rule as an explanation if it does. If no such rule exists, we use rules from Theorem 6. Other strategies for improving the rule quality are discussed in Appendix B.2. + +Results Mean-GNN performance on the test set is shown in Table 1. For completeness, results across all aggregation functions are given in Tables 5 to 7 of Appendix C. We see similar behaviour for mean-GNNs as for sum and max-GNNs when they are restricted to non-negative weights, which matches the behaviour seen by Morris et al. [25] for sum-GNNs and Tena Cucala et al. [35] for max-GNNs. On LUBM, the restriction yields a small decrease in performance. For the monotonic LogInfer datasets, there is a slight increase in performance, and for the non-monotonic one, a substantial decrease, since the dataset explicitly penalises monotonic model behaviour. Finally, on the benchmark datasets, the restriction improves performance, sometimes substantially. This demonstrates that the + +
DatasetExamples of Sound Rules
LUBM□ Publication +∃advisorOf. □ AssociateProfessor +Department □ UndergraduateStudent □ ∃advisorOf. □ FullProfessor
WN-hiermember_of_domain_use(X, Y) → member_meronym(X, Y)
WN-symhas_part(X, Y) ∧ similar_to(X, Y) → derivationally_related_form(X, Y)
WN-hier_nmhierhypernym(X, Y) ∧ synset_domain主題_of(X, Y) → also Seeing(X, Y)
FB237v1T → /music/instrument/instrumentalists(X, Y) +/base/biblioness/bibs_location/state(X, Y) → /film/film/genre(X, Y)
NELLv1organizationterminatedperson(X, Y) ∧ topmemberoforganization(X, Y) → organizationhiredperson(X, Y)
+ +Table 3: Randomly sampled sound monotonic rules of the form given in Theorem 3. + +
Fact RulePublication(http://www.Department10.University0.edu/FullProfessor5/Publication3) +T ⊆ Publication
Fact RuleResearchAssistant(http://www.Department1.University0.edu/GraduateStudent14) +GraduateStudent ⊆ ResearchAssistant
Fact RuleGraduateStudent(http://www.Department12.University0.edu/GraduateStudent48) +ResearchAssistant ⊆ GraduateStudent
Fact RuleGraduateStudent(http://www.Department3.University0.edu/GraduateStudent44) +ResearchAssistant ∩ ∃authorOfPublication. T ⊆ GraduateStudent
Fact RuleCourse(http://www.Department12.University0.edu/Course1) +∃courseTakenBy. T ⊆ ∃hasTeacher. T ⊆ Course
Fact RuleAssistantProfessor(http://www.Department13.University0.edu/AssistantProfessor10) +∃advisorOf. T ⊆ AssistantProfessor
Fact RuleUniversity(http://www.University772.edu) +∃grantedUndergraduateDegreeTo. T ⊆ University
+ +Table 4: Randomly sampled explanatory rules that have successfully been reduced in size, shown with the corresponding facts produced by the MAGNN on LUBM. + +restriction of mean-GNNs to MAGNNs can be done in practice without sacrificing significant model performance, while enabling the extraction of sound rules. We hypothesize that the occasional performance increase seen when restricting the models to non-negative weights is due to (1) it regularizing the model and (2) the dataset patterns not requiring negative weights to be captured. + +In Table 2, we present counts of the number of sound monotonic rules that were obtained on each dataset. On LUBM, WN-hier, WN-hier_nmhier, and FB237v1, we find a number of sound rules with a varying number of body concepts. On average, we found 2 sound rules on LUBM that have empty bodies, and 29 on FB237v1; these rules are clearly absurd, but are what the GNN has learned, highlighting the need for sound rules so that such issues in the models can be exposed. Further to this, we note that almost no sound monotonic rules were obtained on WN-sym, WN18RRv1, or NELLv1. This is especially concerning for WN-sym, which was populated with the consequences of monotonic rules. This behaviour is due to the very restricted form that sound monotonic rules can have when used with the link prediction encoding of Tena Cucala et al. [37], as discussed in Section 3.1. Morris et al. [25], on the other hand, found that all of the rules used to create WN-sym were sound when using sum-GNNs with non-negative weights. We show some sample sound monotonic rules in Table 3, with more given in Table 8 of Appendix C. + +In Table 9 of Appendix C, we give samples of explanatory rules of the form given in Theorem 6. On LUBM, these explanatory rules had 25 body concepts on average. When using the improved + +procedure of first checking for explanatory rules of the form given in Theorem 3, the rules averaged only 11 body concepts. On average, out of 1990 true positive predictions, the process only failed 22 times to explain the predicted fact using a rule from Theorem 3. In Table 4, we give samples of such explanatory rules, which are useful for providing insights into the data, as well as exposing issues with the trained MAGNN. For example, the explanatory rule $\top \sqsubseteq$ Publication exposes a nonsensical rule learned by the MAGNN, which arises as a consequence of publications in LUBM having no incoming edges. Likewise, it is sensible that the MAGNN has learned that research assistants are graduate students, but it has also erroneously learned that all graduate students are research assistants. Finally, note that the fourth rule in the table is subsumed by the third rule — this is because they come from two different MAGNNs, which have learned different sound rules. Table 10 of Appendix C provides a longer list of explanatory rules that have been reduced in size. + +# 5 Discussion + +Our Contribution We considered mean-GNNs with non-negative weights (MAGNNs), showing first that they can express logical functions that go beyond FOL. We proved which ELUQ rules (a class of monotonic ALCQ rules) can be sound for MAGNNs, which turned out to be very limited. The resulting rule space is finite, which enabled us to define a procedure to check for all sound ELUQ rules. We also provided a restricted fragment of FOL, from which sound rules can be constructed to explain any prediction of a MAGNN. In our experiments, we found that mean-GNNs still perform well on benchmark datasets when restricted to having non-negative weights. We also found a variety of sound monotonic rules on half the datasets, with provably almost no sound monotonic rules on the other half. This, alongside several nonsensical sound rules, raise questions about how mean-GNNs can be encouraged to recover the underlying rules in the datasets, and suggests that max- or sum-GNNs may be preferable when provable soundness is required. It also shows the importance of rule extraction techniques, given the good empirical performance of MAGNNs on the test datasets but nonsensical rules they have learned (for example, that everything is a publication). Finally, we computed explanatory rules for predictions made by MAGNNs on the LUBM dataset. + +Logical Explanations for GNNs Other works also extract logical explanations for GNN predictions [4, 5, 20, 28]. However, none of these provide theoretical guarantees that the explanations are sound for the model, only empirical evidence that they are approximately faithful. Pluska et al. [28], for example, learn decision trees from GNNs and then extract logical rules from the decision trees — they are not equivalent to the GNNs they are derived from, so the extracted logical rules are not provably sound. Furthermore, Kohler and Heindorf [20] only consider explanations in EL, which is a fragment of ELUQ. + +Mean-GNN Theory Further works also provide theoretical analyses of GNNs with mean aggregation. Vasileiou et al. [39] provide a generalisation bound for mean-GNNs. Adam-Day et al. [2] show that mean-GNNs used for graph classification tend to a constant function as the sizes of the input graphs tend to infinity (under certain random graph models). Their approach of tending the size to infinity is similar to our technique used in the proof of Theorem 3. In contemporaneous work, Schonherr and Lutz [33] identify the FOL expressivity of mean-GNNs in the uniform (standard) and non-uniform (only on graphs up to a bounded number of nodes) settings. In the uniform setting, they prove that any continuous mean-GNN (that is equivalent to a FOL classifier) has an equivalent logical classifier expressible in alternation-free modal logic (AFML). However, they do not provide a construction of the AFML classifier, a demonstration that such a construction would be obtainable in practice, or a means to recover sound rules if the mean-GNN is not equivalent to a FOL classifier. + +**Limitations** Our work is limited in several ways. First, despite being able to check the infinite space of ELUQ rules for soundness by instead checking finitely many rules of the form given in Theorem 3, the size of this finite space grows exponentially in the number of predicates, making the search intractable as the number of predicates increases (as seen on FB237v1, for example). Also, it is an open question as to whether ELUQ is a maximal monotonic fragment of ALCQ, so there may be other monotonic rules that can be sound for mean-GNNs. Finally, we only consider the $\leq$ operator for the mean-GNN step classification function $c\mathbf{1}s_t$ due to its standard use in prior work [25, 35, 37] and the fact that using $<$ yields an entirely different set of theoretical results (for example, rules with a body of $\exists P.A$ could be sound for such a mean-GNN, without $\exists P.\top$ being sound). + +# Acknowledgments and Disclosure of Funding + +Matthew Morris is funded by an EPSRC scholarship (CS2122_EPSRC_1339049). This work was also supported by Samsung Research UK, the EPSRC projects UKFIRES (EP/S019111/1) and ConCur (EP/V050869/1). The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work: http://dx.doi.org/10. 5281/zenodo.22558. + +For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission. + +# References + +[1] Ralph Abboud, Ismail Ceylan, Thomas Lukasiewicz, and Tommaso Salvatori. Boxe: A box embedding model for knowledge base completion. Advances in Neural Information Processing Systems, 33:9649-9661, 2020. +[2] Sam Adam-Day, Michael Benedikt, Ismail Ceylan, and Ben Finkelshtein. Almost surely asymptotically constant graph neural networks. Advances in Neural Information Processing Systems, 37:124843-124886, 2024. +[3] Veeti Ahvonen, Damian Heiman, Antti Kuusisto, and Carsten Lutz. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: LLMs were not used in the paper. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Proofs + +# A.1 Proposition 1 + +Proposition. There exists a MAGNN $\mathcal{M}$ such that for any dataset $D$ and constant $a\in \mathrm{con}(D)$ , $U(a)\in T_{\mathcal{M}}(D)$ if and only at least half of the neighbours $b$ of $a$ in $D$ are such that $U(b)\in D$ . This logical function cannot be defined in FOL [21]. + +Proof. Consider a MAGNN $\mathcal{M}$ over a signature with only one unary predicate $(U)$ and only one binary predicate $(P)$ , corresponding to colour $c$ in the canonical encoding. Let $\mathcal{M}$ have $L = 1$ (one layer), with $\mathbf{A}_1 = (0)$ , $\mathbf{b}_1 = (0)$ , and $\mathbf{B}_1^c = (1)$ . Finally, let $\sigma_1$ be the identity function and $t_{\mathcal{M}} = 0.5$ . + +Consider a dataset $D$ and constant $a$ in $D$ . Then $\mathbf{v}_L^a$ , the labelling of $v^a$ after $\mathcal{M}$ is applied to $\mathrm{enc}(D)$ , depends only on the number and type of the direct neighbours of $v^a$ . Each neighbour $v^b$ either has $U(b) \in D$ or $U(b) \notin D$ . We end up with + +$$ +\begin{array}{l} \mathbf {v} _ {L} ^ {a} [ 1 ] = \frac {\left| \left\{v ^ {b} \mid \left(v ^ {a} , v ^ {b}\right) \in E ^ {c} \text {a n d} \mathbf {v} _ {L} ^ {b} [ 1 ] = 1 \right\} \right|}{\left| N _ {c} \left(v ^ {a}\right) \right|} \\ = \frac {\left| \left\{b \mid P (a , b) \in D \text {a n d} U (b) \in D \right\} \right|}{\left| \left\{b \mid P (a , b) \in D \right\} \right|} \\ \end{array} +$$ + +Now $\mathbf{v}_L^a [1]\geq 0.5 = t_\mathcal{M}$ if and only if $|\{b\mid P(a,b)\in D$ and $U(b)\in D\} |\geq |\{b\mid P(a,b)\in$ $D$ and $U(b)\notin D\} |.$ I.e. $U(a)\in T_{\mathcal{M}}(D)$ if and only at least half of the neighbours $b$ of $a$ in $D$ are such that $U(b)\in D$ + +However, this logical function cannot be expressed in FOL. To prove this, we consider Libkin [21], who defines the majority predicate $\mathrm{MAJ}(\varphi ,\psi)$ , which tests if all constants satisfying $\varphi$ contains at least half of the constants satisfying $\psi$ . We find that, for every dataset $D$ and constant $a\in \operatorname {con}(D)$ we have $(D,a)\models \mathbf{MAJ}(R(a,x)\land U(x),R(a,x))$ if and only if "at least half of the neighbours $b$ of $a$ in $D$ are such that $U(b)\in D$ " if and only if $U(a)\in T_{\mathcal{M}}(D)$ . + +However, Libkin [21, Corollary 13.17] proves that MAJ queries cannot be defined in an extension of FOL, from which it trivially follows that they cannot be defined in FOL. Thus, $\mathcal{M}$ is equivalent to a logical function that cannot be defined in FOL. + +# A.2 Theorem 3 + +Theorem. Let $\mathcal{M}$ be a MAGNN and $r$ an ELUQ rule that is sound for $\mathcal{M}$ . Then there exists a finite set of rules $R$ such that: + +1. Each $r' \in R$ has the form $\exists P_1. \top \cap \dots \cap \exists P_j. \top \cap A_1 \cap \dots \cap A_k \cap \top \subseteq A_{k+1}$ , where each $P_i$ is a binary predicate, $A_i$ a unary predicate, and $j, k \in \mathbb{N}_0$ . +2. Each $r' \in R$ is sound for $\mathcal{M}$ . +3. $R$ subsumes $r$ . + +Proof. First, notice that $r$ is of the form $C^1 \sqcup \ldots \sqcup C^m \subseteq A$ for ELUQ concepts $C^1, \ldots, C^m$ containing no disjunction in the outer scope (i.e. disjunction is only nested within existential quantification), $m \in \mathbb{N}$ , and atomic concept $A$ . Note that $r$ not containing any disjunctions in the outer scope is covered by $m = 1$ . + +Now, let $R' := \{C^1 \sqsubseteq A, \dots, C^m \sqsubseteq A\}$ . By the operation of rule application, $T_{R'}(D) = T_r(D)$ for all datasets $D$ . Also, each rule in $R'$ is sound for $\mathcal{M}$ , as a consequence of $r$ being sound for $\mathcal{M}$ . We will construct $R$ by including in it a rule for each rule in $R'$ . + +Consider each $r_1 \in R'$ . If $r_1$ contains no $\exists$ or $\geq_n$ operators, then it is already trivially of the form specified in the theorem, so let $r_k = r_1$ and include it in $R$ . Also, as stated above, $r_1$ is sound for $\mathcal{M}$ . Otherwise, we have that $r_1$ is of the form $\geq_n P.C_1 \sqcap C_2 \subseteq A$ , where $P$ is a binary predicate, $C_1$ an ELUQ concept, $C_2$ an ELUQ concept without disjunction, and $n$ a positive integer. + +We now inductively define a sequence $r_2, \ldots, r_k$ of ELUQ rules that such for all $i \in \{1, \ldots, k-1\}$ , + +1. $r_{i + 1}$ is sound for $\mathcal{M}$ +2. $r_{i + 1}$ subsumes $r_i$ + +# Inductive Step + +For this purpose, consider the last defined rule $r_i$ : we define $r_{i+1}$ and prove the above properties. + +If $r_i$ contains no $\exists$ or $\geq_n$ operators, then let $k := i$ and cease the inductive construction. As assumed in the inductive hypothesis, $r_i$ is sound for $\mathcal{M}$ . Otherwise, we have that $r_i$ is of the form $\geq_n P.C_1 \sqcap C_2 \sqsubseteq A$ , where $P$ is a binary predicate, $C_1$ an ELUQ concept, $C_2$ an ELUQ concept without disjunction, and $n$ a positive integer. + +Now let $r_{i+1}$ be the rule $\exists P. \top \square C_2 \subseteq A$ . Trivially, $r_{i+1}$ subsumes $r_i$ , since they have the same head and whenever the body of $r_{i+1}$ is satisfied, so too is the body of $r_i$ . It remains to prove that $r_{i+1}$ is sound for $\mathcal{M}$ . For this purpose, assume to the contrary that $r_{i+1}$ is not sound for $\mathcal{M}$ . + +If for all datasets $D_{1}$ such that $P(a,b) \in D_{1}$ and $(D_{1},a) \models C_{2}$ for some constant $b$ , we have $A(a) \in T_{\mathcal{M}}(D_1)$ , then $r_{i + 1}$ would be sound for $\mathcal{M}$ . This follows from these conditions covering all datasets that satisfy the body of $r_{i + 1}$ . + +Likewise, if for all datasets $D_{1}$ such that $P(a,b)\in D_1$ , $(D_{1},a)\models C_{2}$ , $b\neq a$ , and $b$ is not mentioned in any other fact of $D_{1}$ , we have $A(a)\in T_{\mathcal{M}}(D_1)$ , then $r_{i + 1}$ would be sound for $\mathcal{M}$ . This is because neither mentioning $b$ in another fact of $D_{1}$ nor having $a = b$ will decrease the output of $\mathcal{M}(D_1)$ for node $v^{a}$ , + +So instead there exists some dataset $D_{1}$ such that $P(a,b) \in D_{1}$ , $(D_{1},a) \models C_{2}$ , and $A(a) \notin T_{\mathcal{M}}(D_1)$ for some constants $a \neq b$ , where $b$ is not mentioned in any fact of $D_{1}$ besides $P(a,b)$ . Here follows a representation of $D_{1}$ : + +![](images/679201054148294254ff1b633a1ecfbe12ce7fe47e97499dd630459d547976eb.jpg) + +Now consider $\mathbf{v}_L^a$ , the output for node $v^a$ when $\mathcal{M}$ is applied to $\mathrm{enc}(D_1)$ , and let $1 \leq p \leq \delta$ be in the index of the unary predicate $A$ in the canonical encoding. Since $A(a) \notin T_{\mathcal{M}}(D_1)$ , we must have $\mathbf{v}_L^a[p] < t_{\mathcal{M}}$ . Let $t \coloneqq \frac{t_{\mathcal{M}} - \mathbf{v}_L^a[p]}{2}$ . + +We now define a dataset $D_{m}$ , parameterized by a non-negative integer $m$ . We define $D_{m} = D_{1} \setminus \{P(a, b)\} \cup \{P(a, c_{1}), \dots, P(a, c_{n})\} \cup \{P(a, b_{1}), \dots, P(a, b_{m})\}$ for distinct constants $c_{1}, \dots, c_{n}, b_{1}, \dots, b_{m}$ (all also distinct from $a$ ) and further require that $(D_{m}, c_{1}) \models C_{1}, \dots, (D_{m}, c_{n}) \models C_{1}$ , by including other facts necessary for the satisfaction to hold. We also require that $b_{1}, \dots, b_{m}$ are not mentioned in any other facts of $D_{m}$ . These further requirements may be satisfied in several different ways, but any of them will suffice for the definition of $D_{m}$ . Here follows a representation of $D_{m}$ : + +![](images/978b828be58fe195c4287f22d28d1c8932c260f3c9defba37ad750a4d17dd369.jpg) + +Notice that $(D_{m},a)\models C_{2}$ and $(D_{m},a)\models \geq_{n}P.C_{1}$ , so $A(a)\in T_{r_i}(D_m)$ . But from the soundness of $r_i$ , we also have $A(a)\in T_{\mathcal{M}}(D_m)$ . However, as $m\to \infty$ , the computation in node $v^{a}$ when $\mathcal{M}$ is applied to $\mathsf{enc}(D_m)$ tends to that of $\mathcal{M}$ being applied to $\mathsf{enc}(D_1)$ — this is a consequence of the definition of MAGNNs. Thus, there exists some $m$ such that when considering $\mathbf{v}_L^a$ , the output of $\mathcal{M}$ applied to $\mathsf{enc}(D_m)$ , we have $\mathbf{v}_L^a [p] < t$ (note that this requires the use of continuous activation functions, otherwise it is possible that $\mathbf{v}_L^a [p]\geq t$ for every $m$ ). + +But since $t < t_{\mathcal{M}}$ , we have $\mathbf{v}_L^a[p] < t_{\mathcal{M}}$ . And since $A(a) \in T_{\mathcal{M}}(D_m)$ , we have $\mathbf{v}_L^a[p] \geq t_{\mathcal{M}}$ . This is a contradiction, so $r_{i+1}$ is sound for $\mathcal{M}$ . + +# End of Inductive Construction + +Since $r_1$ contains a finite number of $\exists$ or $\geq_n$ operators and each $r_{i+1}$ has one fewer $\exists$ or $\geq_n$ operator than $r_i$ , this inductive construction is guaranteed to terminate for some $k$ . + +We thus have a sequence $r_1, r_2, \ldots, r_k$ of ELUQ rules such that for all $i \in \{1, \ldots, k-1\}$ , $r_{i+1}$ subsumes $r_i$ . Since $r_{i+1}$ subsumes $r_i$ , for all datasets $D$ we have $T_{r_i}(D) \subseteq T_{r_{i+1}}(D)$ . So $T_{r_1}(D) \subseteq T_{r_2}(D) \subseteq \ldots \subseteq T_{r_k}(D)$ , and thus $T_{r_1}(D) \subseteq T_{r_k}(D)$ . Also, notice that $r_k$ contains no existential quantifiers or disjunction, so it is of the form specified in the theorem. Furthermore, $r_k$ is sound for $\mathcal{M}$ , as was shown above. We include $r_k$ in $R$ . + +# Conclusion + +We now have a set $R$ of rules of the form specified in the theorem, such that each is sound for $\mathcal{M}$ . It remains to be shown that $R$ subsumes $r$ . + +Let $D$ be a dataset. Recall that $T_{r}(D) = T_{R^{\prime}}(D)$ . But for each $r_1 \in R'$ , there exists $r_k \in R$ such that $T_{r_1}(D) \subseteq T_{r_k}(D)$ . Thus $\bigcup_{r_1 \in R'} T_{r_1}(D) \subseteq \bigcup_{r_k \in R} T_{r_k}(D)$ , and so $T_{R^{\prime}}(D) \subseteq T_{R}(D)$ . Hence, we have $T_{r}(D) \subseteq T_{R}(D)$ , so $R$ subsumes $r$ , as required. + +![](images/f197aae14e006c5d3fc9d127e0ea43c653057085eacbc7520a29fd8f1661aa6d.jpg) + +# A.3 Proposition 4 + +Proposition. Let $\mathcal{M}$ be a MAGNN and $r:\exists P_1.\top \sqcap \ldots \sqcap \exists P_j.\top \sqcap A_1\sqcap \ldots \sqcap A_k\subseteq A_{k + 1}$ be a rule of the form given in Theorem 3. Define $D_{base}:= \{A_1(a),\dots,A_k(a),P_1(a,b),\dots,P_j(a,b)\}$ for constants $a\neq b$ . Then $r$ is sound for $\mathcal{M}$ if and only if $A_{k + 1}(a)\in T_{\mathcal{M}}(D_{base})$ . + +Proof. We prove both directions of the if and only if. First, assume $r$ is sound for $\mathcal{M}$ . Notice that $(D_{\mathrm{base}},a)\models B$ , where $B$ denotes the body concepts of $r$ , so we have $A_{k + 1}(a)\in T_r(D_{\mathrm{base}})$ . But since $r$ is sound for $\mathcal{M}$ , $T_{r}(D_{\mathrm{base}})\subseteq T_{\mathcal{M}}(D_{\mathrm{base}})$ and thus $A_{k + 1}(a)\in T_{\mathcal{M}}(D_{\mathrm{base}})$ as required. + +Now assume that $A_{k + 1}(a) \in T_{\mathcal{M}}(D_{\mathrm{base}})$ . We prove that $r$ is sound for $\mathcal{M}$ . For this purpose, let $D$ be an arbitrary dataset: we prove that $T_r(D) \subseteq T_{\mathcal{M}}(D)$ . So let $A_{k + 1}(c) \in T_r(D)$ . + +# Extending The Base Dataset + +Consider the set $D_{\mathrm{splits}}$ , consisting of every dataset $\{A_1(a),\dots,A_k(a),P_1(a,b_1),\dots,P_j(a,b_j)\}$ for (not necessarily mutually distinct) constants $b_1,\ldots ,b_j$ that are distinct to $a$ . Then for each such $D_{1}\in D_{\mathrm{splits}}$ , the computation of $\mathcal{M}(D_1)$ is identical to that of $\mathcal{M}(D_{\mathrm{base}})$ . So $A_{k + 1}(a)\in T_{\mathcal{M}}(D_1)$ . + +Now consider the set $D_{\mathrm{merges}}$ , consisting of every dataset $\{A_1(a), \dots, A_k(a), P_1(a, b_1), \dots, P_j(a, b_j)\}$ for (not necessarily mutually distinct) constants $b_1, \dots, b_j$ that are not necessarily distinct to $a$ . Each $D_2 \in D_{\mathrm{merges}}$ either has $a$ distinct to all $b_1, \dots, b_j$ , or it does not. If it does, then $D_2 \in D_{\mathrm{splits}}$ , in which case $A_{k+1}(a) \in T_{\mathcal{M}}(D_2)$ . Otherwise, $D_2$ has a corresponding $D_1 \in D_{\mathrm{splits}}$ where each $b_i$ such that $b_i = a$ is swapped out for another constant $b_i' \neq a$ . + +Consider $\mathbf{v}_L^a$ and $\mathbf{v}_L^{a'}$ , the outputs for node $v^a$ when $\mathcal{M}$ is applied to $D_{1}$ and $D_{2}$ respectively. Notice that, by the operations of MAGNNs, for all $p \in \{1, \dots, \delta\}$ we have $\mathbf{v}_L^a[p] \leq \mathbf{v}_L^{a'}[p]$ . This is because merging each of the neighbours $b_i'$ of $a$ into $b_i = a$ will never decrease the output of the MAGNN. + +Now let $p$ be the index of $A_{k + 1}$ in the canonical encoding. Then since $\mathbf{v}_L^a[p] \leq \mathbf{v}_L^{a'}[p]$ and $A_{k + 1}(a) \in T_{\mathcal{M}}(D_1)$ , we have $t_{\mathcal{M}} \leq \mathbf{v}_L^a[p] \leq \mathbf{v}_L^{a'}[p]$ , and thus $A_{k + 1}(a) \in T_{\mathcal{M}}(D_2)$ . So for every $D_2 \in D_{\mathrm{merges}}$ , we have $A_{k + 1}(a) \in T_{\mathcal{M}}(D_2)$ . + +# Conclusion + +We now prove that $A_{k + 1}(c) \in T_{\mathcal{M}}(D)$ . Since $A_{k + 1}(c) \in T_r(D)$ , there exists (not necessarily distinct) constants $d_1, \ldots, d_j$ in $D$ such that $D' := \{A_1(c), \ldots, A_k(c), P_1(c, d_1), \ldots, P_j(c, d_j)\} \subseteq D$ . There are potentially multiple such constants $d_i$ that could be used for $D' \subseteq D$ . For each $i \in \{1, \ldots, j\}$ , if there exists constant $d_i \neq a$ such that $P_i(c, d_i) \in D$ , then choose this $d_i$ to be in $D'$ . + +But then, up to the relabelling of the constant $c \mapsto a$ , we have $D' \in D_{\mathrm{merges}}$ , so $A_{k+1}(c) \in T_{\mathcal{M}}(D')$ , since MAGNNs are agnostic to the specific constants used. + +$D$ is a superset of $D'$ . Consider $v^c \in \mathsf{enc}(D')$ and the application of $\mathcal{M}$ to $D'$ and $D$ , resulting in $\mathbf{v}_L^{c'}$ and $\mathbf{v}_L^c$ , respectively. Any unary facts in $D$ that are not in $D'$ will preserve $\mathbf{v}_L^{c'} \leq \mathbf{v}_L^c$ , from the definition of MAGNNs and the canonical encoding. The only binary facts in $D$ that are not in $D'$ that could decrease the computation of $\mathbf{v}_L^{c'}$ (leading to $\mathbf{v}_L^{c'}[p] > \mathbf{v}_L^c[q]$ for some index $q$ ) are of the form $P_i(c, d_i)$ for $c \neq d_i$ , where $D'$ contains some fact $P_i(c, c)$ . However, as was stated in the definition of $D'$ , if $D$ contains a fact $P_i(c, d_i)$ with $c \neq d_i$ , then $D'$ will not contain $P_i(c, c)$ . So, no facts added when extending $D'$ to $D$ will decrease the computation of $\mathbf{v}_L^{c'}$ , and we obtain $\mathbf{v}_L^{c'} \leq \mathbf{v}_L^c$ . + +Now let $p$ be the index of $A_{k + 1}$ in the canonical encoding. Then since $A_{k + 1}(c) \in T_{\mathcal{M}}(D')$ , we have $t_{\mathcal{M}} \leq \mathbf{v}_L^{c'}[p] \leq \mathbf{v}_L^c[p]$ , and thus $A_{k + 1}(c) \in T_{\mathcal{M}}(D)$ , as required. + +![](images/50ab97d94c69628c99ca6d70292a0ca60c779008ef23bfd91d2dcff6908bcd2a.jpg) + +# A.4 Lemma 7 + +We provide a lemma that is used in the proof of Theorem 6. + +Lemma 7. Let $\mathcal{M}$ be a MAGNN, $D_{a}$ a dataset, and $a\in \mathrm{con}(D_a)$ a constant. For any $\ell \in \mathbb{N}_0$ and $c\in \mathrm{con}(D_a)$ , we have $(D_{a},c)\models C_{\ell}^{c}$ , where $C_\ell^c$ is dependent on $D_{a}$ . + +Proof. We prove this by induction on $\ell$ . + +# Base Case + +For $\ell = 0$ and some $c\in \mathrm{con}(D_a)$ , we have $C_0^c = \top \sqcap A_1\sqcap \ldots \sqcap A_k$ , where $A_{1},\dots,A_{k}$ are all atomic concepts $A_{i}$ such that $A_{i}(c)\in D_{a}$ . But then $(D_{a},c)\models A_{i}$ for every $A_{i}$ in the conjunction of $C_0^c$ , so $(D_{a},c)\models C_{0}^{c}$ . + +# Inductive Step + +Consider some $\ell \in \mathbb{N}$ , and assume that the claim holds for $\ell - 1$ . Let $c \in \mathrm{con}(D_a)$ . We have $C_{\ell}^{c} = A_{1} \sqcap \ldots \sqcap A_{k} \sqcap C'$ , where $A_{1}, \ldots, A_{k}$ are all atomic concepts $A_{i}$ such that $A_{i}(c) \in D_{a}$ and $C'$ denotes the remainder of the concept. Similarly to the base case, we obtain $(D_{a}, c) \models A_{1} \sqcap \ldots \sqcap A_{k}$ . + +$C'$ is a conjunction of $\top$ with a concept $C_P$ for each binary predicate $P$ such that there exists $P(c, c_i) \in D_a$ . So consider such a $P$ : we prove that $(D_a, c) \models C_P$ . + +Let $c_{1},\ldots ,c_{n}$ be all the constants $c_{i}\in \mathrm{con}(D_{a})$ such that $P(c,c_i)\in D$ . Then we have $C_P\coloneqq \exists_nP.(C_{\ell -1}^{c_1},\dots,C_{\ell -1}^{c_n})\sqcap \leq_nP.\top$ . Since $c$ has exactly $n$ $P$ -neighbours in $D_{a}$ , we have $(D_a,c)\models \leq n$ $P.\top$ . By induction, $(D_{a},c_{i})\models C_{\ell -1}^{c_{i}}$ for every $i\in \{1,\dots,n\}$ . Also, all $c_{1},\ldots ,c_{n}$ are distinct. So from the semantics of $\exists_{n}$ , $(D_{a},c)\models \exists_{n}P.(C_{\ell -1}^{c_{1}},\dots,C_{\ell -1}^{c_{n}})$ . Thus, we obtain $(D_a,c)\models C_P$ . + +Now since $(D_{a},c)\models C_{P}$ holds for every binary predicate $P$ , we have $(D_{a},c)\models C^{\prime}$ , and thus $(D_{a},c)\models C_{\ell}^{c}$ , as required. + +# A.5 Theorem 6 + +Theorem. For any MAGNN $\mathcal{M}$ , dataset $D_{a}$ , and fact $A(a) \in T_{\mathcal{M}}(D_a)$ , the rule $r:C_L^a \subseteq A$ (dependent on $D_{a}$ ) is sound for $\mathcal{M}$ , and $A(a) \in T_r(D_a)$ . + +Proof. First, we have $A(a) \in T_r(D_a)$ , since $A$ is the head of $r$ and $(D_a, a) \models C_L^a$ follows directly from Lemma 7. It remains to be shown that $r$ is sound for $\mathcal{M}$ . Let $D_b$ be a dataset: to show soundness, we prove that $T_r(D_b) \subseteq T_{\mathcal{M}}(D_b)$ . For this purpose, let $A(b) \in T_r(D_b)$ be an arbitrary fact; we prove $A(b) \in T_{\mathcal{M}}(D_b)$ . + +# Minimal Satisfaction Dataset + +We define a dataset $D_L^a$ in an analogous fashion to the definition of $C_L^a$ , to be the $L$ -hop neighbourhood of the constant $a$ in $D_a$ . + +Given $c \in \operatorname{con}(D_a)$ , $\ell \in \mathbb{N}_0$ , we inductively define a dataset $D_{\ell}^{c}$ by $D_{\ell}^{c} \coloneqq \{A_i(c) \mid A_i(c) \in D_a\}$ . Then, if $\ell = 0$ , we are done: $D_{\ell}^{c}$ has been defined. Otherwise, for each binary predicate $P$ , let $c_1, \ldots, c_n$ be all the constants $c_i \in \operatorname{con}(D_a)$ such that $P(c, c_i) \in D_a$ . Then, if $n > 0$ , extend $D_{\ell}^{c}$ as follows: $D_{\ell}^{c} \coloneqq D_{\ell}^{c} \cup D_{\ell - 1}^{c_1} \cup D_{\ell - 1}^{c_2} \cup \ldots \cup D_{\ell - 1}^{c_n} \cup \{P(c, c_i) \mid P(c, c_i) \in D_a\}$ . $D_{\ell}^{c}$ is extended for each binary predicate $P$ . Note that each $D_{\ell - 1}^{c_i}$ is defined inductively. + +At this point in the proof, we have three datasets to consider: + +- $D_{a}$ — the original input dataset from which a fact $A(a)$ was produced by the MAGNN $T_{\mathcal{M}}$ +- $D_{b}$ — an arbitrary dataset instantiated for the proof of soundness +- $D_L^a$ — the $L$ -hop neighbourhood of the constant $a$ in $D_a$ + +In the remainder of the proof, we show the following in sequence: + +(1) $A(a) \in T_{\mathcal{M}}(D_L^a)$ , since $A(a) \in T_{\mathcal{M}}(D_a)$ — follows from definition of $D_L^a$ as the $L$ -hop neighbourhood. +(2) $A(b) \in T_{\mathcal{M}}(D_b)$ , since $A(a) \in T_{\mathcal{M}}(D_L^a)$ — follows from the fact that any differences between $D_b$ and $D_L^a$ cannot decrease the MAGNN output for $v^a / v^b$ , plus a mapping between constants of $D_b$ and $D_L^a$ . + +(1) Prove $A(a)\in T_{\mathcal{M}}(D_L^a)$ Using $A(a)\in T_{\mathcal{M}}(D_a)$ + +First note that $D_L^a \subseteq D_a$ , since only facts from $D_a$ are included in its definition. Consider $\mathbf{v}_L^a, \mathbf{v}_L^{a'}$ , the output of $\mathcal{M}$ on $D_a, D_L^a$ respectively for the vertex corresponding to $a$ , and let $p$ be the index of $A$ in the canonical encoding. Since $A(a) \in T_{\mathcal{M}}(D_a), \mathbf{v}_L^a[p] \geq t_{\mathcal{M}}$ . However, since $\mathcal{M}$ has only $L$ layers, the computation of $\mathbf{v}_L^a$ is affected only by vertices in the $L$ -hop neighbourhood of $v^a$ . + +Since $D_L^a$ contains exactly the $L$ -hop neighbourhood of $a$ , we thus have $\mathbf{v}_L^a = \mathbf{v}_L^{a'}$ . This implies $\mathbf{v}_L^{a'}[p] \geq t_{\mathcal{M}}$ and thus $A(a) \in T_{\mathcal{M}}(D_L^a)$ . + +(2) Prove $A(b)\in T_{\mathcal{M}}(D_b)$ Using $A(a)\in T_{\mathcal{M}}(D_L^a)$ + +First, we inductively construct a mapping $\pi : \operatorname{con}(D_L^a) \to \operatorname{con}(D_b)$ from constants to constants. Intuitively, this will track "corresponding" constants from the two datasets in the computation of $\mathcal{M}$ . We also assign a "level" to each constant of $\operatorname{con}(D_L^a)$ from the set $\{0, \dots, L\}$ . We perform the induction on $\ell$ from $L$ to 0, simultaneously proving that for each constant $c \in \operatorname{con}(D_L^a)$ of level $\ell$ , we have: + +$$ +\left(D _ {L} ^ {a}, c\right) \vDash C _ {\ell} ^ {c} \text {a n d} \left(D _ {b}, \pi (c)\right) \vDash C _ {\ell} ^ {c} \tag {3} +$$ + +For the base case, we define $\pi(a) = b$ , and let the level of $a$ be $L$ . Since $A(b) \in T_r(D_b)$ , $(D_b, b) \models C_L^a$ , the body of $r$ . So we have $(D_b, \pi(a)) \models C_L^a$ and also trivially $(D_L^a, a) \models C_L^a$ . + +Now consider each constant $c \in \mathrm{con}(D_L^a)$ of level $\ell \geq 1$ ; we define $\pi$ for constants of level $\ell - 1$ . For each binary predicate $P$ , let $c_1, \ldots, c_n$ be all the constants $c_i \in \mathrm{con}(D_a)$ such that $P(c, c_i) \in D_a$ . Let the level of each $c_1, \ldots, c_n$ be $\ell - 1$ . + +Let $d \coloneqq \pi(c)$ . By the inductive assumption, $(D_b, \pi(c)) \models C_\ell^c$ , so we have that $(D_b, d) \models \exists_n P.(C_{\ell-1}^{c_1}, \ldots, C_{\ell-1}^{c_n}) \sqcap \leq_n P. \top$ . So there exist distinct constants $d_1, \ldots, d_n \in \operatorname{con}(D_b)$ such that $(D_b, d_1) \models C_{\ell-1}^{c_1}, \ldots, (D_b, d_n) \models C_{\ell-1}^{c_n}$ and $P(d, d_1) \in D_b, \ldots, P(d, d_n) \in D_b$ . We define $\pi(c_1) = d_1, \ldots, \pi(c_n) = d_n$ . Notice that we have that $(D_b, \pi(c_1)) \models C_{\ell-1}^{c_1}, \ldots, (D_b, \pi(c_n)) \models C_{\ell-1}^{c_n}$ , and trivially from the construction of $D_L^a$ that $(D_L^a, c_1) \models C_{\ell-1}^{c_1}, \ldots, (D_b, c_n) \models C_{\ell-1}^{c_n}$ , as required for the induction proof. + +# Monotonicity + +For each constant $c \in \mathrm{con}(D_L^a)$ and $d \in \mathrm{con}(D_b)$ , let $\mathbf{v}_L^c$ , $\bar{\mathbf{v}}_L^d$ be the outputs when $\mathcal{M}$ is applied to $D_L^a$ and $D_b$ , respectively. Likewise, let $E^o$ , $\bar{E}^o$ denote the edges and $N_o$ , $\bar{N}_o$ the neighbours in $\mathrm{enc}(D_L^a)$ and $\mathrm{enc}(D_b)$ for colour $o$ , respectively. We prove the following claim by induction on $\ell$ from 0 to $L$ . + +Claim: For each $\ell \in \{0,\dots,L\}$ , constant $c\in \mathrm{con}(D_L^a)$ of level $\ell^{\prime}\in \{\ell ,\dots ,L\}$ , and $i\in \{1,\dots ,\delta_{\ell}\}$ , we have $\mathbf{v}_{\ell}^{c}[i]\leq \bar{\mathbf{v}}_{\ell}^{\pi (c)}[i]$ . + +# Base Case + +For the base case $(\ell = 0)$ , let $i \in \{1, \dots, \delta_0\}$ , let $\ell' \in \{0, \dots, L\}$ , and consider each constant $c \in \mathrm{con}(D_L^a)$ of level $\ell'$ . + +As proven in Equation (3), we have $(D_L^a, c) \models C_{\ell'}^c$ and $(D_b, \pi(c)) \models C_{\ell'}^c$ . Recall that $C_{\ell'}^c$ includes $\top \sqcap A_1 \sqcap \ldots \sqcap A_k$ , where $A_1, \ldots, A_k$ are all atomic concepts $A'$ such that $A'(c) \in D_a$ . From the definition of $D_L^a$ , the unary facts in $D_L^a$ that mention $c$ are precisely $A_1(c), \ldots, A_k(c)$ . Then since $(D_b, \pi(c)) \models C_{\ell'}^c$ , we have $A_1(\pi(c)), \ldots, A_k(\pi(c)) \in D_b$ . + +So if $\mathbf{v}_0^c [i] = 1$ , then $A_{i}(c)\in D_{L}^{a}$ , where $A_{i}$ is the unary predicate with index $i$ in the canonical encoding. This implies that $A_{i}(\pi (c))\in D_{b}$ , and thus $\bar{\mathbf{v}}_0^{\pi (c)}[i] = 1$ . This proves the base case, since $\mathbf{v}_0^c [i] = 1\Rightarrow \bar{\mathbf{v}}_0^{\pi (c)}[i] = 1$ and the definition of the canonical encoding let us conclude that $\mathbf{v}_0^c [i]\leq \bar{\mathbf{v}}_0^{\pi (c)}[i]$ . + +# Inductive Step + +For the inductive step, we prove the claim for $\ell \geq 1$ ; to do so, assume the claim holds for $\ell - 1$ . Let $\ell' \in \{\ell, \dots, L\}$ , let $i \in \{1, \dots, \delta_{\ell}\}$ , and consider each constant $c \in \mathrm{con}(D_L^a)$ of level $\ell'$ . We prove that $\mathbf{v}_{\ell}^{c}[i] \leq \bar{\mathbf{v}}_{\ell}^{\pi(c)}[i]$ . Consider the computation of $\mathbf{v}_{\ell}^{c}[i]$ and $\bar{\mathbf{v}}_{\ell}^{\pi(c)}[i]$ : + +$$ +\mathbf {v} _ {\ell} ^ {c} [ i ] = \sigma_ {\ell} \left(\mathbf {b} _ {\ell} [ i ] + \sum_ {j = 1} ^ {\delta_ {\ell - 1}} \mathbf {A} _ {\ell} [ i, j ] \mathbf {v} _ {\ell - 1} ^ {c} [ j ] + \sum_ {o \in \operatorname {C o l}} \sum_ {j = 1} ^ {\delta_ {\ell - 1}} \mathbf {B} _ {\ell} [ i, j ] \text {m e a n} (\{\{\mathbf {u} _ {\ell - 1} [ j ] \mid (v ^ {c}, u) \in E ^ {o} \})\right) \tag {4} +$$ + +$$ +\bar {\mathbf {v}} _ {\ell} ^ {\pi (c)} [ i ] = \sigma_ {\ell} \left(\mathbf {b} _ {\ell} [ i ] + \sum_ {j = 1} ^ {\delta_ {\ell - 1}} \mathbf {A} _ {\ell} [ i, j ] \bar {\mathbf {v}} _ {\ell - 1} ^ {\pi (c)} [ j ] + \sum_ {o \in \operatorname {C o l}} \sum_ {j = 1} ^ {\delta_ {\ell - 1}} \mathbf {B} _ {\ell} [ i, j ] \text {m e a n} (\{\{\bar {\mathbf {u}} _ {\ell - 1} [ j ] \mid (v ^ {\pi (c)}, u) \in \bar {E} ^ {o} \})\right) \tag {5} +$$ + +From the induction hypothesis, since $c$ is of level $\ell' \geq \ell - 1$ , we have $\mathbf{v}_{\ell - 1}^c[j] \leq \bar{\mathbf{v}}_{\ell - 1}^{\pi(c)}[j]$ . Since all else in the above equations is equal, it remains to be shown that, for all colours $o \in \mathrm{Col}$ and $j \in \{1, \dots, \delta_{\ell - 1}\}$ : + +$$ +\operatorname {m e a n} \left(\left\{\left. \left\{\mathbf {u} _ {\ell - 1} [ j ] \right| (v ^ {c}, u) \in E ^ {o} \right\} \right\}\right) \leq \operatorname {m e a n} \left(\left\{\left. \left\{\bar {\mathbf {u}} _ {\ell - 1} [ j ] \right| (v ^ {\pi (c)}, u) \in \bar {E} ^ {o} \right\} \right\}\right) +$$ + +So let $o \in \mathrm{Col}$ , $j \in \{1, \dots, \delta_{\ell-1}\}$ , and let $P$ be the binary predict corresponding to colour $o$ . Now if there exists no $c_k \in \mathrm{con}(D_a)$ such that $P(c, c_k) \in D_a$ , then by the definition of $E^o$ as the edges from $\mathsf{enc}(D_L^a)$ , we have $E^o = \emptyset$ and thus mean $(\{\{\mathbf{u}_{\ell-1}[j] \mid (v^c, u) \in E^o\}\}) = 0$ . So the inequality trivially holds, since each $\bar{\mathbf{u}}_{\ell-1}[j]$ is non-negative. + +Thus, instead consider all $c_{1},\ldots ,c_{n}\in \mathrm{con}(D_{a})$ such that $P(c,c_k)\in D_a$ . Note that each such $c_{k}$ has level $\ell^{\prime} - 1$ . Then since $\ell^{\prime} - 1\geq \ell -1$ , from the inductive hypothesis, we have $\mathbf{v}_{\ell -1}^{c_k}[j]\leq \bar{\mathbf{v}}_{\ell -1}^{\pi (c_k)}[j]$ for each $c_{k}$ . + +Furthermore, since $c \in \mathrm{con}(D_L^a)$ has level $\ell'$ , we obtain $(D_b, \pi(c)) \models C_{\ell'}^c$ from Equation (3). Thus, we have $(D_b, \pi(c)) \models \exists_n P.(C_{\ell' - 1}^{c_1}, \dots, C_{\ell' - 1}^{c_n}) \cap \leq_n P. \top$ . So there exist exactly $n$ distinct constants $d_1, \dots, d_n \in \mathrm{con}(D_b)$ such that $P(\pi(c), d_1) \in D_b, \dots, P(\pi(c), d_n) \in D_b$ . However, recall from the definition of $\pi$ , that these constants are precisely $d_1 = \pi(c_1), \dots, d_n = \pi(c_n)$ . + +From the canonical encoding, we thus have a one-to-one correspondence between the sets $\{v^{c_k} \mid (v^c, v^{c_k}) \in E^o\}$ and $\{v^{\pi(c_k)} \mid (v^{\pi(c_k)}, v^{\pi(c_k)}) \in \bar{E}^o\} = \{v^{d_k} \mid (v^{\pi(c_k)}, v^{d_k}) \in \bar{E}^o\}$ . Also, as stated above, from the inductive hypothesis, we have $\mathbf{v}_{\ell-1}^{c_k}[j] \leq \bar{\mathbf{v}}_{\ell-1}^{\pi(c_k)}[j]$ for each $c_k$ . Combining these, we obtain the inequality $\text{mean}(\{\{\mathbf{u}_{\ell-1}[j] \mid (v^c, u) \in E^o\}\}) \leq \text{mean}(\{\{\bar{\mathbf{u}}_{\ell-1}[j] \mid (v^{\pi(c)}, u) \in \bar{E}^o\}\})$ . This allows us to conclude $\mathbf{v}_{\ell}^{c}[i] \leq \bar{\mathbf{v}}_{\ell}^{\pi(c)}[i]$ , as required. + +# Conclusion + +We have shown that for each $\ell \in \{0,\dots,L\}$ , constant $c\in \mathrm{con}(D_L^a)$ of level $\ell^{\prime}\in \{\ell ,\dots,L\}$ , and $i\in \{1,\dots,\delta_{\ell}\}$ , we have $\mathbf{v}_{\ell}^{c}[i]\leq \bar{\mathbf{v}}_{\ell}^{\pi (c)}[i]$ . + +Now let $p$ be the index of $A$ in the canonical encoding. We obtain $\mathbf{v}_L^a[p] \leq \bar{\mathbf{v}}_L^{\pi(a)}[p]$ , and thus $\mathbf{v}_L^a[p] \leq \bar{\mathbf{v}}_L^b[p]$ , since $b = \pi(a)$ and $a$ is of level $L$ . But since $A(a) \in T_{\mathcal{M}}(D_L^a)$ , $\mathbf{v}_L^a[p] \geq t_{\mathcal{M}}$ , thus $\bar{\mathbf{v}}_L^b[p] \geq t_{\mathcal{M}}$ , and finally $A(b) \in T_{\mathcal{M}}(D_b)$ . + +# B Algorithms + +# B.1 Checking Soundness of ELUQ rules + +Algorithm 1: Check the soundness of all rules of the form given in Theorem 3 for MAGNN $\mathcal{M}$ +Input : MAGNN $\mathcal{M}$ +Output: $S$ : set of sound rules for $\mathcal{M}$ of the form given in Theorem 3 that together subsume every ELUQ rule that is sound for $\mathcal{M}$ +1 $S \gets \emptyset$ // result set +2 foreach $n \in \{1, \ldots, \delta + |\mathrm{Col}|\}$ do +3 // consider rules whose body contains exactly $n$ concept atoms +foreach candidate rule $r$ of form + $r: \exists P_1. \top \square \dots \square \exists P_j. \top \square A_1 \square \dots \square A_k \sqsubseteq A_{k+1}$ +with exactly $n$ body concepts do +4 // check subsumption by a sound (smaller) rule +if $\exists r' \in S$ with head $A_{k+1}$ whose body concepts are contained in those of $r$ then +5 $S \gets S \cup \{r\}$ +6 continue +7 // construct dataset with fresh constants $a \neq b$ $D_{\text{base}} \gets \{A_1(a), \dots, A_k(a), P_1(a,b), \dots, P_j(a,b)\}$ +8 if $A_{k+1}(a) \in T_{\mathcal{M}}(D_{\text{base}})$ then +9 $S \gets S \cup \{r\}$ +10 return $S$ + +# B.2 Refining Explanations + +There are two primary issues with the rules defined in Theorem 6. + +First, the size of the rules can be large in practice, making them difficult to read, and also limiting their ability to generalise to other datasets. The size of the rule can be iteratively refined by pruning facts from $D_{a}$ and then recomputing $C_L^a$ . + +More precisely, for $D_a' \coloneqq D_a \setminus \{\alpha\}$ for some unary fact $\alpha \in D_a$ such that $A(a) \in T_{\mathcal{M}}(D_a')$ , the rule $r': C_L^a \subseteq A$ (dependent on $D_a'$ ) is still sound for $\mathcal{M}$ , by the same argument made in Theorem 6. This can be done iteratively, until we find that $A(a) \notin T_{\mathcal{M}}(D_a')$ . This works similarly for $D_a' \coloneqq D_a \setminus D_\alpha$ , where $D_\alpha = \{P(c, c_1), \dots, P(c, c_n)\}$ for some binary predicate $P$ and constants $c, c_1, \dots, c_n$ , such that there is no fact $P(c, d) \in D_a$ with $P(c, d) \notin D_\alpha$ , and $A(a) \in T_{\mathcal{M}}(D_a')$ . Note that there are many different choices for $\alpha$ and $D_\alpha$ in each iteration, so the space should be searched to find explanatory rules with minimal size. Only facts $\alpha$ and sets $D_\alpha$ within the $L$ -hop neighbourhood of $a$ should be considered for pruning, as this is what will decrease the number of concepts in the body $C_L^a$ . + +The second issue is that the inclusion of the $\leq_{n} P \top$ term limits their ability to produce facts on other datasets with higher numbers of neighbours. As a possible solution, we can substitute $n$ for some $m > n$ . Consider a dataset $D_{a}$ and fact $A(a) \in T_{\mathcal{M}}(D_{a})$ . Instead of defining $C_{\ell}^{c} \coloneqq \dots \sqcap \leq_{n} P \top$ in Definition 5, we can define $C_{\ell}^{c} \coloneqq \dots \sqcap \leq_{m} P \top$ , for some $m \geq n$ . Any $m$ can be chosen such that extending $D_{a}$ with $D_{\ell}^{c} \coloneqq \{P(c, d_{1}), \dots, P(c, d_{m - n})\}$ for fresh constants $d_{1}, \dots, d_{m - n}$ still leads to $A(a) \in T_{\mathcal{M}}(D_{a} \cup D_{\ell}^{c})$ . If the above is done for multiple constants $c_{1}, \dots, c_{k}$ and $\ell_{1}, \dots, \ell_{k}$ , it must be ensured that $A(a) \in T_{\mathcal{M}}(D_{a} \cup D_{\ell_{1}}^{c_{1}} \cup \dots \cup D_{\ell_{k}}^{c_{k}})$ . + +The simple approach we take in this paper is as follows. Given a predicted fact $A(a)$ on a dataset $D$ , we first identify all atoms $A_{1},\ldots ,A_{n}$ such that $A_{i}(a)\in D$ , and check if the rule $A_{1}\sqcap \dots \sqcap A_{n}\sqsubseteq A$ is sound using Proposition 4. If it is not, we then identify all binary predicates $P_{1},\ldots ,P_{m}$ such that $P_{i}(a,b)\in D$ for any constant $b$ , and check if the rule $A_{1}\sqcap \dots \sqcap A_{n}\sqcap \exists P_{1}.\top \sqcap \dots \sqcap \exists P_{m}.\top \sqsubseteq A$ is sound using Proposition 4. If either of these rules is sound, it explains the prediction $A(a)$ on $D$ . + +
DatasetAggWeights%Acc%Prec%RecF1
LUBMMeanStandard97.1 ± 0.096.9 ± 0.097.2 ± 0.097.1 ± 0.0
Non-Neg91.5 ± 0.087.8 ± 0.096.4 ± 0.091.9 ± 0.0
MaxStandard95.7 ± 0.095.3 ± 0.096.3 ± 0.095.8 ± 0.0
Non-Neg91.6 ± 0.088 ± 0.096.3 ± 0.092 ± 0.0
SumStandard96.1 ± 0.095.9 ± 0.096.2 ± 0.096.1 ± 0.0
Non-Neg91 ± 0.088.1 ± 0.094.9 ± 0.091.4 ± 0.0
+ +Table 5: Results for GNNs across various aggregation functions and standard / non-negative weights, for the LUBM dataset. Metrics are computed on the test set and shown with a 95% CI. + +
DatasetAggWeights%Acc%Prec%RecF1
WN-hierMeanStandard99.5 ± 0.099.5 ± 0.099.5 ± 0.099.5 ± 0.0
Non-Neg99.8 ± 0.099.7 ± 0.099.9 ± 0.099.8 ± 0.0
MaxStandard99.3 ± 0.099.5 ± 0.099.1 ± 0.099.3 ± 0.0
Non-Neg99.9 ± 0.099.8 ± 0.0100 ± 0.099.9 ± 0.0
SumStandard98.9 ± 0.099.5 ± 0.098.3 ± 0.098.9 ± 0.0
Non-Neg98.8 ± 0.097.6 ± 0.0100 ± 0.098.8 ± 0.0
WN-symMeanStandard99.4 ± 0.099.5 ± 0.099.3 ± 0.099.4 ± 0.0
Non-Neg100 ± 0.0100 ± 0.0100 ± 0.0100 ± 0.0
MaxStandard99.3 ± 0.099.1 ± 0.099.5 ± 0.099.3 ± 0.0
Non-Neg100 ± 0.0100 ± 0.0100 ± 0.0100 ± 0.0
SumStandard99.1 ± 0.099.3 ± 0.098.8 ± 0.099.1 ± 0.0
Non-Neg100 ± 0.0100 ± 0.0100 ± 0.0100 ± 0.0
WN-hier_nmhierMeanStandard86.2 ± 0.084.7 ± 0.088.4 ± 0.086.5 ± 0.0
Non-Neg71.6 ± 0.080.1 ± 0.158.8 ± 0.167.2 ± 0.0
MaxStandard86.9 ± 0.085.1 ± 0.089.5 ± 0.087.2 ± 0.0
Non-Neg69.5 ± 0.070.1 ± 0.068.4 ± 0.069.2 ± 0.0
SumStandard89.5 ± 0.088.1 ± 0.091.4 ± 0.089.7 ± 0.0
Non-Neg66.9 ± 0.073.5 ± 0.159.3 ± 0.262.7 ± 0.1
+ +Table 6: Results for GNNs across various aggregation functions and standard / non-negative weights, for the LogInfer datasets. Metrics are computed on the test set and shown with a $95\%$ CI. + +# C Full Results + +Full results for mean, sum, and max aggregation are given in Tables 5 to 7. Randomly sampled sound monotonic rules of the form given in Theorem 3 are shown in Table 8. Randomly sampled explanatory rules for predictions on LUBM of the form given in Theorem 6 are shown in Table 9. + +Randomly sampled explanatory rules that have been reduced in size are shown in Table 10. The final entry in the table is an example of a predicted fact where the size reduction process failed and fell back to using Theorem 6. + +On average, out of 1990 true positive predictions, the model only failed 22 times to explain the predicted fact using a rule from Theorem 3. For 4 of our 5 trained MAGNNs, it only failed 10 times each. For the other MAGNN, it failed 72 times. One factor that brought up the average number of body concepts is that some of the explanations were very large: for example, the rule that explained the prediction Department(http://www.Department2.University0.edu) had 1914 body concepts. + +
DatasetAggWeights%Acc%Prec%RecF1
FB237v1MeanStandard68.7 ± 0.095.4 ± 0.039.3 ± 0.055.7 ± 0.0
Non-Neg71.8 ± 0.075.4 ± 0.064.8 ± 0.069.7 ± 0.0
MaxStandard67 ± 0.094.9 ± 0.035.9 ± 0.052 ± 0.0
Non-Neg75.8 ± 0.085.9 ± 0.163.4 ± 0.072.6 ± 0.0
SumStandard68.4 ± 0.099.7 ± 0.036.8 ± 0.053.8 ± 0.0
Non-Neg73.8 ± 0.079.8 ± 0.164.5 ± 0.071.2 ± 0.0
WN18RRv1MeanStandard93.7 ± 0.098.5 ± 0.088.8 ± 0.093.4 ± 0.0
Non-Neg95.5 ± 0.098.1 ± 0.092.7 ± 0.095.3 ± 0.0
MaxStandard93.6 ± 0.097 ± 0.090.1 ± 0.093.4 ± 0.0
Non-Neg95.5 ± 0.098.7 ± 0.092.1 ± 0.095.3 ± 0.0
SumStandard93.9 ± 0.096.1 ± 0.091.6 ± 0.093.8 ± 0.0
Non-Neg94.9 ± 0.097 ± 0.092.7 ± 0.094.8 ± 0.0
NELLv1MeanStandard75.2 ± 0.193.8 ± 0.053.4 ± 0.265.7 ± 0.2
Non-Neg93.4 ± 0.088.8 ± 0.099.4 ± 0.093.8 ± 0.0
MaxStandard51.2 ± 0.018.8 ± 0.43.5 ± 0.15.9 ± 0.1
Non-Neg93.2 ± 0.092.4 ± 0.094.1 ± 0.093.2 ± 0.0
SumStandard51.8 ± 0.018.3 ± 0.44.9 ± 0.17.8 ± 0.2
Non-Neg92 ± 0.086.3 ± 0.0100 ± 0.092.6 ± 0.0
+ +Table 7: Results for GNNs across various aggregation functions and standard / non-negative weights, for the standard benchmark datasets. Metrics are computed on the test set and shown with a $95\%$ CI. + +
DatasetSample Sound Rules
LUBM□ Publication +□ UndergraduateStudent +authorOfPublication. □ ResearchAssistant +advisorOf. □ AssociateProfessor +courseTakenBy. □ hasTeacher. □ Course +Department □ UndergraduateStudent □ advisorOf. □ FullProfessor
WN-hiermember_of_domain_usage(X, Y) → member_meronym(X, Y) +derivationally_related_form(X, Y) ∧ _similar_to(X, Y) → +.synset_domain_topic_of(X, Y) +also_see(X, Y) ∧ _similar_to(X, Y) ∧ _verb_group(X, Y) → +instance_hypernym(X, Y)
WN-symhas_part(X, Y) ∧ _similar_to(X, Y) +derivationally_related_form(X, Y) →
WN-hier_nmhierhas_part(X, Y) ∧ _member_meronym(X, Y) +member_of_domain_usage(X, Y) +has_part(X, Y) ∧ _verb_group(X, Y) → +member_of_domain_usage(X, Y) +hypernym(X, Y) ∧ synset_domain_topic_of(X, Y) → also_see(X, Y)
FB237v1T → /music/instrument/instrumentalists(X, Y) +T → /people/person/sibling_s./people/sibling relatonship/sibling(X, Y) +base/biblioness/bibs_location/state(X, Y) → /film/film/genre(X, Y) +olympics/olympic Games/sports(X, Y) → +base/popstra/celebrity/dated/.base/popstra/dated/participant(X, Y)
NELLv1organizationterminatedperson(X, Y) ∧topmemberoforganization(X, Y) → organizationhiredperson(X, Y) +worksfor(X, Y) ∧ organizationterminatedperson(X, Y) → organizationhiredperson(X, Y)
+ +Table 8: Randomly sampled sound monotonic rules of the form given in Theorem 3. + +
Fact RulePublication(http://www.Department2.University0.edu/AssociateProfessor7/Publication0) +T □ Publication
Fact RuleUniversity(http://www.University370.edu) +T □ ∃31grantedUndergraduateDegreeTo.(GraduateStudent □ ∃1authorOfPublication.(Publication) □ ≤1authorOfPublication. T) □ ≤1grantedUndergraduateDegreeTo. T □ University
Fact RuleGraduateStudent(http://www.Department6.University0.edu/GraduateStudent29) +T □ ∃4authorOfPublication. (T, T, Publication, Publication) □ ≤4authorOfPublication. T □ GraduateStudent
Fact RuleTeachingAssistant(http://www.Department10.University0.edu/GraduateStudent46) +GraduateStudent □ ∃3authorOfPublication. (T, T, Publication) □ ≤3authorOfPublication. T □ TeachingAssistant
Fact RuleGraduateStudent(http://www.Department14.University0.edu/GraduateStudent2) +ResearchAssistant □ ∃5authorOfPublication. (T, Publication, Publication, Publication, Publication) □ ≤5authorOfPublication. T □ GraduateStudent
Fact RuleUndergraduateStudent(http://www.Department5.University0.edu/UndergraduateStudent87 +T □ UndergraduateStudent
Fact RuleTeachingAssistant(http://www.Department6.University0.edu/GraduateStudent48) +T □ ∃3authorOfPublication. (T, Publication, T) □ ≤3authorOfPublication. T □ TeachingAssistant
Fact RuleGraduateStudent(http://www.Department6.University0.edu/GraduateStudent1) +T □ ∃1authorOfPublication. (T) □ ≤1authorOfPublication. T □ GraduateStudent
Fact RuleCourse(http://www.Department14.University0.edu/Course39) +T □ ∃1courseTakenBy. (T, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, UndergraduateStudent, Undergraduatestudent. (AssistantProfessor □ ∃7advisorOf. (UndergraduateStudent, UndergraduateStudent, T, T, TeachingAssistant □ GraduateStudent GraduateStudent, T) □ ≤7advisorOf. T □ ∃5authorOfPublication. (Publication, Publication, Publication, Publication) □ ≤5authorOfPublication. T) □ ≤1hasTeacher. T □ Course
Fact RuleResearchAssistant(http://www.Department11.University0.edu/GraduateStudent71) +T □ ∃4authorOfPublication. (Publication, Publication, Publication, Publication) □ ≤4authorOfPublication. T □ ResearchAssistant
Fact RuleCourse(http://www.Department14.University0.edu/Course46) +T □ ∃9courseTakenBy. (T, T, UndergraduateStudent, T, UndergraduateStudent, T, UndergraduateStudent, T, UndergraduateStudent) □ ≤9courseTakenBy. T □ ∃1hasTeacher. (Lecturer) □ ≤1hasTeacher. T □ ∃1hasTeachingAssistant. (TeachingAssistant □ GraduateStudent □ ∃5authorOfPublication. (Publication, T, Publication, Publication, T) □ ≤5authorOfPublication. T) □ ≤1hasTeachingAssistant. T □ Course
Fact RuleUniversity(http://www.University138.edu) +T □ ∃2grantedUndergraduateDegreeTo. (GraduateStudent □ ∃1authorOfPublication. (Publication) □ ≤1authorOfPublication. T, GraduateStudent □ ∃5authorOfPublication. (T, Publication, T, Publication, Publication) □ ≤5authorOfPublication. T) □ ≤2grantedUndergraduateDegreeTo. T □ University
+ +Table 9: Randomly sampled explanatory rules of the form given in Theorem 6, shown with the corresponding facts produced by the MAGNN. + +
Fact +RulePublication(http://www.Department10.University0.edu/FullProfessor5/Publication3) +T □ Publication
Fact +RuleUndergraduateStudent(http://www.Department8.University0.edu/UndergraduateStudent170) +T □ UndergraduateStudent
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+ +Table 10: Randomly sampled explanatory rules that have been reduced in size, shown with the corresponding facts produced by the MAGNN. The rules are grouped by similar bodies / heads. 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Prior work has focused on minimizing the price of explainability for specific clustering objectives, lacking a general method to fit an explanation tree to any given clustering, without restrictions. In this work, we propose a new and generic approach to explainable clustering, based on spectral graph partitioning. With it, we design an explainable clustering algorithm that can fit an explanation tree to any given non-explainable clustering, or directly to the dataset itself. Moreover, we show that prior algorithms can also be interpreted as graph partitioning, through a generalized framework due to [44] wherein cuts are optimized in two graphs simultaneously. Our experiments show the favorable performance of our method compared to baselines on a range of datasets. + +# 1 Introduction + +As machine learning increasingly permeates daily life and forms the basis for consequential decision making in the real world, explaining its outputs in a manner that is interpretable to humans often becomes imperative. In a recent influential work, Moshkovitz et al. [33] proposed a model of explainability in clustering. In their model, a clustering of points in a feature space $\mathbb{R}^d$ is explainable if it is described by a binary decision tree, where each internal node corresponds to thresholding the points along a single coordinate. Thus, the assignment of points to clusters can be described by a sequence of individual feature thresholds, arguably making it easy to explain and interpret. See Figure 1 for illustration. + +![](images/52965bed30537987f257720c78809f6330d5504b73dc47f33b221169bf988f9d.jpg) +Figure 1: Illustration of explainable clustering. Clusters are generated from three gaussians. The dashed lines on the left and the decision tree on the right define the explainable clustering regions, with some points attributed to the wrong cluster. + +![](images/08653502e36637ffce1dee9040b84891883a0b4464d87e0628990b03fed6593e.jpg) + +Moshkovitz et al. [33] presented the Iterative Mistake Minimization (IMM) algorithm, which takes as input an already computed $k$ -medians or $k$ -means clustering of the data, called the reference clustering, and "rounds" into an explainable clustering. It works by fitting the reference clustering with a decision tree that greedily minimizes wrong point-to-cluster assignments. They proved that the loss in clustering cost, called the price of explainability, can be bounded as a function of $k$ . This has led to surge of theoretical work on bounding the price of explainability, culminating in tight bounds for $k$ -medians and nearly tight bounds for $k$ -means [26, 28, 30, 31, 32, 12, 7, 19, 20, 4, 27]. + +This voluminous body of work has so far mostly focused on $k$ -medians and $k$ -means. These methods require the reference clustering to be endowed with centroids in order to work. This fails to capture widely used notions of clustering that do not produce centroids, like kernel $k$ -means or spectral clustering [39, 10, 46], which one might wish to use as the reference clustering. [15] recently took + +![](images/6f9b4adb64904a8647bbc0b56b298ec34ccba0a8c3e3903696cd1f6e31e6dc8a.jpg) +Figure 2: Left: The two-moons dataset admits an explainable clustering with small error. $k$ -Means clustering will fail to capture the moons if used as reference clustering. Kernel $k$ -means and spectral clustering capture the moons correctly, and can be rounded to the optimal explainable clustering. Right: A 3-way clustering example from [33]. The horizontal cut leads to an error-free explainable clustering if chosen at the first step, but CART will select the error-heavy vertical cut first. + +![](images/6782a46c363b98076ee571f9f7bfb3c067dd39083df7de78913a101233390acd.jpg) + +a first step in this direction, introducing the Kernel IMM algorithm, which extends IMM to handle kernel $k$ -means as the reference clustering, for certain types of kernels. + +Ideally, one would want a generic explainable clustering method that can be composed over any given reference clustering, irrespective of how it was computed, whether it has centroids, or any other constraint. The only such approach at present is through repurposing classical methods for supervised classification with decision trees, which predate the definition of explainable clustering by decades. CART [6, 36] is a method for decision tree learning over a labeled dataset, which outputs a tree with nodes corresponding to feature thresholds and leaves corresponding to class labels, with the aim of minimizing classification errors with respect to supervised labels. Thus, given any reference clustering, its cluster assignments can be viewed as supervised labels, and CART can be used to round it into an explainable clustering. + +Limitations of current methods. As discussed above, explainable clustering methods that require a centroid-based reference clustering, like $k$ -means, are unable to handle cluster structures not captured by centroids. This was the motivation in [15], who gave the example of the classical "two moons" dataset, depicted in Figure 2 (left). Since $k$ -means fails to capture the moon structure, any explainable clustering method built on it as the reference will fail as well. On the other hand, kernel $k$ -means and spectral clustering capture the moons correctly, and can be rounded to the explainable rounding depicted in the figure, with a small number of errors. + +While the Kernel IMM algorithm does not require centroids, it is limited to kernel $k$ -means with specific kernels, and varies by the specific kernel. Hence, it cannot be used as a generic method. Furthermore, the method is rather complex, and is evaluated in [15] on datasets of size only up to hundreds of points, leaving its scalability unclear. Our larger scale experiments indeed show that Kernel IMM becomes infeasible on bigger data. + +The CART algorithm, as mentioned above, can be repurposed for explainable clustering even though it is not designed for this task. To demonstrate its drawbacks, [33] gave a toy example where CART fails to find a simple error-free explainable clustering. It is shown in Figure 2 (right), and we revisit this example in more detail in Section 3.3. There have been some questions whether this failure mode is representative of behavior on real data, or merely a pathological case not encountered in practice. Preliminary small scale results in [15] implied that CART performs well empirically. However, our larger scale experiments will show that CART indeed suffers low performance on real data. + +The upshot is that despite a large body of work, there is no generic method for explainable clustering, which is oblivious to the type of reference clustering and robust across datasets. This is the gap that we address in this work. + +Our results. We introduce a new approach to explainable clustering, based on spectral graph techniques. Cheeger's inequality relates the eigenvalues of a graph to its optimal cut conductance, a fundamental graph partitioning objective that yields high-quality cuts. This useful fact is widely used + +in algorithms. A geometric consequence of it, stated in Theorem 2.2, relates coordinate cuts in $\mathbb{R}^d$ to cut conductances in the graph, provided that the graph usefully captures geometric relations between the data points in $\mathbb{R}^d$ . This hints at a relevance to explainable clustering. + +We use this connection in two ways: + +- SPEx-Clique: We may describe any given reference clustering with a graph that contains a clique over each cluster. Applying the spectral result iteratively to this graph leads to an algorithm that can round any given reference clustering, without limitations, into an explainable clustering. +- SPEx-kNN: We may also use the spectral result on a graph built directly on the dataset in $\mathbb{R}^d$ , like the $k$ -nearest neighbor graph. This leads to a "reference-free" algorithm, which computes an explainable clustering directly from the data, without using a reference clustering at all. + +To gain more insight into existing methods and their relation to ours, we show they are captured by a generalized graph partitioning framework of non-uniform sparse cuts [44], allowing us to view them through a unified analytic lens. Namely, prior methods can be seen as different choices of graphs to describe the reference clustering. + +Our experiments on a range of datasets and baseline shed light on the empirical performance of explainable clustering methods and showcases the advantage of our approach. + +# 1.1 Related Work + +Much work has focused on proving multiplicative bounds on the price of explainability for $k$ -medians and $k$ -means. In deterministic algorithms, IMM [33] achieved $O(k)$ for $k$ -medians and $O(k^2)$ for $k$ -means. [12] improved the $k$ -means bound to $O(k \log k)$ , with an algorithm we will call EMN. + +Randomized methods have enabled better bounds. For $k$ -medians, a sequence of works analyzed a natural randomized procedure and ultimately obtained a tight bound of $(1 + o(1))\ln k$ [30, 32, 12, 19, 20]. For $k$ -means, [20] achieved $O(k\ln \ln k)$ , which is tight up to the $\ln \ln k$ term. Some works have also shown better bounds in the low-dimensional data regime [26, 12, 7], and studied the computational complexity of approximating the optimal explainable clustering [4, 20, 27]. + +Some works have used modified or extended definitions of tree-based explainable clustering [16, 18, 25, 40, 9, 45, 35]. In particular, [18] generalized the definition far beyond [33], allowing oblique (hyperplane) cuts that involve multiple coordinates rather than just single coordinate cuts. This greatly increases the expressive capacity of the resulting clustering—in fact, [18] show it can exactly capture any reference $k$ -means clustering, without any "price of explainability"—though arguably at the price of rendering the clustering less explainable. While defining explainability in clustering remains an open-ended question, in this work we focus on the original explainability model of [33]. + +In the broader context of explainable machine learning, the model of [33] is an example of an intrinsically explainability method, where the given model (in this case, the reference clustering) is approximated by a different model (in this case, the explainable clustering tree) on which structural explainability constraints are imposed. This is in contrast with post-hoc explainability methods, which aim to endow the given reference model with explanations without modifying it. + +# 2 Spectral Explainable Clustering + +We begin with some necessary background and notation on graph partitioning. Let $G(X, E, w)$ be an undirected graph. Given a strict non-empty subset $S \subset X$ , let $e_G(S, X \setminus S)$ denote the sum of edge weights crossing between $S$ and $X \setminus S$ , and let $\mathrm{vol}_G(S)$ denote the sum of weighted degrees of nodes in $S$ (called the volume of $S$ ). + +There are two standard notions for quantifying the outer connectivity of $S$ within $G$ (see [46]): + +$$ +\phi_ {G} (S) = \frac {e _ {G} (S , X \setminus S)}{| S |}; \psi_ {G} (S) = \frac {e _ {G} (S , X \setminus S)}{\operatorname {v o l} _ {G} (S)} \tag {1} +$$ + +$\phi_G(S)$ is called the sparsity of $S$ , while $\psi_G(S)$ is called the conductance of $S$ . Both notions have analogues for considering $S$ and its complement $X \setminus S$ as a two-way cut: + +$$ +\Phi_ {G} (S) = \frac {e _ {G} (S , X \setminus S)}{\frac {1}{| X |} \cdot | S | \cdot | X \setminus S |}; \Psi_ {G} (S) = \frac {e _ {G} (S , X \setminus S)}{\frac {1}{\operatorname {v o l} _ {G} (X)} \cdot \operatorname {v o l} _ {G} (S) \cdot \operatorname {v o l} _ {G} (X \setminus S)}. \tag {2} +$$ + +$\Phi_G(S)$ is sometimes called the ratio cut objective [47], while $\Psi_G(S)$ is called the normalized cut objective [41]. These objectives sometimes appear in the literature slightly differently, with $\min \{|S|, |X \setminus S|\}$ and $\min \{\mathrm{vol}_G(S), \mathrm{vol}_G(X \setminus S)\}$ as their respective denominators. These are the same as Equation (2) up to a factor of 2, since $\frac{1}{2} \min \{s, n - s\} \leq \frac{1}{n} \cdot s \cdot (n - s) \leq \min \{s, n - s\}$ for all $0 < s < n$ , which applies here with $n = |X|$ , $s = |S|$ in $\Phi_G(S)$ , and with $n = \mathrm{vol}_G(X)$ , $s = \mathrm{vol}_G(S)$ in $\Psi_G(S)$ . + +# 2.1 The Spectral Approach + +Our approach is grounded in the following theorem, which is a generalization of a theorem given in [11], based on a framework developed by [2, 3], in the context of nearest neighbor search. It relates coordinate cuts in $\mathbb{R}^d$ to the Cheeger inequality [1, 8], a fundamental result in spectral graph theory with a myriad of algorithmic implications. While they stated the theorem specifically for the nearest neighbor graph of a point set in $\mathbb{R}^d$ , it holds for general weighted graphs over points in $\mathbb{R}^d$ , as we now state. We provide a proof in Section A. + +Definition 2.1. Let $X \subset \mathbb{R}^d$ . The coordinate cut given by coordinate $j \in \{1, \dots, d\}$ and threshold $\tau \in \mathbb{R}$ is $S_{j,\tau}(X) \coloneqq \{x \in X : x_j \leq \tau\}$ . The cut is valid if $S_{j,\tau}(X) \neq \emptyset$ and $S_{j,\tau}(X) \neq X$ . + +Theorem 2.2. Let $X \subset \mathbb{R}^d$ be a set of points, where $x \in X$ has coordinates $x = (x_1, \ldots, x_d)$ . Let $G(X, E, w)$ be a graph with vertex set $X$ . Consider two distributions over pairs of points in $X$ : + +- $\mathcal{D}_{\mathrm{adj}}$ is the distribution over adjacent pairs in $G$ , where a pair $x, y \in E$ is sampled with probability proportional to the edge weight between them. +- $\mathcal{D}_{\mathrm{all}}$ is the distribution over all pairs $x, y \in X$ , where $x$ and $y$ are sampled independently, each with probability proportional to its weighted degree in $G$ . + +Then, there is a valid coordinate cut $j,\tau$ such that + +$$ +\Psi_ {G} (S _ {j, \tau} (X)) \leq \sqrt {\frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \| x - y \| _ {2} ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \| x - y \| _ {2} ^ {2}}}. +$$ + +Intuitively, the theorem draws a connection between combinatorial graph cuts and geometric properties of the set of points: if the graph nodes graph are associated with embeddings in $\mathbb{R}^d$ , then there is a "good quality" graph cut, whose conductance is upper-bounded in terms of the squared Euclidean distances between the node embeddings. Furthermore, that cut is a coordinate cut. + +The conductance bound is governed by the ratio of expected squared distances in $X$ according to two distributions over pairs of points: the numerator samples pairs of points adjacent in $G$ , while the denominator samples any pair of points. Thus, the ratio (and hence the cut conductance) is smaller when $G$ captures a geometrically meaningful structure in the dataset, wherein adjacent pairs of points are expected to be nearer to each other than general pairs of points. + +This implies an approach to explainable clustering. Given a reference clustering, we can describe it with a suitable graph $G$ , and iteratively look for the coordinate cut with minimum conductance in each node of the decision tree. If the reference clustering is of good quality, in the sense that a pair of points are expected to be nearer to each other if they are clustered together, then Theorem 2.2 guarantees the existence of a low-conductance cut. At the same time, since the nearness of points in the same cluster is a "soft" requirement (it only needs to hold in expectation), it renders the theorem robust to various types of clustering, allowing for unconstrained cluster shapes, outliers, etc. + +Another possibility that arises is to dispense with the reference clustering, and construct a graph that captures nearness/farness directly from the dataset. In Section 2.2 we describe the explainable tree construction with a general graph, and in Section 2.3 we discuss graph selection. + +# 2.2 Iterative Tree Construction + +Given a graph $G(X, E, w)$ over the dataset $X \subset \mathbb{R}^d$ , our task is to construct an explainable clustering decision tree $T$ . Let $\ell > 0$ be the desired number of leaves. + +An explainable clustering tree is a decision tree in which every internal node $v$ is associated with a coordinate $j_v \in \{1, \dots, d\}$ and a threshold $\tau_v \in \mathbb{R}$ , inducing the condition $x_{j_v} \leq \tau_v$ given a point $x \in \mathbb{R}$ . The coordinate thresholds associate a subset $X_v \subset X$ with each node $v$ : the root is associated + +with $X$ , and every non-root $v$ with parent $u$ is associated with $X_{v} = S_{j_{u},\tau_{u}}(X_{u})$ if $v$ is the left child of $u$ , or $X_{v} = X_{u} \setminus S_{j_{u},\tau_{u}}(X_{u})$ if it $v$ the right child of $u$ . + +Let $\mathcal{L}(T)$ denote the set of leaves in $T$ . They induce partition of $X$ into clusters, $X = \cup_{v \in \mathcal{L}(T)} X_v$ . We measure the quality of $T$ by a generalization of the normalized cut objective to multi-way partitions [46]: + +$$ +\bar {\Psi} _ {G} (T) = \sum_ {v \in \mathcal {L} (T)} \psi_ {G} \left(X _ {v}\right). \tag {3} +$$ + +The smaller $\bar{\Psi}_G(T)$ , the better the partition induced by its leaves. Note this generalizes the two-way cut objective $\Psi_G(S)$ (Equation (2)), since $\Psi_G(S) = \psi_G(S) + \psi_G(X\setminus S)$ . + +To build the tree, we start with $T$ containing only a root. As long as $T$ does not yet have the requisite number of leaves $\ell$ , we choose the leaf $v$ to split next to greedily minimize $\bar{\Psi}_G(T)$ . Splitting a leaf $v$ with a cut $S \subset X_v$ replaces the summand $\psi_G(X_v)$ in Equation (3) with $\psi_G(S) + \psi_G(X_v \setminus S)$ , and thus $v$ is chosen to maximize the reduction in $\bar{\Psi}_T(G)$ its split would yield, which is + +$$ +\psi_ {G} \left(X _ {v}\right) - \min _ {S} \left(\psi_ {G} (S) + \psi_ {G} \left(X _ {v} \backslash S\right)\right), \tag {4} +$$ + +where $S$ ranges over all valid coordinate cuts. We associate $v$ with the $j_v, \tau_v$ corresponding to the minimizer $S$ , split $v$ into two new leaves along this cut, and repeat. See Algorithm 1. + +Note that Algorithm 1 has the flexibility to produce a tree $T$ with any desired number of leaves, regardless of the number of clusters in the reference clustering. In contrast, centroid- + +based methods like IMM, EMN and Kernel IMM are bound to produce the same number of leaves in $T$ as the number of centroids in the reference $k$ -median or (kernel) $k$ -means clustering, and increasing the number of leaves requires separate techniques [17, 31, 15]. Increasing the number of leaves is helpful for attaining a smaller price of explainability due to the increased expressivity of $T$ , albeit at the expense of being less explainable due to its bigger size. We discuss this further in Appendix B. Our main evaluation will focus on producing trees with the same number of leaves as clusters in the reference clustering. + +# 2.3 SPEx-Clique and SPEx-kNN + +To capture a given reference clustering with a graph, there are several natural choices: + +- Clique graph: points are adjacent if and only if they are in the same cluster (thus, every cluster becomes a clique). +- Star graph: if the reference clustering is endowed with centroids, each point can be made adjacent to its cluster centroid (thus, every cluster becomes a star). +- Independent set (IS) graph: points are adjacent if and only if they are not in the same cluster (thus, every cluster becomes an independent set). This is the complement of the clique graph. Here, of course, one would wish to maximize rather than minimize the cut objective. +if they are in the same cluster (thus, every cluster +red with centroids, each point can be made adjacent (times a star). +if and only if they are not in the same cluster (thus, this is the complement of the clique graph. Here, of course minimize the cut objective. + +The clique graph may seem like the most natural choice, and this is indeed the one we make. Perhaps surprisingly, as we will show in Section 3, when previous methods (IMM, EMN, CART) are interpreted through this graphical lens, they turn out to correspond to either the star graph or the IS graph. This directly relates to their limitations, like requiring centroids (in the case of the star graph) or failing the toy example from Figure 2 (in the case of the IS graph). + +# Algorithm 1 SPEx + +input: Dataset $X\subset \mathbb{R}^d$ , graph $G(X,E,w)$ , target number of clusters $\ell$ + +output: Decision tree $T$ where every internal node is associated with a coordinate $j$ and threshold $\tau$ + +BUILDTREE(X,G, $\ell$ + +$T\gets$ initialize a tree with a single node $v$ + +$j,\tau \gets \mathrm{argmin}_{j,\tau}\mathbf{CUTSCORE}(X,j,\tau)$ + +$Q \gets$ initialize a maximum priority for the tree leaves, with priorities given by LEAFSCORE + +$Q.$ push(v,X,j,τ) + +while $T$ has less than $\ell$ leaves do + +$v,X_{v},j_{v},\tau_{v}\gets Q.pop()$ + +Associate $v$ with the cut $j,\tau$ + +Split $v$ into two new leaves $v_{L}, v_{R}$ + +$$ +X _ {v _ {L}} \leftarrow S _ {j _ {v}, \tau_ {v}} (X _ {v}) +$$ + +$$ +X _ {v _ {L}} \leftarrow X _ {v} \backslash X _ {v _ {L}} +$$ + +$$ +j _ {L}, \tau_ {L} \leftarrow \operatorname {a r g m i n} _ {j, \tau} \mathrm {C U T S C O R E} \left(X _ {v _ {L}}, j, \tau\right) +$$ + +$$ +j _ {R}, \tau_ {R} \leftarrow \operatorname {a r g m i n} _ {j, \tau} \operatorname {C U T S C O R E} \left(X _ {v _ {R}}, j, \tau\right) +$$ + +$$ +Q. p u s h \left(v _ {L}, X _ {v _ {L}}, j _ {L}, \tau_ {L}\right) +$$ + +$$ +Q. p u s h \left(v _ {R}, X _ {v _ {R}}, j _ {R}, \tau_ {R}\right) +$$ + +$$ +\text {r e t u r n} T +$$ + +CUTSCORE $(X^{\prime},j,\tau)$ ： + +if $S_{j,\tau}(X) = \emptyset$ or $S_{j,\tau}(X) = X$ then + +$$ +\mathbf {\Delta \Pi} ^ {\mathrm {r e t u r n}} \propto +$$ + +$$ +\text {r e t u r n} \psi_ {G} \left(S _ {j, \tau} \left(X ^ {\prime}\right)\right) + \psi_ {G} \left(X _ {v} \backslash S _ {j, \tau} \left(X ^ {\prime}\right)\right) +$$ + +LEAFSCORE $(v,X^{\prime},j,\tau)$ ： + +$$ +\text {r e t u r n} \psi_ {G} \left(X ^ {\prime}\right) - \operatorname {C U T S C O R E} \left(X ^ {\prime}, j, \tau\right) +$$ + +# + +Given any reference clustering, using the clique graph as $G$ in Algorithm 1 yields the algorithm we call SPEx-Clique. We also consider an variant that uses no reference clustering, by constructing the nearest neighbor graph directly on the points in $X$ . This yields the algorithm we call SPEx-kNN. + +# 2.4 Computational Efficiency + +SPEx as well as the baselines we consider share the following high-level structure. In each tree node $u$ with $n_u$ points, for each coordinate, they sort the points by that coordinate (time $O(n_u\log n_u)$ ) and perform a sweep-line procedure that iterates over the $n_u - 1$ prefix/suffix cuts by moving nodes from the suffix to the prefix one at a time. As each node is moved, the cut score is updated accordingly (let $S$ denote the time it takes to update), and eventually the cut with the overall optimal score is selected. Repeating this for all coordinates takes time $O(dn_u(\log n_u + S))$ . In each tree level, the sum $\sum_{u}n_{u}$ is $n$ for SPEx and CART and $n + k\leq 2n$ for IMM and EMN, therefore the time per level is $O(dn(\log n + S))$ . Summing over up to $k - 1$ levels in the tree, the total time is $O(kdn(\log n + S))$ . + +The algorithms may differ on the time $S$ it takes update cut scores during sweep-line. In SPEx-Clique, we need not store the entire clique graph; rather, we only store point-cluster assignments $(O(n)$ memory). During sweep-line, we maintain two cluster histograms for the prefix and the suffix $(O(k)$ memory), from which the cut score can be updated in $S = O(1)$ time. In SPEx-kNN, letting $\kappa$ denote the number of neighbors per node (note that this is a different parameter than the number of clusters $k$ ), we can store the kNN graph in $O(n\kappa)$ memory and update cut scores in $S = O(\kappa)$ time. + +# 3 Lens: Non-Uniform Sparse Cuts + +While our spectral approach to explainable clustering may seem rather different from previous methods, in this section we show a generalized graph partitioning framework that captures them in a unified way. It is based on non-uniform sparse cuts as defined by Trevisan [44], where cuts are optimized simultaneously in two graphs that share the same set of nodes. + +Trevisan [44] defined the Non-Uniform Sparsest Cut problem as follows. Let $G(X, E_G, w_G)$ and $H(X, E_H, w_H)$ be two graphs on the same set of nodes $X$ . The $(G, H)$ -sparsity of a cut $(S, X \setminus S)$ is defined as the (normalized) ratio of edges cut in $G$ to edges cut in $H$ : + +$$ +\Psi_ {G, H} (S) = \frac {\frac {1}{\operatorname {v o l} _ {G} (X)} \cdot e _ {G} (S , X \setminus S)}{\frac {1}{\operatorname {v o l} _ {H} (X)} \cdot e _ {H} (S , X \setminus S)}. +$$ + +The goal in the Non-Uniform Sparsest Cut problem is to find the cut with the smallest $\Psi_{G,H}(S)$ . + +In [44] it is observed that this generalized graph partitioning problem captures several classical problems defined on a single graph $G$ as special cases: + +- The classical ("uniform") Sparsest Cut problem, of minimizing $\Phi_G(S)$ from Equation (2), is the case of $\Psi_{G,H}(S)$ where $H$ is an unweighted clique over $X$ . +- The Normalized Cut problem, of minimizing $\Psi_G(S)$ from Equation (2), is the case of $\Psi_{G,H}(S)$ where $H$ is the $G$ -degree weighted clique over $X$ , in which the edge weight in $H$ between every pair $x,y\in X$ is the product of their weighted degrees in $G$ . +- The Minimum $st$ -Cut problem is the case of $\Psi_{G,H}(S)$ where $H$ contains a single edge between a "distinguished" pair $s, t \in X$ . + +Here, we further observe that this framework is useful in capturing prior methods for explainable clustering (or close variants of those methods), as they in fact produce coordinate cuts that minimize $\Psi_{G,H}(S)$ with particular choices of graphs $G$ and $H$ , even though neither method is originally given in terms of graphical sparse cut terms. + +- IMM [33] corresponds to $G$ being the star graph over a given reference clustering endowed with centroids, and $H$ containing a single edge that connects a pair of diametrical (furthest) centroids. +- EMN [12] also corresponds to $G$ being the star graph, but with $H$ being an unweighted clique over the $k$ cluster centroids. +- CART closely corresponds to $H$ being the independent set (IS) graph over the reference clustering (see Section 2.3), and $G$ being the $H$ -degree weighted clique. + +We now discuss each algorithm in turn, and highlight useful implications of this graph-theoretic lens. + +# 3.1 IMM as Non-Uniform Sparse Cut + +Given a reference clustering $\mathcal{C} = (C_1, \ldots, C_k)$ endowed with cluster centroids $M = \{\mu^{(1)}, \ldots, \mu^{(k)}\}$ , the IMM algorithm builds a decision tree in which every node $u$ is associated with a subset $X_u \subset X \cup M$ of points and centroids. In each node, it chooses the coordinate cut that minimizes the number of "mistakes", i.e., of points placed at a different side of the cut than their centroid, while separating at least one pair of centroids. Formally, for $x \in X$ , let $\mu(x)$ denote the centroid of the cluster containing $x$ . The cut $S \subset X_u$ in a node $u$ is chosen by IMM to minimize, + +$$ +\operatorname {m i s} _ {u} (S) = | \{x \in S: \mu (x) \in X _ {u} \setminus S \} \cup \{x \in X _ {u} \setminus S: \mu (x) \in S \} |, +$$ + +among all coordinate cuts $S$ that satisfy $\mu^{(i)} \in S$ and $\mu^{(j)} \in X \setminus S$ for at least one pair $\mu^{(i)}, \mu^{(j)} \in M \cap X_u$ . The construction terminates when each leaf in the tree contains exactly one centroid. + +To cast this as non-uniform sparse cut, consider a slightly modified variant of IMM. For each node $u$ , Let $\mu', \mu'' \in X_u$ be a pair of centroids at maximal distance among the centroids in $M \cap X_u$ . Choose the coordinate cut $S \subset X_u$ that minimizes $\mathrm{mis}_u(S)$ among those that separate $\mu', \mu''$ . This is a subset of the cuts in the original IMM. For this variant, let $G$ be the star graph over the reference clustering, and $G_u = G[X_u]$ be its restriction to the subset $X_u \subset X \cup M$ associated with each tree node $u$ . Then $\mathrm{mis}_u(S) = e_{G_u}(S, X_u \setminus S)$ , since each cut edge marks a point separated from its centroid. Let $H_u$ be the single edge graph over $X_u$ that contains only an edge connecting $\mu'$ and $\mu''$ . Then we have, $e_{H_u}(S, X_u \setminus S) = 1$ if the cut $(S, X_u \setminus S)$ separates $\mu', \mu''$ , and $e_{H_u}(S, X_u \setminus S) = 0$ otherwise. Thus, minimizing $\mathrm{mis}_u(S)$ subject to a cut separating $\mu', \mu''$ is equivalent to minimizing the ratio $e_{G_u}(S, X_u \setminus S) / e_{H_u}(S, X_u \setminus S)$ . This is equal (up to normalization) to the non-uniform cut sparsity, $\Psi_{G_u, H_u}(S)$ . Even though this modified IMM considers less cuts than the original IMM, it attains the same price of explainability for $k$ -medians, by a different proof based on graph partitioning, that avoids the intricate potential function analysis in [33]. We show this in Section A.3. + +# 3.2 EMN as Non-Uniform Sparse Cut + +The EMN algorithm [12] is an improvement of IMM by the following modification: in every tree node $u$ , instead of minimizing $\mathrm{mis}_u(S)$ , it chooses the threshold coordinate cut $S \subset X_u$ that minimizes the ratio $\mathrm{mis}_u(S) / f_u(S)$ , where [12] define $f_u(S) = \min \{|S \cap M|, |(X_u \setminus S) \cap M|\}$ . As with IMM, letting $G_u$ be the star graph over the reference clustering when restricted to $X_u$ , we have $\mathrm{mis}_u(S) = e_{G_u}(S, X_u \setminus S)$ . Let $H_u$ be the graph with vertex set $X_u$ whose edges form an unweighted clique over $M \cap X_u$ (the rest of the vertices in $X_u$ are isolated in $H_u$ ). Then, + +$$ +e _ {H _ {u}} (S, X _ {u} \setminus S) = | S \cap M | \cdot | (X _ {u} \setminus S) \cap M |. +$$ + +Recall from Section 2 that $\frac{1}{2} f_u(S) \leq \frac{1}{|M \cap X_u|} \cdot e_{H_u}(S, X_u \setminus S) \leq f_u(S)$ . Thus, minimizing the ratio $\mathrm{mis}_u(S) / f_u(S)$ is equivalent, up to a factor of 2, to minimizing the ratio $e_{G_u}(S, X_u \setminus S) / e_{H_u}(S, X_u \setminus S)$ . This is equivalent to minimizing the non-uniform cut sparsity $\Psi_{G_u, H_u}(S)$ . + +# 3.3 CART as Non-Uniform Sparse Cut + +We first review the CART algorithm in the form used in explainable clustering. The Gini impurity of a distribution $(p_1, \ldots, p_k)$ over $k$ elements is defined as $v(p) = \sum_{i=1}^{k} p_i (1 - p_i)$ . Given a point set $X \subset \mathbb{R}^d$ , a reference clustering $\mathcal{C} = \{C_1, \ldots, C_k\}$ , and a subset $S \subset X$ , we may define a distribution $p_{\mathcal{C}}(S)$ over the clusters as $p_{\mathcal{C}}(S) = \left(\frac{|S \cap C_1|}{|S|}, \frac{|S \cap C_2|}{|S|}, \ldots, \frac{|S \cap C_k|}{|S|}\right)$ , which corresponds to sampling a uniformly random point from $S$ and returning its cluster ID. The impurity of $S$ is defined as $v_{\mathcal{C}}(S) = v(p_{\mathcal{C}}(S))$ , and the impurity of $(S, X \setminus S)$ as a two-way cut as, + +$$ +\Upsilon_ {\mathcal {C}} (S, X \setminus S) = \frac {| S |}{| X |} \cdot v _ {\mathcal {C}} (S) + \frac {| X \setminus S |}{| X |} \cdot v _ {\mathcal {C}} (X \setminus S). +$$ + +The CART algorithm builds a decision tree starting with a root node associated with all of $X$ . It then iteratively chooses a leaf node whose associated subset $X'$ maximizes $\text{score}(X') = v_{\mathcal{C}}(X') - \min_{S \subset X'} \Upsilon_{\mathcal{C}}(S, X' \setminus S)$ , where $S$ ranges over all coordinates threshold cuts, and splits that leaf across that cut. The score captures the impurity reduction gained by splitting $X'$ along its best cut. Note how this is analogous to Equation (4) in SPEx. + +Table 1: Datasets. *Training set only. †Embedded with CLIP [37]. ‡Embedded with SBERT [38]. + +
CIFAR-10*20Newsgroups*MNIST*Caltech 101BeansBreast CancerEcoliIris
# Points50,00011,31460,000867713,611569336150
# Dimensions512†768‡512†512†163074
# Classes1020101017283
Reference[24][5][29][13][23][43][21][14]
+ +Now, we show that a small modification to CART is equivalent to a case of non-uniform sparsest cut. Let $H$ be the independent set graph over the reference clustering $\mathcal{C}$ (see Section 2.3). For a subset $S \subset X$ , let $\alpha_{H}(S)$ denote the sum of edge weights with both endpoints in $S$ . We can now interpret the impurity $v_{\mathcal{C}}(S)$ of a subset $S \subset X$ as, + +$$ +v _ {\mathcal {C}} (S) = \sum_ {i = 1} ^ {k} \frac {| S \cap C _ {i} |}{| S |} \left(1 - \frac {| S \cap C _ {i} |}{| S |}\right) = \frac {1}{| S | ^ {2}} \sum_ {i = 1} ^ {k} | S \cap C _ {i} | (| S | - | S \cap C _ {i} |) = \frac {2 \cdot \alpha_ {H} (S)}{| S | ^ {2}}. +$$ + +Thus, the two-way cut impurity becomes + +$$ +\Upsilon_ {\mathcal {C}} (S, X \backslash S) = \frac {2}{| X |} \left(\frac {\alpha_ {H} (S)}{| S |} + \frac {\alpha_ {H} (X \backslash S)}{| X \backslash S |}\right). \tag {5} +$$ + +To proceed, we make the modification of replacing set cardinalities with their volumes in $H$ , as in the transition from sparsity to conductance in Equations (1) and (2). We get the modified two-way cut impurity, instead of Equation (5): + +$$ +\widetilde {\Upsilon} _ {\mathcal {C}} (S, X \setminus S) = \frac {2}{\operatorname {v o l} _ {H} (X)} \left(\frac {\alpha_ {H} (S)}{\operatorname {v o l} _ {H} (S)} + \frac {\alpha_ {H} (X \setminus S)}{\operatorname {v o l} _ {H} (X \setminus S)}\right). +$$ + +This is precisely (up to normalization terms) the Nassoc graph cut objective defined in [41, eq. (3)].3 As they observed, it is directly related to the normalized cut objective, i.e., to the cut conductance: + +$$ +\Psi_ {H} (S) = 2 - \frac {\operatorname {v o l} _ {H} (X)}{2} \cdot \widetilde {\Upsilon} _ {\mathcal {C}} (S, X \setminus S). +$$ + +Thus, finding a coordinate cut $S$ that minimizes our modified impurity $\widetilde{\Upsilon}$ is equivalent to maximizing the usual cut conductance $\Psi_H(S)$ in $H$ . This, in turn, is equivalent to maximizing the non-uniform cut sparsity $\Psi_{H,G}(S)$ , where $H$ is the independent set graph and $G$ is the $H$ -degree weighted clique over $X$ , since (as observed in [44] and mentioned in the beginning of this section), for this choice of graphs it holds that $\Psi_{H,G}(S) = \Psi_H(S)$ , up to normalization. Finally, since $\Psi_{G,H}(S) = 1 / \Psi_{H,G}(S)$ by definition, this is equivalent minimizing the non-uniform cut sparsity $\Psi_{H,G}(S)$ . + +This point of view clarifies the failure of CART in Figure 2. The IS graph over a reference clustering incentivizes cuts that separate pairs of points residing in different clusters, but has no incentive to not separate points residing in the same cluster, since they are not connected with any edges and hence have no effect on the cut value. Since CART maximizes cut conductance, it opts to cut the larger number of edges between the bottom clusters (vertical cut) over the smaller number of edges between the top cluster and each of the bottom clusters (horizontal cut), which is incompatible with clustering. + +# 4 Experimental Evaluation + +We evaluate our methods compared to baselines, on eight public real-world datasets of various sizes and dimensions, detailed in Table 1. Our code is available online. $^4$ + +Our algorithms. We focus on two instantiations of our method: SPEx-Clique with spectral clustering as the reference, and SPEx-kNNwhere the $k$ -NN graph is constructed with $k = 20$ . Section B includes additional results for SPEx-Clique with $k$ -means and kernel $k$ -means as the reference clustering, and for SPEx-kNN with other values of $k$ . + +Baselines. We evaluate the following baselines: + +![](images/5cf09437fb5751e7ae4b0f9f95ec2075f414717663fbe9231bd89705bc33c543.jpg) + +![](images/3c6bf5a6974d96cfe51f2315f30df7cf17fc6467113fa0db4b0d0158633f7394.jpg) + +![](images/d7c19a5598ebe724d0968757649b4e0f001f165ed8c607c0ad2c45acfc65d440.jpg) + +![](images/70663b5ea14d2922469ddeb359fd93ba4e919d2d0a23659b20a351c54300c93b.jpg) + +![](images/3d2a0c68ea34933c5d1c113a16687bbe56a67d5089894856aa260760cd70f6bd.jpg) + +![](images/f20e162a102098de517efe7cd904f1c31ef61c96fc83466db6944c3706151936.jpg) + +![](images/6f8f5882fc098cef3e9fb491c543252990c9200e6e10ab325da6612522e25d8e.jpg) + +![](images/5c6239f4105991c64a0da2ce946d300f0665b8a6dae2656b1317df7f534d4f38.jpg) + +![](images/3c610b2ca050b0e0e2b237ba4601fbf2e99cccd1c6d4c0cc3c3f5edf191f8880.jpg) + +![](images/4ede78b0d24c76837464453f0a50f53f6403a6e0324ec3b2c9519fba3e295809.jpg) + +![](images/d7b06fe458b90252376ad1e9dc76adb8c41c67236e36eff0a1c39b22acc1183b.jpg) + +![](images/314ee423d873ef1bf3efbd0ec209f40f53f76ab5ef45fd18f430978c902711c8.jpg) + +![](images/ab25793fba0eadce0ed3b7d81e02649d2dd00c81757231d80f6d3e505649f7f8.jpg) + +![](images/f390a7119105d2de3ed3108a3b422f61736cafa3de9b5849d98ab3d29d1c8390.jpg) + +![](images/39f988c09f2b63b68f6eafdf48dfbd2a56208e73629a51c49f36d846879e804a.jpg) + +![](images/24905f953854640bd96134a252e7274833adc6dcba3e70308c1c38d04387b859.jpg) + +![](images/d1a63d2db6970a3db1bc25a6abdf25b5220640bd6108a5056e078f521663f2f5.jpg) +SpEx-Clause (ours) + +![](images/5239829b2bc6b7cad0a4f8f72ecc83149a3dd7095f4c1ec24cc3a8ef19ab3512.jpg) +SpEx-kNN (ours) + +![](images/46f38e7935cf1e6ff6405f1e4634949e19f027b11a1f06fdef70aa16de718e21.jpg) +EMN +Figure 3: Adjusted Rand Index (ARI) and Adjusted Mutual Information (AMI), higher is better. + +![](images/7ba84cec30736dca0d9f597def444fc3622da0584710e28d47d7a14a1bfa4faa.jpg) +CART-k-means + +![](images/0222dc5f7da0c0222d22b256988f11f402bf30aa4dcdbd817d393fafcc1550c7.jpg) +CART-spectral + +![](images/916f9b8ad3f95457843ed50d0a1d4a2bfb1b82aa76c12c16e19da4ffbf28ae96.jpg) +Kernel-IMM + +- EMN [12], the state of the art for explainable clustering with $k$ -means as the reference clustering. Note that EMN cannot work with a spectral or a kernel $k$ -means reference clustering, as its operation requires the reference clustering to have centroids. +- CART, with both $k$ -means and spectral clustering as the reference clustering. +- Kernel IMM [15], which uses kernel $k$ -means as the reference clustering. + +Our experiments indicated that Kernel IMM does not scale well, failing to run on the larger scale datasets. We therefore report its results only for the three smaller scale datasets (Breast Cancer, Ecoli, Iris), which are the ones used in [15]. See more in Section B. + +Evaluation measures. The classes of each dataset are used as groundtruth clusters. To evaluate each method, we report two standard (and incomparable) measures of clustering agreement between two clusterings of the same data: Adjusted Rand Index (ARI) [22] and Adjusted Mutual Information (AMI) [34]. Each is explainable clustering is evaluated for its agreement with the groundtruth clustering through these measures. + +# 4.1 Results + +Main results are reported in Figure 3. Tables 2 and 3 augment them with additional choices of reference clustering for SPEx-Clique and CART and list ARIs with respect to the reference clustering. They also contain two synthetic small-scale datasets, R15 and Pathbased, used in [15]. Appendix B includes additional results. + +The results show that SPEx-Clique is the most consistently high-performing algorithm. It outperforms each baseline in most cases in both evaluation measures, and is never exceeded by more than one baseline at the same time. SPEx-kNN is highly effective particularly on the low dimensional datasets + +Table 2: Results on smaller datasets. Rows are grouped by reference clustering, shown as the first row in each group. The REF column lists the ARI with respect to the reference clustering (rather than the ground truth as in ARI column). Best scores per reference clustering are in bold. + +
AlgorithmR15PathbasedEcoliIrisCancer
ARIAMIREFARIAMIREFARIAMIREFARIAMIREFARIAMIREF
REF: Spectral.993.9941..526.5701..711.6531..630.6611..743.6261.
SPEx-kNN.982.987.989.332.410.551.679.642.863.450.647.450.507.490.562
SPEx-Clique.986.989.993.441.517.824.662.621.847.576.629.787.694.588.785
CART.986.989.993.441.517.824.672.618.886.576.629.787.683.560.811
REF: k-means.993.9941..461.5431..489.6091..641.6691..491.4641.
EMN.986.989.993.461.5431..456.559.873.576.629.772.491.4641.
SPEx-Clique.986.989.993.461.5431..456.559.873.576.629.772.491.4641.
CART.986.989.993.421.507.897.454.543.840.576.629.772.491.4641.
REF: Kernel k-means.908.9671..919.8881..538.6121..731.7671..116.2281.
Kernel IMM.904.962.986.614.614.583.522.560.848.732.788.924.127.241.93
SPEx-Clique.869.941.951.479.553.450.529.573.851.732.788.924.406.414.511
CART.682.876.759.479.553.450.500.558.824.732.788.924.406.414.511
+ +Table 3: Results on larger datasets. + +
AlgorithmMNISTCaltech 101NewsgroupsBeansCifar
ARIAMIREFARIAMIREFARIAMIREFARIAMIREFARIAMIREF
REF: Spectral.745.8201..563.8591..431.6711..586.6771..712.8011.
SpEx-kNN.092.338.150.121.497.168.042.170.09.574.690.649.342.434.373
SpEx-Clique.217.384.282.247.521.303.078.223.189.564.671.743.320.438.394
CART.027.148.030-.010.166.005.008.078-.013.542.658.705.036.169.035
REF: k-means.364.4811..405.8221..502.6601..572.6891..636.7381.
EMN.25.342.42.249.548.416.115.219.163.563.688.780.314.402.387
SpEx-Clique.209.336.403.122.495.195.098.249.128.562.687.773.288.410.370
CART.124.299.229-.017.082.003.005.092.006.536.669.757.045.180.037
+ +(Beans, Ecoli and Iris).5 On these datasets, it outperforms all baselines as well as SPEx-Clique, with the exception of Kernel IMM on Iris.6 + +CART performs well on the smaller datasets (confirming similar results reported in [15]), however, it performs poorly on the larger datasets. Recall that CART is not originally intended for explainable clustering, but can be repurposed for it since it produces a structurally compatible output (a binary decision tree with coordinate cuts). In Sections 1 and 3.3, we discussed a toy example from [33] that demonstrates a failure mode of CART for explainable clustering due to its incompatible objective. Our experimental results indicate this incompatibility may also impede its performance on real data. + +Conclusion and limitations. Prior work on explainable clustering has focused on a few specific reference objectives, allowing it to prove worst-case approximation bounds tailored to them (cf. Section 1.1). SPEx, on the other hand, fills the gap of a generic, reference-oblivious method, currently lacking in the literature. Thus, while theoretically well-grounded (Theorem 2.2, Section 3), it cannot offer such bounds. Other methods may be better choices for some specific reference objectives – e.g., EMN is known to be near-optimal for $k$ -means, and Tables 2 and 3 indeed show that it is generally better than SPEx for that reference objective. SPEx is better for data where objectives already well-studied in explainable clustering (like $k$ -means) fall short, and more versatile objectives (like spectral clustering) are needed. As future work, it would be interesting to explore if the theoretical framework we develop can yet lead to formal approximation bounds for some of these objectives. + +# Acknowledgements + +This work was supported by Len Blavatnik and the Blavatnik Family foundation and by an Alon Scholarship of the Israeli Council for Higher Education. 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IEEE, 1989. + +# Impact Statement + +This paper is concerned with fitting interpretable explanations to decisions made by machine learning algorithms, in this case, grouping of individual points into clusters. While explainability in AI is intended to promote safe and ethical use in machine learning, any deployment of automated systems in settings consequential to human individuals may have unforeseen consequences. While our work is not associated with any particular outstanding or new risks, every real-world use of an explainability mechanism in AI should be used with caution and incorporate human oversight, testing, and other use case specific safeguards to mitigate any associated risks. + +# A Proofs + +In Section A.1, we provide a proof of Theorem 2.2. In Section A.2, as a concrete demonstration, we show an application of it that directly relates the $k$ -means reference clustering cost to the graph conductance in a certain graph that describes the reference clustering. In Section A.3, we provide the modified IMM analysis discussed in Section 3.1. + +# A.1 Proof of Theorem 2.2 + +We restate the theorem: + +Theorem A.1 (Theorem 2.2, restated). Let $X \subset \mathbb{R}^d$ be a set of points, where $x \in X$ has coordinates $x = (x_1, \ldots, x_d)$ . Let $G(X, E, w)$ be a graph with vertex set $X$ . Consider the following two distributions over pairs in of points in $X$ : + +- $\mathcal{D}_{\mathrm{adj}}$ is the distribution over adjacent pairs in $G$ , where a pair $x, y \in E$ is sampled with probability proportional to the edge weight between them. +- $\mathcal{D}_{\mathrm{all}}$ is the distribution over all pairs $x, y \in X$ , where $x$ and $y$ are sampled independently, each with probability proportional to its weighted degree in $G$ . + +Then, there is a valid coordinate cut $j,\tau$ such that + +$$ +\Psi_ {G} (S _ {j, \tau} (X)) \leq \sqrt {\frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \| x - y \| _ {2} ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \| x - y \| _ {2} ^ {2}}}. +$$ + +The proof uses arguments that are standard in spectral graph theory, and follows [11] in a generalized form (with an arbitrary graph). We include it here for completeness and clarity. + +Recall the following standard useful fact, + +Fact A.2. Let $a_1, \ldots, a_n \geq 0$ and $b_1, \ldots, b_n \geq 0$ such that $\sum_i b_i > 0$ . Then + +$$ +\min _ {i} \frac {a _ {i}}{b _ {i}} \leq \frac {\sum_ {i} a _ {i}}{\sum_ {i} b _ {i}} \leq \max _ {i} \frac {a _ {i}}{b _ {i}} +$$ + +Using it we have, + +$$ +\begin{array}{l} \frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \| x - y \| _ {2} ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \| x - y \| _ {2} ^ {2}} = \frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \sum_ {i = 1} ^ {d} | x _ {i} - y _ {i} | ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \sum_ {i = 1} ^ {d} | x _ {i} - y _ {i} | ^ {2}} \\ = \frac {\sum_ {i = 1} ^ {d} \mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} | x _ {i} - y _ {i} | ^ {2}}{\sum_ {i = 1} ^ {d} \mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} | x _ {i} - y _ {i} | ^ {2}} \\ \geq \min _ {i ^ {*} \in \{1, \dots , d \}} \frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \left| x _ {i ^ {*}} - y _ {i ^ {*}} \right| ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \left| x _ {i ^ {*}} - y _ {i ^ {*}} \right| ^ {2}}. \tag {6} \\ \end{array} +$$ + +Let $n = |X|$ be the number of nodes in $G$ . For convenience we arbitrarily label them $1, \ldots, n$ so we can index the entries a vector $z \in \mathbb{R}^n$ by points in $X$ , so the coordinates of $z$ are $(z(x))_{x \in X}$ . We do so similarly for matrices in $\mathbb{R}^{n \times n}$ . + +Recall that $w(x,y)$ is the edge weight between $x$ and $y$ in $G$ . Let $d_G(x)$ denote the weighted degree of $x$ in $G$ . Let $\Delta_G = \sum_{x,y} w(x,y)$ be the sum of all edge weights. Let $A_G \in \mathbb{R}^{n \times n}$ be its + +weighted adjacency matrix, $A_G(x,y) = w(x,y)$ . Let $D_G \in \mathbb{R}^{n \times n}$ be its diagonal matrix of degrees, $D_G(x,x) = d_G(x)$ . Let $L_G = D_G - A_G$ be its Laplacian matrix. A standard fact in spectral graph theory is the identity $z^T L_Gz = \sum_{x,y} w(x,y)|z(x) - z(y)|^2$ for every $z \in \mathbb{R}^n$ . + +Let $z_{*} \in \mathbb{R}^{n}$ be the vector with entries defined by $z_{*}(x) = x_{i^{*}}$ , where $i^{*} \in \{1, \dots, d\}$ is the minimizer from Equation (6). Recalling that $\mathcal{D}_{\mathrm{adj}}$ is the distribution over pairs $x, y$ with probability mass $\frac{w(x, y)}{\Delta_G}$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {x, y \sim \mathcal {D} _ {\mathrm {a d j}}} | x _ {i ^ {*}} - y _ {i ^ {*}} | ^ {2} = \sum_ {x, y} \frac {w (x , y)}{\Delta_ {G}} | x _ {i ^ {*}} - y _ {i ^ {*}} | ^ {2} \\ = \sum_ {x, y} \frac {w (x , y)}{\Delta_ {G}} \left| z _ {*} (x) - z _ {*} (y) \right| ^ {2} \\ = \frac {z _ {*} ^ {T} L _ {G} z _ {*}}{\Delta_ {G}}. \tag {7} \\ \end{array} +$$ + +Let $H$ be a weighted clique over $X$ in which the edge weight between every pair $x, y \in X$ is + +$$ +w _ {H} (x, y) = \frac {d _ {G} (x) \cdot d _ {G} (y)}{2 \Delta_ {G}}. \tag {8} +$$ + +This is the $G$ -degree weighted clique over $X$ mentioned in Section 3, except we scale all weights down by $2\Delta_G$ . The weighted degree of $x$ in $H$ is + +$$ +\begin{array}{l} d _ {H} (x) = \sum_ {y \neq x} w _ {H} (x, y) \\ = \frac {d _ {G} (x)}{\Delta_ {G}} \sum_ {y \neq x} d _ {G} (y) \\ = \frac {d _ {G} (x)}{2 \Delta_ {G}} \left(2 \Delta_ {G} - d _ {G} (x)\right) \quad \text {s i n c e} \Delta_ {G} = \sum_ {x, y} w (x, y) = \frac {1}{2} \sum_ {y \in X} d _ {G} (y) \\ = d _ {G} (x) - \frac {\left(d _ {G} (x)\right) ^ {2}}{2 \Delta_ {G}}. \tag {9} \\ \end{array} +$$ + +Let $A_H$ , $D_H$ and $L_H = D_H - A_H$ be the weighted adjacency matrix of $H$ , its diagonal degree matrix, and its Laplacian matrix, respectively. Let $v_G \in \mathbb{R}^n$ be the vector of weighted degrees in $G$ , scaled down by $\sqrt{2\Delta_G}$ : + +$$ +v _ {G} (x) = \frac {d _ {G} (x)}{\sqrt {2 \Delta_ {G}}}. +$$ + +Observe that, by Equations (8) and (9), + +$$ +D _ {H} (x, y) = \left\{ \begin{array}{l l} d _ {G} (x) - \frac {(d _ {G} (x)) ^ {2}}{2 \Delta_ {G}} & \text {i f} x = y \\ 0 & \text {i f} x \neq y \end{array} \right. \quad \text {a n d} \quad A _ {H} (x, y) = \left\{ \begin{array}{l l} 0 & \text {i f} x = y \\ \frac {d _ {G} (x) \cdot d _ {H} (y)}{2 \Delta_ {G}} & \text {i f} x \neq y \end{array} \right. +$$ + +Therefore, + +$$ +L _ {H} = D _ {H} - A _ {H} = D _ {G} - v _ {G} v _ {G} ^ {T}. \tag {10} +$$ + +Now, recall that $\mathcal{D}_{\mathrm{all}}$ is the distribution over pairs $x,y$ , where $x$ and $y$ are i.i.d., each with probability proportional to its degree in $G$ . Hence, + +$$ +\begin{array}{l} \mathbb {E} _ {x, y \sim \mathcal {D} _ {\mathrm {a l l}}} | x _ {i ^ {*}} - y _ {i ^ {*}} | ^ {2} = \sum_ {x, y} \frac {d _ {G} (x)}{2 \Delta_ {G}} \cdot \frac {d _ {G} (y)}{2 \Delta_ {G}} \cdot | x _ {i ^ {*}} - y _ {i ^ {*}} | ^ {2} \\ = \frac {1}{2 \Delta_ {G}} \sum_ {x, y} w _ {H} (x, y) \left| z _ {*} (x) - z _ {*} (y) \right| ^ {2} \quad \text {b y} (8) \\ = \frac {z _ {*} ^ {T} L _ {H} z _ {*}}{2 \Delta_ {G}}. (11) \\ \end{array} +$$ + +Plugging Equations (7) and (11) into Equation (6), we have obtained, + +$$ +\frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \| x - y \| _ {2} ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \| x - y \| _ {2} ^ {2}} \geq \frac {2 \cdot z _ {*} ^ {T} L _ {G} z _ {*}}{z _ {*} ^ {T} L _ {H} z _ {*}}. \tag {12} +$$ + +Finally, let $\mathbf{0}$ and $\mathbf{1}$ denote the all-0 and all-1 vectors in $\mathbb{R}^n$ . Let $\gamma_G$ be the constant $\gamma_G = v_G^T z_* / v_G^T\mathbf{1}$ . Letting + +$$ +\tilde {z} = z _ {*} - \gamma_ {G} \mathbf {1}, +$$ + +we have + +$$ +v _ {G} ^ {T} \tilde {z} = v _ {G} ^ {T} \left(z _ {*} - \gamma_ {G} \mathbf {1}\right) = v _ {G} ^ {T} z _ {*} - \frac {v _ {G} ^ {T} z _ {*}}{v _ {G} ^ {T} \mathbf {1}} \cdot v _ {G} ^ {T} \mathbf {1} = 0. \tag {13} +$$ + +It is a standard fact that every graph Laplacian $L$ satisfies $L\mathbf{1} = \mathbf{0}$ . Therefore, + +$$ +L _ {G} z _ {*} = L _ {G} (\tilde {z} + \gamma_ {G} {\bf 1}) = L _ {G} \tilde {z}, +$$ + +and, using Equations (10) and (13), + +$$ +L _ {H} z _ {*} = L _ {H} (\tilde {z} + \gamma_ {G} \mathbf {1}) = L _ {H} \tilde {z} = \left(D _ {G} - v _ {G} v _ {G} ^ {T}\right) \tilde {z} = D _ {G} \tilde {z}. +$$ + +Plugging these into Equation (12), we obtain + +$$ +\frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \| x - y \| _ {2} ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \| x - y \| _ {2} ^ {2}} \geq 2 \cdot \frac {\tilde {z} ^ {T} L _ {G} \tilde {z}}{\tilde {z} ^ {T} D _ {G} \tilde {z}}. \tag {14} +$$ + +Now we can state the Cheeger inequality for graphs, or rather, a somewhat more general and useful form of it (see, e.g., [42, Theorem 21.1.3]): + +Theorem A.3. Let $G(X,E,w)$ be an undirected weighted graph with $n$ nodes. Let $A_G$ be its weighted adjacency matrix, $v_G$ the vector weighted degrees, $D_G$ the diagonal matrix of weighted degrees, and $L_G = D_G - A_G$ the Laplacian. + +Suppose we have a vector $\tilde{z} \in \mathbb{R}^n$ that satisfies $\tilde{z}^T v_G = 0$ . Then, there is a threshold $\tilde{\tau} \in \mathbb{R}$ such that if we consider the cut $S = \{x \in X : \tilde{z}(x) \leq \tilde{\tau}\}$ , it satisfies $S \neq \emptyset$ , $S \neq X$ , and + +$$ +\Psi_ {G} (S) \leq \sqrt {2 \cdot \frac {\tilde {z} ^ {T} L _ {G} \tilde {z}}{\tilde {z} ^ {T} D _ {G} \tilde {z}}}. \tag {15} +$$ + +Theorem A.3 together with Equation (14) imply a cut with conductance + +$$ +\Psi_ {G} (S) \leq \sqrt {\frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a d j}}} \| x - y \| _ {2} ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {\mathrm {a l l}}} \| x - y \| _ {2} ^ {2}}}, +$$ + +given by thresholding the coordinates of $\tilde{z}$ at $\tilde{\tau}$ . Recalling that for every $x\in X$ it holds that + +$$ +\tilde {z} (x) = z _ {*} (x) - \gamma_ {G} = x _ {i ^ {*}} - \gamma_ {G}, +$$ + +the cut $S$ can be equivalently described as $S = \{x\in X:x_{i^*}\leq \tilde{\tau} +\gamma_G\}$ . Therefore, it is a coordinate cut with coordinate $i^{*}$ and threshold $\tau = \tilde{\tau} +\gamma_{G}$ , and Theorem 2.2 is proven. + +Remark A.4. For context, we relate Theorem A.3 to the more familiar form of Cheeger's inequality. Let $\hat{L}_G = D^{-1/2} L_G D^{-1/2}$ be the normalized Laplacian matrix of $G$ . It is positive semi-definite and its smallest eigenvalue is 0. Let $\lambda_G$ be its second smallest eigenvalue, and $z_G$ a corresponding eigenvector. It can be shown that the quotient at right-hand side of Equation (15) is minimized by choosing $\tilde{z} = D^{1/2} z_G$ , and that the minimal value it takes is $\lambda_G$ . Hence the more standard form of Cheeger's inequality, $\Psi_G(S) \leq \sqrt{2 \lambda_G}$ . + +# A.2 $k$ -Means Example of Theorem 2.2 + +As a demonstrative example, let us show a special case of Theorem 2.2, in which a reference $k$ -means clustering is described by a suitable graph, and observe the conductance bounds it yields. Let $\mathcal{C} = \{C_1, \ldots, C_k\}$ be a partition of $X$ into $k$ clusters with corresponding centroids $\mu^{(1)}, \ldots, \mu^{(k)}$ . + +In $k$ -means, the centroids are given by cluster means, $\mu^{(i)} = \frac{1}{|C_i|}\sum_{x\in C_i}x$ . Denote the cost of cluster $C_i$ by $\mathrm{cost}(C_i) = \sum_{x\in C_i}\| x - \mu^{(i)}\| _2^2$ , and the total reference clustering cost by + +$$ +\operatorname {c o s t} (\mathcal {C}) = \sum_ {i = 1} ^ {k} \operatorname {c o s t} (C _ {i}) = \sum_ {i = 1} ^ {k} \sum_ {x \in C _ {i}} \| x - \mu^ {(i)} \| _ {2} ^ {2}, +$$ + +We instantiate Theorem 2.2 with a clique graph with edge weights inversely proportional to their cluster sizes. + +Corollary A.5. In the weighted clique graph where every edge connecting $x, y \in C_i$ has weight $1 / (|C_i| - 1)$ , there is a coordinate cut with conductance at most + +$$ +\sqrt {2 \cdot \frac {\operatorname {c o s t} (\mathcal {C})}{\frac {1}{| X |} \sum_ {x , y \in X} \| x - y \| _ {2} ^ {2}}}. +$$ + +The proof uses the following fact which can be verified by direct substitution. + +Fact A.6. For every set of points $C$ with mean $\mu = \frac{1}{|C|}\sum_{x\in C}x$ + +$$ +\sum_ {x \in C} \| x - \mu \| _ {2} ^ {2} = \frac {1}{| C |} \sum_ {x \neq y \in C} \| x - y \| _ {2} ^ {2}. +$$ + +Proof. Denote the weighted clique graph by $G(X, X^2, w)$ . For this graph: + +$$ +w (x, y) = \left\{ \begin{array}{l l} \frac {1}{| C _ {i} | - 1} & \exists i x, y \in C _ {i} \\ 0 & e l s e \end{array} \right. +$$ + +(Note that if $|C_i| = 1$ then the only node in $C_i$ has no incident edges, thus the above setting of edge weights is well-defined.) Thus, it holds that: + +$$ +\forall i,x\in C_{i} ; d_{G}(x) = \sum_{\substack{y\in C_{i}\\ y\neq x}}\frac{1}{|C_{i}| - 1} = 1. +$$ + +Hence, Theorem 2.2 implies the existence of a coordinate cut for which the conductance is at most + +$$ +\begin{array}{l} \sqrt {\frac {\mathbb {E} _ {x , y \sim \mathcal {D} _ {a d j}} \| x - y \| _ {2} ^ {2}}{\mathbb {E} _ {x , y \sim \mathcal {D} _ {a l l}} \| x - y \| _ {2} ^ {2}}} = \sqrt {\frac {\sum_ {x , y \in X} \frac {w (x , y)}{\sum_ {x ^ {\prime} , y ^ {\prime} \in X} w \left(x ^ {\prime} , y ^ {\prime}\right)} \| x - y \| _ {2} ^ {2}}{\sum_ {x , y \in X} \frac {d _ {G} (x) d _ {G} (y)}{\sum_ {x ^ {\prime} , y ^ {\prime} \in X} d _ {G} (x) d _ {G} (y)}} \| x - y \| _ {2} ^ {2}}} \\ = \sqrt {\frac {\frac {1}{| X |} \sum_ {i = 1} ^ {k} \frac {1}{| C _ {i} | - 1} \sum_ {\substack {x \neq y \\ x , y \in C _ {i}}} \| x - y \| _ {2} ^ {2}}{\frac {1}{| X | ^ {2}} \sum_ {x , y \in X} \| x - y \| _ {2} ^ {2}}} \\ = \underset {T h e o r e m A. 6} {\overset {=} {\sqrt {\frac {\sum_ {i = 1} ^ {k} \frac {1}{| C _ {i} | - 1} \left| C _ {i} \right| \sum_ {x \in C _ {i}} \| x - \mu^ {(i)} \| _ {2} ^ {2}}{\frac {1}{| X |} \sum_ {x , y \in X} \| x - y \| _ {2} ^ {2}}}} \\ \leq \sqrt {\frac {2 \cdot \sum_ {i = 1} ^ {k} \sum_ {x \in C _ {i}} \| x - \mu^ {(i)} \| _ {2} ^ {2}}{\frac {1}{| X |} \sum_ {x , y \in X} \| x - y \| _ {2} ^ {2}}} \\ = \sqrt {2 \cdot \frac {\operatorname {c o s t} (\mathcal {C})}{\frac {1}{| X |} \sum_ {x , y \in X} \| x - y \| _ {2} ^ {2}}}. \\ \end{array} +$$ + +![](images/e388f14f99d87f6c792fba6c6f52cfd2fbd5077b174adcd5a77c53c4a1b2f75f.jpg) + +Note that the denominator term $\frac{1}{|X|}\sum_{x,y\in X}\| x - y\| _2^2$ is a fixed size for the dataset and is independent of the reference clustering $\mathcal{C}$ . Thus, with an appropriate choice of graph to describe the reference clustering, Theorem 2.2 yields coordinate cuts with conductance directly related to the reference clustering cost. + +# A.3 IMM Alternative Analysis + +In this section we focus on $k$ -medians clustering in the $\ell_1$ norm. For every subset $C \subset \mathbb{R}^d$ , denote its median by + +$$ +\operatorname {m e d} (C) = \min _ {z \in \mathbb {R} ^ {d}} \sum_ {x \in C} \| x - y \| _ {1}. +$$ + +Let $\mathcal{C} = C_1, \ldots, C_k$ be a reference $k$ -medians clustering of $X$ with respective cluster centroids $\mu^{(i)} = \mathrm{med}(C_i)$ . The $k$ -medians clustering cost of $\mathcal{C}$ is defined as + +$$ +\operatorname {c o s t} _ {1} (\mathcal {C}) = \sum_ {i = 1} ^ {k} \sum_ {x \in C _ {i}} \| x - \mu^ {(i)} \| _ {1}. +$$ + +Let $T_{\mathrm{IMM}}(\mathcal{C})$ be the explainable clustering tree returned by IMM algorithm. [33] proved the following theorem: + +Theorem A.7. For every reference clustering $\mathcal{C}$ with centroids, $\mathrm{cost}_1(T_{\mathrm{IMM}}(\mathcal{C}))\leq O(k)\cdot \mathrm{cost}_1(\mathcal{C})$ + +In the terminology from Section 1, the "price of explainability" of $k$ -medians clustering is $O(k)$ . This remains the best bound to date for a deterministic algorithm. + +In this section we prove the same theorem for our slightly modified variant from Section 3.1, where it was cast in terms of minimizing the non-uniform cut sparsity. The goal is to demonstrate how the graph partitioning framework can be used analytically, and present an alternative and arguably simpler proof for the same price of explainability upper bound. Let $\tilde{T}_{\mathrm{IMM}}(\mathcal{C})$ denote the tree returned by our modified IMM. + +Theorem A.8. For every reference clustering $\mathcal{C}$ with centroids, $\mathrm{cost}_1(\tilde{T}_{\mathrm{IMM}}(\mathcal{C})) \leq O(k) \cdot \mathrm{cost}_1(\mathcal{C})$ . + +For the proof, we introduce some notation aligned with [33] for better comparison and readability. Recall that the IMM algorithm builds an explainable decision tree on both the points $X$ and the centroids $M = \{\mu^{(1)},\ldots ,\mu^{(k)}\}$ , such that each tree leaf contains exactly one centroid, and partition of $X$ into the $k$ leaves forms the explainable clustering. For every $x\in X$ , denote by $\mu (x)\in M$ its centroid in the reference clustering $\mathcal{C}$ . For every tree node $u$ , let $X^u\subset X$ and $M^u\subset M$ be the subsets of points and centroids, respectively, contained in $u$ , and let $Y^{u} = X^{u}\cup M^{u}$ . A mistake in $u$ is a pair $x\in X^u$ , $c(x)\in M^u$ that are separated by the threshold coordinate cut in $u$ . Let $t^u$ denote the number of mistakes in $u$ . Also, recall the in our modified IMM, in each node $u$ we fix a pair of centroids $\mu_u^\prime ,\mu_u^{\prime \prime}$ at maximal distance among the centroids in $u$ : + +$$ +\| \mu_ {u} ^ {\prime} - \mu_ {u} ^ {\prime \prime} \| _ {1} = \min _ {i, j: \tilde {\mu} ^ {\prime}, \tilde {\mu} ^ {\prime \prime} \in M ^ {u}} \| \tilde {\mu} ^ {\prime} - \tilde {\mu} ^ {\prime \prime} \| _ {1}. +$$ + +The proof of Theorem A.8 has two steps, analogous to the proof of Theorem A.7. The first step is a straightforward consequence of the triangle inequality. The detailed proof appears in [33] and is omitted here. + +Lemma A.9 (Lemma 5.5 in [33]). + +$$ +\operatorname {c o s t} _ {1} (\tilde {T} _ {\mathrm {I M M}} (\mathcal {C})) \leq \operatorname {c o s t} _ {1} (\mathcal {C}) + \sum_ {u \in T _ {\mathrm {I M M}} ^ {\prime} (\mathcal {C})} t _ {u} \cdot \| \mu_ {u} ^ {\prime} - \mu_ {u} ^ {\prime \prime} \| _ {1}. +$$ + +The second step is the more involved part of the proof, and the part where the graph-based analysis presented here departs from [33]. + +Lemma A.10 (Lemma 5.6 in [33]). Let $H$ be the height of $\tilde{T}_{\mathrm{IMM}}(\mathcal{C})$ . Then, + +$$ +\sum_ {u \in T _ {\mathrm {I M M}} ^ {\prime} (\mathcal {C})} t _ {u} \cdot \| \mu_ {u} ^ {\prime} - \mu_ {u} ^ {\prime \prime} \| _ {1} \leq H \cdot \operatorname {c o s t} _ {1} (\tilde {T} _ {\mathrm {I M M}} (\mathcal {C})). +$$ + +Since $\tilde{T}_{\mathrm{IMM}}(\mathcal{C})$ has $k$ leaves, its height is at most $k$ , thus Theorems A.9 and A.10 together immediately imply Theorem A.8. + +Proof of Theorem A.10. Fix a node $u$ in $\tilde{T}_{\mathrm{IMM}}(\mathcal{C})$ . Let $\times_{j=1}^{d}[a_j^u, b_j^u]$ be a bounding box containing all of $Y^u$ . Define the following distribution $\mathcal{D}$ over coordinate cuts $j, \tau$ : first, pick $j \in \{1, \ldots, d\}$ with probability proportional to $|b_j^u - a_j^u|$ ; then, pick $\tau$ uniformly at random in $[a_j^u, b_j^u]$ . + +Let $\tilde{G}(Y^u, \tilde{E})$ be any undirected graph over $Y^u$ . The probability that an edge $xy \in \tilde{E}$ is cut by $(j, \tau) \sim \mathcal{D}$ is + +$$ +\sum_ {j = 1} ^ {d} \frac {\left| b _ {j} ^ {u} - a _ {j} ^ {u} \right|}{\left\| b ^ {u} - a ^ {u} \right\| _ {1}} \cdot \frac {\left| x _ {j} - y _ {j} \right|}{\left| b _ {j} ^ {u} - a _ {j} ^ {u} \right|} = \frac {\left\| x - y \right\| _ {1}}{\left\| b ^ {u} - a ^ {u} \right\| _ {1}}, +$$ + +and therefore, + +$$ +\mathbb {E} _ {(j, \tau) \sim \mathcal {D}} \left[ e _ {\tilde {G}} \left(S _ {j, \tau}, Y _ {u} \backslash S _ {j, \tau}\right) \right] = \frac {1}{\| b ^ {u} - a ^ {u} \| _ {1}} \sum_ {x y \in \tilde {E}} \| x - y \| _ {1}. \tag {16} +$$ + +Let $S_{u}$ be the set of coordinate cuts $(j, \tau)$ that satisfy $\tau \in [a_{j}, b_{j}]$ . Let $m_{u}(j, \tau)$ denote the number of mistakes that a cut $j, \tau$ makes (i.e., the number of points in $x \in X^{u}$ such that $c(x) \in M^{u}$ , but the cut separates $x$ and $c(x)$ ). Recall that our modified IMM chooses the cut $j_{u}, \tau_{u}$ in the node among the cuts in $S_{u}$ that separate the pair $\mu_{u}^{\prime}, \mu_{u}^{\prime \prime}$ , and $t_{u} = m_{u}(j_{u}, \tau_{u})$ . + +Let $G(Y^u, E_G)$ be the star graph on $Y^u$ , where each $x \in X^u$ is adjacent to $c(x)$ if $c(x) \in M^u$ , and is an isolated node otherwise. Observe that $m_u(j, \tau) = e_G(S_{j,\tau}, Y_u \setminus S_{j,\tau})$ . Let $H(Y^u, E_H)$ be the graph that contains a single edge between $\mu_u', \mu_u''$ . Observe that $e_H(S_{j,\tau}, Y_u \setminus S_{j,\tau})$ equals 1 if the cut $j, \tau$ separates $\mu_u', \mu_u''$ , and 0 otherwise. We denote this as $e_H(S_{j,\tau}, Y_u \setminus S_{j,\tau}) = 1\{(j, \tau) \text{ separates } \mu_u', \mu_u''\}$ . Therefore, + +$$ +\begin{array}{l} t_{u} = m_{u}(j_{u},\tau_{u}) = \min_{\substack{j,\tau \in \mathcal{S}_{u}:\\ (j,\tau)\text{separates}\mu^{\prime}_{u},\mu^{\prime \prime}_{u}}}m_{u}(j,\tau) \\ = \min _ {j, \tau \in S _ {u}} \frac {m _ {u} (j , \tau)}{\mathbf {1} \left\{(j , \tau) \text {s e p a r a t e s} \mu_ {u} ^ {\prime} , \mu_ {u} ^ {\prime \prime} \right\}} \\ = \min _ {j, \tau \in \mathcal {S} _ {u}} \frac {e _ {G} \left(S _ {j , \tau} , Y _ {u} \backslash S _ {j , \tau}\right)}{e _ {H} \left(S _ {j , \tau} , Y _ {u} \backslash S _ {j , \tau}\right)} \\ \leq \frac {\mathbb {E} _ {(j , \tau) \sim \mathcal {D}} \left[ e _ {G} \left(S _ {j , \tau} , Y _ {u} \setminus S _ {j , \tau}\right) \right]}{\mathbb {E} _ {(j , \tau) \sim \mathcal {D}} \left[ e _ {H} \left(S _ {j , \tau} , Y _ {u} \setminus S _ {j , \tau}\right) \right]} \\ = \frac {\sum_ {x y \in E _ {G}} \| x - y \| _ {1}}{\sum_ {x y \in E _ {H}} \| x - y \| _ {1}} \\ = \frac {\sum_ {x \in X ^ {u}} \| x - c (x) \| _ {1}}{\| \mu_ {u} ^ {\prime} - \mu_ {u} ^ {\prime \prime} \| _ {1}} \\ \end{array} +$$ + +Theorem A.2 + +Equation (16) + +definition of $G$ and $H$ . + +Rearranging, $t_{u} \cdot \| \mu_{u}^{\prime} - \mu_{u}^{\prime \prime}\|_{1} \leq \sum_{x\in X^{u}}\| x - c(x)\|_{1}$ . Since in each level $L$ in the tree the clusters $\{X^{u}:u\in L\}$ form a partition of $X$ , we get + +$$ +\sum_ {u \in L} t _ {u} \cdot \| \mu_ {u} ^ {\prime} - \mu_ {u} ^ {\prime \prime} \| _ {1} \leq \sum_ {u \in L} \sum_ {x \in X ^ {u}} \| x - c (x) \| _ {1} = \sum_ {x \in X} \| x - c (x) \| _ {1} = \operatorname {c o s t} _ {1} (\mathcal {C}). +$$ + +Summing again over the $H$ levels in the tree yields the lemma. + +Note that the ratio $\frac{e_G(S_{j,\tau}, Y_u \backslash S_{j,\tau})}{e_H(S_{j,\tau}, Y_u \backslash S_{j,\tau})}$ that arises in the proof is the non-uniform cut sparsity $\Psi_{G,H}(S_{j,\tau})$ from Section 3. + +# B Additional Experimental Results + +Running times. Table 4 contains running time measurements. For reference-based methods (EMN, CART and SPEx-Clique) we measure the reference clustering step and tree construction step separately. For SPEx-kNN, we measure the $k$ -NN graph construction step and the tree construction step separately. + +In our experiments, Kernel IMM proved feasible to run only on the smaller datasets (R15, Pathbased, Iris, Ecoli, Breast Cancer). Each algorithm was allotted three hours to run on each dataset. On the + +Table 4: Runtime comparison of the non kernel k-means based algorithms. Presented times are the median runtime across five runs. For the Clique and CART algorithms, performance was measured based on a spectral reference. + +
AlgorithmBeansIrisCIFARCaltech 101R15PathbasedEcoliCancerNewsgroupsMNIST
Spectral reference1.8s22.1ms1m12s7.6s91.6ms29ms34.2ms34.1ms8.54s1min53s
k-means reference86.4ms4.21ms2.69s2.1s10.4ms4.16ms5.39ms7.01ms1.04s5.16s
k-NN graph build1.87s7.01ms1m55s2.23s19ms3.57ms9.31ms8.03ms5.79s2min59s
SPEx-kNN3.69s27ms5m53s1m48s93.4ms22.8ms85.3ms90.8ms2min27s2min52s
SPEx-Clique203ms8.36ms28.2s1m12s17ms9.65ms13.3ms16.3ms13.2s39.4s
EMN13.5s6.51ms5m13s15m40s101ms13.3ms20.9ms184ms7min56s5min20s
CART349ms14.8ms32.2s8min51s72.7ms9.4ms33.9ms31.9ms39.2s30s
+ +Table 5: Performance of SPEx-kNN for different $k$ values. The reference compared in the REF column is the reference with the same $k$ applied. + +
kEcoliIrisCancer
ARIAMIREFRSAMIREFARIAMIREF
2.594.571.006.287.370.020.681.603-.019
5.594.589.640.450.647.895.594.546.688
10.682.648.828.450.647.514.507.490.547
15.682.648.854.450.647.445.507.490.567
20.679.642.863.450.647.450.507.490.562
50.679.638.593.600.642.756.507.490.532
+ +larger datasets, Kernel IMM either ran out of memory, or failed to complete running within the allotted time. All other methods finished running within up to 16 minutes (see Table 4). Therefore, we report results for Kernel IMM only for the smaller datasets. + +kNN graph parameters. Table 5 includes additional results for SPEx-kNN, showing how its performance changes as $k$ (the parameter of the $k$ -NN graph) varies on the Ecoli, Iris and Breast Cancer datasets. + +Number of leaves. As mentioned in Section 2, SPEx can produce a tree with any desired number of leaves $\ell$ . Our evaluation so far has focused on the setting $\ell = k$ , i.e., the number of leaves in the output tree is equal to the number of clusters in the reference clustering. To clarify the relation between $k$ and $\ell$ , viewing the reference clustering as a trained model, $k$ is its number of outcomes and $\ell$ is the number of explanations our explainability method can yield. Since each outcome must have its own separate explanation, we must have $\ell \geq k$ , thus $\ell = k$ is the "most explainable" setting. As $\ell$ increases, the expressiveness of the explainable clustering tree grows and it is better able to approximate the reference clustering, albeit at the cost of explainability, since now some model outcomes would have multiple different explanations. + +Table 6 includes result with $\ell = 2k$ for SPEx-Clique, CART-Spectral, CART- $k$ -means, and ExKMC [17], which is an extension of IMM to support more leaves (which IMM does not naturally support, due to its reliance on reference clustering centroids; the same goes for EMN). The results do not point to a clearly superior method. + +Table 6: Results with $\ell = {2k}$ leaves. + +
AlgorithmEcoliIrisCancerMNISTCaltech 101NewsgroupsBeansCifar
ARIAMIARIAMIARIAMIARIAMIARIAMIARIAMIARIAMIARIAMI
SpEx-Clique.469.589.716.739.491.464.197.317.203.565.088.281.370.512.309.484
ExKMC.458.582.730.755.491.464.224.328.249.546.096.263.370.512.323.467
CART-k-means.447.573.716.739.491.464.031.1770.208.026.148.370.512.144.371
CART-Spectral.434.493.610.648.749.631.080.2100.214.006.104.608.695135.381
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Adams $^{2}$ + +$^{1}$ Department of Materials Science & Engineering, University of Illinois at Urbana-Champaign + +$^{2}$ Department of Computer Science, Princeton University + +3Materials Research Laboratory, University of Illinois at Urbana-Champaign + +# Abstract + +Accelerating inverse design of crystalline materials with generative models has significant implications for a range of technologies. Unlike other atomic systems, 3D crystals are invariant to discrete groups of isometries called the space groups. Crucially, these space group symmetries are known to heavily influence materials properties. We propose SGEquiDiff, a crystal generative model which naturally handles space group constraints with space group invariant likelihoods. SGEquiDiff consists of an SE(3)-invariant, telescoping discrete sampler of crystal lattices; permutation-invariant, transformer-based autoregressive sampling of Wyckoff positions, elements, and numbers of symmetrically unique atoms; and space group equivariant diffusion of atomic coordinates. We show that space group equivariant vector fields automatically live in the tangent spaces of the Wyckoff positions. SGEquiDiff achieves state-of-the-art performance on standard benchmark datasets as assessed by quantitative proxy metrics and quantum mechanical calculations. Our code is available at https://github.com/rees-c/sgequidiff. + +# 1 Introduction + +Crystals comprise critical technologies like batteries [66], topological materials [90], electronic devices [96], photovoltaics [28], and more. Materials scientists have catalogued $\mathcal{O}(10^{5})$ crystals experimentally [7] and $\mathcal{O}(10^{6})$ in silico with density functional theory (DFT) simulation [15, 39, 79]. In contrast, the number of stable crystalline materials with five elements or less is estimated to exceed $10^{13}$ , and even higher-order compositions are common in real materials [16]. Generative models offer a promising path to rapidly explore the vast space of crystals [100, 40, 62, 10]. + +Unlike molecules, crystals span the periodic table and exhibit discrete spatial symmetries according to one of 230 space groups [2]. Specifically, crystals have invariances to discrete translations, rotations, reflections, and sequences thereof that transform atoms into themselves or into identical atoms. The list of space group actions which map a point into itself is called a stabilizer group. When a set of points in $\mathbb{R}^3$ have conjugate stabilizer groups, the set is called a Wyckoff position. Importantly, as shown in Figure 1, Wyckoff positions can have zero volume, comprising points, lines, or planes. These zero volume sets are referred to as special Wyckoff positions. We give a more formal treatment of space groups and Wyckoff positions in Sec. 3. + +Despite the fact that most existing crystal generative models ignore space groups and Wyckoff positions, they are critical for modeling real materials. Firstly, space groups and Wyckoff positions correlate strongly with materials properties; Neumann's principle states that all crystal properties share the same invariances as the crystal itself [65]. Thus even slightly perturbing atoms out of special Wyckoff positions will reduce the crystal's space group symmetry and can subsequently cause significant (even discontinuous) changes to its macroscopic properties [14, 11, 82, 9]. Secondly, we + +![](images/1e1f6feb7330164075a36d1e59920511d47d638c37817a01e241abe8a005abda.jpg) +Figure 1: (a) The asymmetric unit (II) and special Wyckoff positions labeled by letter in the conventional unit cell of space group 10. (b-c) Histograms of occupied space groups and Wyckoff dimensionalities by crystals in the MP20 [100, 39, 7] training dataset. Space groups and Wyckoff positions were determined by the SpaceGroupAnalyzer module in pymatgen [70, 91] using tolerances of $0.1\AA$ and $5^{\circ}$ . These tolerances help account for the moderate convergence criteria of the Materials Project DFT relaxations. + +![](images/8593f3a12303ab704315a2aca4778a3ebf4b00660fc37d1a86bcbf56add34cbe.jpg) + +![](images/b2c90a538620d4aca6183fcccb4dd71b1c56b2e907ad0ab8d8dcb43826803d39.jpg) + +show empirically in Figure 1 that known materials are usually invariant to high symmetry space groups with atoms in zero-volume Wyckoff positions. An intuitive explanation for this phenomenon is that atoms of the same species in a given crystal are most stable in a specific neighboring environment, and the tendency to attain this environment naturally leads to symmetry [32]. Yet, most existing crystal generative models learn continuous distributions over all three spatial dimensions of atom positions, assigning zero probability measure to placing atoms in special Wyckoff positions. One might argue that crystals from these models can be relaxed into high symmetry positions with DFT or machine learned force fields. However, besides the apparent difficulty of generating atoms sufficiently close to high symmetry positions for relaxation [107, 31, 42], such a framework assigns different model probabilities to crystals that relax into the same structure, obfuscating training and evaluation. + +In this paper, we propose SGEquiDiff, which enforces space group constraints during generation and yields explicit space group invariant model likelihoods. SGEquiDiff combines SE(3)-invariant, discretized lattice sampling; permutation-invariant, transformer-based generation of occupied Wyckoff positions and elements; and space group equivariant diffusion of atomic coordinates. For the latter, we built a space group equivariant graph neural network and trained it with space group equivariant scores from a Space Group Wrapped Normal distribution. + +The main contributions of our work are summarized as follows: + +- We built a space group equivariant denoising graph neural network and regressed it against scores from a novel Space Group Wrapped Normal distribution. +- We prove that space group equivariant vector fields live in the tangent space of each Wyckoff position, ensuring atoms never leave their Wyckoff positions during diffusion. This result obviates the need to project atoms onto Wyckoff positions, which can lead to indeterminate probability distributions [41] or a discontinuous generation process [55, 78]. +- SGEquiDiff proposes an explicit-likelihood autoregressive alternative to diffusion-based sampling of space group-constrained lattice parameters. The autoregressive sampling also enables masked in-filling. This may be useful, e.g., to generate optimal substrate materials with low lattice mismatch for epitaxial crystal growth [110, 19], commonly used for semiconductors and magnetic devices. +- We verified the efficacy of our model on standard crystal datasets with proxy metrics and rigorous DFT calculations. + +# 2 Related Work + +Early crystal generative models represented crystals as voxelized images [67, 34] or padded tensors of 3D coordinates [49, 75] to train variational autoencoders (VAE) [18] or generative adversarial networks [26]. Several works have enforced the SE(3) and periodic translational invariance of crystals by leveraging graph neural networks. One popular approach is to use diffusion models [83, 84, 33] + +on crystal lattices, atom types, and atom positions [100, 40, 107, 57]. These models have also been extended with the flow matching framework [58], accelerating sampling [62]; the stochastic interpolants framework [1], expanding the model design space [35]; and the Bayesian Flow Networks framework [27, 97] which iteratively refines model parameters instead of samples. Other works have aimed to learn the SE(3) and periodic invariances through data augmentations [104, 31] or data canonicalization [101] with large language models (LLMs) [31, 101] or with diffusion of raw attributes [104] or of latents [42]. Recent works have used LLMs to generate noisy crystals or textual context which is passed as input to graph-based diffusion or flow matching [86, 103, 71]. We view many of these existing works as orthogonal to ours since our handling of space group symmetries can be combined with their modeling frameworks. + +Two of the aforementioned works attempted to learn space group-conditioned generation without hard constraints. The graph diffusion model MatterGen [107] was fine-tuned on 14 space groups and used ground truth numbers of atoms per unit cell per space group to initialize generation. However, they could only generate stable, unique, and novel crystals in target space groups with $16\%$ accuracy. Similarly, CrystalLLM [31] only managed $24\%$ accuracy despite a generous SpaceGroupAnalyzer [91, 70] tolerance of $0.2\mathring{\mathrm{A}}$ . + +Relevant to our model, a growing number of works have considered hard space group constraints during generation; however, to our knowledge, they either produce crystals without space group invariant likelihoods, thus assigning different likelihoods to symmetrically equivalent atoms, or rely on local structure relaxations using machine learning interatomic potentials (MLIPs) or DFT. WyCryst [108] trained a VAE to generate elements and Wyckoff position occupations, but rely on expensive DFT relaxations of atoms from uniform random locations in the Wyckoff positions. Crystal-GFN [64] considered space group constraints for the task of distribution matching under the GFlowNet framework [6] but did not address how to sample atom coordinates. DiffCSP++ [41] trained a graph-based diffusion model with masked diffusion of the unit cell lattice, continuous element diffusion with a post-hoc argmax, and projected diffusion of atom positions on the Wyckoff subspaces. However, DiffCSP++ is not space group equivariant to the best of our knowledge and relies on fixed templates from the training data of space group, number of atoms per unit cell, and occupied Wyckoff positions. SymmCD [55] is a non-equivariant diffusion model which leverages asymmetric units to reduce memory footprints; they use discrete diffusion of Wyckoff positions and elements and post-hoc projections of atomic coordinates onto Wyckoff positions. SymmBFN [78] recently extended SymmCD to the Bayesian Flow Networks framework [27], also requiring post-hoc projections of atom positions to the Wyckoff subspaces. CrystalFormer [10] trained an autoregressive transformer, canonicalizing crystals as a sequence of atoms ordered lexicographically by Wyckoff letter and then fractional coordinates. However, they rely on von Mises distributions which are not space group invariant. WyckoffTransformer [45] also autoregressively predicts atom types and Wyckoff positions but relies on DiffCSP++ or MLIPs to refine atoms from uniform random coordinates in the Wyckoff positions. We note that MLIPs are hindered by kinetic barriers and only preserve space group symmetry if they predict conservative forces of an invariant energy, generally requiring an expensive backward pass for inference [24, 77, 51]. WyckoffDiff [47] generates Wyckoff positions and elements with D3PM-based diffusion [3] but similarly relies on MLIPs to determine atom coordinates. Unlike these existing works, our model learns to generate complete crystals from scratch and produces space group invariant likelihoods via space group equivariant diffusion. + +# 3 Preliminaries + +Space groups Formally, a space group $G \in \mathbb{G}$ , where $\mathbb{G}$ denotes the set of 230 space groups, is a group of isometries that tiles $\mathbb{R}^3$ . In particular, $G$ is generated by an infinite subgroup of discrete lattice translations $T_L = \{n_1L_1, n_2L_2, n_3L_3 | n_i \in \mathbb{Z}, L_i \in \mathbb{R}^3\}$ as well as a collection of other symmetry operations $g(\cdot) = \{R_g(\cdot) + v_g | R_g \in \mathrm{O}(3), v_g \in \mathbb{R}^3\} \in G$ , where $R_g$ is a point group operation (rotation, reflection, or identity) and $v_g$ is a translation. + +Wyckoff positions Given a space group and a point $x \in \mathbb{R}^3$ , the stabilizer group $G_x \equiv \{g | gx = x\} \subset G$ (also called the site symmetry group in materials science) is the finite subgroup of $G$ that leaves $x$ invariant. A Wyckoff position is then defined as the set of points with conjugate stabilizer groups, i.e., $\{x' | \exists g \in G : G_{x'} = gG_xg^{-1}\}$ . Conceptually, if $g$ is a point group operation, this means that all points in a Wyckoff position are invariant to the same space group operations up to a change + +of basis. By convention, when $x$ is described with respect to the lattice basis $\{L_1, L_2, L_3\}$ , the size of the orbit of $x$ in the unit cell, $|\{gx | g \in G, gx \in [0,1)^3\}|$ , is called the Wyckoff multiplicity. Wyckoff positions whose stabilizer groups are non-trivial, i.e., include more than the identity operation, are referred to as special Wyckoff positions as opposed to the general Wyckoff position defined by the identity stabilizer group. Wyckoff positions are labeled by multiplicity and Wyckoff letter, where the lexicographic ordering gives the Wyckoff positions in (partial) order of increasing multiplicity. + +Unit cells The infinite translational periodicity of a crystal can be represented with a parallelipiped $\Gamma$ called the unit cell. The unit cell reduces infinite crystals by removing redundancy induced by $T_{L}$ , the group of discrete lattice translations. In this way, crystals are represented by the tuple $M = (A,X,L)$ where $A = (a_1',\ldots,a_N') \in \mathbb{A}^N$ are the atom types, $\mathbb{A}$ is the set of all chemical elements, and $N$ is the number of atoms in the unit cell; $X = \{(x_1',\ldots,x_N') | x_i' \in \Gamma\}$ are the Cartesian atom coordinates; and $L = (L_1,L_2,L_3) \in \mathbb{R}^{3\times 3}$ are the unit cell basis vectors. Given $M$ , the infinite periodic structure can be reconstructed by applying the actions of $T_{L}$ as $\{(a_i',x_i' + n_1L_1 + n_2L_2 + n_3L_3)_i = 1|n_j \in \mathbb{Z}\}$ . Alternatively, the atom coordinates can be given in the lattice basis instead of the Cartesian basis. Coordinates in the lattice basis are commonly referred to as fractional coordinates. In fractional coordinates, the infinite crystal is reconstructed as $\{(a_i',x_i' + n_1e_1 + n_2e_2 + n_3e_3)_i = 1|n_j \in \mathbb{Z}\}$ . For the rest of this paper, we assume atom coordinates are always in fractional coordinates. While the choice of unit cell is not unique, prior crystal generative models [100, 40, 62] either canonicalize it with a minimum-volume primitive cell determined by the Niggli algorithm [29] or a conventional cell [2] which contains all the symmetries of the space group. + +Asymmetric units In our work, we represented crystals with a convex polytope $\Pi \in \mathbb{R}^3$ , the asymmetric unit (ASU), which maximally reduces infinite crystals by removing all redundancies induced by the space group $G \supseteq T_L$ . Under this formulation, we consider atoms in the ASU with fractional coordinates $X = \{(x_1, \dots, x_n) | x_i \in \Pi\} \in \mathbb{R}^{n \times 3}$ , atom types $A = (a_1, \dots, a_n) \in \mathbb{A}^n$ and Wyckoff positions $W = \{(w_1, \dots, w_n) | w_i = G_{x_i}\} \in \mathbb{W}^n$ where $n \leq N$ . The infinite periodic structure of a crystal can be reconstructed by applying the actions of $G$ to $\Pi$ , i.e., + +$$ +\left\{\left(a _ {i}, g _ {i j} x _ {i}\right) \mid x _ {i} \in \Pi , g _ {i j} \in G / w _ {i}, i \in (1, \dots , n) \right\}. +$$ + +By focusing on these $n$ symmetrically inequivalent atoms, we reduced our model's memory footprint and minimized the dimensionality of the generative modeling task. We canonicalized the non-unique choice of ASU using those listed in the International Tables for Crystallography [2] with additional conditions on faces, edges, and vertices from Grosse-Kunstleve et al. [30] to ensure that the ASUs are exact, i.e., that $\Pi$ tiles $\mathbb{R}^3$ without overlaps at the boundaries $\partial \Pi$ . + +Diffusion models Diffusion-based generative models [33, 84] construct samples $\mathbf{x}_0\sim p_{t = 0}$ from a noisy prior $\mathbf{x}_T\sim p_{t = T}$ by evolving $\mathbf{x}_T$ through time $t$ with a learned denoising process. The training process generally involves iteratively adding noise to data samples $\mathbf{x}_0$ via the forward stochastic differential equation (SDE), + +$$ +d \mathbf {x} = \mathbf {f} (\mathbf {x}, t) d t + \eta (t) d \mathbf {w}, \tag {1} +$$ + +where $\mathbf{f}(\mathbf{x},t):\mathbb{R}^d\to \mathbb{R}^d$ is the drift coefficient, $\mathbf{w}$ is the standard Wiener process, and $\eta (t)\in \mathbb{R}$ is the diffusion coefficient. Samples are generated by running the process backwards in time from $T$ to 0 via the reverse SDE, + +$$ +d \mathbf {x} = [ \mathbf {f} (\mathbf {x}, t) - \eta (t) ^ {2} \nabla_ {\mathbf {x}} \log p _ {t} (\mathbf {x}) ] d t + \eta (t) d \bar {\mathbf {w}}, \tag {2} +$$ + +where $\bar{\mathbf{w}}$ is another standard Wiener process. The generative model aims to learn the score of the marginal distribution $\nabla_{\mathbf{x}}\log p_t(\mathbf{x})$ , and then simulate Eq. 2 to sample from $p_0$ . In our model, we have $\mathbf{f}(\mathbf{x},t) = 0$ . + +Space group equivariant functions For a group of isometries $G'$ , Rezende et al. [76] and Köhler et al. [52] showed that equivariant flows preserve invariant probability densities. Conversely, the gradient of a $G'$ -invariant function is equivariant [44]. We are interested in using this result for the space groups. A function $f: \mathbb{R}^3 \to \mathbb{R}^3$ is space group equivariant if it satisfies $R_g f(x) = f(gx)$ for all $g \equiv (R_g, v_g) \in G$ , where $R_g \in O(3)$ is a point group element and $v_g \in \mathbb{R}^3$ is a translation. Symmetrization via summing over group elements has become a popular method to build invariant and equivariant neural networks in the last few years [105, 74]. Recently, Mirramezani et al. [63] + +![](images/50b8d30be3c5ebee256ce1718cd75916652e4f63c73785be4d1f8530c7fa98f9.jpg) +Figure 2: Illustration of our crystal generation process. + +![](images/f71a9f4887d784f70136ed05e01185836f8866fbcc5e3f1feccdf23446a26e29.jpg) + +showed how symmetrization can be applied to space groups despite their countably infinite size. In particular, given the finite quotient group $G / T_{L}$ and a $T_{L}$ -invariant function $\hat{f} : \mathbb{R}^3 \to \mathbb{R}^3$ , the following function is space group equivariant: + +$$ +f (x) = \frac {1}{\left| G / T _ {L} \right|} \sum_ {g \in G / T _ {L}} R _ {g} ^ {- 1} \hat {f} (g x). \tag {3} +$$ + +Besides faithfully modeling the space group symmetries of crystals, we discovered that space group equivariant vector fields conveniently live in the tangent spaces of the Wyckoff positions. We formalize this in the following theorem. + +Theorem 1 Let $G$ be a space group, $x \in \mathbb{R}^3$ be a point residing in a Wyckoff position with stabilizer group $G_x$ , and $f: \mathbb{R}^3 \to \mathbb{R}^3$ be a space group equivariant function. Then, for arbitrary constant $c \in \mathbb{R}$ and any space group element $g \in G_x$ , + +$$ +g (x + c f (x)) = x + c f (x) \Rightarrow G _ {x} \subseteq G _ {x + c f (x)}. \tag {4} +$$ + +See proof in Sec. A.1. This result implies that a space group equivariant flow is automatically restricted to the manifold of its Wyckoff position, preventing the need for projections that lead to indeterminate probability distributions [41] or a discontinuous generation process [55, 78]. The result also implies that if, for example, a 0D Wyckoff position (a point) lies in a 1D Wyckoff position (a line), then a space group equivariant vector field cannot move an atom along the line through the point. This is problematic because we observed examples where traversing between symmetrically unique regions of a 1D Wyckoff position requires moving through a 0D Wyckoff position (e.g., in space group 192, the 1D Wyckoff position $12\mathrm{k}$ is split by the 0D Wyckoff position 4c. See Fig. 3). This issue motivates SDE-based sampling since Gaussian noise (projected onto the tangent space of the Wyckoff position) can stochastically perturb atoms through 0D Wyckoff positions while maintaining that the marginal distribution over atom coordinates is still space group invariant. + +# 4 Proposed method: SGEquiDiff + +# 4.1 Crystal representation + +Using sympy [61] and PyXtal [22], we removed redundancy induced by space group symmetry from every Wyckoff position by intersecting each one with its exact asymmetric unit. Each zero-dimensional Wyckoff position was reduced to a single point, each one-dimensional position to a set of line segments, each two-dimensional position to a set of convex polygonal ASU facets, and each three-dimensional position to the ASU interior. We will refer to these convex sets as Wyckoff shapes. Intersections for space group 192 are shown as an example in Figure 3. We used Wyckoff shapes for memory-efficient crystal representation and sampling as will be described in the following sections. + +We decomposed each crystal into $M = (G, L, A, W, X) \in (\mathbb{G}, \mathbb{R}^{3 \times 3}, \mathbb{A}^n, \mathbb{W}^n, \mathbb{R}^{n \times 3})$ , where $n$ is the number of atoms in the asymmetric unit, $G \in \mathbb{G}$ is the space group, $L \in \mathbb{R}^{3 \times 3}$ is the + +conventional lattice basis, $A \in \mathbb{A}^n$ are elements, $W \in \mathbb{W}^n$ are Wyckoff positions, and $X \in \mathbb{R}^{n \times 3}$ are fractional atom coordinates. Several variables in $M$ impose hard constraints on each other. To handle these dependencies, the model sequentially samples each constrained variable after its constraining variable(s). We factorized the generation process as follows: + +$$ +p (M) = p (G) \times p (L | G) \times p (W, A | L, G) \times p (X | W, A, L, G). \tag {5} +$$ + +Structuring the generation process in this way rather than jointly modeling all crystal components avoids tuning per-component loss weights [100], memory-intensive mask states [104], and post-hoc projections that are intractable to marginalize over [55]. We set $p(G)$ to the empirical training distribution. Now we describe our method for the remaining conditional distributions. + +# 4.2 Telescoping discrete sampling of $L$ + +We parameterized univariate conditionals to autoregressively sample the 3 lattice lengths $(a, b, c)$ and 3 angles between them $(\alpha, \beta, \gamma)$ for conventional unit cells, applying physical constraints to each conditional. Canonicalizing the lattice with the conventional lattice parameters makes the lattice sampling automatically SE(3) invariant. Denoting $l = (a, b, c, \alpha, \beta, \gamma)$ , the model learned + +$$ +p (L | G) = \prod_ {i = 1} ^ {6} p \left(l _ {i} \mid l _ {< i}, G\right), \tag {6} +$$ + +where $p(l_i|l_{ 0. +$$ + +Besides physical constraints, lattice generation requires the flexibility to learn highly peaked distributions since small perturbations to a crystal lattice can significantly alter materials properties. In $\mathrm{BaTiO_3}$ for example, $0.03\AA$ strain was found to increase the ferroelectric transition temperature by $500^{\circ}\mathrm{C}$ and the remnant polarization by $250\%$ [13]. In contrast, the range of conventional lattice lengths in the MP20 dataset is over $100\AA$ . To address this challenge, we chose to discretize the lattice parameters to a resolution of $0.01\AA$ . Naively, this resolution requires a softmax over $N_{l} = \mathcal{O}(10^{4})$ classes per lattice parameter to achieve a $100\AA$ range. We overcame this poor scaling by telescoping the categorical distribution. See Figure 2 for a visual explanation. At a high level, the range of lattice parameters was first binned very coarsely, and a class $b_{1}$ was sampled from $p(b_{1}|l_{ 0$ . As $\sigma_{t}$ gets large, the SGWN approaches the uniform distribution. Noisy samples $x_{t}$ were created by sampling Gaussian noise $\epsilon \sim \mathcal{N}(0,I)$ and reparameterizing as $x_{t} = x_{0} + \mathcal{P}_{x_{0}}(\sigma_{t}\epsilon)$ where $\mathcal{P}_{x_0}(\cdot)$ is an orthogonal projection to the tangent space of $x_0$ 's Wyckoff shape. For the backward process, we constructed the uniform prior $p(x_{T})$ on each Wyckoff position. To do so without redundantly assigning probability mass to symmetrically equivalent points, we placed uniform distributions on the relevant Wyckoff shapes. See Sec. A.3 for details. We used predictor-corrector sampling [85] to sample $x_0$ . + +To learn the space group equivariant scores of the SGWN distribution, we built a space group equivariant graph neural network $s_{\theta}$ (see Sec. A.4 for architectural details) and trained it with the following score matching loss: + +$$ +L _ {X} = \mathbb {E} _ {x _ {t} \sim p _ {t} (x _ {t} | x _ {0}) p (x _ {0}), t \sim \mathcal {U} (0, T)} \left[ | | s _ {\theta} (x _ {t}, t) - \lambda_ {t} \nabla_ {x _ {t}} \log p (x _ {t} | x _ {0}) | | _ {2} ^ {2} \right] +$$ + +where $\lambda_{t} = \mathbb{E}\big[||\nabla_{x_{t}}\log p(x_{t}|x_{0})||\big]^{-1}$ was employed for training stability. Approximate computation of $\lambda_{t}$ and $\nabla_{x_t}\log p(x_t|x_0)$ is described in Sec. A.2. + +To save memory when symmetrizing $s_{\theta}$ with Equation 3, we note that only $|G| / |G_x| \leq |G| / |T_L|$ elements in the set of points $\{gx|g \in G / T_L\}$ are unique. Thus we only predicted $\hat{f}_{\theta}(gx)$ on unique points and subsequently indexed into these predictions to compute the full sum over the $|G| / |T_L|$ summands in Equation 3. For points $x \in \mathbb{R}^3$ in special Wyckoff positions with large stabilizer groups $G_x$ , this significantly reduced the number of forward passes through the model. + +Table 1: MP-20 dataset. We report metrics of validity for 10,000 sampled crystals; uniqueness and novelty for 1,000 valid crystals; diversity and goodness-of-fit distributional distances for valid, unique, and novel crystals; and stability for valid, unique, and novel crystals. Stability is defined as having a DFT-predicted $E_{\mathrm{hull}} < 0.1$ eV/atom with respect to the Materials Project v2022.8.23. $T$ represents the inference-time temperature of discrete distributions in SGEquiDiff. Unless noted otherwise, $T = 1.0$ . Best result is bolded, second best is underlined. Standard deviations over three training runs are reported in parentheses as errors of rightmost digits. + +
Time ↓ (s / batch)Validity (%) ↑U.N. rate (%) ↑WpDistribution distance ↓CMD ↓Diversity ↑S.U.N. (%) ↑ MP-2024
StructureCompositionTemplateStructureWN,tJSDGJSDdWyclofollStructureStructureComposition
CDVAE [100]90699.9985.6615.598.20.65901.4230.69570.45900.48210.653913.7014.2
DiffCSP [40]15499.9282.2112.285.60.14540.40000.46380.23280.17660.958815.69-
DiffCSP++ [41]48499.9285.941.184.70.16580.50020.1608*0.0449*0.10790.932915.2323.4
SymmCD [55]13988.2486.769.687.70.16400.32130.16690.03440.32330.911115.62-
FlowMM1[62]-96.8583.19----------
MatterGen [107]3,689100.082.615.286.30.20590.24160.43310.21290.13380.988915.6724.3
SGEquiDiff226(12)99.25(28)86.16(32)18.6(16)80.0(16)0.1938(1049)0.2092(678)0.1733(53)0.0252(00)0.1718(58)0.9190(204)15.64(7)24.4
SGEquiDiff (T=1.5)-99.87(18)83.36(52)31.9(15)90.1(14)0.3810(1440)0.3000(148)0.2183(73)0.0430(43)0.2932(92)0.8556(101)15.82(18)-
SGEquiDiff (T=2.0)-98.40(37)81.08(22)42.8(31)95.0(2)0.6524(937)0.3685(409)0.2698(108)0.0564(61)0.3760(115)0.8439(137)16.28(7)-
SGEquiDiff (T=3.0)-97.84(53)78.60(35)57.1(32)95.6(5)1.473(267)0.4962(177)0.3367(204)0.0751(41)0.4656(132)0.8177(101)17.16(30)-
+ +* Uses fixed templates from the training data. † Values reported by Miller et al. [62]. + +# 5 Experiments + +**Evaluations.** We evaluated SGEquiDiff on two benchmark datasets: MP20 [100], containing 45,231 experimentally known crystals with up to 20 atoms per unit cell, and the more challenging MPTS52 dataset [4], containing 40,476 experimentally known crystals with up to 52 atoms per unit cell. Evaluations were conducted on 10,000 generated crystals. Following Xie et al. [100], we computed structural and compositional validity percentages using heuristics of interatomic distances and charge, respectively. We note that only $\sim 90\%$ of crystals in MP20 pass the composition validity checker based on SMACT [16]. Of the 10,000 generated crystals, we randomly sampled 1,000 determined to be both structurally and compositionally valid. Of these 1,000 crystals, we determined how many were unique and novel with respect to the training data using pymatgen's StructureMatcher [70] with stol=0.3, angle_tol=5, and ltol=0.2 (U.N. structures). We also followed SymmCD and calculated the fraction of unique and novel templates (U.N. templates), where a template is defined as a space group and multiset of occupied Wyckoff positions. U.N. structures were used to compute (1) distribution distances between ground truth test and generated materials properties, including Wasserstein distances for atomic density $\rho$ and number of unique elements $N_{el}$ as well as Jensen-Shannon divergences (JSD) for space group $G$ and occupied Wyckoff dimensionalities $d_{\mathrm{Wyckoff}}$ ; (2) Central Moment Discrepancy (CMD) [106] up to 50 moments between ground truth test and generated CrystalNN structural fingerprints [109]; and (3) structural and compositional diversity as measured by average pairwise $L_{2}$ -distances between CrystalNN and Magpie [93] fingerprints, respectively. Average sampling times per batch of 500 crystals were measured on an NVIDIA A40 GPU. We list hyperparameters and training times for SGEquiDiff in Sec. A.7. + +Finally, for all U.N. structures out of 1,000 random valid crystals, we conducted structure relaxations with expensive density functional theory (DFT) calculations to assess thermodynamic stability. Aligned with analyses of DFT-calculated energies for experimentally observed crystals [88] and of DFT errors relative to experiment [87, 5], stability was defined as having a predicted energy above the hull less than $0.1\mathrm{eV}$ /atom with respect to crystals in the Materials Project v2022.8.23 [39]. Further details on the DFT calculations are in Sec. A.8. The fraction of stable materials was multiplied by the fraction of U.N. structures to calculate the stable, unique, and novel (S.U.N.) rate. We qualify that a low energy above the hull as predicted by DFT has been found to be a necessary but insufficient condition for synthesizeability [88]; developing physical theories for predictive synthesis is an active area of research in the materials science community [60, 25, 43, 12, 68, 48]. Due to the computational expense of DFT calculations, we only computed S.U.N. rates on a subset of baseline models. While prior works pre-relax generated crystals with foundational machine learning interatomic potentials (MLIPs) before conducting DFT relaxations [62, 86, 107], we opted not to use MLIPs to avoid confating their biases [17, 53] with our evaluations of the generative models. + +Baselines. We compared our model to several prior methods. CDVAE (4.9M parameters) [100] predicts lattices and numbers of atoms per unit cell with a VAE and then samples elements and coordinates with denoising diffusion. DiffCSP (12.3M parameters) [40] jointly diffuses the lattice with the atoms using fractional atom coordinates. MatterGen (44.6M parameters) [107] similarly diffuses the lattice, atom types, and atom positions, but leverages Cartesian atom coordinates. FlowMM [62] extends DiffCSP with the flow matching framework. DiffCSP++ (12.3M parameters) [41] extends DiffCSP by enforcing space group constraints with projected diffusion of the lattice and + +Table 2: MPTS-52 dataset. We report metrics of validity for 10,000 sampled crystals; uniqueness and novelty for 1,000 valid crystals; and diversity and distributional distances for valid, unique, and novel crystals. Best result is bolded, second best is underlined. For DiffCSP and DiffCSP++, we used the same diffusion corrector step sizes as for MP-20, and the DiffCSP++ batch size was decreased to 96 to avoid OOMs. CDVAE failed to train due to graph construction errors from isolated atoms. Standard deviations over three training runs are reported in parentheses as errors of rightmost digits. + +
Time ↓ (s / batch)Validity (%) ↑U.N. rate (%) ↑Distribution distance ↓CMD ↓ StructureDiversity ↑ Structure
StructureCompositionTemplateStructureWρWNclJSDGJSDdyckoffStructure
DiffCSP [40]46767.4755.819.879.81.1890.50060.69000.08180.8621
DiffCSP++ [41]123099.8777.521.086.90.82440.36920.27590.11540.8457
SymmCD [55]21087.1178.1814.890.21.1260.35060.27300.11460.7843
SGEquiDiff530(40)97.79(37)79.83(96)38.7(25)89.8(7)0.6110(1592)0.1736(428)0.3104(62)0.1300(142)0.8613(292)
+ +atom coordinates; however, they do so without space group equivariant scores. SymmCD (60.4M parameters) [55] enforces space group constraints with non-equivariant diffusion of atoms in the asymmetric unit followed by post-hoc projections to the Wyckoff positions. + +Results. On MP20, we found SGEquiDiff was competitive on all metrics and achieved the highest S.U.N. rate (Table 1). While MatterGen achieved a comparable S.U.N. rate, we note that it is a significantly larger model (44.6M parameters) than SGEquiDiff (5.5M parameters), sampled crystals $\sim 16$ times slower, and yielded worse JSD metrics for space groups and Wyckoff dimensions. Compared to DiffCSP++, SGEquiDiff achieved a slightly higher S.U.N. rate and significantly higher U.N. template rate with $\sim 2.2$ times fewer model parameters. Unsurprisingly, SGEquiDiff and other space group-constrained models (DiffCSP++, SymmCD) strongly outperformed baselines on the JSD metric for space groups and Wyckoff dimensions. We also show that by increasing the inference-time temperature of SGEquiDiff's categorical distributions over space group, Wyckoff position, element, and the stop token, the U.N. rates and composition diversity metric can be significantly improved at the expense of slightly lower validity rates and higher distributional distances to the test set. Inspecting normalized root mean square Cartesian displacements between as-generated and DFT-relaxed structures, we found all models performed similarly, with MatterGen being the best on average (Table 4). Examples of S.U.N. crystals from SGEquiDiff are shown in Fig. 4. + +We also benchmarked SGEquiDiff on the more difficult MPTS52 dataset (Table 2), which most other models do not present results on. Using the same hyperparameters as used on MP20, we found that SGEquiDiff was able to scale to more atoms while maintaining a high rate of valid, unique, and novel crystals. SGEquiDiff notably achieved the highest U.N. template rate, composition validity, and Wasserstein distances for density and number of elements. Interestingly, all space group-constrained models significantly outperformed DiffCSP in the validity metrics, highlighting the advantage of enforcing nontrivial space group symmetries as an inductive bias. + +Ablations. To isolate effects of various components of SGEquiDiff, we performed ablations on the MP-20 dataset. First, we replaced our autoregressive lattice sampler with the DDPM-based [33] sampling of E(3)-invariant lattice matrix representations proposed in DiffCSP++ and used in SymmCD (Table 6). We used the same cosine noise scheduler as DiffCSP++, jointly diffusing lattices and atom coordinates. We term this variant of SGEquiDiff as $+LDiff$ . We observed that $+LDiff$ improved the Wasserstein distance for number of elements (possibly because conditioning of the Wyckoff-Element transformer on lattices was removed) and CMD of structural fingerprints. However, our autoregressive lattice sampler yielded better validity, JSD of Wyckoff dimensionalities, and S.U.N. rate. We also observed that in roughly $1\%$ of samples, the $+LDiff$ model produced lattice matrices with negative determinant, resulting in a spurious inversion that does not preserve the symmetry of the 22 chiral space groups. We show examples of S.U.N. crystals from $+LDiff$ in Fig. 5 + +We also replaced our space group and Wyckoff position features with those introduced in SymmCD [55] (Table 5). We found our featurizations yielded better structure validity, composition validity, Wasserstein distance for atomic density, structural divergence, and validation log-likelihoods of ground truth Wyckoff positions. In contrast, SymmCD features yielded more diverse samples, evidenced by higher U.N. rates and composition diversity. + +Since space group symmetrization via Eq. 3 can be applied to any periodic translation invariant model, we replaced our FAENet-inspired [20] graph neural network (GNN) with the EGNN-inspired [80] model (CSPNet) used by other crystal generative models [40, 41, 62]. Under a fixed number of trainable parameters (2.2M), we found our GNN yielded better metrics for structural validity, composition validity, and CMD of structural fingerprints (Table 5). + +Crystal structure prediction (CSP). Consistent with previous works [40, 41, 62], we evaluated SGEquiDiff on its ability to predict crystals from a test set using elemental composition as input. Performance was assessed by calculating match rate (MR) and root mean squared error (RMSE) of atomic fractional coordinates based on a single generated crystal per composition. RMSE was normalized by the average free length per atom. Matching was done with pymatgen StructureMatcher [70] using stol=0.5, angle_tol=10, and 1tol=0.3. To adapt SGEquiDiff to this task, we combined the pretrained $+LDiff$ variant of SGEquiDiff with the metric learning-based approach proposed in DiffCSP++ (CSPML) for selecting templates from training data and performing element substitutions to satisfy the given composition. Without CSP-specific training (unlike baselines), SGEquiDiff+LDiff achieved competitive performance (Table 3). We note SGEquiDiff could alternatively be adapted for CSP by reordering the factorization in Eq. 5 or by supplying composition as conditioning during training. While more challenging, learning instead of strictly enforcing composition may be desirable since an arbitrary composition is not guaranteed to host a stable crystal (accordingly, experimentally observed compositions can be weighted averages of multiple phases). + +We highlight that the current CSP task is slightly misaligned with real-world application. Many existing models assume the number of atoms per unit cell will be known beforehand, which is not necessarily true in practice. Additionally, a single composition can host multiple stable crystal structures depending on experimental conditions. For example, iron can exist as a body-centered cubic (2 atoms per conventional cell), face-centered cubic (4 atoms per conventional cell), or at high pressures, hexagonal close-packed structure (6 atoms per conventional cell). Furthermore, materials discovery campaigns are often aimed at composition spaces with sparse training data, violating the i.i.d. assumption of the MP20 dataset splits [100]. + +Table 3: Crystal structure prediction. Leveraging the template selection and element substitution approach developed in DiffCSP++ [41], we evaluated SGEquiDiff+LDiff on the crystal structure prediction task introduced by DiffCSP [40]. + +
MP-20MR (%)↑RMSE↓
CDVAE* [100]33.900.1045
DiffCSP* [40]51.490.0631
FlowMM* [62]61.390.0566
DiffCSP++ (+CSPML) [41]70.580.0272
SGEquiDiff (+LDiff,CSPML)69.420.0416
+ +* Uses ground truth number of atoms per unit cell. + +# 6 Conclusion + +In this paper, we proposed SGEquiDiff to generate crystals with space group invariant likelihoods by leveraging equivariant diffusion. We showed that space group equivariant flows automatically live on the manifolds of the Wyckoff positions. Significantly, SGEquiDiff achieved state-of-the-art generation rates of stable, unique, and novel crystals as evaluated by rigorous quantum mechanical simulations. We proposed an efficient autoregressive method for sampling space group-constrained lattices. We also showed that tuning the inference-time temperature for autoregressive sampling of discrete crystal attributes provides control over the novelty of generated crystals. + +Limitations and future directions. Due to computational constraints, we were only able to calculate S.U.N. rates on 1,000 valid crystals per model, adding unknown variance to reported metrics. SGEquiDiff's training data was pre-processed to assign space groups to crystals with nonzero tolerances over atom positions and lattice angles; it is possible that some materials were incorrectly symmetrized to higher symmetry space groups than representative of reality. SGEquiDiff also does not provide guidance for how to synthesize generated materials. Other symmetry groups relevant to crystals were ignored, including magnetic space groups [94], spin space groups [98], and layer groups [23]. Practically relevant deviations of real materials from perfect crystals were not considered, including defects, compositional disorder, surface effects, and interfacial effects. Future work might include application to a broader set of tasks such as those derived from nonscalar properties [102, 59, 95] or materials spectroscopy [56]. Furthermore, training non-equivariant models with space group invariant target distributions and/or applying inference-time space group symmetrization may enable a broader diversity of model architectures to generate high symmetry crystals. + +Broader impacts. This work has the potential to accelerate discovery of advanced materials for energy, electronics, optics, catalysis, aerospace, and more. Possible negative impacts include development of materials that are toxic, require energy-intensive processing, lead to depletion of raw minerals, or are used for military applications. + +# Acknowledgments and Disclosure of Funding + +We acknowledge helpful discussions with Cindy Zhang, Barry Bradlyn, Alex M. Ganose, and Eric Toberer. This research used the Delta advanced computing and data resource (award OAC 2005572) and the Illinois Campus Cluster, operated by the Illinois Campus Cluster Program in conjunction with the National Center for Supercomputing Applications. The research was supported by the National Science Foundation under Grant Nos. DGE 21-46756 and 2118201. + +# References + +[1] Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions, 2023. URL https://arxiv.org/abs/2303.08797. +[2] M.I. Aroyo, H. Burzlaff, G. Chapuis, W. Fischer, H.D. Flack, A.M. Glazer, H. Grimmer, B. Gruber, Th. Hahn, H. Klapper, E. Koch, P. Konstantinov, V. Kopsky, D.B. Litvin, A. Looijenga-Vos, K. Momma, U. Mueller, U. Shmueli, B. Souvignier, J.C.H. Spence, P.M. de Wolff, H. Wondratschek, and H. Zimmerman. International Tables for Crystallography: Space-group symmetry, volume A. International Union of Crystallography, Chester, England, 2 edition, 2016. ISBN 978-0-470-97423-0. doi: 10.1107/97809553602060000114. URL https://it.iucr.org/Ac/. +[3] Jacob Austin, Daniel D. Johnson, Jonathan Ho, Daniel Tarlow, and Rianne van den Berg. 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Local structure order parameters and site fingerprints for quantification of coordination environment and crystal structure similarity. RSC Advances, 10(10):6063-6081, 2020. ISSN 20462069. doi: 10.1039/c9ra07755c. URL https://pubs.rsc.org/en/content/articlehtml/2020/ra/c9ra07755chtps://pubs.rsc.org/en/content/articlelanding/2020/ra/c9ra07755c. +[110] A. Zur and T. C. McGill. Lattice match: An application to heteroepitaxy. Journal of Applied Physics, 55(2):378-386, 01 1984. ISSN 0021-8979. doi: 10.1063/1.333084. URL https://doi.org/10.1063/1.333084. + +# A Appendix + +![](images/52676e527882d37a7398c3ced8992a1387ca7ffa321febca6f6a50048121df24.jpg) +Figure 3: Special Wyckoff positions and the asymmetric unit in the conventional unit cell of hexagonal space group 192. Closed asymmetric unit boundary edges and facets $(\partial \Pi)$ are in orange. + +Table 4: Normalized root mean square displacements (RMSD) in Angstroms and their standard deviations calculated with pymatgen StructureMatcher between as-generated and DFT-relaxed crystals. RMSD is normalized by $(V / N)^{1 / 3}$ for unit cell volume $V$ and number of atoms per cell $N$ . + +
CDVAE [100]DiffCSP++ [41]MatterGen [107]SGEquiDiffSGEquiDiff+LDiff
0.0642±0.05680.0543±0.08120.0270±0.03870.0619±0.08040.0860±0.0810
+ +# A.1 Proofs + +Theorem 1. Let $G$ be a space group, $x \in \mathbb{R}^3$ be a point residing in a Wyckoff position with stabilizer group $G_x$ , and $f: \mathbb{R}^3 \to \mathbb{R}^3$ be a space group equivariant function. Then, for arbitrary constant $c \in \mathbb{R}$ and space group element $g \in G_x$ , + +$$ +g (x + c f (x)) = x + c f (x), \tag {9} +$$ + +or equivalently, $G_{x}\subseteq G_{x + cf(x)}$ + +Proof. Let $\{R_{g_x}, v_{g_x}\} = g_x \in G_x$ be a stabilizer group element where $R_{g_x} \in O(3)$ is a point group operation, $v_{g_x} \in [0,1)^3$ is a fractional lattice translation, and the action of $g_x$ is $R_{g_x}(\cdot) + v_{g_x}$ . The definition of a stabilizer group requires that + +$$ +R _ {g _ {x}} x + v _ {g _ {x}} = x \Rightarrow v _ {g _ {x}} = 0. +$$ + +Thus, we have + +$$ +f \left(R _ {g _ {x}} x\right) = f (x). \tag {10} +$$ + +The definition of space group equivariance requires that for any $R \in G$ , + +$$ +f (R x) = R f (x). \tag {11} +$$ + +Combining equations 10 and 11, we see + +$$ +R _ {g _ {x}} f (x) = f (x), \tag {12} +$$ + +i.e., $f(x)$ is also stabilized by $G_{x}$ . Finally, we have + +$$ +R _ {g _ {x}} (x + c f (x)) = R _ {g _ {x}} x + c R _ {g _ {x}} f (x) \tag {13} +$$ + +$$ += x + c f (x). \tag {14} +$$ + +Theorem 2. The score of the SGWN distribution is space group equivariant, i.e., given $g = \{R,v\} \in G$ , + +$$ +\nabla_ {x _ {t}} \log p (R x _ {t} + v | x _ {0}) = R \nabla_ {x _ {t}} \log p (x _ {t} | x _ {0}). +$$ + +Proof. The score of the SGWN is given as + +$$ +\nabla_ {x _ {t}} \log p (x _ {t} | x _ {0}) = \frac {\sum_ {g _ {j} = \{R _ {j} , t _ {j} \} \in G} \exp \left(\frac {- | | x _ {t} - (R _ {j} x _ {0} + t _ {j}) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) (R _ {j} x _ {0} + t _ {j} - x _ {t})}{\sigma_ {t} ^ {2} p (x _ {t} | x _ {0})}. +$$ + +Now, we transform the input with $g = \{R,v\}$ + +$$ +\nabla_ {x _ {t}} \log p (R x _ {t} + v | x _ {0}) = \frac {\sum_ {g _ {j} = \{R _ {j} , t _ {j} \} \in G} \exp \left(\frac {- | | R x _ {t} + v - (R _ {j} x _ {0} + t _ {j}) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) (R _ {j} x _ {0} + t _ {j} - R x _ {t} - v)}{\sigma_ {t} ^ {2} p (R x _ {t} + v | x _ {0})}. +$$ + +We will prove the equivariance of the numerator, which we denote as $\phi(Rx_{t} + v|x_{0})$ . For the denominator, it can be similarly proved that the SGWN is $G$ -invariant, i.e., $p(Rx_{t} + v|x_{0}) = p(x_{t}|x_{0})$ . + +$$ +\begin{array}{l} \phi (R x _ {t} + v | x _ {0}) = \sum_ {\{R _ {j}, t _ {j} \}} \exp \left(\frac {- | | R x _ {t} + v - (R _ {j} x _ {0} + t _ {j}) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) (R _ {j} x _ {0} + t _ {j} - R x _ {t} - v) \\ = \sum_ {\{R _ {j}, t _ {j} \}} \exp \left(\frac {- | | R (x _ {t} - R ^ {T} (R _ {j} x _ {0} + t _ {j} - v)) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) \Big [ R (R ^ {T} (R _ {j} x _ {0} + t _ {j} - v) - x _ {t}) \Big ] \\ = \sum_ {\{R _ {j}, t _ {j} \}} \exp \left(\frac {- | | x _ {t} - R ^ {T} (R _ {j} x _ {0} + t _ {j} - v) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) \Big [ R \big (R ^ {T} (R _ {j} x _ {0} + t _ {j} - v) - x _ {t} \big) \Big ] \\ = \sum_ {\{R _ {j}, t _ {j} \}} \exp \left(\frac {- | | x _ {t} - g ^ {- 1} (R _ {j} x _ {0} + t _ {j}) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) \Big [ R \big (g ^ {- 1} (R _ {j} x _ {0} + t _ {j}) - x _ {t} \big) \Big ] \\ = \sum_ {\{R _ {j ^ {\prime}}, t _ {j ^ {\prime}} \}} \exp \left(\frac {- | | x _ {t} - (R _ {j ^ {\prime}} x _ {0} + t _ {j ^ {\prime}}) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) \Big [ R (R _ {j ^ {\prime}} x _ {0} + t _ {j ^ {\prime}} - x _ {t}) \Big ] \\ = R \phi (x _ {t} | x _ {0}). \\ \end{array} +$$ + +# A.2 Implementation details + +We approximated the SGWN distribution with a truncated sum over lattice translations as + +$$ +q (x _ {t} | x _ {0}) \propto \sum_ {t _ {L} \in \mathbb {Z} ^ {3} \cap [ - m, m ] ^ {3}} \sum_ {\{R _ {j}, t _ {j} \} \in G / T _ {L}} \exp \left(\frac {- | | x _ {t} - (R _ {j} x _ {0} + t _ {j} + t _ {L}) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) +$$ + +where $m \in \mathbb{Z}_{+}$ . We have dropped the dependence of $q$ on $G$ and $\sigma_{t}$ from the notation for clarity. The space group-equivariant score of $q(x_{t}|x_{0})$ is given as + +$$ +\begin{array}{l} \nabla_ {x _ {t}} \log q (x _ {t} | x _ {0}) = \nabla_ {x _ {t}} \log \left[ \sum_ {t _ {L} \in \mathbb {Z} ^ {3} \cap [ - m, m ] ^ {3}} \sum_ {\{R _ {j}, t _ {j} \}} \exp \left(\frac {- | | x _ {t} - (R _ {j} x _ {0} + t _ {j} + t _ {L}) | | _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) \right] \\ = \frac {\sum_ {t _ {L} \in \mathbb {Z} ^ {3} \cap [ - m , m ] ^ {3}} \sum_ {\left\{R _ {j} , t _ {j} \right\}} \exp \left(\frac {- \left| \left| x _ {t} - \left(R _ {j} x _ {0} + t _ {j} + t _ {L}\right) \right| \right| _ {F} ^ {2}}{2 \sigma_ {t} ^ {2}}\right) \left(R _ {j} x _ {0} + t _ {j} + t _ {L} - x _ {t}\right)}{\sigma_ {t} ^ {2} q \left(x _ {t} \mid x _ {0}\right)}. \tag {15} \\ \end{array} +$$ + +For each Wyckoff position $w$ , we pre-computed approximate values of $\lambda_t = \mathbb{E}_{x_t \sim p(x_t | x_0), x_0 \sim \mathcal{P}_w(\text{Uniform}(\cdot))} \left[ ||\nabla_{x_t} \log p(x_t | x_0)|| \right]^{-1}$ with Monte Carlo sampling. We denote $\mathcal{P}_w(\text{Uniform}(\cdot))$ as the uniform distribution on the Wyckoff shapes (see Sec. 4.1 and A.3) corresponding to Wyckoff position $w$ . First, we sampled $s$ points $\{x_0^i\}_{j=1}^s$ from $\mathcal{P}_w(\text{Uniform}(\cdot))$ . Then, for each sample $x_0^i$ , we sampled Gaussian noise $\epsilon \sim \mathcal{N}(0, \sigma_t^2)$ and reparameterized as $x_t^i = x_0^i + \mathcal{P}_{x_0^i}(\epsilon)$ . The approximation of $\lambda_t$ for Wyckoff position $w$ in space group $G$ was then computed as + +$$ +\tilde {\lambda} _ {t} = \left[ \frac {1}{s} \sum_ {i = 1} ^ {s} \nabla_ {x _ {t} ^ {i}} \log q (x _ {t} ^ {i} | x _ {0} ^ {i}; G, w, \sigma_ {t}) \right] ^ {- 1} +$$ + +In our experiments, we set $s = 2500$ , $m = 3$ , $\sigma_{1} = 0.002$ , $\sigma_{T} = 0.5$ , and $T = 1000$ . + +# A.3 Uniform priors on Wyckoff positions + +To sample from the uniform distribution in a given Wyckoff position, we sampled uniformly on the Wyckoff shapes in the ASU. The likelihoods of these uniform priors are simply given as 1 for 0D Wyckoff positions, and the reciprocal of the total length, area, and volume of the relevant Wyckoff shapes for the 1-, 2-, and 3-D Wyckoff positions, respectively. For a 0D Wyckoff position, coordinates are fully determined and 'sampling' simply returns a point. For a 1D Wyckoff position, we sampled a line segment with probability proportional to its length, and then sampled uniformly on the line segment. For a 2D Wyckoff position, we pre-computed Delaunay triangulations of the polygonal facets belonging to the Wyckoff position, sampled a triangle proportionally to its area, and then sampled from a uniform Dirichlet distribution in the triangle. For a 3D Wyckoff position, we used rejection sampling from a bounding box around the ASU. + +# A.4 Architecture + +# A.4.1 Wyckoff-Element Transformer + +Our encoder-decoder Transformer architecture can be summarized as follows: + +$$ +z ^ {0} \leftarrow \operatorname {M L P} \left(e _ {S G} \| e _ {W} \| e _ {A}\right) +$$ + +$$ +z ^ {l + 1} \leftarrow \operatorname {E n c o d e r L a y e r} \left(z _ {i} ^ {l}, m _ {\text {c a u s a l}}\right) +$$ + +$$ +z _ {W}, z _ {A} \leftarrow \operatorname {S p l i t} \left(z ^ {l _ {\max }}\right) +$$ + +$$ +z _ {A} \leftarrow \operatorname {M L P} \left(z _ {A}, e _ {W}\right) +$$ + +$$ +p _ {W, \text {s t o p}} \leftarrow \text {A t t e n t i o n} (K = [ e _ {W} ^ {\text {a l l}} | | e _ {\text {s t o p}} ], V = [ e _ {W} ^ {\text {a l l}} | | e _ {\text {s t o p}} ], Q = z _ {W}, \text {m a s k} = m _ {W}) +$$ + +$$ +p _ {A} \leftarrow \operatorname {A t t e n t i o n} \left(K = e _ {A} ^ {\text {a l l}}, V = e _ {A} ^ {\text {a l l}}, Q = z _ {A}, \operatorname {m a s k} = m _ {A}\right) +$$ + +where $e_{SG}$ , $e_W$ , and $e_A$ are predicted embeddings of the crystal's space group, occupied Wyckoff positions, and atomic elements, respectively; $l_{\mathrm{max}}$ is the number of encoder layers; $m_{\mathrm{causal}}$ is a causal attention mask enforcing atom orderings; $e_W^{\mathrm{all}}$ , $e_{\mathrm{stop}}$ , and $e_A^{\mathrm{all}}$ are predicted embeddings of all sampleable Wyckoff positions, the stop token, and all sampleable elements, respectively; $m_W$ and $m_A$ are attention masks enforcing lexicographic atom orderings; $p_{W,\mathrm{stop}}$ and $p_A$ are the model probabilities over Wyckoff positions, the stop token, and elements; and EncoderLayer(x, m) is a module summarized by the following: + +$$ +x _ {\text {n o r m}} \leftarrow \operatorname {L N} (x) +$$ + +$$ +x \leftarrow x + \text {M u l t i h e a d S e l f A t t e n t i o n} \left(x _ {\text {n o r m}}, m\right) +$$ + +$$ +x \leftarrow \operatorname {D r o p o u t} (x) +$$ + +$$ +x \leftarrow \operatorname {D r o p o u t} \left(x + \operatorname {L i n e a r} (\operatorname {D r o p o u t} (\operatorname {M L P} (x)))\right) +$$ + +# A.4.2 Graph Neural Network + +Our graph neural network architecture can be summarized as follows. Our GNN was a modified version of FAENet [20], replacing sum pooling with variance-preserving aggregation [81] and removing frame averaging since we trivially achieve SE(3) invariance by canonicalizing crystals with the ASU representation. We constructed fully connected atom graphs $\mathcal{G}$ wherein each atom in the conventional unit cell was connected to every other atom in the conventional unit cell by their minimum-length edge under periodic boundary conditions. If ties existed, all corresponding edges were included. At initialization, to build features which were invariant to lattice translations, we sampled frequencies $\nu_{k} \in \mathbb{Z}^{3} \cap [-512, 512]^{3} \setminus \mathbf{0}$ without replacement with probabilities drawn from + +a discretized standard normal distribution. The architecture is summarized as follows: + +$$ +t _ {e m b} = \operatorname {M L P} (t) +$$ + +PlaneWaveEmbedding $(\cdot)\gets \big[\cos (2\pi (\cdot)\nu_{1})||\sin (2\pi (\cdot)\nu_{1})||\cos (2\pi (\cdot)\nu_{2})||\sin (2\pi (\cdot)\nu_{2})||\dots \big]$ + +$$ +h _ {i} ^ {0} \leftarrow \operatorname {M L P} \left(e _ {a _ {i}} | | \text {P l a n e W a v e E m b d i d g i n g} (x _ {i}) | | t _ {e m b}\right) +$$ + +$$ +e _ {i j} \leftarrow \operatorname {M L P} \left(\operatorname {P l a n e W a v e E m b d e d i n g} \left(\mathrm {x} _ {\mathrm {j}} - \mathrm {x} _ {\mathrm {i}}\right) | | \operatorname {R B F} \left(\mathrm {d} _ {\mathrm {i j}}\right) | | \operatorname {N o r m} (\mathbf {l})\right) +$$ + +$$ +f _ {i j} ^ {l} \leftarrow \operatorname {M L P} \left(e _ {i j} \| h _ {i} ^ {l} \| h _ {j} ^ {l}\right) +$$ + +$$ +h _ {i} ^ {l + 1} \leftarrow h _ {i} ^ {l} + a \cdot \mathrm {M L P} \circ \operatorname {G r a p h N o r m} \left(\frac {1}{\sqrt {| \mathcal {N} _ {\mathrm {i}} |}} \sum_ {\mathrm {j} \in \mathcal {N} _ {\mathrm {i}}} \mathrm {h} _ {\mathrm {j}} ^ {\mathrm {l}} \odot \mathrm {f} _ {\mathrm {i j}} ^ {\mathrm {l}}\right) +$$ + +$$ +h _ {i} ^ {l _ {\max }} \leftarrow \operatorname {M L P} \left(h _ {i} ^ {0} | | \dots | | h _ {i} ^ {l _ {\max }}\right) +$$ + +$$ +\hat {f} _ {i} \leftarrow \operatorname {M L P} \left(h _ {i} ^ {l _ {\max}}\right) +$$ + +where $t$ is the diffusion timestep; $x_{i}\in [0,1)^{3}$ is the fractional coordinate of atom $i$ ; $d_{ij}$ is the minimum-image pairwise Cartesian distance between atoms $i$ and $j$ , i.e., for lattice matrix $L\in \mathbb{R}^{3\times 3}$ , + +$$ +d _ {i j} = \min _ {n _ {1}, n _ {2}, n _ {3} \in \mathbb {Z}} | | L x _ {i} - L (x _ {j} + n _ {1} + n _ {2} + n _ {3}) | |; +$$ + +$h_i^l$ is the node feature of the $i$ th atom after $l$ rounds of message passing; $\mathbf{l} \in \mathbb{R}^6$ are the conventional lattice parameters; $l_{\mathrm{max}}$ is the number of message passing layers; $a$ is a learnable scalar initialized to zero; $e_{a_i}$ is the embedding of atom $i$ 's element; and $\hat{f}_i \in \mathbb{R}^3$ is the non-symmetrized score prediction (see Eq. 3) for atom $i$ . For the LDiff variant of our model, we additionally had + +$$ +\epsilon_ {L} \leftarrow \operatorname {M L P} \left(\frac {1}{N} \sum_ {i = 1} ^ {N} h _ {i} ^ {l _ {\max}}\right) +$$ + +where $\epsilon_L\in \mathbb{R}^6$ is the predicted noise for the lattice $k$ -vector (see [41]). + +We see that $\hat{f}_i$ is invariant to lattice translations since the model's spatial inputs are the minimum-image distances $d_{ij}$ under periodic lattice translations and translation invariant plane wave embeddings of relative and absolute atomic fractional coordinates ( $x_j - x_i$ and $x_i$ , respectively). We symmetrized $\hat{f}_i$ with space group equivariance according to Equation 3. + +# A.4.3 Space group and Wyckoff position features + +To represent a space group, we created one-hot features indicating the 6 lattice centering types, 6 crystal families, 32 crystallographic point groups, whether the space group is chiral, and whether the space group is centrosymmetric. Since this representation ignores fractional translations from glide and screw symmetries, we counted the maximum number of collisions between space group representations (16), arbitrarily indexed colliding space groups, and created additional one-hot features to avoid collisions. This yielded $6 + 6 + 32 + 1 + 1 + 16 = 62$ features. While the last 16 features are not physically motivated, they give the model the capacity to differentiate space groups and learn their relationships. These features are more abstracted than those of SymmCD [55], which used binary features indicating the 26 space group symmetry operations, 15 symmetry axes, and 7 lattice systems, totaling $7 + 26 \times 15 = 397$ features. + +To featurize a Wyckoff position, we used our 62 space group features; one-hot features for the 100 unique site symmetry symbols across space groups, the 4 numbers of degrees of freedom in atom coordinates (0, 1, 2, or 3), and the 17 unique Wyckoff multiplicities; and 48 Fourier features of Wyckoff shapes' midpoints and vertices (see Sec 4.1). This yielded $62 + 100 + 4 + 17 + 48 = 231$ Wyckoff position features. To compute Fourier features $\nu \in \mathbb{R}^{48}$ , we collected the midpoint and vertices of each Wyckoff shape belonging to the Wyckoff position. We used these points + +$\{p_i \in \mathbb{R}^{3 \times 1}\}_{i=1}^{n_w}$ to compute the following: + +$$ +f = 2 \pi \cdot \operatorname {l i n s p a c e} (\operatorname {s t a r t} = 1, \operatorname {e n d} = 3 2, \operatorname {s t e p s} = 8) \in \mathbb {R} ^ {1 \times 8} +$$ + +$$ +\nu_ {i} = \operatorname {f l a t t e n} (p _ {i} f) \in \mathbb {R} ^ {2 4} +$$ + +$$ +\nu = \frac {1}{n _ {w}} \sum_ {i} ^ {n _ {w}} [ \sin (\nu_ {i}), \cos (\nu_ {i}) ] \in \mathbb {R} ^ {4 8} +$$ + +We confirmed that this representation yielded no collisions between featurizations of different Wyckoff positions. Our representation differs from that of SymmCD, which represents each Wyckoff position by one-hot features of the 15 symmetry axes and 13 symmetry operations (totaling $15 \times 13 = 195$ features) corresponding to the Wyckoff position's stabilizer group. + +# A.5 Ablation results + +Table 5: SGEquiDiff Ablations on MP-20. We replaced our space group and Wyckoff position features with those of SymmCD [55] (+SymmCD features) and our denoising graph neural network with that of DiffCSP [40], DiffCSP++ [41], and FlowMM [62] (+CSPNet). For fair comparison in the latter experiment, we fixed the number of message passing steps (5), the number of Gaussian edge frequencies (96), and the number of trainable parameters (2.2M) by reducing CSPNet's hidden dimension from 512 to 224. Standard deviations over three training runs are reported in parentheses as errors of rightmost digits. + +
Validation Wyckoff log-likelihood ↑Validity (%) ↑U.N. rate (%) ↑Distribution distance ↓CMD ↓ StructureDiversity ↑
StructureCompositionTemplateStructureWρWNv,tJSDdwyckoffStructureComposition
SGEquiDiff-0.3206(83)99.25(28)86.16(32)18.6(16)80.0(16)0.1938(1049)0.2092(678)0.0252(90)0.1718(58)0.9190(204)15.64(7)
+SymmCD features-0.4925(30)98.55(21)82.82(28)30.0(21)84.9(18)0.4217(1525)0.1706(279)0.0395(138)0.3307(125)0.9078(52)16.10(30)
+CSPNet-99.00(5)--80.6(18)---0.1823(21)0.9252(155)-
+ +Table 6: LDiff ablation on MP-20. We replaced our autoregressive lattice sampling method with joint diffusion of atom coordinates and E(3)-invariant lattice matrix representations proposed in DiffCSP++ [41] (+LDiff). We report metrics of validity for 10,000 sampled crystals (including the percentage of generated lattices with positive determinant, a requirement to enforce symmetry of chiral space groups); uniqueness and novelty for 1,000 valid crystals; diversity and goodness-of-fit distributional distances for valid, unique, and novel crystals; and stability for valid, unique, and novel crystals. Stability is defined as having a DFT-predicted $E_{\mathrm{hull}} < 0.1$ eV/atom with respect to the Materials Project v2022.8.23. Best result is bolded, second best is underlined. Standard deviations over three training runs are reported in parentheses as errors of rightmost digits. + +
Time ↓(s / batch)|L| > 0Validity (%) ↑U.N. rate (%) ↑WpDistribution distance ↓CMD ↓StructureDiversity ↑S.U.N. (%) ↑
StructureCompositionTemplateStructureWNetJSDGJSDdWyclockoffStructureCompositionMP-2024
SGEquiDiff226(12)100(0)99.25(28)86.16(32)18.6(16)80.0(16)0.1938(1049)0.2092(678)0.1733(553)0.0252(90)0.1718(58)0.9190(204)15.64(7)
+LDiff227(12)98.95(49)97.02(97)84.10(63)20.3(20)80.3(10)0.1435(613)0.1292(86)0.1847(122)0.0579(151)0.1549(72)0.9474(131)15.87(17)
+ +* For efficiency, stability calculations were conducted on 100 random valid, unique, and novel structures. + +# A.6 Examples of S.U.N. crystals + +![](images/60c368ffdbdde71b837c7a5f88126a7d705cf0e54e1739953b49c5df6abb5950.jpg) +LaTmTl2 + +![](images/91d46d8a6b974b2c11cec94dd73f5787ba035f97c2891c7321edcd5d8a5e036e.jpg) +NaNdTl2 + +![](images/9dc27c4c5a2ec2d29cdbdeb634fda7f4158066a654d1235be68db8bd3e65a093.jpg) +TIPPt5 + +![](images/e16a8898d73431da2887a356cb9891985bfc9bdd1f2dde683c06c8313ff6a1e8.jpg) + +![](images/4ea6c4d97e60714c2b606ab973323ca46aca02971436835bbced955ac4a3e623.jpg) +$\mathrm{Nd}_2\mathrm{Cu}_2\mathrm{As}_4$ + +![](images/260eafcf099d26a372bf1f9f067933bdaac523023ac3a3a5b809fcc11253dab0.jpg) +$\mathrm{Y}_2\mathrm{AgIr}$ + +![](images/ed1e6a75ad8fe8dcf280db29fd1f71a68fe803ebcbae4d80e6417babec16659b.jpg) +BeCoSi + +![](images/cc4c45c3a1422e1774b9041ff3db2e7ca858fc496daff601b75d42a1812c2a7a.jpg) +Ta2Cr4 + +![](images/42eee8cc6769fdc482cc628273fc3072bf14fda57bea0865d422c5bac010ba93.jpg) +$\mathrm{Ca_4Sn_4Pt_4}$ +$\mathrm{La_8Ce_4Ir_4}$ + +![](images/3bccc01d6ca861ea32268eaa1162a0e9a411757b92cfd54a6999b2f208655e2b.jpg) +$\mathrm{Er}_2\mathrm{Fe}_2\mathrm{Co}_2$ + +![](images/cd92af1e0741f248e79dddb193a565892d62d62cb81218f8d8a146742bebebef.jpg) +Figure 4: S.U.N. crystals generated by SGEquiDiff trained on MP20. +Li2ThBi2 + +![](images/f0c664d7d1bd78a9b8f39cdb1e3ff92bcf675516ccd6472a5d1404f4fa654ae5.jpg) + +![](images/3472bd05ee8c84db92394ae091af72b98f83754f14392fb6ae9687d4a9a11dba.jpg) + +![](images/63479e0c1354f7b4a7c94d65f32cbea4626eee971b0fa195d55a14bede2d8d2d.jpg) +$\mathrm{Y}_2\mathrm{Tm}_2\mathrm{In}_2$ +Figure 5: S.U.N. crystals generated by the $+LDiff$ variant of SGEquiDiff trained on MP20. + +![](images/6a3fc7575807026728e8b5476f30dcbabcf36a70769c622674d02e65519a31fd.jpg) +Ti5Nb3Mo +LuPuRu2 + +![](images/0e8961648c47e50eb8c67c8759ebb512f66996384246db049dcb693364a061ea.jpg) +EuAs2Pt2 +YbAg2P2 + +# A.7 Training and hyperparameters + +Our code was written with PyTorch [72] and PyTorch Geometric [21]. The model was trained with the Adam optimizer [50] on a single NVIDIA A40 GPU. Training SGEquiDiff took approximately 10 hours on MP20 and 12 hours on MPTS52. Data splits were the same as provided by Xie et al. [100] and Baird et al. [4]. Hyperparameters were tuned manually. + +We adopted a modular architecture, simultaneously but independently training the space group sampler, lattice sampler, Wyckoff-Element transformer, and denoising graph neural network. Early stopping based on validation performance was applied to each module separately, eliminating the need for gradient balancing between modules. + +
HyperparameterValue
Batch size256
Number of epochs1000
Teacher forced lattice length noise range0.3
Teacher forced lattice angle noise range5.0
Lattice sampler hidden dimension256
Lattice length Fourier scale5.0
Lattice angle Fourier scale1.0
Lattice length bin edges Fourier scale2.0
Lattice angle bin edges Fourier scale1.0
Transformer dropout rate0.1
Transformer hidden dimension256
Transformer number of hidden layers4
Transformer number of heads2
SGWN lattice translations3
SGWN Monte Carlo samples2500
Time embedding dimension128
Number of plane wave frequencies96
Number of Cartesian distance radial basis functions96
Radial basis functions cutoff distance10.0
Edge embedder hidden dimension128
Node embedder hidden dimension256
Number of message passing steps5
Space group learning rate\( 1 \times 10^{-3} \)
Lattice learning rate\( 1 \times 10^{-4} \)
Transformer learning rate\( 1 \times 10^{-4} \)
GNN learning rate\( 1 \times 10^{-3} \)
Weight decay0.0
Learning rate schedulerReduceLROOnPlateau
Scheduler factor0.6
Scheduler patience30
Minimum learning rate\( 1 \times 10^{-5} \)
Gradient clipping by value0.5
Number of parameters5,513,396
+ +# A.8 Density Functional Theory Calculations + +Total energy calculations were performed with density functional theory (DFT) using the Vienna Ab Initio Simulation Package (VASP) [54] with Projector Augmented Wave (PAW) pseudopotentials [8]. The Perdew-Burke-Ernzerhof (PBE) [73] exchange-correlation functional was used for structural relaxations. All input parameters, plane wave cutoffs, convergence criteria, and k-point densities were determined by pymatgen MPRelaxSet [69]. + +Relaxations which don't converge with default settings from VASP (version 6.4) and MPRelaxSet do not necessarily indicate issues with generated crystal structures. Specifically, we encountered and attempted to resolve three failure modes during relaxations: errors related to symmetry-finding, issues converging energy and electronic charge density within DFT's self-consistent field (SCF) loops, and issues converging atomic positions. To resolve symmetry-finding errors (reported as SGRCON or INVGRP in VASP's output files), we tightened the tolerance of the symmetry detection algorithm by setting SYMPREC=1E-6 from the default 1E-5. To resolve SCF convergence issues, we changed the electronic minimization algorithm to the slower but more stable ALGO=NORMAL from the default ALGO=FAST. If SCF-related issues persisted, we additionally set AMIX=0.2 from the default 0.4, which results in slower but more stable convergence of the charge density, and increased the number of electronic optimization steps with NELM=200 (default 100). Finally, to resolve issues related to convergence of atomic positions, we changed the structure optimization algorithm to the slower but more precise IBRION=1 (default 2). We emphasize that these changes in VASP parameters will change the relaxation trajectory but not change the minimal energy structure or loosen convergence criteria relative to MPRelaxSet's default settings. + +For direct comparison with Materials Project (MP) data, we applied the MP's energy correction scheme for anions and mixing GGA/GGA+U calculations [37, 38]. Crystal stabilities were determined through convex hull analyses using all competing phases available in MP v2022.8.23 [39]. + +# NeurIPS Paper Checklist + +# 1. 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Spatial Multiplexing of Complementary Encoders and Decoders for Multimodal LLMs/full.md b/NeurIPS/2025/SpaceServe_ Spatial Multiplexing of Complementary Encoders and Decoders for Multimodal LLMs/full.md new file mode 100644 index 0000000000000000000000000000000000000000..b46e9a72b9abd95c34fa5da7fec1145e566c831d --- /dev/null +++ b/NeurIPS/2025/SpaceServe_ Spatial Multiplexing of Complementary Encoders and Decoders for Multimodal LLMs/full.md @@ -0,0 +1,788 @@ +# SpaceServe: Spatial Multiplexing of Complementary Encoders and Decoders for Multimodal LLMs + +Zhicheng Li $^{1,2}$ Shuoming Zhang $^{1,2}$ Jiacheng Zhao $^{1,2*}$ Siqi Li $^{4}$ Xiyu Shi $^{1,2}$ + +Yangyu Zhang $^{1,2}$ Shuaijiang Li $^{1,2}$ Donglin Yu $^{5}$ Zheming Yang $^{1,2}$ Yuan Wen $^{3}$ Huimin Cui $^{1,2,6}$ + +1 State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences, China + +$^{2}$ University of the Chinese Academy of Sciences + +3 University of Aberdeen 4 Beijing University of Technology + +5 The University of Illinois Urbana-Champaign 6 XCORESIGMA CO.,LTD. + +# Abstract + +Recent multimodal large language models (MLLMs) marry modality-specific vision or audio encoders with a shared text decoder. While the encoder is compute-intensive but memory-light, the decoder is the opposite, yet state-of-the-art serving stacks still time-multiplex these complementary kernels, idling SMs or HBM in turn. We introduce SpaceServe, a serving system that space-multiplexes MLLMs: it decouples all modality encoders from the decoder, and co-locates them on the same GPU using fine-grained SM partitioning available in modern runtimes. A cost-model-guided Space-Inference Scheduler (SIS) dynamically assigns SM slices, while a Time-Windowed Shortest-Remaining-First (TWSRFT) policy batches encoder requests to minimise completion latency and smooth decoder arrivals. Evaluation shows that SpaceServe reduces time-per-output-token by $4.81 \times$ on average and up to $28.9 \times$ on Nvidia A100 GPUs. SpaceServe is available at https://github.com/gofreelee/SpaceServe + +# 1 Introduction + +Multimodal large language models (MLLMs), for example Qwen2-VL[36] and other recent systems[36, 3, 33, 27], have graduated from lab demos to production services that field image-grounded questions, interpret charts, and reason over audio or video snippets. Conceptually, a MLLM marries the unprecedented linguistic capability of a large text-only foundation model with a set of modality-specific encoders that translate pixels, waveforms, or frames into the token space the decoder already understands. This "encoder + shared-text-decoder" blueprint is now standard across state-of-the-art models[3, 33, 18, 36]. + +As foundation-model decoders continue to improve "for free" (e.g., via larger pre-training corpora [21, 32, 17, 34, 9] or better post-training [26, 25, 28]), attention is shifting toward the encoders. These front-end components are becoming both more powerful and more complex [33, 36, 3, 27]—handling higher-resolution images, longer audio clips, and even variable-length video. Consequently, an MLLM is effectively two decoupled yet complementary neural networks: a compute-intensive, memory-light encoder paired with a memory-hungry, compute-light decoder. This pronounced resource asymmetry remains largely unaddressed by today's serving stacks. + +Serving MLLMs inherits every headache of text-only LLM serving—tight tail-latency [15, 5] targets, massive KV-cache footprints [35, 29, 13], dynamic batching[7, 38], and adds a new one: modality-specific encoders. Most production stacks [13, 41, 10, 1] remain tuned for single-modality workloads. Optimizations such as prefetching decoding disaggregation [11, 42], chunked prefetch [1] boost through- + +put for text models assuming a homogeneous stream of decoder kernels. That assumption breaks once a vision or audio encoder, with a completely different resource profile, enters the pipeline. + +Lacking encoder-aware machinery, state-of-the-art systems such as vLLM [13] or TGI [10] revert to time-multiplexing when they serve MLLMs: the GPU runs the vision/audio encoder first, then switches context and launches the text decoder, and so on. Encoder and decoder kernels therefore contend for the accelerator sequentially, losing the complementary compute-versus-memory balance that could otherwise be exploited. + +Our key insight is that encoders and decoders have complementary resource footprints: the compute cycles a decoder leaves idle are exactly what an encoder can exploit, while the memory an encoder scarcely touches is what the decoder hungers for to house its KV-cache. We therefore advocate space-multiplexing instead of traditional time-multiplexing—running encoder and decoder kernels simultaneously on the same GPU, each claiming just the compute and memory it truly needs. + +We embody this idea in SpaceServe, a split-encoder serving that: 1) decouples all modality-specific encoders from the shared text-decoder; 2) dynamically routes multimodal requests through a modality-aware scheduler that adapts to workload mix; 3) achieves space-multiplexing by co-locating encoder and decoder kernels on each GPU whenever their complementary compute-memory footprints align. Our contributions are as follows: + +- Split-encoder serving architecture: We introduce SpaceServe, the first system that decouples all modality-specific encoders from the shared text decoder and provides a GPU runtime that co-locates their kernels via space-multiplexing. A lightweight, cost-model-guided co-location policy packs encoder and decoder kernels whenever their resources footprints fit. +- TWSRFT encoder scheduler: To tackle bursty multimodal workloads, we design a TimeWindowed Shortest-Remaining-First (TWSRFT) scheduler that orders encoder requests within each time window by their remaining work. This preemptive policy minimizes encoder completion latency and smooths the arrival pattern seen by the shared decoder, improving end-to-end tail latency without harming encoder performance. +- Comprehensive evaluation: We evaluate SpaceServe with 4 size of state-of-the-art MLLMs [36, 3] on three GPU combinations (1*A100, 4*A100, 8*A100) using mixed workloads sampled from two well-known benchmarks for image [40] and video [8]. Empirical results demonstrate that SpaceServe cuts time-per-output-token (TPOT) by 4.81x on average and up to 28.9x. + +# 2 Motivation: From time-Multiplexing to space-Multiplexing + +# 2.1 Preliminary + +The canonical MLLM stack is organized around two core modules: 1) Modality-specific encoders that translate raw visual, audio, or other non-text inputs into high-dimensional embeddings. 2) A shared text-LLM decoder that performs autoregression generation over these embeddings. A lightweight pre-processing stage precedes the encoders, normalising each modality into the expected tensor format—tokenising text, slicing images into fixed-size patches, or converting audio into waveforms, thus the encoders can operate on a consistent representation. For instance, a vision encoder typically use ViT-based [6] structure to encode visual patches into visual tokens, which are aligned with text tokens in a unified representation space for integration and then processed by the LLM for reasoning services and coherent outputs generation. A classic MLLM architecture is depicted by Figure 1a. + +# 2.2 Resource footprint complementary: A quantitative analysis + +Our major insight is a clear resource-footprint complementarity between modality encoders and the shared text decoder. We substantiate this with a quantitative study of arithmetic intensity (AI)—floating-point operations per transferred byte (FLOPs / Byte)—showing that encoders are compute-rich but memory-light, whereas decoders are memory-hungry yet compute-light. This complementarity underpins the space-multiplexing design that follows. + +Due to space limit, the detailed computation of AI for MLLMs are put in Appendix D. + +Table 1 reports the arithmetic intensity (AI, FLOPs/byte) of Qwen 2-VL-7B at two image resolutions— $512 \times 512$ and $2048 \times 2048$ . Processing such dynamic input sizes is crucial for accuracy, yet it complicates serving. Two key take-aways emerge: + +Figure 1: Architecture of MLLMs and how vllm and SpaceServe serves MLLMs +![](images/698ed1d0c0a72b4d21947d442a13b47765783732a67b13f36cdc10d9ae8afc7f.jpg) +(a) Core components of MLLM: encoder and decoder. + +![](images/c21dbdc2a623d603ab278313ee91ba9f537afa4894976b1e100b914f0a16d131.jpg) +(b) From time-Multiplexing to space-Multiplexing + +Table 1: Arithmetic intensity (FLOPs / byte) of each serving stage in Qwen2-VL-7B on A100 GPU as image resolution increases from $512 \times 512$ to $2048 \times 2048$ . Higher AI means the kernel is more compute-bound; lower AI implies it is more memory-bound. + +
StageAI @ 512 × 512AI @ 2048 × 2048
Encoder encoding338.218826.97
Decoder prefilling72.131212.43
Decoder decoding0.8870.887
+ +1. Encoders and decoders are complementary. At both resolutions the encoder encoding remains compute-bound—AI increases from 338.21 to 8826.97 FLOPs/byte as resolution increases, but even 338.21 still far exceeds the decoder's AI. By contrast, the decoder decoding stage stays memory-bound at 0.887 FLOPs/byte. Consequently, an encoder's surplus memory bandwidth can satisfy a decoder's KV-cache needs, while a decoder's idle SM cycles can be reclaimed by encoder kernels—precisely the synergy exploited by our space-multiplexing scheduler. +2. Input dynamics reshape the resource footprint. Increasing resolution quadruples the number of $14 \times 14$ vision patches (e.g., $256 \rightarrow 1024$ patches), inflating memory traffic slower than compute and increasing encoder AI by roughly $4 \times$ , as the AI of encoder is quadratically related to the number of patches, as detailed in Appendix C. The Encoder phase changes from $338.21 \rightarrow 8826.97$ , and the decode phase is nearly flat, so the relative gap between encoder and decoder shrinks as images grow. Any serving stack that statically partitions GPU resources will mis-size one stage or the other once the workload shifts; a scheduler must adapt per request, co-locating kernels according to their current AI rather than a fixed worst-case estimate. + +In short, dynamic, high-resolution inputs amplify the encoder-decoder asymmetry and underscore the need for an adaptive, space-multiplexed serving strategy. + +# 2.3 Toward Space-Multiplexed Serving + +Existing serving stacks are modality-agnostic, so their schedulers treat vision/audio encoders and text decoders as unrelated jobs that simply take turns on the GPU. The result is classic time-multiplexing (Figure 1b, top): the system either runs a batch of encoders or a batch of decoders, never both, squandering the complementary SM-versus-HBM footprints we documented earlier. + +State-of-the-art tweaks such as chunked prefetch[1] (Figure 1b, middle) do achieve within-decoder space-multiplexing—interleaving the prefetch and decode phases of text requests—but they rely on a critical assumption: both phases operate on the same modality (tokens) with a shared architecture. Vision and audio encoders violate that assumption; they ingest image patches or spectrogram windows and execute a completely different kernel mix, so chunked prefetch cannot be repurposed to overlap them with decoder stages. + +We therefore argue for true space-multiplexing across modalities (Figure 1b, bottom). Through a GPU-level partitioning runtime and a cost-aware scheduler that allocates sub-GPU slices of SMs (streaming multiprocessors) to each request, our approach co-locates encoders and decoders concurrently, turning their complementary resource profiles into tangible throughput and latency gains. + +# 3 SpaceServe: Design + +![](images/4bfff275806e50d718907abe6b9847d0e850ac0d81e7b47fa1a06ef12da6ca65.jpg) +Figure 2:SpaceServe System Overview + +SpaceServe converts the traditional time-multiplexed serving pipeline into a space-multiplexed one by decoupling modality encoders from the shared text decoder and then co-locating their kernels on the same GPUs. By co-locating kernels, hardware resources are optimized, as different kernels demand distinct resource types. For example, compute-intensive kernel utilize more stream multiprocessors (SMs) but require less memory bandwidth, while memory-bound kernels can use the remaining less SMs without losing bandwidth. Strategic sharing of these kernels on the same GPU enhances resource utilization and boosts performance. Figure 2 sketches the high-level architecture of our approach, which decouples encoders from the LLM and co-locate them on the same GPU. SpaceServe achieves space-multiplexing of GPU resources through two complementary design pillars that operate in a split-share fashion: $①$ Disaggregated architecture. SpaceServe cleanly decouples modality encoders from the shared text decoder, enabling each component to execute and be scheduled independently. $②$ GPU runtime with a Space-Inference Scheduler (SIS). A novel runtime characterises the resource footprints of encoder and decoder kernels per request and co-locates them on the same GPU set. As illustrated in Figure 2, the dedicated encoder pool is fronted by a time-window scheduler that minimises encoding latency under fluctuating loads. At the heart of the system, the SIS ( $\S\S$ 3.3) dynamically partitions streaming multiprocessors (SMs) between encoders and decoders, guided by a cost model ( $\S\S$ 3.1) trained offline from extensive profiling of representative encoder and LLM workloads. During runtime, the SIS continuously monitors GPU utilisation; whenever a new request arrives, it rebalances the SM allocation according to the model's recommendation, sustaining high utilisation while respecting tail-latency constraints. + +# 3.1 Hierarchical Operator-level Profiling for Resource-Utility Curves + +Efficient sub-GPU sharing requires a precise performance model that can predict how both encoders and decoders behave under (i) dynamic inputs and (ii) dynamic resource allocations. Two challenges emerge: 1) Input-dependent behaviour: As shown in § 2 and §§ 2.2, changes in image or video resolution alter the arithmetic intensity (AI) of both encoders and decoders, shifting their compute-memory balance. 2) Non-linear scaling with SM slices. The latency or throughput of a kernel does not scale linearly with the fraction of streaming multiprocessors (SMs) it receives, making naive proportional models inaccurate. + +These realities motivate the notion of Resource-Utility Curves: explicit mappings from the GPU resources granted to a component (e.g., SM percentage) to the utility it delivers (e.g., latency, tokens). Such curves are the foundation of an adaptive serving stack. We construct them via a hierarchical, operator-level profiling workflow ( $\S\S 3.1$ ), which samples the performance of individual operators under controlled SM partitions and then aggregates the results to produce end-to-end utility curves for each encoder and decoder variant. The details of controlling SMs allocation are in $\S\S\S 3.3.2$ . Thanks to the convergence of operators, we mainly focus on two types of operators: + +Algorithm 1 Compose Model-Level Resource-Utility Curve +Input: ModelComponentType MCT +Input: ModelArchitecture Arch (details $N_{layers}$ , $d_{model}$ , etc.) +Input: PrimaryInput $P_{in}$ (a specific $L_{patches}$ or $S_{len}$ ) +Input: SMCountRange $R_{SM}$ +Input: OpPerfDB +Output: ModelCurve $[N_{SM}] \rightarrow$ TotalModelLatency for $P_{in}$ +1: ModelCurve $\leftarrow \emptyset$ +2: for $N_{SM} \in R_{SM}$ do +3: TotalModelLatency $\leftarrow 0$ +4: for $i \leftarrow 1$ to Arch.Nlayers do +5: latency $\leftarrow 0$ {Latency counter for the $i^{th}$ layer} +6: for $O_{layer} \in$ Arch.getLayerOps(MCT, i) do +7: $D_{layerOp} \leftarrow$ Arch.getOpDimensions(Olayer, MCT, Pin, i) +8: latency $\leftarrow$ latency + OpPerfDB[Olayer.Type][ $D_{layerOp}]$ [NSM] +9: end for +10: LayerOverheads $\leftarrow$ Arch.getLayerOverheads(MCT, i, Pin, NSM) +11: latency $\leftarrow$ latency + LatOverheads +12: TotalModelLatency $\leftarrow$ TotalModelLatency + latency +13: end for +14: ModelCurve $[N_{SM}] \leftarrow$ TotalModelLatency +15: end for +16: +17: return ModelCurve + +Profiling GEMM Resource Utility Curve : (1)For Encoder: Profile various GEMMs involved in FFN layers (e.g., matrices of size $L$ patches $\times d_{enc}$ by $d_{enc} \times 4d_{enc}$ ) and attention projections (e.g., $L$ patches $\times d_{enc}$ by $d_{enc} \times d_{enc}$ ). For each matrix size, a curve of Latency vs. Nsm is generated. (2)For Decoder: Profile GEMMs for FFN layers (e.g., $1 \times d_{llm}$ by $d_{llm} \times 4d_{llm}$ ), attention projections for the new token, and the LM head. Each results in a Latency vs. Nsm curve. + +Profiling Attention Resouce Utility Curve:(1)For Encoder: Profile the encoder's self-attention mechanism for different Lpatches. This also yields a set of Latency vs. Nsm curves, one for each Lpatches. (2) For Decoder: Profile the attention mechanism computing a new token's attention against a KV cache of length $Slen$ . This generates Latency vs. Nsm curves for various $Slen$ . + +After the profile, we compose the Model-level resource-utility curves by Algorithm 1. + +# 3.2 Encoder Stage: TWSRFT Scheduler + +In line with dynamic batching [38] for text only decoders, SpaceServe also incorporate a modality-aware scheduler for batching encoder requests. The choice of requests to batch significantly affects latency metrics, such as Time to First Token (TTFT) and Time per Output Token (TPOT). For instance, encoding a high-resolution image typically takes longer than processing lower-resolution image embeddings. To minimize latency and boost throughput, batching smaller requests is effective. However, this approach faces two constraints. First, GPU computational capacity limits batching, as encoders are compute-intensive. For example, the NVIDIA A100 has an optimal arithmetic intensity of approximately 161 FLOPs/Byte. The threshold varies across GPU models. Second, prioritizing small requests can starve larger requests, delaying their processing. + +To address these challenges, SpaceServe introduces the Time-Windowed Shortest Remaining Time First (TWSRTF) scheduler, the modality scheduler shown in Figure 2. The TWSRTF scheduler uses + +a time-window approach to manage request batching. Requests are collected in an input queue within a fixed time window. Within this window, requests are batched by size, prioritizing the smallest ones first. The scheduler also monitors batch capacity to ensure its stay within the GPU's limits. The detailed algorithm implementation is presented in Algorithm 2 + +Algorithm 2 Time-Windowed Shortest Remaining Time First Scheduling for Encoder +Input: Request queue $Q$ , window size $w$ , maximum patches $seqlength_{batchsuitable}$ +Output: Encoded batches of requests +1: while $Q$ is not empty do +2: $W \gets Q.\mathrm{get}(w)$ {Fetch a window of requests} +3: Sort $W$ by ascending patches +4: $i \gets 0$ ; $B \gets [W[i]$ ; $s \gets W[i]$ .patches +5: $i \gets i + 1$ +6: while $i < |W|$ do +7: if $s + W[i]$ .patches < seqlength_{batchsuitable} then +8: Append $W[i]$ to $B$ +9: $s \gets s + W[i]$ .patches +10: $i \gets i + 1$ +11: else +12: encoder $(B)$ {Process batch with encoder} +13: Remove $B$ from $Q$ +14: end if +15: end while +16: end while + +# 3.3 Co-Locating Stage: Space Inference Scheduler for Space Multiplexing + +The Online Space Inference Scheduler is a cornerstone of the Co-Locating Stage, dynamically managing request execution flows and the partitions of Streaming Multiprocessors (SMs) between encoders and the LLM decoders based on runtime conditions. It oversees the scheduling of incoming requests and ensures seamless coordination with the Modality Encoder Scheduler. When the Modality Encoder Scheduler, using Time-Windowed Shortest Remaining Time First (TWSRTF), dispatches a new encoding request, it triggers an action in the Online Space-Multiplex Scheduler. The scheduler utilizes the request's metadata, e.g. input sequence length and image resolution, alongside the cost model derived from offline profiling (detailed in §§ 3.1). This enables dynamic allocation of GPU resources of the same processor, particularly SMs, between the encoder task and current LLM decoding operations. Such adaptive, request-aware resource allocation optimizes performance and maximizes hardware utilization for diverse multimodal workloads. + +# 3.3.1 GPU Runtime: Fine-grained Resource Partition + +Effective space-multiplexing hinges on the ability to partition a single GPU and allocate resources below the device boundary. Modern accelerators now expose precisely this functionality. The AMD HIP runtime, for example, supports CU masks [2] that assign disjoint CU sets to different streams, while NVIDIA GPUs offer comparable streaming-multiprocessor control via libsmctrl [4]. SpaceServe builds on these sub-GPU partitioning primitives to realise fine-grained resource allocation. + +# 3.3.2 Request-aware SM Partition + +During the serving phase of the inference engine, we design an online allocation mechanism based on our profiler to dynamically optimize Streaming Multiprocessor (SM) partitioning. + +SpaceServe decomposes the inference service of multimodal large models into three independent processes: a CPU process, an Encoder process, and a LLM process. The CPU process handles user requests and performs preprocessing of multimodal data. Request information is then dispatched separately to both the Encoder process and the LLM process. Given that the Encoder and LLM have distinct resource preferences and requirements, SpaceServe applies different scheduling strategies tailored to each. SpaceServe performs dynamic and efficient GPU resource partitioning tailored to incoming requests. This adaptive allocation is underpinned by our comprehensive Model Resource + +Utility Curves in §3.1. The underlying Streaming Multiprocessor (SM) partitioning algorithm, which is elaborated in Appendix E operates on the core principle of finding the resource configuration that minimizes the sum of the Encoder's and Decoder's execution times. + +As for the LLM Decoder, SpaceServe employs a ChunkedPrefill [1] with vision cache scheduling strategy for the LLM processing, which is formally described in Algorithm 3. + +# 4 Evaluation + +# 4.1 Experimental Setup + +Hardware: We evaluate SpaceServe on a server which is equipped with 8 NVIDIA A100 SXM GPUS and Intel Xeon(R) Gold 6430 CPU. + +Models and workloads: We select 4 state-of-the-art MLLMs with varying model size, including Qwen2-VL-2B [36], Qwen2-VL-7B [36], Qwen2.5-VL-32B [3] and Qwen2-VL-72B [36]. These models were chosen for their proficiency in handling high-resolution images with arbitrary aspect ratios, understanding long contextual sequences. Due to GPU-memory constraints, we run Qwen2-VL-2B and Qwen2-VL-7B on a single GPU, Qwen2.5-VL-32B on four GPUs, and the largest Qwen2-VL-72B on eight GPUs. For the multi-GPU configurations we employ tensor parallelism. + +Datasets: We build a 1,740-example evaluation set by sampling from MMMU-Pro [40] and Video-MME [8], preserving an $8:2$ ratio of image to video items. + +Baseline Systems: We adopt vLLM [13]—a SOTA serving framework—as our baseline. In particular, we evaluate against the latest vLLM v1 architecture, which integrates recent advances such as zero-overhead scheduling and a suite of MLLM-specific optimizations. + +Metrics: LLM inference efficiency centers on two critical latency metrics: Time To First Token (TTFT) and Time Per Output Token (TPOT), which we use as primary metrics. Lower values for both TTFT and TPOT indicate better performance. + +# 4.2 End-to-End Evaluation Results + +![](images/97f6c08e3a1258ce2433f5e4696c408f52f91093a9b38a5601adf03d1ecdc067.jpg) +Figure 3: TPOT of SpaceServe and vLLM on various models, lower TPOT $\downarrow$ is better + +![](images/dce9d44ec8a28c8c0b65bde22b297cab0834bc1460d01fa1bfe7d1c9e0ad9904.jpg) + +![](images/ca2c4be304601f416ce45b21ec1ad0f1fec54545c71cdcd68e79ea5abb5cd12e.jpg) + +![](images/20edc41852f8869284df26d6d5f93f2ce6d8f1d44c6f2c7a70abee303b9fbecb.jpg) + +In the experiment, we increased the request rate from 2 to 10 requests per second. Across various model sizes, ranging from 7B to 72B, SpaceServe consistently and significantly outperforms the vLLM baseline. For instance, while serving the Qwen2-VL-2B model, vLLM's average TPOT increased from 101ms at 2 RPS to 365ms at 10 RPS, indicating a significant drop in efficiency. In contrast, SpaceServe demonstrated robust performance, with TPOT increasing only slightly from 8.85ms (2 RPS) to 12.62ms (10 RPS). This represented a substantial improvement, achieving a 28.9x reduction in TPOT compared to vLLM at 10 RPS and thereby enhancing service throughput and user experience. + +This trend continued across other models at 10 RPS. For Qwen2-VL-7B, SpaceServe recorded a TPOT of 40ms compared to vLLM v1's 489ms, delivering a speedup of 12.3x. For larger models with greater resource demands, such as Qwen2.5-VL-32B on four NVIDIA A100 GPUs, SpaceServe's TPOT was 144ms compared to vLLM's 522ms (10 RPS), a speedup of 3.62x. Similarly, when serving Qwen2-VL-72B on eight NVIDIA A100 GPUs at 10 RPS, SpaceServe maintained its advantage with a TPOT of 155.8ms versus vLLM v1's 489ms, achieving a 3.14x speedup. + +Why do we have such an excellent token generation speed: The inherent latency for generating a single token during the decoder phase is typically low, around 10 milliseconds for Qwen2VL-2B. However, vLLM-v1's time-division multiplexing architecture causes decoding requests to be blocked when the GPU is occupied by an encoder task, which can take several hundred milliseconds, e.g. 671ms for Qwen2VL-2B. This forces decoding operations into a wait state until the encoder task completes, significantly inflating the Time Per Output Token (TPOT) in high request-per-second (RPS) scenarios due to frequent encoder invocations. + +In stark contrast, SpaceServe employs a spatial multiplexing strategy (or a functionally equivalent mechanism that allows for concurrent or dedicated resource allocation for encoder and decoder stages). This architectural design effectively decouples the high-throughput decoder operations from the more time-intensive encoder tasks. By mitigating these inter-dependencies, SpaceServe ensures that the token generation rate for the decoder remains stable and consistently high, achieving approximately 100 tokens per second, even during frequent encoder activity. This resilience of the decoding pipeline to encoder-induced stalls is the primary contributor to SpaceServe's superior performance under demanding, mixed encoder-decoder workloads when measured in TPOT. + +Due to page constraints, we put the TTFT results on Appendix B. Unlike the sharp TPOT gains, the time-to-first-token (TTFT) remains virtually unchanged between SpaceServe and vLLM for all four models. This is expected: SpaceServe's design focuses on harvesting the memory-bandwidth slack present while an encoder is running to accelerate the memory-bound decoder, boosting steady-state throughput rather than the initial-token latency captured by TTFT. + +# 4.3 Ablation Study + +The performance advantages of SpaceServe compared to vLLM are attributed to two primary innovations: (1) the efficient space-multiplexing methodology. (2) the specialized Time-Windowed Shortest Remaining Time First (TW-SRTF) scheduler for the vision encoder. To delineate the individual contributions of these elements, we performed targeted ablation studies, adhering to the experimental setup described earlier. + +# 4.3.1 Ablation Study: Space Inference Scheduler vs. MPS + +We ablate our space-multiplexing design against NVIDIA Multi-Process Service (MPS) [20], using Qwen2-VL-7B as the test model. + +![](images/9d0c3ee4762f4c9b5656d1fde60191ab12f99f208bf26ec7d80f240695f4bf69.jpg) +(a) TPOT compared with MPS + +![](images/53f55782b911c9435687299f3af6927dc780e972684fad6552801d713d45b0bb.jpg) +(b) TPOT compared with vLLM scheduler +Figure 4: Ablation study of Qwen-2-VL-7B with TPOT + +As shown in Figure 4, the ablation study confirms our hypothesis: Compared to vLLM, the MPS (Multi-Process Service) version of SpaceServe accelerates the TPOT metric while keeping a largely unaffected TTFT. Our ablation study further evaluates our SpaceServe model, which employs a fine-grained allocation strategy, against this MPS version. At an input rate of 2 Requests Per Second (RPS), the MPS version of SpaceServe shows a TPOT of $90~\mathrm{ms}$ . This latency increases with rising RPS, reaching a peak of $132~\mathrm{ms}$ . In contrast, our SpaceServe (with the fine-grained strategy) achieves a TPOT of $28.25~\mathrm{ms}$ at 2 RPS, and its TPOT only reaches a maximum of $40.68~\mathrm{ms}$ as RPS increases. This comparison highlights that our fine-grained allocation strategy maintains significantly lower TPOT latency and greater stability under high concurrent requests, achieving a TPOT speedup of up to $3.3\mathrm{x}$ relative to the MPS version. Moreover, as depicted in Figure 5a and Figure 5b, the TTFT for both approaches is nearly identical, indicating that these TPOT enhancements do not compromise first-token latency. + +The substantial performance improvement stems from alleviating resource contention at the SM microarchitectural level. When Encoder and LLM processes are co-located using only NVIDIA MPS, their respective CUDA kernels, comprised of multiple warps, may be concurrently scheduled by the MPS server onto any available SM. Within a single SM, these distinct workloads then vie for limited resources such as arithmetic units, register file space, L1 cache, and shared memory bandwidth. The disparate nature of these tasks—the Encoder being typically compute-bound and the LLM decoding phase often exhibiting bursty memory access patterns and varied computational demands—can lead to suboptimal SM utilization. For example, high register usage by one task could limit the number of active warps (occupancy) for the other, or differing memory access patterns could lead to L1 cache pollution, increasing effective memory latency. + +By contrast, SpaceServe's SM partitioning dedicates distinct sets of SMs to the Encoder and LLM processes. This spatial isolation minimizes direct inter-process contention for intra-SM resources. Each process can therefore more effectively exploit the full capacity of its allocated SMs, leading to improved instruction issue rates and better sustained occupancy for its specific workload characteristics. This dedicated execution environment allows for more streamlined processing within each SM partition, reducing stalls and ultimately translating to a lower TPOT and an enhanced throughput. + +# 4.3.2 Ablation Study: TWSRFT Scheduler vs. vLLM Default Scheduler + +We ablate our redesigned encoder scheduler—built for split-encoder architectures—using the Qwen2-VL-7B model. + +The results demonstrate a clear advantage for our custom scheduling approach under increasing load. At a low concurrency of 2 Requests Per Second (RPS), the SpaceServe-vLLMscheduler exhibited a TPOT of $30.73\mathrm{ms}$ , which was only slightly higher than our full SpaceServe system's TPOT of $28.25\mathrm{ms}$ . However, as the concurrency increased to 4 RPS, the TPOT for SpaceServe-vLLMscheduler rose more sharply to $55.88\mathrm{ms}$ , whereas our SpaceServe system maintained a TPOT of $40.18\mathrm{ms}$ . This performance disparity became significantly more pronounced at 10 RPS: the TPOT for SpaceServe-vLLMscheduler surged to $114.22\mathrm{ms}$ , while our SpaceServe system impressively sustained a low TPOT latency of $39.8\mathrm{ms}$ , showcasing its stability and efficiency at higher throughputs. The observed degradation in TPOT for the SpaceServe-vLLMscheduler configuration can be attributed to the vLLM scheduler's lack of an independent batching strategy for the encoder component. When the aggregated batch size of vision tokens processed by the encoder reaches a critical threshold, any further increase in the number of concurrent requests leads to a substantial rise in the overall execution time for the encoder stage. This, in turn, inflates the average processing time per request. Consequently, the LLM process does not receive a sufficient or timely stream of vision tokens, which acts as a bottleneck and significantly slows down the generation of subsequent output tokens, thereby increasing the TPOT. + +# 4.4 Performance on Modern MoE-based Architectures + +To demonstrate SpaceServe's effectiveness on the latest generation of MLLMs, we evaluated its performance on cutting-edge models featuring sparse MoE layers: DeepSeek-VL2 and Kimi-VL. The experiments were run under a high-concurrency load of 10 requests per second. + +The results, presented in Table 2, show that SpaceServe delivers substantial performance gains. We achieve a $4.08\mathbf{x}$ TPOT speedup on DeepSeek-VL2 and a remarkable $9.84\mathbf{x}$ speedup on Kimi-VL. + +This dramatic performance improvement, especially on Kimi-VL, is attributed to its exceptionally compute-intensive vision encoder, which creates a significant imbalance with its memory-bound MoE decoder. This is precisely the scenario where SpaceServe's spatial multiplexing provides the greatest benefit, by co-locating the complementary kernels to maximize GPU utilization. + +# 4.5 Performance Scaling with Input Resolution + +Modern MLLMs increasingly use high-resolution inputs to improve understanding. This makes the vision encoder substantially more compute-intensive, creating a major bottleneck for traditional serving systems. We evaluated SpaceServe from 224x224 to 2Kx2K under high concurrency (10 RPS). As shown in Table 3, vLLM's TPOT rises sharply with resolution because the long-running encoder blocks the decoder, whereas SpaceServe's TPOT stays low and stable. + +Table 2: TPOT speedup on MoE-based MLLMs under a high-concurrency workload (10 RPS). + +
Model (MoE-based)FrameworkTTFT (ms)TPOT (ms)TPOT Speedup
DeepSeek-VL2vLLM-v15597122.5Baseline
SpaceServe (Ours)449530.04.08x
Kimi-VLvLLM-v157900482.0Baseline
SpaceServe (Ours)5632349.09.84x
+ +Crucially, the performance advantage of SpaceServe scales with the computational load. As illustrated in Table 3, the TPOT speedup escalates from $1.37\mathbf{x}$ on low-resolution inputs to an impressive $12.39\mathbf{x}$ at $2\mathrm{K}$ resolution. This demonstrates that spatial multiplexing is most effective precisely where it is most needed: mitigating the latency of compute-heavy tasks. Far from being a limitation, high-resolution inputs highlight SpaceServe's fundamental strength in harnessing resource heterogeneity to deliver efficient performance for the next generation of high-fidelity MLLMs. + +Table 3: Performance comparison across varying input resolutions at 10 RPS. As resolution increases, SpaceServe maintains a stable, low TPOT, while the baseline's latency degrades significantly. + +
ResolutionFrameworkTTFT (s)TPOT (ms)TPOT Speedup
224 × 224vLLM-v10.0923.6Baseline
SpaceServe (Ours)0.0917.21.37x
512 × 512vLLM-v10.2138.6Baseline
SpaceServe (Ours)0.2125.91.49x
1K × 1KvLLM-v115.31212.6Baseline
SpaceServe (Ours)15.5435.65.97x
2K × 2KvLLM-v1139.35470.7Baseline
SpaceServe (Ours)135.7638.012.39x
+ +# 5 Related Work + +Disaggregated Serving. Promising for large models, disaggregated serving (e.g., SplitWise [22], DistServe [42], DéjàVu [31]) decouples prefill/decode stages, mitigating interference for better TTFT/TPOT control. These LLM-focused systems often neglect the MLLM-specific encoding step. Even recent extensions (Pensieve [39], Mooncake [23], PD-Serve [11]) with advanced KV cache management offer limited MLLM applicability. + +Multi-modality Model Serving. Serving multimodal models (MLLMs) typically relies on adapting LLM systems (e.g., vLLM [13], SGLang [41]) or using recent MLLM inference code [24]. Adapting LLM systems hits LMM encoding bottlenecks with rich multimedia. Techniques like KV cache eviction [14, 19] and compression [12] are often model/scenario-specific (e.g., Inf-MLLM's [19] single-GPU streaming) and may miss cloud SLOs. Early disaggregation ideas (e.g., EPD [30]) lack SpaceServe's comprehensive scheduling and partitioning. + +Complementarity with Sparse MLLM Architectures. SpaceServe complements sparse multimodal architectures, including Mixture-of-Experts designs [33, 37, 16]. Sparsity can lower per-request compute, but it also increases resource heterogeneity by leaving hardware assigned to inactive experts underutilized. SpaceServe exploits this opportunity by co-locating the compute-intensive encoder of one request with the sparse, memory-bound decoder of another, using spatial multiplexing to fill idle resources and raise overall utilization + +# 6 Conclusion + +This paper introduces SpaceServe, a novel MLLM serving system featuring two key innovations: fine-grained GPU resource management and an advanced scheduling framework. This design enables concurrent low-latency inference and high-throughput for multimodal LLMs, outperforming SOTA systems by up to $28.9 \times$ + +# Acknowledgments + +This work was supported by the National Key R&D Program of China (Grant No. 2024YFB4505603) and by the National Natural Science Foundation of China (Grant Nos. U23B2020, 62090024, and 62302479). + +# References + +[1] Amey Agrawal, Nitin Kedia, Ashish Panwar, Jayashree Mohan, Nipun Kwatra, Bhargav Gulavani, Alexey Tumanov, and Ramachandran Ramjee. Taming Throughput-Latency tradeoff in LLM inference with Sarathi-Serve. In 18th USENIX Symposium on Operating Systems Design and Implementation (OSDI 24), pages 117-134, Santa Clara, CA, July 2024. USENIX Association. +[2] AMD Corporation. Setting the number of compute units. https://rocm.docs.amd.com/en/latest/how-to/setting-cus.html. 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In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. +[42] Yinmin Zhong, Shengyu Liu, Junda Chen, Jianbo Hu, Yibo Zhu, Xuanzhe Liu, Xin Jin, and Hao Zhang. DistServe: Disaggregating prefill and decoding for goodput-optimized large language model serving. In 18th USENIX Symposium on Operating Systems Design and Implementation (OSDI 24), pages 193–210, Santa Clara, CA, July 2024. USENIX Association. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We have clearly stated the main claims made in the abstract and introduction, which accurately reflect the paper's contributions and scope. The claims are supported by theoretical and experimental results, and we have discussed the limitations of our work. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We discuss the limitation of SpaceServe in Appendix A + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. 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Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [No] + +Justification: We will open source the SpaceServe after paper acceptance. + +Guidelines: + +- The answer NA means that paper does not include experiments requiring code. +- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). +- The instructions should contain the exact command and environment needed to run to reproduce the results. 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Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +Justification: We do not use crowdsourcing or human subjects in our research. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. + +We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Limitations and future work + +While SpaceServe effectively addresses the challenge of extended Time Per Output Token (TPOT) in multimodal large language model inference, another critical performance indicator, Time To First Token (TTFT), remains a significant consideration. TTFT directly reflects the system's initial responsiveness and is paramount for a positive user experience. In its current state, SpaceServe maintains TTFT performance comparable to existing state-of-the-art frameworks, such as vLLM-v1. However, substantially reducing the execution latency of a single encoder pass—the primary determinant of TTFT—continues to be a demanding task. Optimizing this single-encoder latency to further enhance TTFT represents an important avenue for our future research. + +# B Experiments for TTFT metric + +Table 4: Time To First Token (TTFT) comparison across different models with varying Request Per Second (RPS), lower TTFT $\downarrow$ is better + +
MethodTime To First Token (TTFT) (s) ↓
RPS246810
Qwen-2-VL-2B
vLLM12.06816.18018.65523.03526.038
SpaceServe13.32417.51521.11424.72627.437
Qwen-2-VL-7B
vLLM19.49222.03330.30634.74538.795
SpaceServe18.83222.41932.05036.53339.086
Qwen-2.5-VL-32B
vLLM16.55332.63141.84946.44047.547
SpaceServe17.13133.24242.48947.14448.966
Qwen-2-VL-72B
vLLM14.87226.64735.71739.78042.067
SpaceServe14.81026.63336.32138.37843.834
+ +![](images/4597041fc8eb40a37040212cd1448f916b4f8959605b7475a3aedca7e16e4a14.jpg) +(a) TTFT compared with MPS +Figure 5: Ablation study of Qwen-2-VL-7B with TTFT + +![](images/4159e8d23a7ae4aa68b0610d762c0dc0b14134a1c1e446e180ac71af6d70e29e.jpg) +(b) TTFT compared with vLLM scheduler + +# C Arithmetic Intensity Modeling for Encoder + +Taking vision encoders for example, the patch generation characteristics can be formulated as: + +$$ +H ^ {\prime} = S \left\lceil \frac {H}{S} \right\rceil , W ^ {\prime} = S \left\lceil \frac {W}{S} \right\rceil \Rightarrow L _ {\text {p a t c h e s}} = \left\lceil \frac {H ^ {\prime}}{L} \right\rceil \times \left\lceil \frac {W ^ {\prime}}{L} \right\rceil \tag {1} +$$ + +where S denotes image padding strides, H denotes the original image height, H' denotes the normalized height, W denotes the original image width, and W' denotes the normalized width. And L denotes the patch size, $L_{\mathrm{patches}}$ denotes the total patches after preprocessing. During the encoder stage, the FLOPs can be formulated as: + +$$ +\mathrm {F L O P s} _ {\text {v i s i o n}} = \left(\underbrace {2 L _ {\text {p a t c h e s}} d _ {\text {v i s i o n}}} _ {\text {E m b e d d i n g}} + \underbrace {8 L _ {\text {p a t c h e s}} d _ {\text {v i s i o n}} ^ {2} + 4 L _ {\text {p a t c h e s}} ^ {2} d _ {\text {v i s i o n}}} _ {\text {A t t e n t i o n}} + \underbrace {1 6 L _ {\text {p a t c h e s}} d _ {\text {v i s i o n}} ^ {2}} _ {\text {F F N}}\right) * \text {L a y e r s} \tag {2} +$$ + +The overall Arithmetic Intensity can be approximated by: + +$$ +\frac {\mathrm {F L O P s} _ {\text {v i s i o n}}}{\mathrm {M e m o r y} _ {\text {w e i g h t}}} \tag {3} +$$ + +# D Arithmetic Intensity Modeling for LLM + +LLM Prefill Stage Before autoregressive token generation, the Large Language Model (LLM) processes the entire input prompt (with sequence length $S_{prompt}$ ) in a single, parallel forward pass. This 'prefill' stage is essential for computing the initial set of Key-Value (KV) states for all tokens in the prompt. To optimize this process, particularly for managing the memory footprint and computational cost of the attention mechanism, techniques like Grouped-Query Attention (GQA) are often employed. These initial KV states, potentially generated using GQA, are then cached and utilized by the subsequent decoding steps. + +LLM Prefix: FLOPs. Let $L_{llm}$ be the number of LLM layers, $d_{llm}$ be the hidden dimension (consistent with the decoding stage notation), $S_{prompt}$ be the input prompt length, and $V$ be the vocabulary size. The FFN expansion factor is $c$ (typically $c = 4$ ). The FLOPs for the Transformer backbone during prefix are primarily from self-attention and FFN computations: + +- Self-Attention per layer: $\approx$ ( $8S_{prompt}6d_{llm}^{2} + 4S_{prompt}^{2}d_{llm}$ ) FLOPs. This encompasses Q,K,V projections ( $6S_{prompt}d_{llm}^{2}$ ), attention score calculations ( $QK^T$ ) and application to Values ( $4S_{prompt}^{2}d_{llm}$ ), and the output projection ( $2S_{prompt}d_{llm}^{2}$ ). +- $FFN$ per layer: $\approx 4cS_{prompt}d_{llm}^2$ FLOPs. For $c = 4$ , this is $16S_{prompt}d_{llm}^2$ . + +The total FLOPs for the Transformer blocks during prefetch (FLOPsPrefill_Transformer) are: + +$$ +F L O P s _ {P r e f i l l _ {-} T r a n s f o r m e r} \approx L _ {l l m} ((8 + 4 c) S _ {p r o m p t} d _ {l l m} ^ {2} + 4 S _ {p r o m p t} ^ {2} d _ {l l m}) \tag {4} +$$ + +Assuming $c = 4$ : + +$$ +F L O P s _ {P r e f i l l \_ T r a n s f o r m e r} \approx L _ {l l m} \left(2 4 S _ {p r o m p t} d _ {l l m} ^ {2} + 4 S _ {p r o m p t} ^ {2} d _ {l l m}\right) \tag {5} +$$ + +If output logits are computed for the entire prompt (e.g., for training or certain inference strategies), the LM head adds $FLOPs_{LM\_Head\_Prefill} \approx 2S_{prompt}d_{llm}V$ . + +LLM Prefix: Memory Access. Memory operations during prefix include: + +- Model Weights (MemoryWeights_LLM): Accessing the model parameters $N_{\text{total.params\_LLM}}$ , where $\text{MemoryWeights\_LLM} \approx N_{\text{total.params\_LLM}} \times$ bytes_per_parameter. +- Input Data & Activations: Reading input prompt embeddings and handling intermediate activations for the $S_{prompt}$ length sequence across layers. +- KV Cache Generation: Writing the KV pairs for all $S_{prompt}$ tokens. The size of this cache is $Memory_{KV\_cache\_prompt} = S_{prompt} \times L_{llm} \times 2 \times d_{llm} \times \text{bytes\_per\_element}$ . + +LLM Prefix: Arithmetic Intensity (AI). The Arithmetic Intensity for prefix is $AI_{Prefill} =$ Equation 5/(Memory Access for Weights, Input, Activations, and KV Cache Write). + +$$ +A I _ {P r e f i l l} = \frac {L _ {l l m} \left(\left(8 + 4 c\right) S _ {p r o m p t} d _ {l l m} ^ {2} + 4 S _ {p r o m p t} ^ {2} d _ {l l m}\right)}{\text {M e m o r y} _ {\text {A c c e s s} _ {-} P r e f i l l}} \tag {6} +$$ + +The $4L_{llm}S_{prompt}^{2}d_{llm}$ term in the FLOPs (Equation 5) means that for long prompts ( $S_{prompt}$ ), the prefetch AI can be substantial, potentially making this phase compute-intensive. This contrasts with the characteristics of the subsequent decoding phase. + +In contrast, the Arithmetic Intensity of the LLM component is significantly lower. The LLM decoding stage operates autoregressively, generating one token at a time. Its computational characteristics are modeled as follows: + +LLM Decoding: FLOPs (Per Token Generation) For the generation of a single token, let $L_{llm}$ be the number of LLM layers, $d_{llm}$ be the hidden dimension, $S_{len}$ be the current sequence length (context from previously generated tokens), and $V$ be the vocabulary size. The primary FLOPs contributions are: + +- Transformer Decoder Layers (with KV Cache): For each token, each layer involves: + +- Self-Attention (new token Q, K, V projections; attention with cached K, V): Approximately $8d_{llm}^2 + 4S_{len}d_{llm}$ FLOPs. This includes Q, K, V projections for the current token $(\approx 6d_{llm}^2)$ , interaction with the $S_{len}$ cached tokens $(\approx 4S_{len}d_{llm}$ for $\mathrm{QK}^\top$ and score-V multiplication), and the output projection $(\approx 22d_{llm}^2)$ . +- Feed-Forward Network (FFN): Typically $16d_{llm}^{2}$ FLOPs (assuming a $4 \times d_{llm}$ intermediate expansion). + +Thus, per layer FLOPs are approximately $24d_{llm}^{2} + 4S_{len}d_{llm}$ . + +- LM Head: Mapping the final hidden state to vocabulary logits: $\approx 2d_{llm}V$ FLOPs. + +The total FLOPs for generating a single token ( $FLOPs_{LLM\_decode_token}$ ) can be expressed as: + +$$ +F L O P s _ {L L M \_ d e c o d e \_ t o k e n} \approx L _ {l l m} \left(2 4 d _ {l m} ^ {2} + 4 S _ {l e n} d _ {l l m}\right) + 2 d _ {l l m} V \tag {7} +$$ + +A common simplified approximation for FLOPs per token (especially when $S_{len}$ is moderate) is $2 \times N_{\text{params\_non\_embedding}}$ , where $N_{\text{params\_non\_embedding}}$ are the non-embedding parameters of the LLM, primarily from FFN and attention projection matrices. + +LLM Decoding: Memory Access (Per Token Generation) Memory access during LLM decoding is dominated by: + +- Model Weights (MemoryWeights_LLM): Parameters of the Transformer layers and LM head. + +$$ +M e m o r y _ {w e i g h t s \_ L L M} \approx N _ {\text {t o t a l} \_ p a r a m s \_ L L M} \times \text {b y t e s} _ {\text {p e r} \_ \text {p a r a m e t e r}} \tag {8} +$$ + +- KV Cache (Memory $K_V$ -cache): Stores Key (K) and Value (V) vectors for $S_{len}$ previous tokens across $L_{llm}$ layers. + +$$ +M e m o r y _ {K V \_ c a c h e} = S _ {l e n} \times L _ {l l m} \times 2 \times d _ {l l m} \times \text {b y t e s} +$$ + +The critical dynamic memory access per token involves reading from and writing to this KV cache, along with accessing the relevant model weights for the current token's computation. + +LLM Decoding: Arithmetic Intensity (AI) The Arithmetic Intensity for LLM decoding is $FLOPs_{LLM\_decode\_token} / Memory_{Access\_per\_token}$ . If we primarily consider the model weights for memory access (as their total size is loaded and parts are accessed for each token generation), a simplified AI is: + +$$ +A I _ {L L M \_ d e c o d e} \approx \frac {2 \times N _ {\text {p a r a m s} \_ \text {n o n} \_ \text {e m b e d d i n g}}}{N _ {\text {p a r a m s} \_ \text {t o t a l} \_ L L M} \times \text {b y t e s} \_ \text {p e r} \_ \text {p a r a m e t e r}} \tag {10} +$$ + +Given that $N_{\text{params\_non\_embedding}}$ is a large fraction of $N_{\text{params\_total\_LLM}}$ , and FLOPs for core matrix multiplies are roughly $2 \times$ parameters involved, if we consider all parameters: + +$$ +A I _ {L L M \_ d e c o d e} \approx \frac {2 \times N _ {\text {p a r a m s} \_ t o t a l \_ L L M}}{N _ {\text {p a r a m s} \_ t o t a l \_ L L M} \times \text {b y t e s} \_ \text {p e r} \_ \text {p a r a m e t e r}} = \frac {2}{\text {b y t e s} \_ \text {p e r} \_ \text {p a r a m e t e r}} \tag {11} +$$ + +For FP16/BF16 (2 bytes/parameter), $AI_{LLM\_decode} \approx 1$ FLOP/Byte. This low AI signifies that LLM decoding is predominantly memory-bandwidth bound. The substantial memory footprint of weights and the dynamically accessed KV cache, relative to the per-token computational work, contributes to this characteristic. + +# E Chunked Prefix Scheduling with Vision Cache Check + +Pre-Scheduling Check: Prior to each scheduling iteration, the scheduler examines the QueueEncoder_result to determine if any requests have completed encoding. Completed encoding results from the Encoder process are cached in the LLM process's vision token cache ChunkedPrefill Execution: + +During scheduling of prefetch-stage requests: The scheduler verifies whether the candidate tokens contain vision tokens. If required vision tokens are not present in the vision cache: Only the preceding text prompt portion is scheduled Vision token processing is deferred until encoding completion + +Scheduling Priority: Strict FCFS ordering maintains request sequence integrity Within the ChunkedPrefill framework: Decoding-stage requests receive scheduling priority over prefetch requests. This ensures optimal throughput while preserving fairness + +Algorithm 3 Chunked Prefix Scheduling with Vision Cache Check +Input: Waiting prefetch requests queue $Q_{prefill}$ During decoding requests queue $Q_{decoding}$ Token budget $B$ (max_num_batched_tokens) Chunk prefetch size $Batch_{chunkssize}$ Vision cache $\mathcal{V}$ Chunking enabled flag $C$ +Output: Scheduled batch $Batch$ containing decode and prefetch chunks +1: Initialize empty batch: $Batch \gets []$ +2: for each request $d$ in $Q_{decoding}$ do +3: Append $d$ to $Batch$ +4: Update $B \gets B -$ tokens needed by $d$ +5: end for +6: while $B > 0$ and $Q_{prefill}$ not empty do +7: $r \gets Q_{prefill}.peek()$ {Next prefetch request} +8: $v_{pos} \gets$ position of first vision token in $r$ +9: if $v_{pos}$ exists and vision tokens at $v_{pos}$ not in $V$ then +10: $t \gets$ number of prompt tokens before $v_{pos}$ +11: Schedule prefetch chunk with tokens $[start_{pos}, v_{pos})$ in $Batch$ +12: else +13: $t \gets$ tokens needed for next chunk of $r$ +14: if $t > B$ and $C$ is enabled then +15: Split $r$ into chunk of size exactly $B$ (take next $B$ tokens) +16: Schedule this chunk for prefetch in $Batch$ +17: Remove these $B$ tokens from $r$ +18: $t \gets B$ +19: break +20: else if $t \leq B$ then +21: Schedule full prefetch request $r$ in $Batch$ +22: Remove $r$ from $Q_{prefill}$ +23: else +24: break {No budget left} +25: end if +26: end if +27: $B \gets B - t$ +28: end while +29: Execute scheduled $Batch$ (decode + chunked prefetch) + +# F SM Partition Algorithm + +Algorithm 4 FindOptimalSMPPartition +Input: +1: $M_E$ {Encoder request metadata} $RCE$ {Encoder Resource-Utility Curve} +2: $L_{ctx}$ {Decoder context length} $C_D$ {Decoder chunk configuration} $RC_D$ {Decoder ResourceUtility Curve} +3: $SM_{total}$ {Total allocatable SMs} $SM_{min\_alloc}$ {Minimum SMs per component (e.g., $\geq 1$ )} +Output: $(sm_{E}^{*},sm_{D}^{*})$ {Optimal SM allocation for Encoder, Decoder} +4: Let $Lat_{E}(s_{E})$ be the latency function for Encoder derived from $RCE$ using $M_E$ +5: Let $Lat_{D}(s_{D})$ be the latency function for Decoder derived from $RC_{D}$ using $C_D,L_{ctx}$ +6: $sm_{E}^{*}\gets$ null +7: $sm_{D}^{*}\gets$ null +8: $L_{makespan}^{*}\gets \infty$ +9: found_valid_allocatio $\leftarrow$ false +10: for $s_E$ from $SM_{min\_alloc}$ to $SM_{total}-SM_{min\_alloc}$ do +11: $s_D\gets SM_{total}-s_E$ +12: if $s_D < SM_{min\_alloc}$ then +13: continue +14: end if +15: current_Lmakespan $\leftarrow$ max(Lat $\mathcal{E}(s_E)$ ,Lat $\mathcal{D}(s_D)$ ) +16: if current_Lmakespan $< L^{*}_{makespan}$ then +17: $L^{*}_{makespan}\gets$ current_Lmakespan +18: $sm_{E}^{*}\gets s_{E}$ +19: $sm_{D}^{*}\gets s_{D}$ +20: found_valid_allocatio $\leftarrow$ true +21: end if +22: end for +23: if found_valid_allocatio then +24: return $(sm_{E}^{*},sm_{D}^{*}),L^{*}_{makespan}$ +25: else +26: return (null,null), $\infty$ {No valid allocation found} +27: end if + +# G Implementation Details + +System stack and versions. SpaceServe is implemented on top of vLLM 0.7.2. We integrate at the scheduler and engine layers, keeping the model execution interface unchanged so that existing MLLM checkpoints can be served without modification. + +Concurrency via CUDA MPs. We enable NVIDIA CUDA Multi-Process Service (MPS) so that the vision encoder and the text decoder run as separate worker processes on the same GPU and submit kernels concurrently.. + +SM partitioning primitive. To realize space multiplexing, SpaceServe uses libsmctrl as the low-level primitive to partition GPU Streaming Multiprocessors (SMs) across worker processes. 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Leveraging our sparse deformable marching cubes (Sparcubes) representation and sparse convolutional VAE (Sparconv-VAE), our method achieves state-of-the-art reconstruction quality on challenging 3D inputs. It robustly handles open surfaces (automatically closed into watertight meshes), recovers hidden interior structures, and faithfully reconstructs highly complex geometries (see zoom-in views, top to bottom). All outputs are fully watertight and 3D-printable, demonstrating the potential of our framework for high-resolution 3D mesh generation. Best viewed with zoom-in. + +# Abstract + +High-fidelity 3D object synthesis remains significantly more challenging than 2D image generation due to the unstructured nature of mesh data and the cubic complexity of dense volumetric grids. Existing two-stage pipelines—compressing meshes with a VAE (using either 2D or 3D supervision), followed by latent diffusion sampling—often suffer from severe detail loss caused by inefficient representations and modality mismatches introduced in VAE. We introduce Sparc3D, a unified framework that combines a sparse deformable marching cubes representation Sparcubes + +with a novel encoder Sparconv-VAE. Sparcubes converts raw meshes into high-resolution $(1024^{3})$ surfaces with arbitrary topology by scattering signed distance and deformation fields onto a sparse cube, allowing differentiable optimization. Sparconv-VAE is the first modality-consistent variational autoencoder built entirely upon sparse convolutional networks, enabling efficient and near-lossless 3D reconstruction suitable for high-resolution generative modeling through latent diffusion. Sparc3D achieves state-of-the-art reconstruction fidelity on challenging inputs, including open surfaces, disconnected components, and intricate geometry. It preserves fine-grained shape details, reduces training and inference cost, and integrates naturally with latent diffusion models for scalable, high-resolution 3D generation. + +# 1 Introduction + +Recent breakthroughs in 3D object generation [2, 15, 17, 24, 29, 32, 39] have enabled applications in virtual domains, such as AR/VR [3, 14, 16, 20] and robotics simulation [34, 35]—as well as in physical contexts like 3D printing [29]. Despite this progress, synthesizing high-fidelity 3D assets remains far more challenging than generating 2D imagery or text stemming from the inherently unstructured nature of 3D data and the cubic scaling of dense volumetric representations. + +Drawing on the success of text-to-image diffusion models [25], many 3D generation pipelines [2, 15, 17, 32, 39] employ a two-stage process: a variational autoencoder (VAE) followed by latent diffusion. Most current VAEs employ either 3D supervision[2, 15, 17, 39]—typically via a global vector set representation—or 2D supervision[32]—commonly via a sparse-voxel representation—but both suffer from limited resolution and a modality mismatch between inputs and outputs. + +3D supervised VAEs [2, 15, 17, 39] require watertight meshes for sampling supervision signals, yet most raw meshes are not watertight and must be remeshed. The common pipeline [2, 15, 39] shown in Fig. 2, first samples an Unsigned Distance Function (UDF) on a voxel grid, then approximates a Signed Distance Function (SDF) by subtracting two voxel sizes—halving the effective resolution and introducing errors that propagate through both VAE reconstruction and diffusion generation. After applying Marching Cubes [19] or Dual Marching Cubes [26], the mesh becomes double-layered, necessitating an additional step to retain only the largest connected component. This step inadvertently discards smaller yet crucial features, misaligning the conditioned input image from raw mesh with the reconstructed 3D shape during diffusion training. + +Most recently, TRELLIS [32] demonstrated the possibility of training a 3D VAE using only 2D supervision. While it avoids degradation from mesh conversion, it still depends on dense volumetric grids for 2D projections, which limit resolution. Moreover, lacking any 3D topological constraints, the generated object's interior geometry may be incorrect and its surfaces can remain open—a critical flaw for applications such as 3D printing. + +All of these VAEs also contend with a fundamental modality gap: 3D supervised methods [2, 15, 17, 39] ingest surface points and normals but decode SDF values, while TRELLIS [32] encodes voxelized DINOv2 [23] features into SDF. Bridging this gap requires heavy attention mechanisms, which increase model complexity and risk of amplifying underlying inconsistencies. + +In this work, we introduce Sparcubes (Sparse Deformable Marching Cubes), a fast, near-lossless pipeline for converting raw meshes into watertight surfaces. Our method begins by identifying a sparse set of activated voxels from the input mesh and performing a flood-fill to assign coarse signed labels. We then optimize grid-vertex deformations via gradient descent and refine them using a view-dependent 2D rendering loss. Sparcubes converts a raw mesh into a watertight $1024^{3}$ grid in under 30 seconds—achieving a threefold speedup over prior watertight conversion methods [2, 15] without sacrificing fine details or small components. + +Building on Sparcubes, we introduce Sparconv-VAE, a modality-consistent VAE composed of a sparse encoder and a self-pruning decoder. By eliminating the modality gap, Sparconv-VAE can employ a lightweight architecture without relying on heavy global attention mechanisms. Experimental results show that our VAE achieves state-of-the-art reconstruction performance and minimal training cost. Furthermore, it seamlessly integrates with existing latent diffusion pipelines such as TRELLIS [32], further enhancing the resolution of the generated 3D objects. + +![](images/288ce5e914a5ce1252f85b10bcb3608f11506973bb54a70c196d974f946b79fd.jpg) +Figure 2: Problems of the previous SDFs extraction pipeline. The widely used SDFs extraction workflow [2, 15, 39] suffers from two critical failures: resolution degradation (show as error) and missing geometry (circled on the right). Converting UDF to SDF by subtracting two voxel sizes effectively halves the spatial resolution. Moreover, the SDF extraction yields a double-layer mesh, from which only the largest connected component is retained, inadvertently discarding smaller but important component. Together, these two deficiencies substantially limit the upper-bound performance of downstream VAEs and generation models. Best viewed with zoom-in. + +Our main contributions can be summarized as: + +- We propose Sparcubes, a fast, near-lossless remeshing algorithm that converts raw meshes into watertight surfaces at $1024^{3}$ resolution in approximately 30s, achieving a $3\times$ speedup over prior methods without sacrificing any components. +- We introduce Sparconv-VAE, a modality-consistent variational autoencoder that employs a sparse convolutional encoder and a self-pruning decoder. By eliminating the input-output modality gap, our architecture achieves high computational efficiency and near-lossless reconstruction, without global attention. +- Experimental results show that, our Sparc3D framework, comprising Sparcubes and Sparconv-VAE, achieves state-of-the-art reconstruction fidelity, reduces training cost, and seamlessly integrates with current latent diffusion frameworks to enhance the resolution of generated 3D objects. + +# 2 Related Work + +# 2.1 3D Shape Representation and Generation + +Mesh and point cloud. Triangle meshes and point clouds are the most common representations of 3D data. Triangle meshes, composed of vertices and triangular faces, offer precise modeling of surface detail and arbitrary topology. However, their irregular graph structure complicates learning, requiring neural networks to handle non-uniform neighborhoods, varying vertex counts, and the absence of a canonical ordering. To address this, recent methods [4, 5, 11, 28, 30] adopt autoregressive models to jointly generate geometry and connectivity, though these suffer from limited context length and slow sampling. In contrast, point clouds represent surfaces as unordered sets of 3D points, making them easy to sample from distributions [21, 31, 33], but difficult to convert directly into watertight surfaces due to the lack of explicit connectivity [9, 36]. + +Isosurface. Isosurface is a continuous surface representing a mesh boundary via a signed distance field (SDF). Most methods [13, 19, 26, 27] subdivide space into voxels, polygonize each cell, then stitch them into a mesh. Marching Cubes uses a fixed lookup table but can suffer topological ambiguities [19]. Dual Marching Cubes (DMC) fixes this by placing vertices on edges where the isosurface crosses and linking them via Dual Contouring, yielding watertight meshes [13, 26]. Both rely on a uniform cube size, limiting detail; FlexiCubes [27] deforms the grid and applies isosurface slope weights to adapt voxel sizes to local geometry, improving accuracy. + +Many 3D generation methods adopt SDF-based supervision [2, 15, 17, 24, 32, 38, 39]. Techniques relying solely on 2D supervision [32] may implicitly learn an SDF, but often produce open surfaces or incorrect interiors due to the lack of volumetric constraints. In contrast, fully 3D-supervised approaches [2, 15, 17, 24, 38, 39] extract explicit SDFs from meshes with arbitrary topology, making accurate and adaptive SDF extraction a critical but challenging component for high-fidelity reconstruction. + +# 2.2 3D Shape VAEs + +VecSet-based VAEs. VecSet-based methods represent 3D shapes as sets of global latent vectors constructed from local surface features. 3DShape2VecSet [37] embeds sampled points and normals into a VecSet using a transformer and supervises decoding via surrounding SDF values. CLAY [38] scales the architecture to larger datasets and model sizes, while TripoSG [18] enhances expressiveness through mixture-of-experts (MoE) modules. Dora [2] and Hunyuan2 [39] improve sampling by prioritizing high-curvature regions. Despite these advances, all approaches face a modality mismatch: local point features are compressed into global latent vectors and decoded back to local fields, forcing the VAE to perform both feature abstraction and modality conversion, which increases reliance on attention and model complexity. + +Sparse voxel-based VAEs. In contrast, sparse voxel-based VAEs preserve spatial structure by converting meshes into sparse voxel grids with feature vectors. XCube [24] replaces the global VecSet in 3DShape2VecSet [37] with voxel-aligned SDF and normal features, improving detail preservation. TRELLIS [32] enriches this representation with aggregated DINOv2 [23] features, enabling joint modeling of 3D geometry and texture. TripoSF [12] further scales the framework to high-resolution reconstructions (see Supplementary Material for details). Nonetheless, these methods still face the challenge of modality conversion—mapping point normals or DINOv2 descriptors to continuous SDF fields remains a key bottleneck. + +# 3 Method + +# 3.1 Preliminaries + +Distance fields. A distance field is a scalar function $\Phi : \mathbb{R}^3 \to \mathbb{R}$ that measures the distance to a surface. The unsigned distance function (UDF) encodes only magnitude, while the signed distance function (SDF) adds sign to distinguish interior and exterior: + +$$ +\operatorname {U D F} (\mathbf {x}, \mathcal {M}) = \min _ {\mathbf {y} \in \mathcal {M}} \| \mathbf {x} - \mathbf {y} \| _ {2}, \quad \operatorname {S D F} (\mathbf {x}, \mathcal {M}) = \operatorname {s i g n} (\mathbf {x}, \mathcal {M}) \cdot \operatorname {U D F} (\mathbf {x}, \mathcal {M}), \tag {1} +$$ + +where $\mathrm{sign}(\mathbf{x},\mathcal{M})\in \{-1, + 1\}$ indicates inside/outside status. For non-watertight or non-manifold meshes, computing sign is non-trivial [1]. + +Marching cubes and sparse variants. The marching cubes algorithm [19] extracts an isosurface mesh from a volumetric field $\Phi$ by interpolating surface positions across a voxel grid: + +$$ +\mathcal {S} = \left\{\mathbf {x} \in \mathbb {R} ^ {3} \mid \Phi (\mathbf {x}) = 0 \right\}. \tag {2} +$$ + +Sparse variants [10] operate on narrow bands $|\Phi(\mathbf{x})| < \epsilon$ to reduce memory usage. We further introduce a sparse cube grid $(V, C)$ , where $V$ is a set of sampled vertices and $C$ contains 8-vertex cubes. Deformable and weighted dual variants, exemplified by FlexiCubes [27], extend this process by modeling the surface as a deformed version of the sparse cube grid. Specifically, each grid node $n_i$ in the initial grid is displaced to a new position $n_i + \Delta n_i$ , forming a refined grid $(N + \Delta N, C, \Phi, W)$ that better conforms to the implicit surface, where the displacements $\Delta n_i$ and per-node weights $w_i \in W$ are learnable during optimization. + +# 3.2 Sparcubes + +Our method, Sparcubes (Sparse Deformable Marching Cubes), reconstructs watertight and geometrically accurate surfaces from arbitrary input meshes through sparse volumetric sampling, coarse-to-fine sign estimation, and deformable refinement. Unlike dense voxel methods, Sparcubes represents geometry using a sparse set of voxel cubes, where each cube vertex carries a signed distance value. This representation enables efficient computation, memory scalability, and supports downstream surface extraction or direct use in learning-based pipelines. + +As shown in Fig. 3, the core pipeline consists of the following steps: + +Step 1: Active voxel extraction and UDF computation. We begin by identifying a sparse set of active voxels within a narrow band around the input surface. These are voxels whose corner vertices + +![](images/8b764a5bbbc2c6d209de9dae11d33d041a410d126851b37b8f0bb1e24a2dc6b2.jpg) +Figure 3: Illustration of our Sparcubes reconstruction pipeline for converting a raw mesh into a watertight mesh. + +lie within a threshold distance $\epsilon$ from the mesh $\mathcal{M}$ . For each corner vertex $\mathbf{x} \in \mathbb{R}^3$ , we compute the unsigned distance to the surface: + +$$ +\operatorname {U D F} (\mathbf {x}) = \min _ {\mathbf {y} \in \mathcal {M}} \| \mathbf {x} - \mathbf {y} \| _ {2}. \tag {3} +$$ + +This yields a sparse volumetric grid $\Phi$ , with distance values concentrated near the surface geometry, suitable for efficient storage and processing. + +Step 2: Flood fill for coarse sign labeling. To convert the unsigned field into a signed distance function (SDF), we apply a volumetric flood fill algorithm [22] starting from known exterior regions (e.g., corners of the bounding box). This produces a binary occupancy label $T(\mathbf{x}) \in \{0,1\}$ , indicating whether point $\mathbf{x}$ is inside or outside the shape. We then construct the coarse signed distance field as: + +$$ +\operatorname {S D F} (\mathbf {x}) = (1 - 2 T (\mathbf {x})) \cdot \operatorname {U D F} (\mathbf {x}), \tag {4} +$$ + +which gives a consistent sign assignment under simple labeling and forms the basis for further refinement. + +Step 3: Gradient-based deformation optimization. Instead of explicitly refining a globally accurate SDF, we directly optimize the geometry of the sparse cube structure to better conform to the underlying surface. Given an initial volumetric representation $(V,C,\Phi_v)$ where $V$ denotes the set of sparse cube corner vertices, $C$ is the set of active cubes, and $\Phi$ is the signed distance field defined at each vertex—we perform a geometric deformation to obtain $(V + \Delta V,C,\Phi_v)$ . This results in a geometry-aware sparse SDF volume that more accurately approximates the zero level set of the implicit surface. Notably, for points where $\Phi (x) > 0$ , the SDF values are often only coarse approximations, particularly in regions far from the observed surface or near topological ambiguities. These regions may exhibit significant errors due to poor connectivity, occlusions, or non-watertight input geometry. As such, rather than refining $\Phi$ globally, we optimize the vertex positions $\Delta V$ to implicitly correct the spatial alignment of the zero level set. To improve the accuracy of sign estimation and geometric alignment, we displace each vertex slightly along the unsigned distance field gradient: + +$$ +\mathbf {x} ^ {\prime} = \mathbf {x} - \eta \cdot \nabla \mathrm {U D F} (\mathbf {x}). \tag {5} +$$ + +This heuristic captures local curvature and topological cues that are not easily recovered through purely topological methods such as flood fill. It also allows us to estimate sign information in regions with ambiguous connectivity, such as thin shells or open surfaces. The final data structure is a sparse cube grid with SDF values on each corner $(V,C,\Phi_v,\Delta V)$ , denoted as Sparcubes. + +Step 4: Rendering-based refinement. Sparcubes supports differentiable mesh extraction, enabling further end-to-end refinement with perceptual signals. When multi-view images, silhouettes, or depth maps are available, we optionally introduce a differentiable rendering loss to further enhance visual realism and geometric alignment. Given a reconstructed mesh $\mathcal{M}_r$ extracted from the deformed Sparcubes, we compute a multi-term rendering loss: + +$$ +\mathcal {L} _ {\text {r e n d e r}} = \left\| \mathcal {R} ^ {D} \left(\mathcal {M} _ {r}\right) - \mathcal {I} _ {\text {o b s}} ^ {D} \right\| _ {2} ^ {2} + \left\| \mathcal {R} ^ {N} \left(\mathcal {M} _ {r}\right) - \mathcal {I} _ {\text {o b s}} ^ {N} \right\| _ {2} ^ {2}, \tag {6} +$$ + +where $\mathcal{R}^D (\mathcal{M}_r)$ denotes the rendered depth image of the mesh under known camera parameters, and $\mathcal{R}^N (\mathcal{M}_r)$ is the corresponding rendered normal map. The terms $\mathcal{I}_{\mathrm{obs}}^D$ and $\mathcal{I}_{\mathrm{obs}}^N$ are the observed depth image and the ground truth normal map derived from the input or canonical mesh. Leveraging our voxel-based data structure, we can easily identify visible voxels and render exclusively within those regions, greatly reducing computational cost. + +# 3.3 Sparconv-VAE + +Building on our Sparcubes representation, we develop Sparconv-VAE, a sparse convolution-based variational autoencoder without high-consuming global attentions, which directly compresses the Sparcubes parameters $\{\phi \in \Phi_v,\delta \in \Delta V\}$ into a sparse latent feature $\mathbf{z}$ and decodes back to the same format without any modality conversion. + +Architecture and Loss Function. Our encoder is a cascade of sparse residual convolutional blocks that progressively downsample the input features. At the coarsest resolution, a lightweight local attention module aggregates neighborhood information. The decoder mirrors this process, interleaving sparse residual convolutions with self-pruning upsample blocks to restore the original resolution and predict the Sparcubes parameters $\{\hat{\phi},\hat{\delta}\}$ . Each self-pruning block first predicts the occupancy of the subdivided voxel occupancy mask $\mathbf{o}$ , supervised by $\mathcal{L}_{\mathrm{occ}} = \mathrm{BCE}(\hat{\mathbf{o}},\mathbf{o})$ , then applies a learned upsampling to refine the voxel features. Because $\phi$ is sign-sensitive (inside vs. outside), we split its prediction into a sign branch and a magnitude branch. The sign branch predicts $\mathrm{sign}(\hat{\phi})$ under $\mathcal{L}_{\phi_{\mathrm{sign}}} = \mathrm{BCE}(\mathrm{sign}(\hat{\phi}),\mathrm{sign}(\phi))$ , while the magnitude branch regresses $\hat{\phi}$ with $\mathcal{L}_{\phi_{\mathrm{mag}}} = \| \hat{\phi},\phi \|_2$ . The deformation vectors are optimized via $\mathcal{L}_{\delta} = \| \hat{\delta},\delta \|_2$ . Finally, we regularize the latent distribution using the VAE's Kullback-Leibler divergence $\mathcal{L}_{\mathrm{KL}} = \mathrm{KL}(q(\mathbf{z}|\delta ,\phi)\| p(\mathbf{z}))$ , yielding a single cohesive training objective that jointly minimizes the occupancy, sign, magnitude, deformation and KL divergence losses: + +$$ +\mathcal {L} = \lambda_ {\text {o c c}} \mathcal {L} _ {\text {o c c}} + \lambda_ {\text {s i g n}} \mathcal {L} _ {\phi_ {\text {s i g n}}} + \lambda_ {\text {m a g}} \mathcal {L} _ {\phi_ {\text {m a g}}} + \lambda_ {\delta} \mathcal {L} _ {\delta} + \lambda_ {\mathrm {K L}} \mathcal {L} _ {\mathrm {K L}}. \tag {7} +$$ + +A detailed description of the module design and the choice of $\lambda$ are in the Supplementary Material. + +Hole filling. Although predicted occupancy may be imperfect and introduce small holes, our inherently watertight Sparcubes representation allows for straightforward hole detection and filling. We first identify boundary half-edges. For each face $\mathbf{f} = \{\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2\}$ , we emit the directed edges $(\mathbf{v}_0\rightarrow \mathbf{v}_1)$ , $(\mathbf{v}_1\rightarrow \mathbf{v}_2)$ , and $(\mathbf{v}_2\rightarrow \mathbf{v}_0)$ . By sorting each pair of vertices into undirected edges and counting occurrences, edges whose undirected counterpart appears only once are marked as boundary edges. We build an outgoing-edge map keyed by source vertex, and then recover closed boundary loops by walking each edge until returning to its start. To triangulate each boundary loop $\mathcal{C} = \{\mathbf{v}_i\}_{i = 1}^n$ , we follow a classic ear-filling pipeline: compute a geometric score at every vertex, fill the "best" ear, and repeat until all open small boundaries vanish. Specifically, the score on each pending filled angle $A_{i}$ is defined by + +$$ +A _ {i} = \operatorname {a t a n 2} \left(\left\| \mathbf {d} _ {i - 1 \rightarrow i} \times \mathbf {d} _ {i \rightarrow i + 1} \right\| _ {2}, - \mathbf {d} _ {i - 1 \rightarrow i} \cdot \mathbf {d} _ {i \rightarrow i + 1}\right). \tag {8} +$$ + +In each iteration, we select the vertex with the smallest $A_{i}$ (i.e., the sharpest convex ear), form the triangle $(\mathbf{v}_{i-1}, \mathbf{v}_{i}, \mathbf{v}_{i+1})$ , and update the boundary. Merging all new triangles with the original face set closes every small hole. + +# 4 Experiments + +# 4.1 Experiment Settings + +Implementation details. We implement all Sparcubes as custom CUDA kernels. Following TRELLIS [32], we train both the Sparconv-VAE and its latent flow model on $500\mathrm{K}$ high-quality assets from Objaverse [8] and Objaverse-XL [7]. The VAE runs on 32 A100 GPUs (batch size 32) with AdamW (initial $\mathrm{LR}1\times 10^{-4}$ ) for two days. We then fine-tune the TRELLIS latent flow model on our VAE latents using 64 A100 GPUs (batch size 64) for ten days. At inference, we sample with a classifier-free guidance scale of 3.5 over 25 steps, matching TRELLIS settings. + +Dataset. Following Dora [2], we curated a VAE test set by selecting the most challenging examples from the ABO [6] and Objaverse [8] datasets—specifically those exhibiting occluded components, intricate geometric details, and open surfaces. To avoid any overlap with training data used in prior work, we additionally assembled a "Wild" dataset with multiple components from online sources that is disjoint from both ABO and Objaverse. For generation, we also benchmarked our method against TRELLIS [32] using the wild dataset. + +Compared methods. We compare our Sparconv-VAE with the previous sate-of-the art vaes, including TRELLIS [32], Craftsman [15], Dora [2] and XCubces [24]. Because our diffusion architecture and + +Table 1: Quantitative comparison of watertight remeshing across the ABO [6], Objaverse [8], and In-the-Wild datasets. Chamfer Distance (CD, $\times 10^4$ ), Absolute Normal Consistency (ANC, $\times 10^2$ ) and F1 score (F1, $\times 10^2$ ) are reported. + +
MethodABO [6]Objverse [8]Wild
CD ↓ANC ↑F1 ↑CD ↓ANC ↑F1 ↑CD ↓ANC ↑F1 ↑
Dora-wt-512 [2]1.1676.9483.184.2575.7761.3567.278.5164.99
Dora-wt-1024 [2]1.0776.9484.564.3575.0463.8463.778.7765.90
Ours-wt-5121.0177.7585.213.0975.3564.810.4788.5896.95
Ours-wt-10241.0077.6685.393.0174.9865.650.4688.5597.06
+ +![](images/00796635021da0913f68ac1bf6164972c5e230cf4713278f17b0cb59cbf58d0c.jpg) + +![](images/7a502a65c268afafb108f41762d52c420427066d4701cd56e1fa8861d67f8203.jpg) + +![](images/640bad59a52bf04efca76074b23e1bb7ff04aaa9cceb9a601f5badc36ae09264.jpg) + +![](images/4c4491ef6cc59b5d1a9e5eee2188e2191ad6bdaff37972eb494ea02d340ca21d.jpg) + +![](images/d6936b301b6cdeafd7d3623cf9332e95e9c11298951cff9659107382df15661a.jpg) + +![](images/35927b27b35a375bdcb8b0f1f12ae82f580199e8e2e0041748350da02fa5ebd9.jpg) +Raw mesh + +![](images/11a4345c788e73fd80b6900ad93129c4d84e4658b4889d1624d98ab23fa965dd.jpg) +Ours-wt-512 + +![](images/d95c5ccf65c96da7a52643b6c60f900b9665217f5fa6aa1d4fef2b42d1794736.jpg) +Dora-wt-512 + +![](images/2ff3922a427fc6fe03ebaa883af605c362c18ad93df7f94afd1fdb93d96e51de.jpg) +Ours-wt-1024 +Figure 4: Qualitative comparison of watertight remeshing pipelines. We evaluate our Sparcubes remeshing pipeline against previous widely used one [2, 15, 39], i.e., Dora-wt [2], at voxel resolutions of 512 and 1024. Compared with the previous method, our Sparcubes preserves crucial components (e.g., the car wheel) and recovers finer geometric details (e.g., the shelving frame). Our wt-512 result even outperforms the wt-1024 remeshed by Dora-wt [2]. Best viewed with zoom-in. + +![](images/eb16b2093793e264c9da1c4976296f875102e64d62aacb0645ff316ee81ff447.jpg) +Dora-wt-1024 + +model size match those of TRELLIS [32], we evaluate our generation results against it to ensure a fair comparison. + +# 4.2 Comparison Results + +Watertight remeshing results. We evaluate both our watertight remeshing (serving as VAE ground truth) on the ABO [6], Objaverse [8], and Wild datasets, using Chamfer distance (CD), Absolute Normal Consistency (ANC), and F1 score (F1) as metrics. As Table 1 shows, our Sparcubes consistently outperforms prior pipelines [2, 15, 39] (reported under "Dora-wt" [2] for brevity) across all datasets and metrics. Remarkably, our wt-512 remeshed outputs even exceed the quality of the wt-1024 results produced by previous methods. Fig. 4 presents qualitative comparisons: our approach faithfully preserves critical components (e.g., the car wheel) and recovers fine geometric details (e.g., shelving frames). + +VAE reconstruction results. We further assess our Sparconv-VAE reconstruction against TREL-LIS [32], Craftsman [15], Dora [2], and XCubes [24] in Table 2. Across the majority of datasets and metrics, our Sparconv-VAE outperforms these prior methods. Qualitative results in Fig. 5 illustrate that our VAE faithfully reconstructs complex shapes with fine details (all columns), converts open surfaces into double-layered watertight meshes (columns 1, 4, and 6), and reveals invisible hidden internal structures (column 6). + +Generation results. We also valid the effectiveness of our Sparconv-VAE for generation by fine-tuning the TRELLIS [32] pretrained model. Under the same diffusion architecture and model size (see Fig. 6), our approach synthesizes watertight 3D shapes with exceptional fidelity and rich detail—capturing, for example, the sharp ridges of pavilion eaves, the subtle facial features of human figures, and the intricate structural elements of robots. + +Table 2: Quantitative comparison of VAE reconstruction across the ABO [6], Objverse [8], and In-the-Wild datasets. Chamfer Distance $(\mathrm{CD}, \times 10^{4})$ , Absolute Normal Consistency (ANC, $\times 10^{2}$ ) and F1 score $(\mathrm{F1}, \times 10^{2})$ are reported. + +
MethodABO [6]Objverse [8]Wild
CD ↓ANC ↑F1 ↑CD ↓ANC ↑F1 ↑CD ↓ANC ↑F1 ↑
TRELLIS [32]1.3275.4880.594.2974.3459.270.7085.6094.04
Craftsman [15]1.5177.4677.472.5377.3755.280.8987.8192.28
Dora [2]1.4577.2178.544.8577.1954.3768.278.7962.07
XCubes [24]1.4265.4577.573.6761.8151.652.0262.2173.74
Ours-5121.0178.0985.333.0975.5964.920.4788.7496.97
Ours-10241.0077.6985.413.0075.1065.750.4688.7097.12
+ +![](images/8d77e59da49675f86483ea53370405663b93c0523e7868f0bd210b57f8dd3ca7.jpg) +Figure 5: Qualitative comparison of VAE reconstructions. Our Sparconv-VAE demonstrates superior performance in reconstructing complex geometries, converting open surfaces into double-layered watertight meshes, and revealing invisible internal structures. Best viewed with zoom-in. + +![](images/7ed42586c333ecf98b54b17b7a2d7861241e58555b6ff011e28c5743ed93827d.jpg) +Figure 6: Qualitative comparison of single-image-to-3D generation. Under the same architecture and model size [32], the generator trained with our Sparconv-VAE yields more detailed reconstructions than TRELLIS [32]. Best viewed with zoom-in. + +Table 3: Ablation study on Steps 3 and 4 in the preprocessing. Chamfer Distance $(\mathrm{CD}, \times 10^{4})$ , Absolute Normal Consistency (ANC, $\times 10^{2}$ ), and F1 score $(\mathrm{F1}, \times 10^{2})$ are reported. + +
w/ Step 3w/ Step 4CD ↓ANC ↑F1 ↑
✓0.9777.3488.13
✓0.9577.8188.73
✓✓0.9278.0889.52
+ +# 4.3 Ablation Studies + +Conversion cost. Compared to existing remeshing methods [2, 15, 39], our Sparcubes achieves substantial speedups: at a 512-voxel resolution, conversion takes only around $15\mathrm{s}$ —half than [2, 15, 39]—and at 1024-voxel resolution, it completes in around $30\mathrm{s}$ versus around $90\mathrm{s}$ by [2, 15, 39]. Moreover, by eliminating modality conversion in our VAE design, we avoid the additional SDF resampling step, which in earlier pipelines adds roughly $20\mathrm{s}$ at 512 resolution and about $70\mathrm{s}$ at 1024 resolution [2, 15, 39]. Detailed performance comparisons can be found in the Supplementary Material. + +Training cost. Thanks to our modality-consistent design, Sparconv-VAE converges less than two days—about four times faster than previous methods, i.e., sparse voxel-based TRELLIS [32] and vecset-based approaches [2, 15] each require roughly seven days to train. + +VAE with 2D rendering supervision. We also investigate the effect of incorporating 2D rendering losses into our VAE by using the mask, depth, and normal rendering objectives. We find that adding 2D rendering supervision yields negligible improvement for our Sparconv-VAE. This observation concurs with Dora [2], where extra 2D rendering losses were likewise deemed unnecessary for a 3D-supervised VAEs. We attribute this to the fact that sufficiently dense 2D renderings encode essentially the same information as the underlying 3D geometry—each view being a projection of the same 3D shape. + +Ablation of each in the preprocessing. We perform an ablation study on Steps 3 and 4 to isolate the contributions of deformation optimization and rendering-based refinement: + +In addition to consistent quantitative gains across all metrics, enabling both steps leads to substantial visual improvements—often beyond what the metrics alone can capture in terms of perceptual quality. + +# 5 Conclusion + +We introduce Sparc3D, a unified framework that tackles two longstanding bottlenecks in 3D generation pipelines: topology-preserving remeshing and modality-consistent latent encoding. At its heart, Sparcubes transforms raw, non-watertight meshes into fully watertight surfaces at high resolution—retaining fine details and small components. Building on this, Sparconv-VAE, a sparse-convolitional variational autoencoder with a self-pruning decoder, directly compresses and reconstructs our sparse representation without resorting to heavyweight attention, achieving state-of-the-art reconstruction fidelity and faster convergence. When coupled with latent diffusion (e.g., TRELLIS), Sparc3D elevates generation resolution for downstream 3D asset synthesis. Together, these contributions establish a robust, scalable foundation for high-fidelity 3D generation in both virtual (AR/VR, robotics simulation) and physical (3D printing) domains. + +Limitations. While our Sparcubes remeshing algorithm excels at preserving fine geometry and exterior components, it shares several drawbacks common to prior methods. First, it does not retain any original texture information. Second, when applied to fully closed meshes with internal structures, hidden elements will be discarded during the remeshing process. + +# References + +[1] M. Atzmon and Y. Lipman. Sal: Sign agnostic learning of shapes from raw data. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2565-2574, 2020. 4 +[2] R. Chen, J. Zhang, Y. Liang, G. Luo, W. Li, J. Liu, X. Li, X. Long, J. Feng, and P. Tan. 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Willcock Suvinay Subramanian Felix Chern Alek Andreev +Shreya Pathak Felix Yu Prateek Jain Henry M. Levy David E. Culler Sanjiv Kumar + +Google + +# Abstract + +The discovery of the lazy neuron phenomenon [54], where fewer than $10\%$ of the feedforward networks (FFN) parameters in trained Transformers are activated per token, has spurred significant interests in activation sparsity for enhancing large model efficiency. While notable progress has been made in translating such sparsity to wall-time benefits across CPUs, GPUs, and TPUs, modern Transformers have moved away from the ReLU activation function crucial to this phenomenon. Existing efforts on re-introducing activation sparsity, e.g., by reverting to ReLU, applying top- $k$ masking or a sparse predictor, often degrade model quality, increase parameter count, complicate training. Sparse attention, the application of sparse activation to the attention mechanism, often face similar challenges. + +This paper introduces the Spark Transformer, a novel architecture that achieves high activation sparsity in both FFN and the attention mechanism while maintaining model quality, parameter count, and standard training procedures. Our method realizes sparsity via top- $k$ masking for explicit control over sparsity level. Crucially, we introduce statistical top- $k$ , a hardware-accelerator-friendly, linear-time approximate algorithm that avoids costly sorting and mitigates significant training slowdown from standard top- $k$ operators. Furthermore, Spark Transformer reallocates existing FFN parameters and attention key embeddings to form a low-cost predictor for identifying activated entries. This design not only mitigates quality loss from enforced sparsity, but also enhances wall-time benefit. Pretrained with the Gemma-2 recipe, Spark Transformer demonstrates competitive performance on standard benchmarks while exhibiting significant sparsity: only $8\%$ of FFN neurons are activated, and each token attends to a maximum of 256 tokens. This translates to a $2.5 \times$ reduction in FLOPs, leading to decoding wall-time speedups of up to $1.79 \times$ on CPU and $1.40 \times$ on GPU. + +# 1 Introduction + +The machine learning field has experienced a rapid expansion in the scale of Transformer models [4, 3, 27, 1], significantly advancing the state-of-the-art in language understanding and generation. However, this pursuit of larger models is often constrained not by inherent limitations in model quality [44], but by the soaring computational costs [76, 68] associated with the number of parameters. This + +*Equal contribution +†Now at xAI +$^\ddagger$ Now at Anthropic + +![](images/3fcbc1c2e808b99251753b0ec63bb8de2bdd23ea469f1c730cf0bf4e1bb555cc.jpg) +(a) FLOPs per token vs. quality (1/6 of full training) +Figure 1: Spark Transformer improves inference efficiency via activation sparsity in both FFN and attention, while being nearly quality neutral. (a) Comparison to prior work in terms of relative FLOPs per token at 8k sequence length (y-axis) vs relative training loss $(\mathrm{x - axis})^4$ . [ ] We use standard Gemma-2 [30] as baseline, which has no activation sparsity. [ ] Methods employing activation sparsity within the FFN layers only. Our Spark FFN achieves the most favorable trade-off compared to ReLU, $ReLU^{2}$ , and Topk, which refer to standard Gemma-2 with activation function switched to ReLU [64], $\mathrm{ReLU}^2$ [92], and the composition of Topk and GELU, respectively. [ ] Combining Spark FFN (with $8 \%$ activated parameters) with Spark Attention (with at most 256 attended tokens), our Spark Transformer achieves performance comparable to Gemma-2 while reducing FLOPs to $40\%$ . (b) Evaluation on standard downstream tasks confirms near-quality neutrality of Spark Transformer. (c) Prefill / decode wall time demonstrate a $1.86\times /1.64\times$ speedup resulting from FLOPs reduction. Results are obtained on a 4-Core CPUs for prompts of 4096 tokens. For prefetch, the prompt is chunked into batches of 64 tokens, following a default setup of gemma .cpp [33]. + +![](images/4b628b4a1826de15aa91c64e404dfa06912df2d12c551a82b3a5afad1815d7f7.jpg) + +(b) Downstream evaluation + +
Prefill speed (ms/token)Decode speed (ms/token)
Gemma-228141
Spark Transformer1586
+ +(c) CPU inference wall time + +challenge is compounded by the trend towards models capable of processing increasingly longer input sequences [72], where computational demands scale proportionally with input length. + +Activation sparsity has emerged as a prominent technique for mitigating the computational burdens associated with both large model sizes and long input sequences. For large models, activation sparsity reduces computational cost by activating only a small fraction of the model's parameters for each input. This approach has attracted considerable attention due to the observed lazy-neuron phenomenon [54] where natural activation sparsity occurs in the feed-forward networks (FFNs) of traditional Transformer models like T5 [71] and ViT [26], without explicit enforcement [91]. Subsequent work has successfully demonstrated wall-time benefits from activation sparsity on multiple hardware platforms, including CPU [90], GPU [82, 58], and TPU [89]. + +Despite the great success, a fundamental challenge arises in the application of sparse activation to the latest and state-of-the-art models. That is, the inherent sparsity, which is pivotal for obtaining efficiency, is largely absent due to the adoption of gated non-ReLU activation functions [19], widely adopted in, e.g., Mistral [41], Gemma [30], LLAMA [27]. This motivates the following question: + +Can we re-introduce a high level of activation sparsity in recent Transformers, without compromising their quality? + +To answer this question, prior work has explored reverting to ReLU variants [64, 92] or incorporating top- $k$ thresholding [89, 81, 83]. However, (Challenge #1) these methods often lead to a degradation in model quality (see Figure 1a). Furthermore, (Challenge #2) top- $k$ thresholding based methods not only introduce non-differentiability that compromises model quality, but also involve sorting to obtain maximally activated neurons, which is inefficient on ML accelerators, such as TPUs [43], with possibly a $10 \times$ training slowdown (see Figure 6). Finally, a low-cost predictor is often introduced to identify activated parameters, critical for maximizing efficiency benefits [58, 90, 82, 89]. However, (Challenge #3) the inclusion of such a predictor increases training pipeline complexity, incurs additional training costs and parameters, and exacerbates model quality loss. + +Sparse attention, the application of activation sparsity to the attention mechanism, presents similar challenges. A common strategy is top- $k$ attention [35], which applies a top- $k$ mask to the attention coefficients. This can be augmented with a low-cost predictor to further enhance efficiency [74, 88, 50]. Nevertheless, achieving both high sparsity and accurate prediction without resorting to complex procedures, training slowdown from top- $k$ , and compromising quality remains an open problem. + +Contributions. This paper introduces the Spark Transformer, a novel architecture that achieves a strong level of activation sparsity in both FFN and attention mechanism with minimal impact on quality, providing a positive answer to the question posed above. + +Spark Transformer comprises Spark FFN and Spark Attention, both of which exploit the interpretation of FFNs and attention mechanisms as key-value lookup tables [32] to provide a unified framework for sparsity and a low-cost predictor (see Figure 2). Our low-cost predictor is constructed by repurposing a subset of the dimensions of the query and key vectors to compute an importance score for each key-value pair (see Section 2). This design addresses Challenge #3 by avoiding the introduction of extra parameters, enabling all model parameters to be trained in a single stage. + +Our key technical tool for introducing sparsity is a statistical top- $k$ operator, which is applied to the predicted scores in Spark FFN and Spark Attention to select the activated keys. In particular, statistical top- $k$ is a linear-complexity algorithm for approximate nearest neighbor search, which addresses the issue of high computation cost of standard top- $k$ algorithms (i.e., Challenge #2). As explained in Section 3, this is achieved by fitting a Gaussian distribution to the activation scores and estimating a threshold that selects the top entries. Moreover, it enjoys continuously differentiability almost everywhere by the usage of a soft-thresholding operator, making the network more amenable to optimization.[5] + +Finally, we demonstrate that Spark Transformer alleviates quality loss (i.e., Challenge #1) by performing a full pretraining using the Gemma-2 recipe [30]. Comparison with prior work shows that Spark Transformer exhibits a more favorable trade-off between FLOPs reduction and quality measured by pretraining loss (Figure 1a). Further evaluation on standard benchmarks confirms that Spark Transformer closely matches the performance of Gemma-2 (see Figure 1b), despite exhibiting a high degree of sparsity: only $8\%$ of FFN neurons are activated, and each token attends to at most 256 tokens. Leveraging this sparsity, we assess the model's inference efficiency on CPUs, demonstrating wall-time speedups of $1.86 \times$ for prefetch and $1.64 \times$ for decode (see Figure 1c). Furthermore, an up to $1.4 \times$ wall-time speedup is obtained on NVIDIA T4 GPU (see Section 4). This enhanced efficiency broadens access to high-quality models for users with limited access to high-FLOP hardware, such as high-end GPUs and TPUs. + +# 2 Spark Transformer + +This section describes Spark FFN and Spark Attention, the two components of Spark Transformer. + +# 2.1 Spark FFN + +FFNs in a standard Transformer are two-layer multi-layer perceptrons that map an input token $\mathbf{q} \in \mathbb{R}^{d_{\mathrm{model}}}$ to an output + +$$ +\text {S t a n d a r d - F F N} (\boldsymbol {q}; \boldsymbol {K}, \boldsymbol {V}) \stackrel {\text {d e f}} {=} \boldsymbol {V} \cdot \sigma \left(\boldsymbol {K} ^ {\top} \boldsymbol {q}\right) \in \mathbb {R} ^ {d _ {\text {m o d e l}}}. \tag {1} +$$ + +In above, $\{\pmb {K},\pmb {V}\} \subseteq \mathbb{R}^{d_{\mathrm{model}}\times d_{\mathrm{ff}}}$ are trainable model parameters, and $\sigma ()$ is a nonlinear activation function. We ignore the dependency on layer index to simplify the notations. + +Each matrix multiplication in eq. (1) has $2d_{\mathrm{model}} \cdot d_{\mathrm{ff}}$ FLOPs hence overall the computation cost is $4d_{\mathrm{model}} \cdot d_{\mathrm{ff}}$ . Previous work shows that when $\sigma(\cdot)$ is ReLU, the activation map $\sigma(\boldsymbol{K}^\top \boldsymbol{q})$ is very sparse after model training. The sparsity can be used trivially to reduce the computation costs in the calculation of its product with the second layer weight matrix $\boldsymbol{V}$ [54], reducing the overall FLOPs count of FFN to $2d_{\mathrm{model}} \cdot (d_{\mathrm{ff}} + k)$ , where $k \ll d_{\mathrm{ff}}$ is the number of nonzero entries in the activation. Note that the sparsity cannot be used to reduce the computation costs associated with $\boldsymbol{K}$ , which constitute half of the total FLOPs in FFN. + +![](images/fbf8b3b965a0c5a9f71b8b25333cf6d7d167dbfa1fa8f774ebe1303037929f7f.jpg) +Figure 2: Architecture of Spark FFN and Spark Attention. (Left) Unified illustration of standard FFN (i.e., eq. (1)) and standard Attention (i.e., eq. (6)). In the case of FFN, $\pmb{q} \in \mathbb{R}^{d_{\mathrm{model}}}$ is the input, $\pmb{K}$ and $\pmb{V}$ are the first and second layer weights, respectively, and $\sigma(\cdot)$ is GELU. In the case of Attention, $\pmb{q} \in \mathbb{R}^{d_{\mathrm{attn}}}$ is the query, $\pmb{K}$ and $\pmb{V}$ are key and value matrices, respectively, and $\sigma(\cdot)$ is softmax. (Right) Unified illustration of Spark FFN (i.e., eq. (2)) and Spark Attention (i.e., eq. (7)). In the case of Spark FFN, $\sigma_1(\cdot)$ is GELU and $\sigma_2(\cdot)$ is identity. In the case of Spark Attention, $\sigma_1(\cdot)$ is softmax and $\sigma_2(\cdot)$ is softmax. In both cases, Statistical-Top $_k$ (i.e., eq. (10)) is applied to introduce sparsity, which enables sparse matrix multiplication with $K_2$ and $V$ that reduces FLOPs count. + +![](images/64b9fde069e921a3ccc5a97241f9074783e500cadef5e49e26764234a72bbbbd.jpg) + +In order to reduce FLOPs count in the first layer of FFN as well, we introduce Spark FFN as follows: + +$$ +\operatorname {S p a r k - F F N} \left(\boldsymbol {q}; \boldsymbol {K} _ {1}, \boldsymbol {K} _ {2}, \boldsymbol {V}, k, r\right) \stackrel {\text {d e f}} {=} \boldsymbol {V} \cdot \left(\sigma \left(\operatorname {T o p} _ {k} \left(\boldsymbol {K} _ {1} ^ {\top} \cdot \boldsymbol {q} [: r ]\right)\right) \odot \left(\boldsymbol {K} _ {2} ^ {\top} \cdot \boldsymbol {q} [ r: ]\right)\right). \tag {2} +$$ + +In above, $K_{1} \subseteq \mathbb{R}^{r \times d_{\mathrm{ff}}}$ , $K_{2} \subseteq \mathbb{R}^{(d_{\mathrm{model}} - r) \times d_{\mathrm{ff}}}$ , and $V \subseteq \mathbb{R}^{d_{\mathrm{model}} \times d_{\mathrm{ff}}}$ are trainable parameters, and the activation $\sigma(\cdot)$ is taken to be GELU [37] following Gemma. $\operatorname{Top}_k$ is introduced for obtaining sparsity, with $k$ being a hyper-parameter specifying the sparsity level. Concretely, $\operatorname{Top}_k$ preserves the largest $k$ values in the activation vector, while setting other values to 0. In this paper, we use the efficient statistical top-k presented in Section 3, which avoids sorting activation values. Finally, the input $q$ is split into $q[:r]$ and $q[r:]$ , which contain the first $r$ and the rest of the dimensions, respectively, with $r$ being a hyper-parameter. It is introduced so that the term $K_{1}^{\top}q[:r]$ serves as a low-rank predictor of the location of the nonzero entries, which allows us to obtain efficiency benefits in computing $K_{2}^{\top}q[r:]$ and the multiplication with $V$ . This is discussed in detail below. + +FLOPs per Token. Direct implementation of the Spark-FFN without exploiting sparsity has the same number of FLOPs as the standard FFN in eq. (1), i.e., + +$$ +2 r \cdot d _ {\mathrm {f f}} + 2 \left(d _ {\mathrm {m o d e l}} - r\right) \cdot d _ {\mathrm {f f}} + 2 d _ {\mathrm {m o d e l}} \cdot d _ {\mathrm {f f}} = 4 d _ {\mathrm {m o d e l}} \times d _ {\mathrm {f f}}, \tag {3} +$$ + +where the three terms are from multiplication with $K_{1}$ , $K_{2}$ , and $V$ , respectively. In Spark-FFN, one may first compute the term $K_{1}^{\top} q[:r]$ as a low-rank predictor. After passing its output through $\mathrm{Top}_k$ , which selects approximately the $k$ most important entries, followed by the activation function $\sigma(\cdot)$ , we obtain a sparse output. Importantly, after obtaining the sparse output there is no need to perform the full computation of the other two matrix multiplications in eq. (2), i.e., $K_{2}^{\top} q[r:]$ and the multiplication with $V$ . Instead, one can perform a sparse matrix multiplication with a drastically reduced FLOPs count: + +$$ +2 r \cdot d _ {\mathrm {f f}} + 2 \left(d _ {\mathrm {m o d e l}} - r\right) \cdot k + 2 d _ {\mathrm {m o d e l}} \cdot k = 2 \left(d _ {\mathrm {f f}} - k\right) \cdot r + 4 d _ {\mathrm {m o d e l}} \cdot k, \tag {4} +$$ + +which is an increasing function of $r$ . In other words, $r$ controls the computation cost. We provide ablation study in Section C.4 to show that the best model quality is obtained when $r \approx \frac{d_{\mathrm{model}}}{2}$ . In this case, the total FLOP count of Spark FFN is approximately $d_{\mathrm{model}} \cdot d_{\mathrm{ff}} + 3 \cdot d_{\mathrm{model}} \cdot k$ , which is a 4-times reduction from eq. (3) when $k$ is small. + +Relation to gated activation. Many of the most recent Transformers, including Gemma, use a variant of the standard FFN in eq. (1) where the activation function is replaced with a gated one: + +$$ +\operatorname {G a t e d - F F N} \left(\boldsymbol {q}; \boldsymbol {K} _ {1}, \boldsymbol {K} _ {2}, \boldsymbol {V}\right) = \boldsymbol {V} \cdot \left(\sigma \left(\boldsymbol {K} _ {1} ^ {\top} \boldsymbol {q}\right) \odot \left(\boldsymbol {K} _ {2} ^ {\top} \boldsymbol {q}\right)\right). \tag {5} +$$ + +In above, $\{\pmb{K}_1,\pmb{K}_2,\pmb{V}\} \subseteq \mathbb{R}^{d_{\mathrm{model}}\times d_{\mathrm{ff}}^{\prime}}$ . Note that when compared with the FFN in eq. (1) for quality studies, $d_{\mathrm{ff}}^{\prime}$ is usually taken to be $2 / 3\cdot d_{\mathrm{ff}}$ to be iso-parameter count [77]. + +Our Spark FFN in eq. (2) bears some resemblance to Gated FFN in that both have two linear maps in the first layer and one in the second layer. The difference lies in that 1) Spark FFN adds a $\mathrm{Top}_k$ to obtain sparsity, and 2) the input to the first layers of Spark FFN are obtained from splitting the dimensions of the input. + +# 2.2 Spark Attention + +In a standard multi-head attention layer, an input $\pmb{x} \in \mathbb{R}^{d_{\mathrm{model}}}$ is mapped to a query, a key, and a value vector of dimension $d_{\mathrm{attn}}$ as $\pmb{q}^{(i)} = \pmb{W}_Q^{(i)}\pmb{x} \in \mathbb{R}^{d_{\mathrm{attn}}}, \pmb{k}^{(i)} = \pmb{W}_K^{(i)}\pmb{x} \in \mathbb{R}^{d_{\mathrm{attn}}}, \pmb{v}^{(i)} = \pmb{W}_V^{(i)}\pmb{x} \in \mathbb{R}^{d_{\mathrm{attn}}}$ for each head $i$ . Here, $\{\pmb{W}_Q^{(i)}, \pmb{W}_K^{(i)}, \pmb{W}_V^{(i)}\} \subseteq \mathbb{R}^{d_{\mathrm{attn}} \times d_{\mathrm{model}}}$ are trainable weights. + +Collecting all the key and value vectors in the context of $\pmb{x}$ into $\pmb{K}^{(i)} = [\pmb{k}_1^{(i)},\dots,\pmb{k}_{n_{\mathrm{ctx}}}^{(i)}]\in \mathbb{R}^{d_{\mathrm{attn}}\times n_{\mathrm{ctx}}}$ and $\pmb{V}^{(i)} = [\pmb{v}_1^{(i)},\dots,\pmb{v}_{n_{\mathrm{ctx}}}^{(i)}]\in \mathbb{R}^{d_{\mathrm{attn}}\times n_{\mathrm{ctx}}}$ , attention conducts the following computation: + +$$ +\text {S t a n d a r d - A t t e n t i o n} (\boldsymbol {q}; \boldsymbol {K}, \boldsymbol {V}) \stackrel {\text {d e f}} {=} \boldsymbol {V} \cdot \operatorname {s o f t m a x} \left(\boldsymbol {K} ^ {\top} \boldsymbol {q}\right) \in \mathbb {R} ^ {d _ {\mathrm {a t t n}}}, \tag {6} +$$ + +where we omit the dependency on $i$ for simplicity. The computation cost associated with eq. (6) is $4d_{\mathrm{attn}} \cdot n_{\mathrm{ctx}}$ for each head. Finally, output from all heads are concatenated followed by a linear map to project to $d_{\mathrm{model}}$ . + +Note that eq. (6) has the same form as FFN in eq. (1) except for the choice of nonlinearity. Hence, following a similar strategy in obtaining Spark FFN, here we present Spark Attention as + +$$ +\operatorname {S p a r k - A t t e n t i o n} (\boldsymbol {q}; \boldsymbol {K}, \boldsymbol {V}, k, r) \stackrel {\text {d e f}} {=} \boldsymbol {V} \cdot \left(\sigma_ {1} \left(\operatorname {T o p} _ {k} ^ {(- \infty)} \left(\boldsymbol {K} _ {1} ^ {\top} \boldsymbol {q} [: r ]\right)\right) \odot \sigma_ {2} \left(\boldsymbol {K} _ {2} ^ {\top} \boldsymbol {q} [ r: ]\right)\right). \tag {7} +$$ + +In above, $K_{1} \in \mathbb{R}^{r \times n_{\mathrm{ctx}}}$ and $K_{2} \in \mathbb{R}^{(d_{\mathrm{attn}} - r) \times n_{\mathrm{ctx}}}$ contain the first $r$ rows and the rest of the rows from $K$ , respectively, and $\sigma_{1}$ and $\sigma_{2}$ are nonlinear functions. Empirically, we find that taking $\sigma_{1} = \text{softmax}$ and $\sigma_{2} = \text{softplus}$ gives good results. Finally, $\operatorname{Top}_{k}^{(-\infty)}$ refers to an operator that keeps the largest $k$ values of the input while setting the rest to $-\infty$ . This operator be implemented using statistical top-k, which we explain in Section 3. + +FLOPs per Token. With a direct implementation the number of FLOPs in eq. (7) is given by $4d_{\mathrm{attn}} \cdot n_{\mathrm{ctx}}$ , which is the same as the FLOPs for eq. (6). However, by noting that the output of the softmax is expected to be sparse with approximately $k$ nonzero entries, the computation costs associated with $K_1^\top q[:r]$ and in the multiplication with $V$ can be drastically reduced. In particular, if we take $r = \frac{d_{\mathrm{attn}}}{2}$ then the FLOPs per token becomes + +$$ +d _ {\text {m o d e l}} n _ {\mathrm {c t x}} + 3 d _ {\text {m o d e l}} \min \left\{k _ {\text {a t t n}}, n _ {\mathrm {c t x}} \right\}, \tag {8} +$$ + +which is nearly a $4 \times$ reduction when $k_{\mathrm{attn}} \ll n_{\mathrm{ctx}}$ . + +# 3 Statistical Top-k + +This section introduces Statistical-Top $_k$ , an efficient algorithm for implementing the Top $_k$ operators in Spark FFN (i.e. eq. (2)) and Spark Attention (i.e. eq. (7)). + +Recall that the soft-thresholding operator is defined for an arbitrary vector $\mathbf{x} \in \mathbb{R}^d$ and a scalar threshold $\theta \in \mathbb{R}$ as + +$$ +\text {S o f t - T h r e s h o l d} (\boldsymbol {x}, \theta) \stackrel {\text {d e f}} {=} \max \left\{\boldsymbol {x} - \theta \cdot \mathbf {1}, \mathbf {0} \right\} \in \mathbb {R} ^ {d}, \tag {9} +$$ + +where $\mathbf{1}$ and $\mathbf{0}$ are $d$ -dimensional vectors with all entries equal to 1 and 0, respectively. The soft-thresholding operator shifts each entry of $\pmb{x}$ to the left by $\theta$ and then thresholds the result at zero. + +We define Statistical-Top $k$ as the following mapping from $\mathbb{R}^d$ to $\mathbb{R}^d$ : + +$$ +\operatorname {S t a t i s t i c a l} - \operatorname {T o p} _ {k} (\boldsymbol {x}) \stackrel {\text {d e f}} {=} \operatorname {S o f t} - \operatorname {T h r e s h o l d} (\boldsymbol {x}, \theta (\boldsymbol {x}, k)), \tag {10} +$$ + +where + +$$ +\theta (\boldsymbol {x}, k) \stackrel {\text {d e f}} {=} \operatorname {m e a n} (\boldsymbol {x}) + \operatorname {s t d} (\boldsymbol {x}) \cdot Q \left(1 - \frac {k}{d}\right). \tag {11} +$$ + +In above, we define $\mathrm{mean}(\pmb{x}) \stackrel{\mathrm{def}}{=} \frac{1}{d} \sum_{i=1}^{d} x_i$ and $\mathrm{std}(\pmb{x}) \stackrel{\mathrm{def}}{=} \sqrt{\frac{1}{d-1} \sum_{i=1}^{d} (x_i - \mathrm{mean}(\pmb{x}))^2}$ , which compute the sample mean and standard deviation of the entries of the input $\pmb{x}$ , respectively. $Q(\cdot)$ is + +the quantile function (i.e., inverse of the cumulative distribution function) of the standard Gaussian distribution. In Spark Transformer, eq. (10) is used as the $\mathrm{Top}_k$ operator in eq. (2). For the operator $\mathrm{Top}_k^{(-\infty)}$ in eq. (7), a slight variant of eq. (10) is used where the entries below the threshold $\theta (\pmb {x},k)$ are set to $-\infty$ instead of 0. + +Statistical-Top $k$ operates by first computing a threshold $\theta (\pmb {x},k)$ such that approximately $k$ entries of $\pmb{x}$ exceed it, and then applying the soft-thresholding operator with this threshold to $\pmb{x}$ to obtain a sparse output. We discuss these two components in the next two subsections. + +# 3.1 Threshold Estimation + +The threshold $\theta (\pmb {x},k)$ in eq. (10) is designed such that, if the entries of $\pmb{x}$ are drawn from a Gaussian distribution, approximately $k$ out of the $d$ entries will exceed this threshold. To understand this, let $\mu$ and $\sigma$ denote the mean and standard deviation of the underlying Gaussian distribution. Its quantile function is given by $\mu +\sigma \cdot Q(p)$ for $p\in (0,1)$ . Consequently, due to the properties of quantile functions, we expect roughly $p\cdot d$ entries of $\pmb{x}$ to exceed $\mu +\sigma \cdot Q(1 - p)$ . In practice, since $\mu$ and $\sigma$ are unknown, they are replaced with the sample mean mean $(\pmb {x})$ and the sample standard deviation $\mathrm{std}(\pmb {x})$ , respectively. + +The following theorem formalizes this argument. + +Theorem 3.1. Let $\pmb{x} \in \mathbb{R}^d$ be a vector with entries drawn i.i.d. from $\mathcal{N}(\mu, \sigma^2)$ . For any $1 \leq k \leq d - 1$ , let $\theta(\pmb{x}, k)$ be a scalar defined in eq. (10). Take any $\delta \in (0, 1)$ and assume $d \geq \max\{2, \log \frac{6}{\delta}\}$ . With a probability of at least $1 - \delta$ , the number of entries of $\pmb{x}$ that are greater than $\theta(\pmb{x}, k)$ , i.e., card $(\{i \in [d] \mid x_i > \theta(\pmb{x}, k)\})$ , satisfies + +$$ +\frac {\left| \mathbf {c a r d} \left(\{i \in [ d ] \mid x _ {i} > \theta (\boldsymbol {x} , k) \}\right) - k \right|}{d} \leq 4 \sqrt {\frac {\log \frac {6}{\delta}}{d}} \left(1 + \sqrt {- 2 \log \min \left\{\frac {k}{d} , 1 - \frac {k}{d} \right\}}\right). \tag {12} +$$ + +Theorem 3.1 provides a relative error bound between $k$ and the true number of entries of $\pmb{x}$ that exceed $k$ . This bound is maximized when $k = 1$ or $k = d - 1$ . Consequently, the worst-case bound is $O\left(\sqrt{\frac{\log d \cdot \log \frac{1}{\delta}}{d}}\right)$ which vanishes as $d$ increases. Notably, the error bound becomes $O\left(\sqrt{\frac{\log \frac{1}{\delta}}{d}}\right)$ when $k = \Theta(d)$ , demonstrating even faster convergence. + +Computation cost. The computation of the threshold $\theta (\mathbf{x},k)$ is highly efficient and resembles the operations used in LayerNorm layers, requiring only $2d$ FLOPs to compute the mean and standard deviation of the samples. This contrasts sharply with a naive sorting-based approach, which has $O(d\log d)$ complexity. + +While the Gaussian quantile function $Q(\cdot)$ lacks a closed-form solution, high-precision piecewise approximation algorithms with constant complexity are available in standard software packages like SciPy [86], readily applicable to our needs. + +# 3.2 Sparsification + +Given the threshold $\theta(x, k)$ , a straightforward approach to obtain a sparse vector is to set all entries of $x$ below the threshold to zero, preserving the remaining values. This operator, sometimes referred to as hard thresholding [10], suffers from discontinuity, potentially hindering its suitability for gradient-descent-based training. + +To address this, Statistical-Top $_k$ employs the soft-thresholding operator defined in eq. (9) [8]. This operator first shrinks all entries of $\pmb{x}$ by the threshold $\theta(\pmb{x}, k)$ and then sets all entries below 0 to 0. Soft thresholding offers the advantages of being continuous and differentiable almost everywhere (except when entries of $\pmb{x}$ coincide with $\theta(\pmb{x}, k)$ ). + +For complete differentiability, one can utilize a smoothing function like the Huber loss [39], defined element-wise on an input $\pmb{x}$ as: + +$$ +\operatorname {H u b e r} (x; \delta) \stackrel {\text {d e f}} {=} \left\{ \begin{array}{l l} \frac {1}{2} x ^ {2} & \text {f o r} | x | < \delta , \\ \delta \cdot \left(| x | - \frac {1}{2} \delta\right) & \text {o t h e r w i s e .} \end{array} \right. \tag {13} +$$ + +The continuous differentiability of the mapping $\pmb{x} \mapsto \mathrm{Huber}(\mathrm{Statistical - Top}_k(\pmb{x});\delta) / \delta$ is established below: + +![](images/c4154c2dbc1a5a2f2692dd5fdf725173b9a8e27beb8422f2ba1aa7143d41b824.jpg) +(a) Sparsity level in Spark FFN + +![](images/2345962afdda7a019e383e2c3f43bb6b93d673f350c778f069b609564d1d9f45.jpg) +(b) Sparsity level in Spark Attention +Figure 3: Sparsity in the intermediate activation of Spark FFN and Spark Attention across 26 layers at selected training steps. For FFN we report the percentage of nonzero entries out of $d_{\mathrm{ff}} = 13824$ entries. For Attention, we report the number of nonzero entries (i.e., attended tokens). Our hyper-parameter choice is to have $8\%$ nonzeros in Spark FFN and at most 256 nonzeros in Spark Attention. + +Theorem 3.2. For any $\delta >0$ , the function $\mathbb{R}^d\to \mathbb{R}^d$ defined as + +$$ +\operatorname {H u b e r} \left(\operatorname {S t a t i s t i c a l} - \operatorname {T o p} _ {k} (\boldsymbol {x}); \delta\right) / \delta \tag {14} +$$ + +is continuously differentiable. + +Note that eq. (14) converges to Statistical-Top $_k(x)$ as $\delta \to 0$ , which can be seen as $\mathrm{Huber}(x; \delta) / \delta \to |x|$ and Statistical-Top $_k(x)$ is always non-negative. In practice, however, we find that using a non-zero $\delta$ does not improve model quality, and therefore we set $\delta = 0$ for simplicity. + +Finally, soft thresholding admits a variational form (see, e.g., [67]): + +$$ +\text {S o f t - T h r e s h o l d} (\boldsymbol {x}, \theta) = \underset {\boldsymbol {z} \geq \mathbf {0}} {\arg \min } \theta \| \boldsymbol {z} \| _ {1} + \frac {1}{2} \| \boldsymbol {x} - \boldsymbol {z} \| _ {2} ^ {2}. \tag {15} +$$ + +This formulation seeks a vector $\mathbf{z}$ that minimizes both its squared $\ell_2$ distance to the input $\pmb{x}$ and its $\ell_1$ norm, with the threshold $\theta$ balancing these terms. Given the sparsity-promoting nature of the $\ell_1$ norm, soft thresholding effectively finds a sparse approximation of the input $\pmb{x}$ . This variational form also connects Statistical-Top $_k$ with other top- $k$ algorithms in the literature; see Section D.3. + +# 4 Experiments + +In this section, we present an experimental evaluation of Spark Transformer using the Gemma-2 2B model. Gemma-2 2B is a decoder-only Transformer with 2 billion parameters, pretrained on 2 trillion tokens of primarily English text data (see [30] for details). To evaluate Spark Transformer, we train a model by substituting the standard FFN and Attention in Gemma-2 2B with their Spark Transformer counterparts (Spark FFN and Spark Attention, respectively). This Spark Transformer model is trained using the same procedure and data as the Gemma-2 2B model. + +Implementation details. Gemma-2 uses a model dimension of $d_{\mathrm{model}} = 2304$ . For FFN, Gemma-2 uses the Gated FFN in eq. (5) with $d_{\mathrm{ff}}' = 9216$ . We replace it with Spark FFN in eq. (2) with $d_{\mathrm{ff}} = 13824$ so that the parameter count keeps the same. In addition, we take $k$ to be 1106, which gives a sparsity level of $8\%$ , and $r = 1024 \approx d_{\mathrm{model}} / 2$ (due to sharding constraints, $r$ can only be a multiple of 256). For Attention, Gemma-2 alternates between a global attention that has a span of 8192 tokens, and a local attention with a 4096 window size, both with $d_{\mathrm{attn}} = 256$ . We replace both with Spark Attention in eq. (6) where for the latter we use the same 4096 window size. For hyper-parameters, we use $k = 256$ , i.e. each token attends to at most 256 tokens, and $r = 128 = d_{\mathrm{attn}} / 2$ . Gemma-2 uses Rotary Position Embedding [84] which is applied to $q$ and the columns of $K$ in eq. (6). For Spark Attention in eq. (7), we apply this position encoding to $q[:r]$ , $q[r:]$ , the columns of $K_1$ , and the columns of $K_2$ . + +# 4.1 Quality + +We evaluate Spark Transformer on a suite of benchmarks that are used in the Gemma-2 paper [30], and report the result in Figure 1b. We observe that Spark Transformer matches the quality of Gemma-2 while having a drastically reduced FLOP count per token. + +The near-quality neutrality of Spark Transformer distinguishes it from related work on enforcing activation sparsity in FFN, which often lead to a quality loss. To demonstrate this, we pretrain variants + +![](images/71009588bcc19cdcfccb0480e02f7637ec08cee9e5a109b7056055a597ad9a6a.jpg) +Red: skipped memory access /compute + +![](images/acbed9de192d76e0e6efefa9f133e30f6310e5a6f6eec8ddb5022765eac00793.jpg) +(a) Vector-Masked Matrix Multiplication +(b) Sparse Vector-Matrix Multiplication +Figure 4: Illustration of the matrix multiplication implementation using sparse activation. (a) Vector-Masked Matrix Multiplication takes a dense vector $\mathbf{q}[r]$ , a dense matrix $K_2^\top$ , and a mask from statistical top-k on $K_1^\top \mathbf{q}[:r]$ to compute $\mathbf{u} := (K_2^\top \mathbf{q}[r]) \odot \mathbf{mask}$ . It skips memory loading and compute associated with the masked columns. (b) Sparse Vector-Matrix Multiplication takes a sparse activation vector $\mathbf{u}$ to compute weighted sum of rows in the dense matrix $V$ . It skips loading and computation of rows corresponding to 0's in $\mathbf{u}$ . To optimize performance, we implement Sparse Vector-Matrix Multiplication using tiling, which helps minimize cross-CPU core synchronization. + +of Gemma-2 2B with 1) activation function switched to ReLU [92], 2) activation function switched to $\mathrm{ReLU}^2$ [64], and 3) Top-k thresholding applied before GELU. Due to high training cost, all models are pretrained for 1/6 of the standard Gemma-2 training iterations. In Figure 1a, we report the training losses and FLOPs per token relative to those of the standard Gemma-2. It can be seen that these models either suffer from a large quality loss (i.e., ReLU and Topk), or do not lead to a sufficient FLOPs reduction (i.e., $\mathrm{ReLU}^2$ ). In contrast, our Spark FFN achieves less quality loss with more FLOPs reduction. Finally, the combination of Spark FFN with Spark Attention introduces additional FLOPs reduction and quality benefits, enabling an almost neutral quality of Spark Transformer with a large overall FLOPs reduction. + +# 4.2 Sparsity + +To verify the effectiveness of statistical top- $k$ , we report the level of sparsity measured in terms of percentage of nonzeros in FFN and the number of nonzeros in Attention. At the beginning of model training, we observe that statistical top- $k$ produces close to $8\%$ nonzeros in FFN (see Figure 3a), which aligns well with our hyper-parameter choice of using $k / d_{\mathrm{ff}} = 8\%$ in Spark FFN. This is expected as the model parameters, particularly $\pmb{K}$ in Spark FFN, are randomly initialized, hence the entries of the activation maps are drawn from a Gaussian distribution which is in accordance with the assumption of statistical top- $k$ . The Gaussian assumption is no longer guaranteed after training, but we empirically observe it to hold approximately (see Section C.1) and statistical top- $k$ reliably produce a sparsity level close to $8\%$ until the end of training at $480k$ steps. Sparsity in attention is reported in Figure 3b, which show that the number of attended tokens is below our hyper-parameter choice of 256 in Spark Attention throughout training. In particular, the numbers are much smaller because the results are from averaging over all tokens many of which have a context length of less than 256. Finally, we observe comparable levels of sparsity during evaluation (see Section C.2). + +# 4.3 Inference Efficiency + +We evaluate the efficiency benefits of Spark Transformer over standard Gemma-2 on both CPUs and GPUs. For CPU evaluation, we use gemma.cpp [33], the official C++ inference engine optimized for CPUs. For GPU evaluation, we use l1ama.cpp [31], a widely-used LLM inference engine which supports running a wide selection of LLM models on GPUs. We modify both frameworks to support sparse matrix multiplication operators, which exploit sparsity in both FFN and Attention layers. Our implementation leverages vector SIMD operations [80] existing in modern CPU, and customized CUDA kernel for GPU. In particular, our implementation reduces not only computational FLOPs but also memory bandwidth requirements, directly accelerating memory-bound scenarios; see Figure 4 for + +![](images/c7ae73ab2e4f43b6d097aae2655436c05d3535e7b59e14792aa3124803446e43.jpg) +(a) 16 Core CPU VM. + +![](images/560e88713a0a8a3264d045374176b0bc3c371457797df4941ae3f87dc1ee9b06.jpg) +(b) 4 Core CPU VM. + +![](images/20632703a1d598ede274ed4b35fe7cc33316bc9cea6b36320fa67535f6d34591.jpg) +(c) NVIDIA T4 GPU. +Figure 5: Spark Transformer decoding speedup from activation sparsity on various hardware platforms. We report decoding speed of Spark Transformer without hardware optimization for sparse activation, with hardware optimization for sparsity in Spark FFN only, and with hardware optimization for sparsity in both Spark FFN and Spark Attention. All experiments use a decode batch size of 1. + +an illustration and Section B in Appendix for details. We show that Spark Transformer significantly improves the efficiency of transformer models, even in highly FLOP-constrained environments such as CPUs. + +CPU results. Figure 5a and 5b report the decoding speed under varying prompt lengths on a 4-Core or a 16-Core CPU. We see that with hardware optimization for sparse activation in both Spark FFN and Spark Attention, a speedup that ranges from $1.35\mathrm{x}$ to $1.79\mathrm{x}$ can be achieved on a 16-Core CPU depending on the prompt length. For short prompts (e.g., 256 tokens) and long prompts (e.g. 4096 tokens), the hardware optimization for Spark FFN and Spark Attention provide the most speedup, respectively. + +Figure 1c further highlights the efficiency of Spark Transformer in both prefetch and decode phases. During the prefetch, the prompt is usually chunked into batches since the process is bounded by memory bandwidth. This may reduce the benefit of activation sparsity as different tokens in a chunk may activate different subsets of parameters (in FFN) and attend to different subsets of tokens (in Attention). However, Figure 1c shows that Spark Transformer maintains strong performance with a chunk size of 64 tokens, following the default setup in gemma.cpp. A more detailed performance analysis of batching/chunking is provided in Section C.3. In addition, Spark Transformer significantly outperforms Gemma-2 during decoding (with batch size=1). + +GPU results. We also evaluate the efficiency gain from sparsity on low-profile GPUs. Figure 5c reports the decoding speed under varying prompt lengths on an NVIDIA T4 GPU. Similar to the CPU case, we see Spark Transformer achieves decode speedup ranging from $1.25 \times$ to $1.40 \times$ . + +# 4.4 Training Efficiency + +The introduction of a top- $k$ operator is expected to lead to a training slowdown due to the extra computational operators. This slowdown can be prohibitively large if one uses the standard approximate top- $k$ operator provided in JAX, namely jax.lax.approx_max_k [14]. Specifically, this JAX top- $k$ operator is optimized to achieve TPU peak performance and has a controllable recall target, which we vary on the x-axis in Figure 6. It can be observed that the JAX top- $k$ leads to more than $10 \times$ slowdown even when operating on a small recall of $50\%$ . In contrast, the slowdown from statistical top- $k$ is with a very small amount, demonstrating its efficiency. Finally, we do not provide the quality of models trained with JAX top- $k$ since such models take a very long time to train. + +![](images/21e6ddde8ac6f113f200e12e32bf45450d42545fa87f0243ecdb9423938715e6.jpg) +Figure 6: Comparison of training slowdown from using our statistical top- $k$ vs the standard top- $k$ (i.e., jax.1ax.approx_max_k [14]) relative to not using any top- $k$ . + +# 5 Discussions + +This paper introduces the Spark Transformer architecture to reduce the FLOPs in both the FFN and Attention of Transformer models. Because the FFN and Attention components dominate the + +computational cost in large Transformers with long contexts, the overall FLOP count for decoding a token is also drastically reduced (see Table C.1), leading to notable computation efficiency benefits on appropriate hardwares. Specifically, the benefit is obtained by selectively activating only part of the model parameters and limiting the attended context for each input. This principle of sparse activation finds a compelling parallel in neuroscience, where studies reveal sparse activity patterns in the brain as a key factor in its remarkable efficiency [5, 7, 48]. While hardware limitations currently hinder the full exploitation of sparse activation, our demonstration of practical wall-time reduction of Spark Transformer on CPU and GPU, together with prior evidence for related techniques [58, 82] including TPU [89], highlight its potential. We hope that this work opens avenues for research into alternative hardware better suited for sparse computations, circumventing the hardware lottery [38] and potentially leading to greater efficiency gains in the future. Additional discussions on related work for activation sparsity, including Mixture-of-Experts, are provided in Section D.1. + +This work also opens promising avenues for future research on efficient inference of large language models, particularly in combining Spark Transformer's architectural efficiency with other leading optimization techniques. We briefly discuss two key synergies here. + +Synergy with Speculative Decoding. Spark Transformer is highly complementary to speculative decoding. As a target model, its faster inference directly accelerates the primary verification bottleneck. As a draft model, its near-quality neutrality and high speed make it an ideal candidate for generating high-quality drafts, potentially leading to higher token acceptance rates and greater overall speedups. A full discussion is provided in Section D.5. + +Synergy with Quantization. We also hypothesize a strong synergy with quantization. The benefits are expected to be multiplicative, as Spark Transformer reduces the number of operations while quantization reduces their cost. More importantly, unlike standard pruning which preserves high-magnitude outliers, our statistical top- $k$ operator uses soft-thresholding. This shrinks the dynamic range of activation distribution, which may reduce sensitivity to activation quantization. A detailed analysis of this mechanism is available in Appendix section D.6. + +# Acknowledgments and Disclosure of Funding + +The authors acknowledge helpful discussion with Zonglin Li (Anthropic), Daliang Li (Anthropic), Amir Yazdanbakhsh (Google), Zeid Samoail (Google), Abhishek Kumar (Amazon), Vlad Feinberg (Google), Veeru Sadhanala (Google) at various stages of this project. + +# References + +[1] Bo Adler, Niket Agarwal, Ashwath Aithal, Dong H Anh, Pallab Bhattacharya, Annika Brundyn, Jared Casper, Bryan Catanzaro, Sharon Clay, Jonathan Cohen, et al. 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Section B contains implementation details of sparse matrix multiplication on CPU. Section C provides details and additional results on experiments. Finally, Section D presents additional discussions on comparing with related work on activation sparsity in FFN (in Section D.2) and on statistical top-k (in Section D.4). + +# Appendix A Proofs + +# A.1 Proof to Theorem 3.1 + +Proof. In this proof we write $\bar{x} \stackrel{\mathrm{def}}{=} \operatorname{mean}(x)$ and $s \stackrel{\mathrm{def}}{=} \operatorname{std}(x)$ for brevity. + +We first establish the concentration bounds that the empirical mean and standard deviation, i.e., $\bar{x}$ and $s$ are close to the true mean and true standard deviation, i.e., $\mu$ and $\sigma$ of the underlying Gaussian, respectively. Recall from the definition of the chi-squared distribution that $(d - 1)\frac{s^2}{\sigma^2} \sim \chi^2(d - 1)$ . Using the Laurent-Massart bound on the tail probability of the chi-squared distribution [47, Corollary of Lemma 1], we have + +$$ +\Pr \left(\left| (d - 1) \frac {s ^ {2}}{\sigma^ {2}} - (d - 1) \right| \geq 2 \sqrt {(d - 1) t} + 2 t\right) \leq 2 e ^ {- t} +$$ + +for every $t > 0$ . We set $t = \log \frac{6}{\delta}$ . Then, with a probability of at least $1 - \delta / 3$ , we have + +$$ +(d - 1) \left| \frac {s ^ {2}}{\sigma^ {2}} - 1 \right| < 2 \sqrt {(d - 1) \log \frac {6}{\delta}} + 2 \log \frac {6}{\delta}, +$$ + +which implies + +$$ +\left| \frac {s ^ {2}}{\sigma^ {2}} - 1 \right| < 2 \sqrt {\frac {\log \frac {6}{\delta}}{d - 1}} + 2 \frac {\log \frac {6}{\delta}}{d - 1} \leq 4 \sqrt {\frac {\log \frac {6}{\delta}}{d}} + 4 \frac {\log \frac {6}{\delta}}{d} \leq 8 \sqrt {\frac {\log \frac {6}{\delta}}{d}}, +$$ + +where the last inequality uses the assumption that $d \geq \max \{2, \log \frac{6}{\delta}\}$ . By rearranging the terms, we get + +$$ +\sigma \left(1 - 8 \sqrt {\frac {\log \frac {6}{\delta}}{d}}\right) \leq \sigma \sqrt {\max \left\{1 - 8 \sqrt {\frac {\log \frac {6}{\delta}}{d}}, 0 \right\}} \leq s \leq \sigma \sqrt {1 + 8 \sqrt {\frac {\log \frac {6}{\delta}}{d}}} \leq \sigma \left(1 + 8 \sqrt {\frac {\log \frac {6}{\delta}}{d}}\right), +$$ + +which simplifies to + +$$ +| s - \sigma | \leq 8 \sigma \sqrt {\frac {\log \frac {6}{\delta}}{d}}. \tag {A.1} +$$ + +eq. (A.1) provides a concentration bound for $s$ . We now proceed to deriving a bound for $\mu$ . Towards that, notice that $\frac{\bar{x} - \mu}{\sigma / \sqrt{d}} \sim \mathcal{N}(0, 1)$ . By using the Mill's inequality that upper bounds the tail probability + +of a standard normal distribution (i.e., if $Z \sim \mathcal{N}(0,1)$ and $t > 0$ , then $\operatorname*{Pr}(|Z| > t) \leq \frac{e^{-t^2 / 2}}{t}$ ), we have + +$$ +\Pr \left(\left| \frac {\bar {x} - \mu}{\sigma / \sqrt {d}} \right| > \sqrt {2 \log \frac {3}{\delta}}\right) \leq \frac {\delta / 3}{\sqrt {2 \log \frac {3}{\delta}}} \leq \delta / 3. +$$ + +Therefore, with probability at least $1 - \delta /3$ we have + +$$ +\left| \frac {\bar {x} - \mu}{\sigma / \sqrt {d}} \right| \leq \sqrt {2 \log \frac {3}{\delta}}, +$$ + +which yields + +$$ +| \bar {x} - \mu | \leq \sigma \sqrt {\frac {2 \log \frac {3}{\delta}}{d}}. \tag {A.2} +$$ + +Combining eq. (A.1) and eq. (A.2), with probability at least $1 - 2\delta /3$ , we have + +$$ +\left| \theta (\boldsymbol {x}, k) - \left(\mu + \sigma Q \left(1 - \frac {k}{d}\right)\right) \right| \tag {A.3} +$$ + +$$ +\leq | \bar {x} - \mu | + | s - \sigma | \left| Q \left(1 - \frac {k}{d}\right) \right| \tag {A.4} +$$ + +$$ +\leq \sigma \sqrt {\frac {2 \log \frac {3}{\delta}}{d}} + 8 \sigma \sqrt {\frac {\log \frac {6}{\delta}}{d}} \left| Q \left(1 - \frac {k}{d}\right) \right|. \tag {A.5} +$$ + +We define the empirical cumulative distribution function (ECDF) of $x_{1}, x_{2}, \ldots, x_{d}$ as + +$$ +\hat {F} _ {d} (x) = \frac {1}{d} \sum_ {i \in [ d ]} \mathbf {1} _ {\{x _ {i} \leq x \}}. +$$ + +Then, the number of the entries of $\pmb{x}$ that are greater than $\theta(\pmb{x}, k)$ may be written as + +$$ +\mathbf {c a r d} \left(\left\{i \in [ d ] \mid x _ {i} > \theta (\boldsymbol {x}, k) \right\}\right) = \sum_ {i \in [ d ]} \mathbf {1} _ {\left\{x _ {i} > \theta (\boldsymbol {x}, k) \right\}} = d \Big (1 - \hat {F} _ {d} (\theta (\boldsymbol {x}, k)) \Big). +$$ + +Let $F$ denote the cumulative distribution function (CDF) of $\mathcal{N}(\mu, \sigma^2)$ . By the Dvoretzky-Kiefer-Wolfowitz inequality [28, 63], we have + +$$ +\Pr \left(\sup _ {u \in \mathbb {R}} \left| \hat {F} _ {d} (u) - F (u) \right| > t\right) \leq 2 e ^ {- 2 d t ^ {2}}. +$$ + +Taking $t = \sqrt{\frac{1}{2d}\log\frac{6}{\delta}}$ and $u = \theta (\pmb {x},k)$ , we obtain + +$$ +\Pr \left(\left| \hat {F} _ {d} (\theta (\boldsymbol {x}, k)) - F (\theta (\boldsymbol {x}, k)) \right| > \sqrt {\frac {1}{2 d} \log \frac {6}{\delta}}\right) \leq \frac {\delta}{3}. \tag {A.6} +$$ + +Applying the union bound on eq. (A.3) and eq. (A.6), we obtain that the following holds with probability at least $1 - \delta$ : + +$$ +\left| \hat {F} _ {d} (\theta (\boldsymbol {x}, k)) - \left(1 - \frac {k}{d}\right) \right| \tag {A.7} +$$ + +$$ += \left| \hat {F} _ {d} \left(\theta (\boldsymbol {x}, k)\right) - F \left(\mu + \sigma Q \left(1 - \frac {k}{d}\right)\right) \right| \tag {A.8} +$$ + +$$ +\leq \left| \hat {F} _ {d} (\theta (\boldsymbol {x}, k)) - F (\theta (\boldsymbol {x}, k)) \right| + \left| F (\theta (\boldsymbol {x}, k)) - F \left(\mu + \sigma Q \left(1 - \frac {k}{d}\right)\right) \right| \tag {A.9} +$$ + +$$ +\leq \sqrt {\frac {1}{2 d} \log \frac {6}{\delta}} + \frac {1}{\sqrt {2 \pi} \sigma} \left| \theta (\boldsymbol {x}, k) - \left(\mu + \sigma Q \left(1 - \frac {k}{d}\right)\right) \right| \tag {A.10} +$$ + +$$ +\leq \sqrt {\frac {1}{2 d} \log \frac {6}{\delta}} + \frac {1}{\sqrt {2 \pi}} \left(\sqrt {\frac {2 \log \frac {3}{\delta}}{d}} + 8 \sqrt {\frac {\log \frac {6}{\delta}}{d}} \left| Q \left(1 - \frac {k}{d}\right) \right|\right) \tag {A.11} +$$ + +$$ +\leq 4 \sqrt {\frac {\log \frac {6}{\delta}}{d}} \left(1 + \left| Q \left(1 - \frac {k}{d}\right) \right|\right). \tag {A.12} +$$ + +In the above expression, the first equality stems directly from the definitions of $F(\cdot)$ and $Q(\cdot)$ , which gives + +$$ +F (\theta (\boldsymbol {x}, k)) = F (\mu + \sigma Q (1 - \frac {k}{d})) = \Phi (Q (1 - \frac {k}{d})) = 1 - \frac {k}{d}, +$$ + +where $\Phi$ denotes the CDF of the standard normal distribution. + +To simplify eq. (A.7), we consider two cases: + +- If $k \leq d / 2$ , by Mill's inequality, we have + +$$ +1 - \Phi (\sqrt {2 \log {\frac {d}{k}}}) \leq \frac {e ^ {- (\sqrt {2 \log {\frac {d}{k}}}) ^ {2} / 2}}{\sqrt {2 \log {\frac {d}{k}}}} = \frac {e ^ {- (\sqrt {2 \log {\frac {d}{k}}}) ^ {2} / 2}}{\sqrt {2 \log {\frac {d}{k}}}} = \frac {k / d}{\sqrt {2 \log {\frac {d}{k}}}} \leq \frac {k}{d}, +$$ + +where the last inequality is because $2\log \frac{d}{k}\geq 1$ . Therefore + +$$ +1 - \frac {k}{d} \leq \Phi (\sqrt {2 \log \frac {d}{k}}), +$$ + +which gives + +$$ +Q \left(1 - \frac {k}{d}\right) \leq \sqrt {2 \log \frac {d}{k}} = \sqrt {- 2 \log \frac {k}{d}}. +$$ + +- If $k > d / 2$ , we have + +$$ +\left| Q (1 - \frac {k}{d}) \right| = \left| Q (1 - \frac {d - k}{d}) \right| \leq \sqrt {- 2 \log \frac {d - k}{d}}. +$$ + +Combining the two cases, we get + +$$ +\left| Q \left(1 - \frac {k}{d}\right) \right| \leq \sqrt {- 2 \log \min \left\{\frac {k}{d} , 1 - \frac {k}{d} \right\}}. +$$ + +Plugging this into eq. (A.7), we obtain + +$$ +\left| \hat {F} _ {d} (\theta (\boldsymbol {x}, k)) - (1 - \frac {k}{d}) \right| \leq 4 \sqrt {\frac {\log \frac {6}{\delta}}{d}} \left(1 + \sqrt {- 2 \log \min \left\{\frac {k}{d} , 1 - \frac {k}{d} \right\}}\right). +$$ + +Recall card $(\{i \in [d] \mid x_i > \theta(\boldsymbol{x}, k)\}) = d\left(1 - \hat{F}_d(\theta(\boldsymbol{x}, k))\right)$ . We conclude that with probability at least $1 - \delta$ , we have + +$$ +\left| \mathbf {c a r d} \left(\left\{i \in [ d ] \mid x _ {i} > \theta (\boldsymbol {x}, k) \right\}\right) - k \right| \leq 4 \sqrt {d \log \frac {6}{\delta}} \left(1 + \sqrt {- 2 \log \min \left\{\frac {k}{d} , 1 - \frac {k}{d} \right\}}\right). +$$ + +![](images/6ce473607b89cbf627c7db27e5c44b980a729ab285ea1ddf96fce22c9ef57464.jpg) + +# A.2 Proof to Theorem 3.2 + +Proof. The Huber statistical top- $k$ in eq. (14) may be written as + +$$ +\operatorname {H u b e r} \left(\text {S t a t i s t i c a l - T o p} _ {k} (\boldsymbol {x}); \delta\right) / \delta = \operatorname {H u b e r} \left(\text {S o f t - T h r e s h o l d} (\boldsymbol {x}, \theta (\boldsymbol {x}, k))\right) / \delta , \tag {A.13} +$$ + +where $\theta (\pmb {x},k)$ is defined in eq. (10). This function is the (multivariate) composition of two functions, namely, $\theta = \theta (\pmb {x},k)$ and Huber(Soft-Threshold $(\pmb {x},\theta))$ . In particular, the former is continuously differentiable (i.e., $C^1$ ) in $\pmb{x}$ , since it is simply a linear combination of sample mean and sample standard deviation both of which are $C^1$ functions. To establish the theorem, we only need to show that Huber(Soft-Threshold $(\pmb {x},\theta))$ is also a $C^1$ function in $(\pmb {x},\theta)$ + +By definition, Huber(Soft-Threshold $(x,\theta))$ is defined entry-wise on $x$ as + +$$ +\operatorname {H u b e r} (\text {S o f t - T h r e s h o l d} (x, \theta)) = \left\{ \begin{array}{l l} \delta x - \delta \theta - \frac {1}{2} \delta , & \text {i f} x > \theta + \delta ; \\ \frac {1}{2} (x - \theta) ^ {2}, & \text {i f} \theta \leq x \leq \theta + \delta ; \\ 0, & \text {i f} x < \theta . \end{array} \right. \tag {A.14} +$$ + +From here it is easy to check that $\mathrm{Huber}(\mathrm{Soft - Threshold}(x,\theta))$ is continuous in $(x,\theta)$ . Its gradient with respect to $(x,\theta)$ is given by + +$$ +\frac {\partial \operatorname {H u b e r} (\text {S o f t - T h r e s h o l d} (x , \theta))}{\partial (x , \theta)} = \left\{ \begin{array}{l l} (\delta , - \delta), & \text {i f} x > \theta + \delta ; \\ (x - \theta , \theta - x) & \text {i f} \theta \leq x \leq \theta + \delta ; \\ (0, 0), & \text {i f} x < \theta , \end{array} \right. \tag {A.15} +$$ + +which is also continuous. This concludes the proof. + +# Appendix B Implementation Details on Sparse Matrix Multiplications + +We describe how we implement sparse matrix multiplications for Spark FFN and Attention in gemma.cpp for CPU and llama.cpp for GPU. We start by focusing on a batch size of one for decoding before expanding our discussion to larger batch sizes and prefetch. + +With batch size of 1, both Spark FFN and Spark Attention utilize two types of sparse vector-matrix multiplication: vector-masked matrix multiplication and sparse vector-matrix multiplication (Figure 4). Given a vector $q$ and a matrix $w$ , vector-masked matrix multiplication multiplies $q$ with the non-masked columns of $w$ based on a masking vector $m$ . Masked columns yield a zero output. Sparse vector-matrix multiplication, on the other hand, involves a vector that contains many zeroes being multiplied by a dense matrix. + +In Spark FFN, we perform vector-masked matrix multiplication for $K_2^\top \mathbf{q}[r]$ (Figure 4a). The masking vector is generated from the output of Statistical-Top $_k$ ( $K_1^\top \mathbf{q}[r]$ ). Based on the mask, Spark FFN skips loading the masked columns of $w$ from DRAM (in CPU setup) or HBM (in GPU setup) and the associated computations. On CPU, Spark FFN utilizes SIMD operations (as in the original Gemma implementation). To further enhance performance, Spark FFN utilizes software CPU prefetching (builtin prefetch) to overlap loading from DRAM to the CPU cache with computations. On GPU, Spark FFN utilizes customized CUDA kernel. + +The same masking vector also identifies the zero entries in the intermediate vector that is multiplied by matrix $V$ (Figure 4b). For this sparse vector-matrix multiplication, we store the matrix in row format. Each CPU thread (or GPU warp) processes a tile of the matrix while skipping the loading and computation of the masked rows. Prefetching and SIMD operations are applied similarly on CPU in this context. + +Spark Attention utilizes these two types of sparse matrix multiplication operators to accelerate qkv computations for each head. + +When extending to decoding with batch sizes greater than one or prefill, we continue to use individual masks to skip computations while using a union of masks from each vector within the batch to create unified masks for memory loading. With larger batches, Spark transformer is expected to save less memory loading (vs. original Gemma), unless there is significant overlap in top-k positions within the same batch. Nonetheless, the Spark transformer consistently reduces FLOP by skipping computations based on individual masks within the batch. + +![](images/b35c2ad8ae852aa5906a94dc77357a2c43b6fe0390d5a97b1c2e513e514e16e2.jpg) +(a) Spark FFN sparsity + +![](images/cab3f4e5da3c58c2702fd523caf60ee2a1a7499705ebf64119db82dd0d1c5599.jpg) +(b) Spark Attention sparsity +Figure C.1: Sparsity in the intermediate activation of Spark FFN and Spark Attention during evaluation (see Figure 3 for results during training). For FFN, we use a simple prompt "test" and report the percentage of nonzero entries in generating the 5th, 10th, and 15th token. For Attention, we report the nuber of nonzero entries at the 512th, 1024th, and 2048th token during prefetch. + +# Appendix C Additional Experimental Results and Details + +# C.1 Distribution of Inputs to Statistical Top-k + +The underlying assumption for statistical top- $k$ is that the activation vector upon which it is applied to, namely, the pre-GELU activation in Spark FFN and the pre-softmax activation in Spark Attention, can be modeled as being drawn from an i.i.d. Gaussian distribution. Here we provide empirical evaluation on the distribution of these activation vectors for Spark Transformer. Results for Spark FFN and Spark Attention are provided in Figure C.4 and Figure C.5, respectively. The results show that the distribution holds close proximity to a Gaussian, hence justifying the use of statistical top- $k$ . + +# C.2 Sparsity Level During Evaluation + +Complementing Figure 3 which reports sparsity level during pretraining, here we report the sparsity level during evaluation to confirm that statistical top- $k$ produces the same level of sparsity during test time. The results are presented in Figure C.1 for some arbitrarily selected tokens. For Attention, in particular, we select tokens at the positions 512, 1024, and 2048 which are all above our choice of $k = 256$ for Spark Attention. + +# C.3 Batching Analysis + +![](images/4fbf87609d9ce46f1e39ff677bad56f753a4bec44bb23412ebd872000c70746f.jpg) +Figure C.2: Spark Transformer vs. Gemma-2 Prefix Token/Sec with Varying Chunk Size. We use a prompt length of 4096 tokens on a 16 core CPU VM. + +Figure C.2 provides the performance comparison between Spark Gemma 2 and Gemma 2, measured in prefetch throughput (tokens/sec) across varying chunk sizes. We use a 4096-token prompt on a 16-core CPU VM. A similar trend is expected during the decoding phase with varying batch sizes. + +Our analysis shows that Spark Transformer provides the highest speedup at batch size 1, and again at large batch sizes (e.g. $>8$ ), where the compute FLOP becomes the primary bottleneck. + +For Gemma-2, as seen in the figure, increasing batch/chunk size leads to a significant improvement in prefetch throughput until the batch size reaches 8. This improvement occurs because batching reduces memory access by reusing weights across multiple tokens in the CPU cache. Once the computation becomes the bottleneck (i.e. batch = 8), further batching provides diminishing returns. + +In contrast, Spark Transformer behaves differently. When the batch size increases from 1 to 2, we observe minimal throughput change. This is due to the lack of overlap in top-k positions between the tokens, resulting from the high sparsity. However, as the batch size increases beyond 4, Spark Transformer starts benefiting from weight reuse, similar to Gemma-2. Spark Transformer continues to show improvements in throughput until the batch size reaches approximately 64, where it eventually becomes FLOP-bound, much later than Gemma-2 due to the reduced FLOP requirements. + +Overall, Spark Gemma demonstrates the most significant gains in two scenarios: when the batch size is 1, a common setting for desktop or mobile devices decoding, and when the batch size is large enough that FLOP becomes the dominant bottleneck. + +# C.4 Effect of $r$ and $k$ in Spark FFN. + +Spark FFN comes with two hyper-parameters, namely $r$ which controls the rank hence FLOP count of the low-cost predictor, and $k$ which controls sparsity of activation hence the FLOP count. In Figure C.3 we provide an ablation study on the effect of these two hyper-parameters, by reporting the training loss curves in the first 25,000 training steps (which is around $5\%$ of full training). From Figure C.3a, the best choice of $r$ is 1024 which is nearly half of $d_{\mathrm{model}} = 2304$ (due to model sharding constraint, $r$ cannot be taken to be exactly a half of $d_{\mathrm{model}}$ ). From Figure C.3b, we see that the model quality is insensitive to choices of $k$ that gives $[5\%, 10\%]$ sparsity, but there is quality loss if we go sparser, e.g. $3\%$ nonzeros. + +![](images/1542d3e2c174d6f8e5461e05a90b49687cbf79de599b079807401271a618b0cd.jpg) +(a) Effect of $r$ + +![](images/9ac007d11e77bf12af81c80249a5b01cefac0392666cb281c1815a02d7e09357.jpg) +(b) Effect of $k$ +Figure C.3: Effect of hyper-parameters $r$ and $k$ in Spark FFN on training loss. A Gaussian filter of $\sigma = 200$ is applied to smooth the loss curves. Models are trained with 1/20 of standard Gemma-2 training iterations. + +# C.5 Additional Ablation Studies + +In this section, we provide ablation studies for understanding the effect of the individual components of Spark Transformer. Towards that, we plot the training loss curves for Gemma-2 and Spark Transformer, see Figure C.6. Here, we restrict to the first 80k training steps out of the 500k total steps since it is costly to fully train all ablation models, and that 80k steps is sufficient for seeing the trend. We can see that Spark Transformer slightly lags behind Gemma-2. However, as demonstrated in Figure 1b, that small difference in training loss does not lead to a substantial difference in evaluation quality. + +In our ablation studies below, we add a single component at a time to Gemma-2 and report the quality impact. + +![](images/3a0302a92881e03dbf1a38595db4fb84580b08c88f1ad4c970e7db8105d0b925.jpg) +Figure C.6: Ablation study in terms of training loss in the first 80k training steps (out of 500k total steps). + +Spark FFN vs Spark Attention. To understand the effect of Spark FFN vs Spark Attention, we conduct an experiment where only attention is switched from a standard one to Spark Attention, + +![](images/acc26bc8ba501be2372e3a92731dd2ca902a478a41e1d09ee64632409467db70.jpg) +(a) Layer 0 + +![](images/434164e98034da3033abc3652c3e136465800a05ef29228b667586e91248d214.jpg) +(b) Layer 8 + +![](images/7c2a68829f3210c3dce80f75e5082bad64b251f3e262d2892d16fcd667e292fc.jpg) +(c) Layer 16 + +![](images/9ad0993caa586694b2f334cc6002054ab5563ed94507ab3165d93a2e4f51e4c1.jpg) +(d) Layer 24 + +![](images/1ab58a44304712bb069052aa6920096c974c1cbaf10e1e499d5c5d175b6cd76c.jpg) +(e) Layer 0 + +![](images/e9c42d63ea56410ed37d3b045041b27cf2c9d19ff3c3a92773a7ae78353b27c5.jpg) +(f) Layer 8 + +![](images/67e5e205b5cf717813bf0bbd0edca51bd79bebe385bceb053a64ecc2bd4026db.jpg) +(g) Layer 16 +Figure C.4: Distribution of the entries of the input activation to statistical top- $k$ in Spark FFN (see Figure C.5 for result of Spark Attention). The two rows correspond to activation at two positions 0 and 1000 of an input, and the columns correspond to activation at four different depth levels $\{0,8,16,24\}$ of the 26-layer pretrained Spark Transformer. The input is the first 1000 tokens of the first essay from https://paulgraham.com/articles.html prepended with the BOS token. We compare the empirical distribution (Empirical) with the Gaussian distribution whose mean and standard deviation (std) are computed as the sample mean and std of the input (Fitted). We see that the Gaussian closely approximates the empirical distribution. We also compare the cutoff value estimated from the Gaussian, i.e., $\theta(\boldsymbol{x}, k)$ used in eq. (10) with $k / d = 5\%$ (Cutoff for fitted), with the cutoff value for obtaining $8\%$ nonzeros on the empirical distribution (Cutoff for empirical). It can be seen that these two cutoff values are close. + +![](images/39ccecc1568e85ef9f936442ebf28977a78969ecc8b532de1a7ec416ed844e2d.jpg) +(h) Layer 24 + +whereas the FFN remains the standard one. The result is illustrated as Gemma-2 + Spark Attention in Figure C.6. It can be seen that Spark Attention provides a minor quality gain over Gemma-2. In comparing Gemma-2 + Spark Attention with Spark Transformer, this also shows that further adding Spark FFN slightly hurts model quality. As noted above, such a small difference does not lead to substantial quality impact on the evaluation tasks. Hence, we conclude here that none of Spark FFN and Spark Attention has significant quality impact. + +Sparsity enforcing vs Low-cost predictor. Sparsity enforcing via statistical top- $k$ and low-cost activation predictor are two relatively independent components of Spark Transformer. This means that, upon the standard Gated FFN (see eq. (5)) that is used in Gemma-2, which we rewrite here for convenience: + +$$ +\operatorname {G a t e d - F F N} \left(\boldsymbol {q}; \boldsymbol {K} _ {1}, \boldsymbol {K} _ {2}, \boldsymbol {V}\right) = \boldsymbol {V} \cdot \left(\sigma \left(\boldsymbol {K} _ {1} ^ {\top} \boldsymbol {q}\right) \odot \left(\boldsymbol {K} _ {2} ^ {\top} \boldsymbol {q}\right)\right), \tag {C.1} +$$ + +we may choose to only apply statistical top- $k$ for enforcing sparsity, i.e., + +$$ +\operatorname {T o p k - G a t e d - F F N} (\boldsymbol {q}; \boldsymbol {K} _ {1}, \boldsymbol {K} _ {2}, \boldsymbol {V}) = \boldsymbol {V} \cdot \left(\sigma \left(\text {S t a t i s t i c a l - T o p} _ {k} \left(\boldsymbol {K} _ {1} ^ {\top} \boldsymbol {q}\right)\right) \odot \left(\boldsymbol {K} _ {2} ^ {\top} \boldsymbol {q}\right)\right). \tag {C.2} +$$ + +Note that applying a sparsifying function on the input to the nonlinear function $\sigma(\cdot)$ as in eq. (C.2) is a common choice in the literature of sparse activations, e.g., [64, 81, 49]; the main difference between these works lies in the specific sparsity enforcing technique, see Table D.1 for a summary. In addition to the sparsifying function, Spark FFN also has another architectural change for the purpose of introducing a low-cost predictor. Here, we rewrite Spark FFN for ease of comparison with eq. (C.2): + +$$ +\operatorname {S p a r k - F F N} (\boldsymbol {q}; \boldsymbol {K}, \boldsymbol {V}, k, r) \stackrel {\text {d e f}} {=} \boldsymbol {V} \cdot \left(\sigma \left(\operatorname {S t a t i s t i c a l - T o p} _ {k} \left(\boldsymbol {K} _ {1} ^ {\top} \boldsymbol {q} [: r ]\right)\right) \odot \left(\boldsymbol {K} _ {2} ^ {\top} \boldsymbol {q} [ r: ]\right)\right). \tag {C.3} +$$ + +Analogous to FFN, we may also only add statistical top- $k$ to attention without the low-cost predictor, i.e., by switching from standard Attention in eq. (6) to the following: + +$$ +\operatorname {T o p k - A t t e n t i o n} (\boldsymbol {q}; \boldsymbol {K}, \boldsymbol {V}) \stackrel {\text {d e f}} {=} \boldsymbol {V} \cdot \operatorname {s o f t m a x} \left(\operatorname {S t a t i s t i c a l - T o p} _ {k} ^ {(- \infty)} \left(\boldsymbol {K} ^ {\top} \boldsymbol {q}\right)\right). \tag {C.4} +$$ + +Here, we aim to understand the effect of introducing statistical top- $k$ without the low-cost predictor. Towards that, we conduct an experiment where FFN and Attention in Gemma-2 are replaced with eq. (C.2) and eq. (C.4), respectively. The result is illustrated as Gemma-2 + Top- $k$ in Figure C.6. + +![](images/cfd3ef324b0063f8eb6eec9407ac8821d32959504f6fabb6df1b0d5561051631.jpg) +(a) Layer 0 + +![](images/ae9168ab09755a1bf8eba620c2997e8335fcdd38f839b540b2cd12e16c191189.jpg) +(b) Layer 8 + +![](images/1bba374f04b4b8d570e3b35358047436c4bd643a55a2c48321ecb45370914100.jpg) +(c) Layer 16 + +![](images/250e07c3c1e3b3b588185a7cb50302fa9818722c0dc9c120da16b5661f94b57d.jpg) +(d) Layer 24 + +![](images/e9ebd87b174bcb95a0120a10f9ed7c6e259b7b28369540860b481eeeeefdb6ab.jpg) +(e) Layer 0 +Table C.1: FLOPs per token comparison: Spark Transformer vs. standard Transformer. In a standard Transformer with model dimension $d_{\mathrm{model}}$ , we assume multi-head attention where the sum of head dimensions equals $d_{\mathrm{model}}$ , and an FFN with non-gated activation and width $d_{\mathrm{ff}}$ . Here, $n_{\mathrm{ctx}}$ represents the context length for the target token. The computational cost is primarily determined by the FFN (assuming $d_{\mathrm{ff}} \gg d_{\mathrm{model}}$ , which is typical) and the attention dot product (assuming a long context length). Spark Transformers introduce sparsity parameters, $k_{\mathrm{ff}}$ and $k_{\mathrm{attn}}$ , to reduce FLOPs. Setting $k_{\mathrm{ff}} = 8\% \times d_{\mathrm{ff}}$ and $k_{\mathrm{attn}} = 256$ achieves a $3.2 \times$ FLOPs reduction in the FFN, a $4 \times$ reduction in the attention dot product, and a $2.5 \times$ reduction overall (assuming $n_{\mathrm{ctx}} = 8k$ ) for Gemma-2B. + +![](images/21cd048e99fff0ebe626addb31a53aef2f3db23311b7beacf8d19cee9faf6cbc.jpg) +(f) Layer 8 + +![](images/6549b3d0fb27df2ba22dfdd325d6cbe2acd7f7ee88f67d723ba72a1b99fa4577.jpg) +(g) Layer 16 +Figure C.5: Distribution of the entries of the input activation to statistical top- $k$ in Spark Attention (see Figure C.4 for result of Spark FFN). The two rows correspond to activation for two different attention heads, and the columns correspond to activation at four different depth levels $\{0, 8, 16, 24\}$ of the 26-layer pretrained Spark Transformer. Model input is the first 1000 tokens of the first essay from https://paulgraham.com/articles.html prepended with the BOS token, and we examine activation of the last token (i.e., inner product between the query embedding of the 1001st token and all 1001 key embeddings). We compare the empirical distribution (Empirical) with the Gaussian distribution whose mean and standard deviation (std) are computed as the sample mean and std of the input (Fitted). We see that the Gaussian closely approximates the empirical distribution. We also compare the cutoff value estimated from the Gaussian, i.e., $\theta(\boldsymbol{x}, k)$ used in eq. (10) with $k = 256$ (Cutoff for fitted), with the cutoff value for obtaining top 256 entries on the empirical distribution (Cutoff for empirical). It can be seen that these two cutoff values are close. + +![](images/d70d2282523da3ea0cfae7e9396a52f93a280aa48e044f499b43a055fd12a60d.jpg) +(h) Layer 24 + +
OperationFLOPs per Token7
Standard TransformerSpark Transformer (Ours)
FFN4dmodeldffdmodeldff + 3dmodelkff
Attention dot product4dmodelnctxdmodelnctx + 3dmodelmin{kattn, nctx}
Attention linear projection8d2model8d2model
+ +It can be seen that the training loss becomes notably larger and the gap compared to Gemma-2 is further increasing with more training steps. This result demonstrates that while the low-cost predictor is introduced to Spark Transformer for reducing the cost in predicting the nonzero entries, it also helps in bridging the gap from the introduction of statistical top- $k$ . In other words, Transformer with low-rank predictors in FFN and Attention is more amenable to activation sparsification without quality loss. + +# C.6 Computing FLOPs per Token + +To understand the FLOPs saving reported in Figure 1a, we provide a comparison between FLOPs of the major components of a Transformer, including FFN, Attention dot product, and Attention projections, with that of Spark Transformer. The results are presented in Table C.1. + +# Appendix D Additional Discussions + +# D.1 Additional Related Work + +In the following, we review a few lines of work closely related to ours. + +Mixture of Experts (MoEs) may be considered as a particular case of sparsely activated models which group the neurons in FFN and activate all neutrons in selected groups [78, 52]. Neuron grouping has the benefit of being better suited for training accelerators compared to unstructured activation sparsity. However, training of MoEs incurs extra complexities in algorithmic design and requires special hardware support [29]. Moreover, the structured nature of sparsity limits the model's flexibility and expressiveness, and recent work advocates the use of a larger number of smaller experts [18, 36]. On the other hand, the discovery of the naturally emerging unstructured activation sparsity has motivated the perspective of naturally emerging experts [91, 25, 16, 70, 85, 95]. + +Sparse activation is a common approach to improve the efficiency of large models and many techniques for a low-cost activation prediction have been developed over the years, such as low-rank factorization [20], quantization [11], product keys [46], hashing [12], etc. With the popularity of modern Transformer models, these techniques become natural choices [40, 90, 58, 82] for reducing their high computation costs. In particular, a lot of the excitement comes from the discovery that the activations in FFNs are naturally sparse [91, 54, 61] and hence efficiency with activation sparsity is obtained without a quality toll. + +Our work falls into the category of the latest work in this direction that aims to bring the benefits to the latest generation large language models that do not have natural sparsity. Early attempts [64, 69, 92] seek to bring back sparsity by switching back to ReLU variants, but it usually incurs a quality loss. The quality gap may be largely bridged by more careful tuning, but the activation becomes less sparse (e.g. $25\%$ nonzeros in LLAMA 7B [81]). Top-k has become a more popular choice for obtaining sparsity recently [83] and is able to maintain neutral quality while offering strong sparsity, but only in selected layers [89]. Moreover, such methods require finetuning to bring sparsity and also obtain a predictor. Without doing finetuning, [49, 56, 94] obtained at most $50\%$ nonzeros under neutral quality. In contrast to these works, our work not only obtains $8\%$ nonzeros in activation of all FFN layers, but also a predictor, all with a single-stage training. We provide a summary of comparison to these methods in Section D.2. + +Finally, the usefulness of activation sparsity goes beyond efficiency. For example, theoretical studies show its benefits for model generalizability and learnability [65, 6]. Moreover, activated neurons may be associated with semantic concepts, which offers understanding of the working mechanism and enables manipulating the output of Transformer models [17, 60]. + +Sparse attention broadly refers to the approach of attending to a selected subset of tokens in the context as a means of reducing computation cost [21, 42, 73]. Works on sparse attention include those that use handcrafted attention patterns [15, 9, 2, 24], which feature simplicity, and learned attention patterns [45, 75, 66] which feature better modeling capacity. However, learning attention patterns often involve learning, e.g., a hash table or k-means centers, which significantly complicates modeling. Closely related to our Spark Attention is the top- $k$ attention [35], which obtains data-adaptive attention simply from top- $k$ thresholding. Our work improves upon top- $k$ attention by introducing a low cost predictor which enables an increased computational benefits from sparsity. Finally, KV pruning approaches drop selected tokens permanently as decoding proceeds [93, 57], and cannot achieve as high compression ratio as sparse attention based approaches. + +# D.2 Comparison with Related Work on Activation Sparsity in FFN + +In Table D.1, we provide a summary of recent work on enabling FFN activation sparsity in the latest LLMs, including ReLUfication [64], ProSpare [81], HiRE [89], and CATS [49]. Please see Section 5 for a discussion of these methods. + +We can see that our Spark Transformer leads to a FLOPs reduction of $-72\%$ , which is more than all the other methods. This comes at the cost of only a $-0.9\%$ quality loss, which is lower than most of the other methods except HiRE. In particular, HiRE is on par with ours in terms of quality loss but achieves less FLOPs reduction. + +Table D.1: Comparison with related work on enforcing activation sparsity in FFN of LLMs. Spark Transformer has the largest FLOPs reduction with one of the smallest quality loss. + +
Main TechniquesMain Results
Enforce sparsityPredict supportBase modelTraining cost8Sparsity (%zeros)Quality9FFN FLOPs
ReLUification10 [64]ReLUNoneOPT 1.3B+0%93%-2%-62%
ReLUNoneFalcon 7B+2%94%-2.5%-62%
Llama 7B+3%62%-1.9%-42%
ProSparse [81]ReLU + ||·||1NoneLlama2 7B+1.8%88%-1.1%-59%
Llama2 13B+6.7%88%-1.4%-59%
2-layer FFNLlama2 7BNA75%NANA
Llama2 13BNA78%NANA
HiRE [89]Group topk + commonpathLow-rank / quantizationPALM2 1B+100%80%-0.8%-60%11
CATS [49]ThresholdingNoneMistral 7B+0%50%-1.5%-33%
Llama2 7B50%-2.4%-33%
Spark TransformerStatistical-topkPartial dimensionsGemma 2B+0%92%-0.9%-72%
+ +Comparing pretraining, finetuning, and "zero-shot" approaches. Existing approaches on enforcing activation sparsity in FFNs can be categorised into three groups depending on when the sparsity is enforced during the training process. + +- Pretraining-based approaches are those whose activation sparsity is enforced from the very beginning of the model pretraining, usually with certain architectural changes upon an established model. Hence, such methods require pretraining a model from scratch. For example, ReLUification [64] is the method of switching the activation function in FFN from a commonly used one, e.g., GELU, back to ReLU. When tested on OPT 1.3B, [64] applied this modification and pretrained a version of OPT 1.3B with ReLU from scratch (Table D.1 contains a summary of relevant results for it). +- Finetuning-based approaches refer to those that take an existing pretrained model without activation sparsity, and perform additional training steps usually with certain architectural changes to induce sparsity. For example, ReLUification discussed above as a pretraining-based approach can also be used as a finetuning-based approach by switching the activation function of a pretrained model to ReLU and performing additional training steps. In [64], this approach is applied to Falcon 7B and Llama 7B, for which the results are summarized in Table D.1. Other methods falling into this category, which we have summarized in Table D.1, include ProSparse [81] and HiRE [89]. +- "Zero-shot" approaches. This refers to methods that enforces activation sparsity upon a pretrained model, without any additional training. CATS [49] is a method of this kind. + +Our Spark Transformer belongs to the category of pretraining-based method, that is, it requires pre-training a model from scratch. It may not be used directly as a finetuning-based approach, due to the drastic difference between the architecture of a standard FFN with gated activation, i.e. eq. (5), and our Spark-FFN. This appears to make Spark Transformer less favorable than finetuning-based approaches, as the latter requires usually a small fraction of the pretraining steps in the finetuning process, see Table D.1 for a summary. However, the long-term serving costs of such models usually dominate overall expenditures, in which case the additional pre-training costs are amortized by the significant efficiency benefits realized from a higher level of sparsity and FLOPs reduction during model inference. + +# D.3 Comparison of Statistical Top- $k$ with Related Top- $k$ Operators + +The variational form in eq. (15) reveals connections of Statistical-Top $k$ with other top- $k$ algorithms in the literature. Specifically, [51] defines a soft top- $k$ as + +$$ +\underset {z} {\arg \min } - \theta \cdot H (z) - \langle z, x \rangle , \text {s . t .} z ^ {\top} \mathbf {1} = k, \mathbf {0} \leq z \leq \mathbf {1}, \tag {D.1} +$$ + +where $H(z)$ is the entropy function. Another work [59] defines the SparseK operator + +$$ +\underset {z} {\arg \min } - H ^ {G} (z) - \langle z, x \rangle , \text {s . t .} z ^ {\top} \mathbf {1} = k, 0 \leq z \leq 1, \tag {D.2} +$$ + +where $H^{G}(z)$ is the generalized Gini entropy. + +Statistical- $\mathrm{Top}_k$ in the form of eq. (15), as well as eq. (D.1) and eq. (D.2), can all be interpreted as finding an output that is close to the input subject to a sparsifying regularization. Their major difference lies in the choice of the sparse regularization. That is, soft top- $k$ and SparseK uses entropy and Gini entropy, respectively, whereas Statistical- $\mathrm{Top}_k$ uses $\ell_1$ . The choice of $\ell_1$ makes Statistical- $\mathrm{Top}_k$ superior in that it has a closed form solution provided by soft-thresholding, which only requires $d$ FLOPs. In contrast, soft top- $k$ and SparseK both do not have closed form solutions and require an iterative algorithm with a FLOP count dependent on the number of iterations. In addition, there is no guarantee that soft top- $k$ and SparseK can obtain (approximately) $k$ nonzero entries as output. + +Finally, we mention that ideas similar to statistical top- $k$ have been used [79, 62] for the problem of distributed training [55]. However, we are the first to introduce, adapt, and verify its effectiveness for activation sparsity. Additional discussions are provided in Section D.4. + +# D.4 Additional Discussions on Statistical Top-k + +Novelty upon existing work. We note that ideas similar to our statistical top- $k$ have appeared in the literature. In particular, [79] introduced the idea of fitting a Gaussian distribution to the entries of an input vector and estimating a threshold from quantile functions. Then, [62] extended the approach to additional distributions. In both cases, the method is used for solving the problem of distributed training. Here, we summarize our contribution upon these works: + +- We are the first to use statistical top- $k$ for enforcing activation sparsity in Transformers. Improving Transformer efficiency via activation sparsity has become a very popular research topic (see Section 5), but may have been suffering from a lack of efficient top- $k$ algorithms for enforcing sparsity. Hence, the introduction of statistical top- $k$ may facilitate the development of this area. +- Synergizing statistical top- $k$ into Transformers is nontrivial. Since the method of statistical top- $k$ is based on fitting a statistical distribution to the activation vector, there is the need to understand the distributions of different activations in order to determine which particular activation vector is suited for the application of statistical top- $k$ and the associated choice of the distribution. In our case, we decide that statistical top- $k$ should be applied to the activation before the nonlinear function (for FFN) and before softmax (for Attention) since entries of this vector provably follow a + +Gaussian distribution at random initialization. We also verify empirically that statistical top- $k$ is still reliable even after initialization. + +- We extended statistical top- $k$ from using the hard-thresholding operator with the estimated statistical threshold to the soft-thresholding operator. This leads to a continuous optimization landscape that may have facilitated the optimization. Empirically, we found this choice to provide quality benefits for Spark Transformer. +- We provide the first theoretical justification for the correctness of statistical top- $k$ , see Theorem 3.1. +- We reveal the conceptual connection between statistical top- $k$ and several related top- $k$ operators in the literature, see Section D.3. Such connections may motivate the development of more powerful top- $k$ algorithms in the future. + +Handling cases when the activation is sharded. The training of modern large Transformer models usually requires sharding certain model weights and activations across multiple computation devices, due to physical limitations on each device's memory. In particular, if sharding is used for the activations upon which the statistical top- $k$ is applied to, i.e., the pre-GELU activation in FFN and the pre-softmax activation in attention, extra care needs to be taken so that statistical top- $k$ is applied correctly. While this has not been the case for our experiment on Gemma-2 2B, here we discuss possible solutions if this case arises, e.g., when training a larger Spark Transformer for which sharding relevant activations may become necessary. + +Assume that an activation vector of length $N$ is sharded over $m$ devices, and we are interested in finding approximately the top- $k$ entries of $N$ with the largest value. There are two ways of applying statistical top- $k$ for this purpose. + +- Global statistical top- $k$ . Here we require each device to compute the mean and variance for entries on itself, then communicate them to all other devices. In this case, each device receives $m - 1$ mean and variance values, which can be combined with mean and variance on its own to obtain global mean and global variance. Then, the rest of the steps in statistical top- $k$ can be carried out on individual devices. In this method, the output is exactly the same as if applying statistical top- $k$ without activation sharding. In terms of cost, there is extra computation coming from aggregating mean and variance from all devices, but the cost is very low as it requires only $O(k)$ FLOPs. The method also introduces a communication cost, but the cost is again small as each device only needs to send / receive $2k - 2$ floating point numbers. +- Local statistical top- $k$ . Here we simply apply statistical top- $k'$ to entries on its own device with $k' = k / m$ . The method is sub-optimal in the sense that it does not necessarily produce the same output as applying the global statistical top- $k$ . However, it has the benefit of not adding any computation and communication cost. + +In cases where $k \ll N$ , the global statistical top- $k$ above is cheap enough and hence could be a natural choice. + +# D.5 Synergy with Speculative Decoding + +Speculative decoding is a prominent technique for accelerating the inference of large autoregressive models [53, 13]. It employs a smaller, faster "draft model" to generate a block of candidate tokens, which are then verified in a single, parallel forward pass by the larger, more accurate "target model". We view Spark Transformer as a highly complementary and synergistic technology, as its architecture enhances efficiency in both roles within the speculative decoding framework. + +Spark Transformer as the target model. The primary computational bottleneck in speculative decoding is the parallel verification step, where the target model processes multiple tokens simultaneously. Spark Transformer is ideally suited to accelerate this step. + +As the target model, Spark Transformer directly reduces the latency of this bottleneck, leveraging the same wall-time speedups demonstrated in single-token decoding. While verifying multiple tokens in parallel (e.g., $k = 2$ to $k = 4$ ) necessarily activates more neurons than a single-token pass, significant sparsity is still maintained. This is for two primary reasons: + +- Small Draft Size: The number of speculative tokens is typically small, meaning the union of activated neurons remains a small fraction of the total. + +- Aggregated Sparsity: Consecutive tokens in a natural language sequence often share a substantial portion of their activated neurons. This phenomenon, termed aggregated sparsity [64], is an extension of the lazy neuron phenomenon [54] and the observed persistence of activation patterns for related inputs. + +Because the union of activated FFN neurons and attended tokens remains sparse, Spark Transformer effectively reduces the computational cost of the most expensive part of the speculative decoding loop. + +Spark Transformer as the Draft Model. An effective draft model must satisfy two criteria: it must be significantly faster than the target model, and it must be accurate (i.e., its predictions must frequently align with the target model's). High accuracy is crucial as it dictates the token acceptance rate, which is the primary driver of the overall speedup. + +Spark Transformer is a strong candidate for a high-performance draft model. Our results demonstrate that it achieves near-quality neutrality with its dense counterpart while operating at a fraction of the inference cost. Using a Spark Transformer as the draft model — either a smaller version or even the same model with a lower sparsity budget — could lead to a much higher acceptance rate than a standard distilled model of equivalent speed. This high fidelity, combined with low latency, makes it an ideal driver for maximizing the efficiency of the speculative decoding process. + +# D.6 Synergy with Quantization Techniques + +We view Spark Transformer and quantization as two orthogonal and highly synergistic optimization axes. Quantization, such as conversion to FP8 or INT8, reduces the computational cost and memory size of each individual operation and parameter. Spark Transformer, in contrast, reduces the total number of operations and memory accesses by dynamically pruning FFN activations and attention tokens. Critically, our sparse implementation also reduces the memory bandwidth bottleneck by skipping the load of masked weights from HBM. The combined benefits are therefore expected to be multiplicative, leading to significant compounded gains in inference throughput and energy efficiency. + +A key question is whether a model that is already sparsely activated is more sensitive to the precision loss from quantization. We hypothesize that, for activation quantization in particular, Spark Transformer may reduce the model's sensitivity and be less prone to accuracy degradation compared to a dense model. This is critical, as recent work has shown that standard sparsity and quantization can be in contradiction: magnitude-based sparsity preserves large-value outliers that are detrimental to quantization-aware scaling [34]. This hypothesis stems directly from the design of our statistical top- $k$ operator. + +- Dynamic Range Compression: The statistical top- $k$ operator in Spark Transformer employs soft-thresholding. This function outputs $\max(x - \theta(x, k), 0)$ , effectively subtracting the learned threshold $\theta(x, k)$ from all activated neurons. This operation inherently compresses the dynamic range of the activation tensor. A primary challenge in LLM activation quantization is the presence of large-magnitude "outlier" values, which skew the quantization scaling factor and lead to significant precision loss for the more common, smaller values [22]. By "pre-conditioning" the distribution and shrinking these outliers, our operator makes the activation tensor significantly more amenable to accurate low-bit representation. +- Zero-Point Stability: The $92\%$ of FFN neurons that are not in the top-k are set to 0. This large volume of true zeros is perfectly and losslessly represented in any quantization format, avoiding the "near-zero" noise that can plague dense models. + +For weight quantization, we hypothesize that the sensitivity would be broadly similar to the dense Gemma-2 counterpart, as Spark Transformer reallocates and reuses the full set of parameters rather than pruning them. + +While a full empirical study of this interaction is a highly promising direction for future work, we hypothesize that Spark Transformer is not only compatible with quantization but may actively facilitate it. This positions our work alongside other recent efforts to create novel sparse-quantized representations [23], offering a robust path to further acceleration." + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction clearly state the Spark Transformer's ability to achieve high activation sparsity in FFN and attention, maintain competitive performance, preserve parameter count, and use standard training, all supported by results. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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+oid sha256:fcc4562e6ea9cc9c208d62bf84664f003e31fd78bbad1c1df3d681755f63c2dc +size 14414637 diff --git a/NeurIPS/2025/Sparse Autoencoders Learn Monosemantic Features in Vision-Language Models/full.md b/NeurIPS/2025/Sparse Autoencoders Learn Monosemantic Features in Vision-Language Models/full.md new file mode 100644 index 0000000000000000000000000000000000000000..bba7f7d8391efc533562899b5d6958dc614aa83b --- /dev/null +++ b/NeurIPS/2025/Sparse Autoencoders Learn Monosemantic Features in Vision-Language Models/full.md @@ -0,0 +1,885 @@ +# Sparse Autoencoders Learn Monosemantic Features in Vision-Language Models + +Mateusz Pach1,2,3,4 Shyamgopal Karthik1,2,3,4,5 Quentin Bouniot1,2,3,4 +Serge Belongie6 Zeynep Akata1,2,3,4 + +$^{1}$ Technical University of Munich $^{2}$ Helmholtz Munich $^{3}$ Munich Center for Machine Learning $^{4}$ Munich Data Science Institute $^{5}$ University of Tübingen $^{6}$ University of Copenhagen mateusz.pach@tum.de + +# Abstract + +Sparse Autoencoders (SAEs) have recently gained attention as a means to improve the interpretability and steerability of Large Language Models (LLMs), both of which are essential for AI safety. In this work, we extend the application of SAEs to Vision-Language Models (VLMs), such as CLIP, and introduce a comprehensive framework for evaluating monoseismicity at the neuron-level in visual representations. To ensure that our evaluation aligns with human perception, we propose a benchmark derived from a large-scale user study. Our experimental results reveal that SAEs trained on VLMs significantly enhance the monoseismicity of individual neurons, with sparsity and wide latents being the most influential factors. Further, we demonstrate that applying SAE interventions on CLIP's vision encoder directly steers multimodal LLM outputs (e.g., LLaVA), without any modifications to the underlying language model. These findings emphasize the practicality and efficacy of SAEs as an unsupervised tool for enhancing both interpretability and control of VLMs. Code and benchmark data are available at https://github.com/ExplainableML/sae-for-vlm. + +# 1 Introduction + +In recent years, Vision-Language Models (VLMs) like CLIP [47] and SigLIP [61] have gained widespread adoption, owing to their capacity for simultaneous reasoning over visual and textual modalities. They have found a surge of applications in various modalities, such as in audio [14, 57] and medicine [62], transferring to new tasks with minimal supervision. Yet our current understanding of VLM internals remains limited, necessitating methods that can systematically probe their representations. Sparse AutoEncoders (SAEs) [37] are an effective approach to probing the internal representations of such models. They efficiently discover concepts (abstract features shared between data points) through their simple architecture learned as a post-hoc reconstruction task. Although analysis with SAEs is popular for Large Language Models (LLMs) [7, 23, 48], for VLMs it has been limited to interpretable classification [35, 49], or the discovery of concepts shared across models [53]. + +Intuitively, SAEs reconstruct activations via a higher-dimensional space to disentangle distinct concepts from their overlapping representations in neural activations [7]. Neurons at different layers within deep neural networks are known to be naturally polysemantic [41], meaning that they can be strongly activated for multiple unrelated concepts such as cellphones and rulers. One common explanation for this behavior is the superposition hypothesis [3, 17], stating that concepts are encoded as linear combination of neurons. SAEs explicitly attempt to solve this issue by separating the entangled concepts into distinct representations. Despite their widespread use in research, the absence of a metric to evaluate SAEs at the neuron-level still hinders their practicality as an interpretation tool. + +![](images/b68c42b213216eac44dae5923ce38146c1f11a6bc3b624171403b4a0316d1e9c.jpg) +Figure 1: Sparse Autoencoder (SAE) in VLM (e.g. CLIP): Top activating images of a neuron in a pretrained VLM layer are polysemantic (left), and those of a neuron in a sparse latent of SAE trained to reconstruct the same layer are monoseismic (right), according to MonoSemanticity score (MS). + +![](images/b00c1df7ecb801f6c7e0d2dc1a0c20d7a595aac95375660c3e8c4fb16f9d7aaf.jpg) + +Discovering neurons encoding human-interpretable concepts requires analyzing them individually, which is even more tedious as layers of both the model and SAE are wide. + +In this work, we quantitatively evaluate SAEs for VLMs through monoseismicity, defined as the similarity between inputs that strongly activate a neuron. We propose the MonoSemanticity score (MS) for vision tasks, that measures the pairwise similarity of images weighted by the activations for a given neuron. Unlike natural language where individual words require surrounding context (such as complete sentences) to clearly identify their meanings, individual images can directly activate neurons without additional context. We define highly activating images as images that strongly fire a particular neuron. Greater similarity among these images suggests that the neuron is more narrowly focused on a single concept, reflecting higher neuron monoseismicity. Using our MS score, we observe and validate that neurons in SAE are significantly more monoseismic (see Figure 1, right) than the original neurons (see Figure 1, left). The neuron of the original VLM typically has a low MS score, it fires for a wide range of objects, from cellphones to rulers. On the other hand, neurons within the SAE are more focused on a single concept, e.g. parrots, obtaining a higher score. This holds even for SAE with the same width as the original layer, implying that the sparse reconstruction objective inherently improves the separability of concepts. We further conduct a large-scale study to quantitatively assess alignment of our proposed MS score with human interpretation of monoseismicity. The results confirm that the difference between scores of two neurons strongly correlates with humans perceiving the higher-scoring neuron as more focused on a single concept. + +Finally, we illustrate applicability of the monoseismicity of vision SAEs by transferring a CLIP-based SAE onto Multimodal LLMs (MLLMs), e.g. LLaVA [36]. Intervening on a single monoseismic SAE neuron in the vision encoder while keeping the LLM untouched allows steering the overall MLLM generated output to either insert or suppress the concept encoded in the selected SAE neuron. We summarize our contributions as follows: + +- We propose the MonoSemanticity score (MS) for SAEs in vision tasks, that computes neuron-wise activation-weighted pairwise similarity of image embeddings. To validate our MS score against human judgment, we conduct a large-scale user study, the results of which can also serve as a benchmark for future research. +- We quantitatively compare MS between SAEs, and across their neurons. We find that Matryoshka SAE [9, 39] achieves overall superior MS, and that wider and sparser latents lead to better scores. +- We leverage the well-separability of concepts in SAE layers to intervene on neuron activations and steer outputs of MLLMs to insert or suppress any discovered concept. + +# 2 Related Work + +Sparse Autoencoders. Recent studies have repurposed traditional dictionary learning to enhance LLM and VLM interpretability [6, 46]. Specifically, there has been success in interpreting and steering LLMs with features learned by SAEs [16, 52]. Several enhancements to SAE mechanisms have been introduced, including new activation functions such as Batch TopK [8] or JumpReLU [48], + +![](images/adb24886c0d4a6190598090b573e6d768f2ee2014e3094034c763e5f18dc18e6.jpg) +(A) Extracting embeddings and activations +(B) Pairwise similarity matrix +(C) Activation-weighted average +Figure 2: Computation of our MonoSemanticity score (MS). (A) Embeddings and activations are extracted for a set of images (B) to compute the pairwise embedding similarities and pairwise neuron activations. (C) MS is the average of embedding similarities weighted by the neuron activations. + +and insights from Matryoshka representation learning [34] for SAEs [9, 39]. We analyze and evaluate neuron-level monoseismicity of SAEs in VLMs and their downstream uses. + +Vision-Language Models. Since Contrastive Language-Image Pretraining (CLIP) [38, 47, 61], many models have emerged that align images and text in a shared embedding space [32, 56] or generate text conditioned on image inputs [12, 19, 36]. They have achieved strong results on benchmarks [59] and found many use-cases [5, 24, 26, 28]. As trust in these models has become a concern [4, 27, 60], understanding decision-making and ensuring safety, for example through steering, is increasingly important [2, 29]. Consequently, prior work has examined their internal representations [21, 22, 40, 45], uncovering interpretable neurons [15]. We demonstrate that SAEs enable more effective interpretation and control than directly operating on raw features. + +SAEs for VLMs. Building on the success of SAEs in interpreting LLMs, researchers have tried applying them to vision and vision-language models, typically on CLIP [20, 35, 49, 50] and other vision encoders (e.g. DINOv2 [44]). There has also been interest in interpreting the denoising diffusion models [25] using SAEs [11, 30, 31, 51], discovering common concepts across different vision encoders [53], and applying them on multimodal LLMs [63]. Concurrently, monoseismicity of multimodal models has been investigated [58], although focusing on inter-modality differences. Our MS score provides a grounded, quantitative measure of the monoseismicity of individual neurons which is empirically aligned with human perception. Overall, we create a rigorous evaluation framework for SAEs in VLMs, as well as demonstrating their use for steering multimodal LLMs. + +# 3 Sparse Autoencoders for VLMs + +# 3.1 Background and Formulation of SAEs + +SAEs implement a form of sparse dictionary learning, where the goal is to learn a sparse decomposition of a signal into an overcomplete dictionary of atoms [42]. More specifically, an SAE consists of linear layers $\mathbf{W}_{\mathrm{enc}} \in \mathbb{R}^{d \times \omega}$ and $\mathbf{W}_{\mathrm{dec}} \in \mathbb{R}^{\omega \times d}$ as encoder and decoder, with a non-linear activation function $\sigma: \mathbb{R}^{\omega} \to \mathbb{R}^{\omega}$ . Both layers share a bias term $\mathbf{b} \in \mathbb{R}^d$ subtracted from the encoder's input and later added to the decoder's output. The width $\omega$ of the latent SAE layer is chosen as a factor of the original dimension, such that $\omega := d \times \varepsilon$ , where $\varepsilon$ is called the expansion factor. + +In general, SAEs are applied on embeddings $\mathbf{v} \in \mathbb{R}^d$ of a given layer $l$ of the model to explain $f: \mathbb{X} \to \mathbb{Y}$ , such that $f_l: \mathbb{X} \to \mathbb{R}^d$ represents the composition of the first $l$ layers and $\mathbb{X} \subset \mathbb{R}^{d_i}$ is the space of input images. A given input image $\mathbf{x} \in \mathbb{X}$ is first transformed into the corresponding embedding vector $\mathbf{v} := f_l(\mathbf{x})$ , before being decomposed by the SAE into a vector of activations $\phi(\mathbf{v}) \in \mathbb{R}^\omega$ , and its reconstruction vector $\hat{\mathbf{v}} \in \mathbb{R}^d$ is obtained by: + +$$ +\phi (\mathbf {v}) := \sigma \left(\mathbf {W} _ {\mathrm {e n c}} ^ {\top} (\mathbf {v} - \mathbf {b})\right), \quad \psi (\mathbf {v}): = \mathbf {W} _ {\mathrm {d e c}} ^ {\top} \mathbf {v} + \mathbf {b}, \quad \hat {\mathbf {v}} := \psi (\phi (\mathbf {v})). \tag {1} +$$ + +The linear layers $\mathbf{W}_{\mathrm{enc}}$ and $\mathbf{W}_{\mathrm{dec}}$ composing the SAE are learned through a reconstruction objective $\mathcal{R}$ and sparsity regularization $\mathcal{S}$ , to minimize the following loss: + +$$ +\mathcal {L} (\mathbf {v}) := \mathcal {R} (\mathbf {v}) + \lambda \mathcal {S} (\mathbf {v}), \tag {2} +$$ + +where $\lambda$ is a hyperparameter governing the overall sparsity of the decomposition. The most simple instantiation [7, 49] uses a ReLU activation, an $L^2$ reconstruction objective and an $L^1$ sparsity penalty, such that + +$$ +\sigma (\cdot) := \operatorname {R e L U} (\cdot), \quad \mathcal {R} (\mathbf {v}) := \| \mathbf {v} - \hat {\mathbf {v}} \| _ {2} ^ {2}, \quad S (\mathbf {v}) := \| \phi (\mathbf {v}) \| _ {1}. \tag {3} +$$ + +The (Batch) TopK SAEs [8, 23, 37] use a TopK activation function governing the sparsity directly through $K$ . Finally, Matryoshka SAEs [9, 39] group neuron activations $\phi^i(\mathbf{v})$ into different levels of sizes $\mathcal{M}$ , to obtain a nested dictionary trained with multiple reconstruction objectives: + +$$ +\mathcal {R} (\mathbf {v}) := \sum_ {m \in \mathcal {M}} \| \mathbf {v} - \mathbf {W} _ {\mathrm {d e c}} ^ {\top} \phi^ {1: m} (\mathbf {v}) \| _ {2}, \tag {4} +$$ + +where $\phi^{1:m}$ corresponds to keeping only the first $m$ neuron activations, and setting the others to zero. It is important to note that Matryoshka SAEs can be combined with any SAE variant, e.g. with BatchTopK [9] or ReLU [39], as only the reconstruction objective is modified. + +# 3.2 Monosemanticity Score + +A neuron's interpretability increases as its representation becomes disentangled into a single, clear concept. Therefore, quantifying the monoseismicity of individual neurons helps identify the most interpretable ones, while aggregating these scores across an entire layer allows assessing the overall semantic clarity and quality of the representations learned by the SAE. We propose measuring monoseismicity by computing pairwise similarities between images that strongly activate a given neuron, where high similarity indicates these images likely represent the same concept. These similarities can be efficiently approximated using deep embeddings from a pretrained image encoder $E$ . Since selecting a fixed number of top-activating images is challenging due to varying levels of specialization across neurons, we instead evaluate monoseismicity over a large, diverse set of unseen images, weighting each image by its activation strength for the neuron. + +We formally describe our proposed MonoSemanticity score (MS) below, with an illustration given in Figure 2. This metric can be computed for each of the $\omega$ neurons extracted from the SAE. Given a diverse set of images $\mathcal{I} = \{\mathbf{x}_n\in \mathbb{X}\}_{n = 1}^N$ , and a pretrained image encoder $E$ , we first extract embeddings to obtain a pairwise similarity matrix $\mathbf{S} = [s_{nm}]_{n,m}\in [-1,1]^{N\times N}$ , which captures semantic similarity between each pair of images. The similarity $s_{nm}$ of the pair $(\mathbf{x}_n,\mathbf{x}_m)$ is computed as the cosine similarity between the corresponding pair of embedding vectors: + +$$ +s _ {n m} := \frac {E (\mathbf {x} _ {n}) \cdot E (\mathbf {x} _ {m})}{| E (\mathbf {x} _ {n}) | | E (\mathbf {x} _ {m}) |}. \tag {5} +$$ + +We then collect activation vectors $\{\mathbf{a}^k = [a_n^k ]_n\in \mathbb{R}^N\}_{k = 1}^\omega$ across all $\omega$ neurons, for all images in the dataset $\mathcal{I}$ . Specifically, for each image $\mathbf{x}_n$ , the activation of the $k$ -th neuron: + +$$ +\mathbf {v} _ {n} := f _ {l} \left(\mathbf {x} _ {n}\right), \quad a _ {n} ^ {k} := \phi^ {k} \left(\mathbf {v} _ {n}\right), \tag {6} +$$ + +where $l$ represents the layer at which the SAE is applied, $f_{l}$ is the composition of the first $l$ layers of the explained model, and $\phi^k$ is the $k$ -th neuron of $\phi (\mathbf{v}_n)$ (or of $\mathbf{v}_n$ when evaluating neurons of the original layer $l$ of $f$ ). To ensure a consistent activation scale, we apply min-max normalization to each $\mathbf{a}^k$ , yielding $\tilde{\mathbf{a}}^k \coloneqq [\tilde{a}_n^k]_n \in [0,1]^N$ , where + +$$ +\tilde {a} _ {n} ^ {k} = \frac {a _ {n} ^ {k} - \min _ {n ^ {\prime}} a _ {n ^ {\prime}} ^ {k}}{\max _ {n ^ {\prime}} a _ {n ^ {\prime}} ^ {k} - \min _ {n ^ {\prime}} a _ {n ^ {\prime}} ^ {k}}. \tag {7} +$$ + +Using these normalized activations, we compute a relevance matrix $\mathbf{R}^k = [r_{nm}^k]_{n,m} \in [0,1]^{N \times N}$ for each one of the $\omega$ neurons, which quantifies the shared neuron activation of each image pair: + +$$ +r _ {n m} ^ {k} := \tilde {a} _ {n} ^ {k} \tilde {a} _ {m} ^ {k}. \tag {8} +$$ + +![](images/fa01ebe38a07b84455f3e08524eea287993e261c777c73f24db8f145837cf325.jpg) +Figure 3: Top activating images of neurons with MonoSemanticsity (MS) scores ranging from high (left) to low (right). Higher scores correlate with more similar images, reflecting monosemanticsity. + +Finally, our proposed score $\mathbf{MS}^k\in [-1,1]$ for the $k$ -th neuron is computed as the average pairwise similarity weighted by the relevance, without considering same image pairs $(\mathbf{x}_n,\mathbf{x}_n)$ : + +$$ +\mathbf {M S} ^ {k} := \frac {1}{N (N - 1)} \sum_ {n = 1} ^ {N} \sum_ {\substack {m = 1 \\ m \neq n}} ^ {N} r _ {n m} ^ {k} s _ {n m}. \tag{9} +$$ + +# 3.3 Steering MLLMs with Vision SAEs + +Finding monosemantic neurons is not only useful for interpretability. The SAEs let us induce controllable semantic biases in the response of MLLMs without modifying the underlying model parameters, and without touching any textual part. In other words, we can steer the model into seeing or not targeted concepts in the image. Our MS score becomes a strong tool to help select the most monosemantic neurons for a precise and efficient steering. + +We first describe LLaVA [36] as an example MLLM architecture. The LLaVA model $g: \mathbb{X} \times \mathbb{T} \to \mathbb{T}$ expects a pair of image and text $(\mathbf{x}, \mathbf{t})$ and outputs a text answer $\mathbf{o}$ , where $\mathbb{T} \subset \mathbb{R}^{d_t}$ is the word embedding space. Internally, it converts the image $\mathbf{x}$ into $t_{\mathbf{x}} \in \mathbb{N}$ token embeddings $\{\mathbf{v}_i\}_{i=1}^{t_{\mathbf{x}}}$ obtained from vision encoder $f_l: \mathbb{X} \to \mathbb{R}^{d \times t_{\mathbf{x}}}$ composed of the first $l$ layers of CLIP [47]. These embeddings are then projected into visual tokens $\mathbf{H}_{\mathbf{x}} \in \mathbb{R}^{d_t \times t_{\mathbf{x}}}$ in the word embedding space, and are finally fed along with tokenized text $\mathbf{H}_{\mathbf{t}} \in \mathbb{R}^{d_t \times t_{\mathbf{t}}}$ into the pretrained LLM (e.g. LLaMA [54] or Vicuna [10]) to obtain the output text $\mathbf{o}$ . + +We modify this architecture by injecting a pretrained SAE $(\phi, \psi)$ of width $\omega$ at the token-level after the vision encoder $f_{l}$ . For all token embeddings $\mathbf{v}_{i} \in \mathbb{R}^{d}, i \in \{1, \dots, t_{\mathbf{x}}\}$ , we first extract the SAE decomposition into activation $\mathbf{a}_{i} := \phi(\mathbf{v}_{i}) \in \mathbb{R}^{\omega}$ across all neurons. After identifying the neuron $k \in \{1, \dots, \omega\}$ representing the targeted concept, to steer the overall model $g$ towards this concept, we manipulate the SAE activations of all token embeddings for the neuron $k$ to obtain $\{\hat{\mathbf{a}}_i \in \mathbb{R}^\omega\}_{i=1}^{t_\mathbf{x}}$ : + +$$ +\forall j \in \{1, \dots , \omega \}, \quad \hat {a} _ {i} ^ {j} = \left\{ \begin{array}{l l} \alpha , & j = k \\ a _ {i} ^ {j}, & j \neq k \end{array} \right. \tag {10} +$$ + +where $\alpha \in \mathbb{R}$ is the intervention value we want to apply to the activation of neuron $k$ . Finally, we decode the manipulated activation vectors for each token $\hat{\mathbf{a}}_i$ back into a manipulated token embedding $\hat{\mathbf{v}}_i = \psi (\hat{\mathbf{a}}_i)\in \mathbb{R}^d$ with the SAE decoder. Token embeddings are then processed as usual to generate the steered LLaVA's response. We include an illustration of the overall process in the Appendix. + +# 4 Experiments + +# 4.1 Experimental Settings + +We apply SAEs to explain fixed and pretrained CLIP ViT-L/14-336px [47], SigLIP SoViT-400m/14-384px [61], AIMv2 L/14-224px [19], and WebSSL MAE-300m/14-224px [18]. The SAEs are trained on activation vectors pre-extracted from the model's responses to ImageNet [13] images. For CLIP, activation vectors are extracted from the classification (CLS) tokens in the residual stream after layers $l \in \{11, 17, 22, 23\}$ , or from the output of the final projection layer. For steering experiments, however, the SAEs are trained on activation vectors corresponding to two random token embeddings per image, taken from layer $l = 22$ . For other encoders, we similarly use the CLS tokens from the final layers, or two random token embeddings if a CLS token is not available. + +In the following sections, we are interested in both BatchTopK [8] and Matryoshka BatchTopK SAEs [9] variants. If not stated otherwise, we set the groups of Matryoshka SAEs as $\mathcal{M} = \{0.0625\omega, 0.1875\omega, 0.4375\omega, \omega\}$ , which roughly corresponds to doubling the size of the number of neurons added with each level down. For the BatchTopK activation, we fix the maximum number of non-zero latent neurons to $K = 20$ . Both SAE types are compared across a wide range of expansion factors $\varepsilon \in \{1, 2, 4, 8, 16, 64\}$ . All SAEs are optimized for $10^{5}$ steps with minibatches of size 4096 using Adam optimizer [33], with the learning rate initialized at $\frac{16}{125\sqrt{\omega}}$ following previous work [23]. To measure SAE performance, we use $R^2$ for reconstruction quality and the $L^0$ norm for the activation sparsity of $\phi(\mathbf{v})$ . Throughout the paper, we quantify MS of neurons using DINOv2 ViT-B [44] as image encoder $E$ , and present more analysis with different encoders in Appendix. Experiments are run on a single NVIDIA A100 GPU. + +# 4.2 Evaluating Interpretability of VLM Neurons + +# 4.2.1 Alignment of MS with human perception + +We first illustrate the correlation between MS score and the underlying monoseismicity of neurons, with examples in Figure 3 of the 16 highest activating images for neurons with decreasing MS from left to right. We observe that the highest scoring neurons (with $\mathrm{MS} = 0.9$ , on the far left) are firing for images representing the same object, i.e. close-up pictures of a koala (on the top) and a hockey (on the bottom). As the score decreases, the corresponding neurons fire for less similar or even completely different objects or scenes. To verify this observation in a more quantitative way, we conducted a large scale user study on the Mechanical Turk platform. This + +![](images/a3c2e2b39362b3b5f331b0c4adf93c1a0751f0da6a79d783c79dd0072741a787.jpg) +Figure 4: Alignment Rate (AR, %) of humans with MS score when judging which neuron in a pair is more monosemantic, grouped by MS difference between the neurons. Bars show AR per interval; dots show cumulative AR up to that interval. + +study resulted in a total of 1000 questions across 71 unique users, with 3 answers per question aggregated through majority voting. Results of this study are presented in Figure 4, and we provide more details on the setup in Appendix. When asked to select the set of more monosemantic images from a pair of sets $(a,b)$ , the users answered in accordance to the MS in $79.0\%$ of the cases, assuming $\delta = |\mathrm{MS}(a) - \mathrm{MS}(b)| \sim \mathcal{U}(0,0.9)$ . This alignment rate monotonically raises from $55.7\%$ for $\delta \in (0.0,0.1)$ to $93.5\%$ for $\delta \in (0.8,0.9)$ , highlighting that users especially agree with MS as the difference in monosemanticity between the two sets becomes more pronounced. This demonstrates that MS can be used as a reliable measure aligning well with human perception of similarity. Detailed results of the user study are released for future benchmark of image similarity and monosemanticity. + +(a) Highest MS scores of neurons in various CLIP ViT-Large [47] layers. + +Table 1: Sparse Autoencoders (SAEs) decompose "No SAE" neurons into more monosemantic units as shown by MonoSemanticity (MS) score. Higher SAE expansion factors yield higher MS scores. + +
SAE typeLayerNo SAEExpansion factor
×1×2×4×8×16×64
BatchTopK [8]110.010.610.730.710.870.901.00
170.010.650.790.860.860.931.00
220.010.660.790.800.880.921.00
230.010.730.720.830.890.931.00
last0.010.570.780.780.810.851.00
Matryoshka [9, 39]110.010.840.900.951.000.891.00
170.010.860.840.930.940.961.00
220.010.830.830.870.941.001.00
230.010.820.840.890.930.961.00
last0.010.820.910.890.930.911.00
+ +(b) Top MS scores of neurons from last layers of different vision encoders. Improvements in MS score from applying Matryoshka SAEs are consistent across all the models. + +
Vision EncoderNo SAEExp. factor ×1×4
WebSSL [18]0.010.790.92
CLIP [47]0.010.820.89
SigLIP [61]0.010.830.88
AIMv2 [19]0.010.590.85
+ +![](images/5556463f6672f0d0a73498ab6b0dd8348b786d1e747020d42fc794a0482c6c2d.jpg) +(a) Impact of expansion factor $\varepsilon$ + +![](images/952959e9338cf41ac637f7b236aa7e3ea321ed89f736ffb06d7ab92235e41693.jpg) +(b) Impact of sparsity factor $K$ +Figure 5: MonoSemanticsity scores in decreasing order across neurons, normalized by width. Results are shown for the last layer of the model, without SAE ("No SAE", in black dashed line), and with SAE in straight lines using either (a) different expansion factors $(\varepsilon = 1, \varepsilon = 2, \varepsilon = 4, \varepsilon = 16, \varepsilon = 64)$ or (b) different sparsity levels $(K = 1, K = 10, K = 20, \text{and} K = 50)$ . + +# 4.2.2 Monosemanticity of SAEs + +In Table 1a, we report MS of the highest scoring neurons of two SAE types (BatchTopK [8] and Matryoshka BatchTopK [9]) trained at different layers with various expansion factors $\varepsilon$ . We also include results for original neurons of the corresponding layer decomposed by SAEs ("No SAE"). We observe that SAEs' neurons consistently have significantly higher MS for their best neuron when compared to original ones, implying that SAEs are better separating and disentangling concepts between their neurons. Interestingly, while the highest MS score is increasing with higher expansion factor $\varepsilon$ , i.e. with increased width $\omega$ of the SAE layer, this holds true already for expansion factor $\varepsilon = 1$ , meaning that the disentanglement of concepts is also linked to the sparse dictionary learning and not only to the increased dimensionality. Finally, comparing SAE variants, we observe that while the Matryoshka reconstruction objective improves the concept separation at same expansion factor, it also achieves about 2 or 3 points lower $R^2$ for the same expansion factors (more details in Appendix). + +Table 1b presents the highest observed MS scores among neurons in the last layers of various image encoders, both before ("No SAE") and after attaching Matryoshka SAEs. We find that the SAE latent neurons outperform the original neurons in every case. As before, increasing the expansion factor $\varepsilon$ helps discover more monosemantic units. This suggests the universality of the SAE approach across vision representations derived from different training objectives. + +To analyze monosemanticity across all neurons, we plot in Figure 5a the scores of both the original neurons and the Matryoshka SAE neurons from the last layer of model $f$ . We ordered neurons by decreasing scores and normalized neuron indices to the [0, 1] interval to better compare SAEs with + +Table 2: Percentage of generations meeting evaluation criteria for concept insertion and suppression. SAE-derived steering directions yield higher success rates than Difference-in-Means (DiffMean) [2]. + +
(a) Concept insertion
OursDiffMean
Desired concept appeared48.753.1
Base prompt followed85.866.2
Both criteria satisfied42.435.8
+ +
(b) Concept suppression
OursDiffMean
Desired concept removed64.464.0
Unrelated concept kept81.438.7
Both criteria satisfied52.533.3
+ +![](images/a1d8e40d41af28256b253941c1d4464989f60c2b029d59228001f73016953cb4.jpg) +Figure 6: Steered LLaVA outputs after clamping the activation values of a chosen neuron, i.e. Neuron #39 of SAE in CLIP corresponding to pencil. Initially, the poem follows the given instruction (the prompt and white image), but as the intervention weight $\alpha$ increases, it becomes increasingly influenced by the neuron's concept, first mentioning the pencil's attributes, then the pencil itself. This shows that our interventions enable new capabilities for the unsupervised steering of these models. + +different widths. These results confirm our analysis above, and demonstrate that the vast majority of neurons within SAEs have improved MS compared to the original neurons. Even when comparing with $\varepsilon = 1$ , i.e. with same width between the SAE and original layers, we can see that about $90\%$ of the neurons within the SAE have better scores than the original neurons, proving once again that the sparse decomposition objective inherently induces a better separation of concepts between neurons. Furthermore, MS scores increase overall with the expansion factor until a certain point $(\varepsilon = 4)$ , after which they decrease overall, reaching even lower values than $\varepsilon = 1$ . Although the relative fraction of neurons at different values of MS is decreasing for very wide latents, the absolute number is still increasing. We refer the reader to Appendix for MS with raw (unnormized) neuron indices. + +The relationship between the sparsity level $K$ used when training Matryoshka SAEs and the scores of the learned neurons is illustrated in Figure 5b. We observe that a stricter sparsity constraint decomposes the representation into more monosemantic features overall. However, this does not imply that the highest sparsity ( $K = 1$ ) is always the best choice, as improvements in MS come at the cost of reduced reconstruction quality. In the same setup, the $R^2$ varies from $31.3\%$ at the lowest $K = 1$ to $74.9\%$ at the highest $K = 50$ . To balance interpretability and reconstruction quality, we set $K = 20$ for which the $R^2$ remains at a reasonable $66.8\%$ . Detailed results are in the Appendix. + +# 4.3 Steering Multimodal LLMs + +We train the Matryoshka BatchTopK SAE [9] with expansion factor $\varepsilon = 64$ on random token embeddings from layer $l = 22$ of the CLIP vision encoder obtained for the ImageNet training data. The trained SAE is plugged after the vision encoder of LLaVA-1.5-7b [36] (uses Vicuna [10] LLM). + +Quantitative Results. We first compare the performance of SAE-based steering for VLMs against Difference-in-Means (DiffMean) [2], a popular approach based on activation steering. For each of + +![](images/00bf61af4ee2c3028cb58051a4413dbb582c593ec241922099df2267cb39281a.jpg) +Figure 7: Overriding neuron activations with negative values allows concept suppressing. Originally, LLaVA describes the images by mentioning all the objects visible on the input images. However, as we decrease the value $\alpha$ of neurons associated with knives and laptops, it first confuses them with wooden utensils and computer monitor, then eventually ignores them completely. At the same time, it continues to faithfully describe other objects like wooden board, strawberries, and pencil holder. + +100 SAE neurons in LLaVA, we identify its top-activating image grid. To perform concept insertion, we boost the neuron's activation and prompt LLaVA with 10 diverse text queries, such as "Propose a math word problem," or "Invent a new holiday," then evaluate if the output contains the concept and still responds to the prompt. For concept suppression, we apply a negative intervention, ask LLaVA to describe the images, and check whether the concept is removed and unrelated images are still described correctly. The evaluation is done with a LLM-as-a-judge setup using GPT-4.1-mini [43]. We provide the full details of the prompts and the evaluation procedure in Appendix B. The results in Table 2 demonstrate that SAE-based directions outperform DiffMean in both concept insertion and suppression. For insertion, SAE effectively introduces the intended concept (48.7% vs. 53.1%) while maintaining much stronger adherence to the base prompt (85.5% vs. 66.2%). For suppression, it performs on par with DiffMean in removing the target concept (64.4% vs. 64.0%), yet far surpasses it in preserving unrelated content (81.4% vs. 38.7%). + +To further confirm the steering capabilities of SAE neurons, we also perform a sanity check on the steering capabilities of the SAE neurons. We directly compare CLIP similarity scores between the top-16 images activating a given neuron and the corresponding text outputs, both before and after steering. We prompt the model with "What is shown on the image? Use exactly one word" and compare its original answer to the one generated after fixing the activation of a specific SAE neuron to $\alpha = 100$ , varying one neuron at a time. In the first setup, a white image is used while intervening on the first 1000 neurons to isolate the neuron manipulation's effect. In the second, 1000 + +Table 3: Mean similarity of neurons' activating images to output word, with and without steering, on white or random ImageNet images. Upper bound with correct image-classname pairs is 0.283, lower bound with random pairs is 0.185. + +
SteeringWhite ImageImageNet
✓0.259 ± 0.0360.263 ± 0.037
✗0.212 ± 0.0210.211 ± 0.028
+ +random ImageNet images are used while steering only 10 neurons to test effects on natural inputs. The results in Table 3 clearly illustrates that steering increases image-text similarity scores. For context, we compute reference similarities in CLIP space: the average similarity between each ImageNet class image and its class name sets an upper bound $(0.283 \pm 0.034)$ , while random image-class pairs set a lower bound $(0.185 \pm 0.028)$ . Neuron steering yields a relative gain of $22\%$ within this range, highlighting the significance of the results. + +Qualitative Examples. Figure 6 illustrates the effectiveness of concept insertion by manipulating a single neuron of the SAE. We prompt the model with the instruction "Write me a short love poem", along with a white image. By intervening on an SAE neuron associated to the pencil concept and increasing the corresponding activation value, we observe the impact on the generated output text. + +While the initial output mentions the "white" color and focuses on the textual instruction, i.e. "love" poem, the output becomes more and more focused on pencil attributes as we manually increase the intervention value $\alpha$ (most highly activating images for the selected neuron is in Appendix) until it only mentions pencils. We provide more examples with a different input prompt ("Generate a scientific article title") in Appendix, for which steered LLaVA exhibits a similar behavior. + +In Figure 7, we show that, by clamping a specific neuron to negative value, we can suppress a concept. LLaVA is asked to answer what is shown on the images given a photo of a cutting board with knives and strawberries and a photo of laptop and pencil holder full of pencils. By default the model generates correct descriptions containing all the objects. However, when we intervene by decreasing the activation value $\alpha$ of the neurons associated with knife and laptop, the resulting descriptions progressively omit these concepts. This provides a promising strategy for filtering out harmful or undesired content at an early stage, before it even reaches the language model. + +# 5 Conclusion + +We introduced the MonoSemanticsity score (MS), a quantitative metric for evaluating monosemanticsity at the neuron level in SAEs trained on VLMs. Our analysis revealed that SAEs primarily increased monosemanticsity through sparsity and wider latents and highlighted the superior performance of Matryoshka SAEs. We further verified the alignment of MS with human perception through a large-scale human study. Leveraging the clear separation of concepts encoded in SAEs, we explored their effectiveness for unsupervised, concept-based steering of multimodal LLMs, highlighting a promising direction for future research. Potential extensions of this work include adapting our metric to text representations and investigating the interplay between specialized (low-level) and broad (high-level) concepts within learned representations. + +Limitations. We focused our evaluation on various SAE architectures, the most common dictionary learning implementations that scale effectively to large VLMs. However, our MS metric is model-agnostic and could be applied to study other comparable approaches as well. High MS score neurons do not always produce precise effects when used to steer MLLM outputs. For example, a golden retriever neuron from an SAE trained on ImageNet can trigger any dog-related output. This could happen because, while SAEs can disentangle detailed classes in the dataset, MLLMs may have limited fine-grained understanding and may not be perfectly aligned with the vision encoder. Moreover, a fraction of the SAE neurons that act as feature detectors do not exhibit any clear steering effect [1]. + +# Acknowledgments and Disclosure of Funding + +This work was partially funded by the ERC (853489 - DEXIM) and the Alfried Krupp von Bohlen und Halbach Foundation, which we thank for their generous support. We are also grateful for partial support from the Pioneer Centre for AI, DNRF grant number P1. Shyamgopal Karthik thanks the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for support. Mateusz Pach would like to thank the European Laboratory for Learning and Intelligent Systems (ELLIS) PhD program for support. + +# References + +[1] Dana Arad, Aaron Mueller, and Yonatan Belinkov. 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Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We present our Monosemanticity score in Section 3.2, the user study validating alignment with human perception in Section 4.2.1, detailed analysis of MS in Section 4.2.2, and results of steering MLLMs in Section 4.3. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We discuss limitations in Section 5. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +# Answer: [NA] + +Justification: The paper does not include theoretical results. + +# Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +# Answer: [Yes] + +Justification: We describe all experimental settings in Section 4.1. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. +- Depending on the contribution, reproducibility can be accomplished in various ways. 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Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [Yes] + +Justification: Our institution does not require IRB approval for these tasks. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [Yes] + +Justification: We evaluate steering on MLLMs, which integrate LLMs as a component. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Contents + +A Broader Impact 22 +B More details on steering 22 +C User study 24 +D Benchmark 29 +E Additional results on monoseismicity 29 + +E.1 Unnormalized plots 29 +E.2 Detailed statistics and more models 30 +E.3 Matryoshka hierarchies 33 + +F Reconstruction of SAEs 34 +G Uniqueness of concepts 35 +H Additional qualitative results 35 + +# A Broader Impact + +Our work contributes to the field of interpretability and alignment, which are essential components for building safe AI systems. Our MonoSemanticity score provides a new way to evaluate the effectiveness of recently popular dictionary learning methods, such as sparse autoencoders (SAEs), by incorporating human judgment into the evaluation process. This makes it easier to assess and build trust in systems that use SAEs. + +In addition, we show that SAEs can be highly effective in steering applications. They can be used to encourage or discourage specific behaviors in models, or to help models recognize or ignore certain concepts, including potentially dangerous ones. This is especially useful for ensuring that models produce desired outputs and remain aligned with human values and goals. + +# B More details on steering + +We illustrate in Figure A1 how we steer LLaVA-like models. We separately train SAEs on top of the pretrained CLIP vision encoder to reconstruct the token embeddings $\mathbf{v}_i$ , and then attach it back after the vision encoder during inference. Intervening on a neuron within the SAE layer steers the reconstructed tokens $\hat{\mathbf{v}}_i$ towards the activated concept, which then steers the LLM's generated output. We present in Figure A2 additional examples of LLaVA prompted to generate scientific titles, and the outputs before and after intervening on SAE neurons. Increasing the activation of specific neurons will modify the outputs to include elements from images highly activating the corresponding neuron. + +![](images/11b9659e357eec385d03c520af0fcf151d9284e6c87c8302b4a94da36a9b7c5a.jpg) +Figure A1: LLaVA-like models can be steered towards seeing a concept (e.g. panda) not present in the input image $\mathbf{x}$ . By attaching SAE after vision encoder and intervening on its neuron representing that concept, we effectively manipulate the LLM's response. Such flexible and precise steering is possible thanks to the extensive concept dictionary identified through the SAE. + +![](images/1f37e9e23f74afd1b832437d640f8aaa88e78364e90c82f515c74b7f4a5b469b.jpg) +Figure A2: Effects of neuron interventions on MLLM-generated scientific article titles. Steering magnitudes are categorized as "0", "medium", and "high" based on the intervention strength. The neurons are visualized with the highest activating images from which we deduce their associated concepts: "polka dots", "shipwreck", and "rainbow". + +The steering capabilities discussed in Section 4.3 are evaluated using an LLM-as-judge setup with the following prompts: + +- "Write me a short love poem," +- "Generate a scientific article title," +"Give me a four-item to-do list," +- "Write me a two-verse rap song," +- "Propose a math word problem," +- "Write a paragraph from a Wikipedia page," +"Invent a new holiday," +- "Write a dialogue," +- "Write a newspaper headline and first paragraph," +- "Give a conversation starter for a party." + +# C User study + +To validate the alignment of our MonoSemanticity score (MS) with human judgment, we conducted a user study. Example questions are shown in Figures A3, A4, and A5. Each question shows two grids of images: + +$$ +\left(\left(x _ {i}\right) _ {i = 1} ^ {1 6}, \left(y _ {i}\right) _ {i = 1} ^ {1 6}\right), +$$ + +where each grid contains the 16 images with the highest activations for two neurons $k_{x}$ and $k_{y}$ , respectively. Formally, for each $i = 1, \ldots, 16$ , + +$$ +x _ {i} := \mathbf {x} _ {n} \in \mathcal {I}, \quad \operatorname {r a n k} _ {n} \left(a _ {n} ^ {k _ {x}}\right) = i, +$$ + +$$ +y _ {i} := \mathbf {x} _ {m} \in \mathcal {I}, \quad \operatorname {r a n k} _ {m} (d _ {m} ^ {k _ {y}}) = i, +$$ + +where $a_{n}^{k}$ is the activation of neuron $k$ on image $\mathbf{x}_n$ , and $\mathrm{rank}_n(a_n^k)$ is the rank of image $\mathbf{x}_n$ when sorting all images by their activation values in descending order. Images $\mathcal{I}$ come from training set of the ImageNet. + +For each neuron pair $(k_x, k_y)$ , we asked three human annotators the question: "Which set of images looks more similar and focused on the same thing?" Each annotator gave an answer $r_j \in \{k_x, k_y\}$ for $j = 1, 2, 3$ . The final human choice was decided by majority vote: + +$$ +R _ {(k _ {x}, k _ {y})} ^ {\text {u s e r}} \in \{k _ {x}, k _ {y} \}. +$$ + +At the same time, we answered the question using Monoseismicity Score: + +$$ +R _ {(k _ {x}, k _ {y})} ^ {\mathrm {M S}} := \left\{ \begin{array}{l l} k _ {x} & \text {i f} \mathrm {M S} ^ {k _ {x}} > \mathrm {M S} ^ {k _ {y}} \\ k _ {y} & \text {o t h e r w i s e} \end{array} \right. +$$ + +We say the MS and users are aligned if their decision is the same: + +$$ +\delta_ {(k _ {x}, k _ {y})} := \left\{ \begin{array}{l l} 1 & \text {i f} R _ {(k _ {x}, k _ {y})} ^ {\text {u s e r}} = R _ {(k _ {x}, k _ {y})} ^ {\text {M S}} \\ 0 & \text {o t h e r w i s e} \end{array} \right. +$$ + +The overall alignment score is the fraction of all neuron pairs where the MS and humans are aligned: + +$$ +\text {A l i g n e m t S c o r e} = \frac {1}{| \mathcal {Q} |} \sum_ {\left(k _ {x}, k _ {y}\right) \in \mathcal {Q}} \delta_ {\left(k _ {x}, k _ {y}\right)}, +$$ + +where $\mathcal{Q}$ is the set of all neuron pairs evaluated. + +In total, we collected 1,000 user pair rankings with the help of 71 annotators on the Mechanical Turk platform. The number of answers per annotator ranged from 1 to 205, with a median of 24. Annotators were compensated at a rate of $0.02 per answer. + +The neurons used in the study were randomly selected from the last layer of CLIP ViT-L, BatchTopK SAE $(\varepsilon = 4, K = 20)$ trained on the last layer of CLIP ViT-L, Matryoshka SAE $(\varepsilon = 4, K = 20)$ trained on the last layer of CLIP ViT-L, and BatchTopK SAE $(\varepsilon = 4, K = 20)$ trained on the last layer of SigLIP SoViT-400m. + +In addition to the plot presenting the user study results in the main paper, we also provide Table A1, which reports the exact values obtained, along with the sizes of each group categorized by MS distances between neuron pairs. When designing the questions, we balanced the number of pairs within each distance interval. Our goal is to evaluate MS computed using embeddings from two different image encoders $E$ , namely DINOv2 ViT-B and CLIP ViT-B. As a result, the group sizes are not perfectly equal due to necessary trade-offs. Nevertheless, all groups are sufficiently large and of comparable size. + +Table A1: Alignment Scores (AS) obtained from user study. To compute the MS, we use embeddings of image encoder $E$ , either DINOv2 ViT-B or CLIP ViT-B. Results are grouped by MS distance between neurons in the question. We made sure that every group is represented by enough pairs. +(a) MS distances computed using DinoV2 embeddings. + +
MS Distance (based on DinoV2)0.0-0.10.1-0.20.2-0.30.3-0.40.4-0.50.5-0.60.6-0.70.7-0.80.8-0.9
Number of pairs1261341161251161141228463
AS (E = DINOv2 ViT-B)0.560.660.710.810.850.930.940.961.00
AS (E = CLIP ViT-B)0.600.660.740.820.870.870.940.961.00
+ +(b) MS distances computed using CLIP embeddings. + +
MS Distance (based on CLIP)0.0-0.10.1-0.20.2-0.30.3-0.40.4-0.5
Number of pairs231224215197132
AS (E = CLIP ViT-B)0.550.810.930.960.93
AS (E = DINOv2 ViT-B)0.530.770.920.960.93
+ +# Click to show/hide instructions + +You will be presented with a pair of image sets, each represented as a grid of 16 images. Your task is to decide which set contains images that look more similar and focused on the same thing. + +Example: + +![](images/64f56bd2017c3dc276ffc0e6258fbf3fdfa9e14336fcec9e7699a1f902c9a4c3.jpg) +Figure A3: Example question used in the user study. Best viewed horizontally. + +Which set of images looks more alike and focused on the same thing? + +Set [A] contains similar images of uniformed people standing in formation. Set [B] contains varied images such as a hodge, baseball, or don. + +__________ + +Set [A] + +![](images/69180a8f695d1ea1085540c37021d112060a36c3947e0b0f92f9f2453954c026.jpg) + +Which + +# + +$\bigcirc$ Set [B] + +Submit + +Which set of images looks more alike and focused on the same thing? + +# Click to show/hide instructions + +You will be presented with a pair of image sets, each represented as a grid of 16 images. Your task is to decide which set contains images that look more similar and focused on the same thing. + +Example: + +![](images/32742e47f3b1e23fe6e077d5a659cb667c98b52f02c1b8eae978fb9fb32980ad.jpg) +Figure A4: Example question used in the user study. Best viewed horizontally. + +Which set of images looks more alike and focused on the same thing? + +Set [A] contains similar images of uniformed people standing in formation. Set [B] contains varied images such as a hoodie, baseball, or dog. + +__________ + +![](images/d2185e66244598036ff09581263cb71fa0ccd42f54a3a1bd0f08a8d0b0e3d93c.jpg) +Set [A] +$\bigcirc$ Set [A] +Submit +Which set of images looks more alike and focused on the same thing? +Set [A] +$\bigcirc$ Set [B] + +$\bigcirc$ Set [B] + +![](images/6f081875ca5049500f73ed72b14d6b85e95249524c8088c76c9613b612d67c4c.jpg) +Set [B] + +# Click to show/hide instructions + +You will be presented with a pair of image sets, each represented as a grid of 16 images. Your task is to decide which set contains images that look more similar and focused on the same thing. + +![](images/f32f697af0ac905102020062b4aa5697af887144addaf2e36f14af0a69e0f96c.jpg) + +Which set of images looks more alike and focused on the same thing? + +Set [A] contains similar images of uniform people standing in formation. Set [B] contains varied images such as a hoodie, baseball, or don. + +__________ + +![](images/8a240fa7b7f55fee5b4bc06ae326068a63593ed84e0e9641bc47b8e5d332acbc.jpg) +Set [A] +Which set of images looks more alike and focused on the same thing? + +$\bigcirc$ Set [B] + +![](images/9768715ae4d65b9752fc134afe6f106ece4438e98dbb27bfdddd607931869190.jpg) +$\bigcirc$ Set [A] +$\bigcirc$ Set [A] +$\bigcirc$ Set [B] + +![](images/6dcb42d679793da8aef7b551b642c50f329d5beae1744a97f9272b2414f21740.jpg) +Set [B] + +Example: +look m +Example +Experiments +Set [A] +1 +Which +Set [A] +$\bigcirc$ Set [B] + +# +# + +# D Benchmark + +While MS shows very good results in our user study, we anticipate the development of improved alternatives in the future. To facilitate such advancements, we will release our collected data as a benchmark for evaluating neuron monoseismicity. + +The benchmark will include the following files: + +- pairs.csv - Contains 1000 pairs of neurons $(r_x, r_y)$ , along with user preferences $R_{(k_x, k_y)}^{\text{user}}$ and MS values computed using two different image encoders: DINOv2 ViT-B and CLIP ViT-B. Each row includes the following columns: k_x, k_y, R_user, MS_x_dino, MS_y_dino, MS_x Clip, MS_y Clip. +- top16_images.csv - Lists the 16 most activating images from the ImageNet training set for each neuron used in the study. Columns: k, x_1, ..., x_16. +- activations.csv - Provides activation values of all 50,000 ImageNet validation images for each neuron. Columns: k, a_1, ..., a_50000. + +With this data and by following our evaluation procedure, researchers will be able to compare their methods directly to MS under same conditions. They will have access to the same underlying information, specifically the complete set of neuron activations on the ImageNet validation set. + +# E Additional results on monoseismicity + +# E.1 Unnormalized plots + +Monosemanticity scores across all neurons, without normalized index, are shown in Figure A6. We observe that neurons cover a wider range of scores as we increase the width of the SAE layer. Furthermore, for a given threshold of monosemanticity, the number of neurons having a score higher than this threshold is also increasing with the width. + +![](images/5983852d77bbc44c28eadcf4ab99f08080d44ad4bc8776f7435bac1518dac3dd.jpg) +Figure A6: MS in decreasing order across neurons. Results are shown for a layer without SAE ("No SAE"), and with SAE using different expansion factors $(\times 1, \times 2, \times 4, \times 16$ and $\times 64)$ . + +# E.2 Detailed statistics and more models + +We report in Tables A2 and A5, the average (± std), best and worst monosemanticity scores across neurons for the two SAE variants, attached at different layers and for increasing expansion factors. Although average scores remain similar when increasing expansion factor, we observe a high increase between the original layer and an SAE with expansion factor $\varepsilon = 1$ . The best scores get consistently better as expansion factor gets increased. + +Until now, our analysis has focused on SAEs trained on CLIP ViT-L activations, evaluated using the MS score computed from embeddings produced by the DINOv2 image encoder $E$ . To broaden this investigation, we now consider SAEs trained on activations from SigLIP SoViT-400m. As an alternative image encoder $E$ , we adopt CLIP ViT-B for evaluation. + +Tables A3 and A6 show average, best and worst MS computed using CLIP ViT-B as the vision encoder $E$ . Even though less distinctively than in original setup, the neurons from SAEs still score better compared to the ones originally found in the model. + +In Tables A4 and A7, we report MS statistics for SAEs trained for SigLIP SoViT-400m model computed using CLIP ViT-B as the vision encoder $E$ . The results highly resemble the ones for CLIP ViT-L model. + +Table A2: The average MS of neurons in a CLIP ViT-L model. DINOv2 ViT-B is used as the image encoder $E$ . + +
SAE typeLayerNo SAEExpansion factor
x1x2x4x8x16x64
BatchTopK110.0135 ± 0.00030.03 ± 0.060.04 ± 0.060.04 ± 0.060.03 ± 0.050.03 ± 0.050.03 ± 0.05
170.0135 ± 0.00040.05 ± 0.070.07 ± 0.090.08 ± 0.110.07 ± 0.100.07 ± 0.100.06 ± 0.10
220.0135 ± 0.00030.14 ± 0.120.18 ± 0.150.20 ± 0.170.21 ± 0.170.21 ± 0.180.17 ± 0.18
230.0135 ± 0.00030.15 ± 0.130.18 ± 0.160.20 ± 0.170.21 ± 0.170.20 ± 0.180.17 ± 0.18
last0.0135 ± 0.00020.12 ± 0.110.17 ± 0.150.19 ± 0.170.19 ± 0.160.16 ± 0.160.13 ± 0.15
Matryoshka110.0135 ± 0.00030.05 ± 0.100.06 ± 0.100.05 ± 0.090.05 ± 0.090.04 ± 0.080.03 ± 0.06
170.0135 ± 0.00040.09 ± 0.140.10 ± 0.150.11 ± 0.160.11 ± 0.150.10 ± 0.150.06 ± 0.10
220.0135 ± 0.00030.17 ± 0.170.21 ± 0.180.23 ± 0.190.23 ± 0.190.23 ± 0.190.18 ± 0.19
230.0135 ± 0.00030.17 ± 0.160.21 ± 0.190.22 ± 0.180.22 ± 0.180.20 ± 0.180.12 ± 0.16
last0.0135 ± 0.00020.16 ± 0.170.20 ± 0.180.23 ± 0.190.22 ± 0.190.19 ± 0.190.13 ± 0.16
+ +Table A3: The average MS of neurons in a CLIP ViT-L model. CLIP ViT-B is used as the image encoder $E$ . + +
SAE typeLayerNo SAEExpansion factor
x1x2x4x8x16x64
BatchTopK110.4837 ± 0.00670.52 ± 0.050.53 ± 0.060.53 ± 0.050.53 ± 0.050.53 ± 0.050.53 ± 0.06
170.4840 ± 0.00790.55 ± 0.070.56 ± 0.080.57 ± 0.080.56 ± 0.050.56 ± 0.080.56 ± 0.09
220.4816 ± 0.00530.60 ± 0.090.61 ± 0.090.62 ± 0.090.63 ± 0.090.62 ± 0.100.60 ± 0.11
230.4814 ± 0.00450.60 ± 0.090.61 ± 0.100.62 ± 0.100.62 ± 0.100.61 ± 0.100.59 ± 0.12
last0.4812 ± 0.00420.59 ± 0.080.60 ± 0.100.61 ± 0.100.61 ± 0.100.59 ± 0.100.56 ± 0.10
Matryoshka110.4837 ± 0.00670.54 ± 0.080.55 ± 0.080.55 ± 0.080.54 ± 0.080.53 ± 0.070.52 ± 0.06
170.4840 ± 0.00790.57 ± 0.090.58 ± 0.090.58 ± 0.100.58 ± 0.100.57 ± 0.100.54 ± 0.09
220.4816 ± 0.00530.61 ± 0.090.62 ± 0.090.63 ± 0.100.62 ± 0.110.62 ± 0.110.59 ± 0.12
230.4814 ± 0.00450.60 ± 0.090.62 ± 0.100.62 ± 0.100.61 ± 0.110.60 ± 0.110.54 ± 0.11
last0.4812 ± 0.00420.59 ± 0.090.61 ± 0.100.62 ± 0.110.61 ± 0.110.59 ± 0.120.54 ± 0.12
+ +Table A4: The average MS of neurons in a SigLIP SoViT-400m model. CLIP ViT-B is used as the image encoder $E$ . + +
SAE typeLayerNo SAEExpansion factor
x1x2x4x8x16x64
BatchTopK110.4805 ± 0.00140.50 ± 0.030.51 ± 0.040.51 ± 0.050.51 ± 0.060.52 ± 0.060.52 ± 0.07
160.4809 ± 0.00240.51 ± 0.040.52 ± 0.050.52 ± 0.060.53 ± 0.070.53 ± 0.070.53 ± 0.08
210.4810 ± 0.00520.52 ± 0.050.53 ± 0.060.53 ± 0.060.53 ± 0.070.54 ± 0.080.53 ± 0.08
last0.4811 ± 0.00480.61 ± 0.090.61 ± 0.090.62 ± 0.090.62 ± 0.090.62 ± 0.100.60 ± 0.11
Matryoshka110.4805 ± 0.00140.50 ± 0.030.50 ± 0.050.50 ± 0.050.50 ± 0.060.51 ± 0.070.51 ± 0.07
160.4809 ± 0.00240.51 ± 0.050.52 ± 0.060.52 ± 0.070.52 ± 0.070.52 ± 0.070.51 ± 0.07
210.4810 ± 0.00520.52 ± 0.050.53 ± 0.060.53 ± 0.060.53 ± 0.070.52 ± 0.070.51 ± 0.07
last0.4811 ± 0.00480.61 ± 0.090.62 ± 0.100.62 ± 0.100.62 ± 0.100.60 ± 0.110.58 ± 0.11
+ +Table A5: Comparison of the best / worst MS of neurons in a CLIP ViT-L model. DINOv2 ViT-B is used as the image encoder $E$ . + +
SAE typeLayerNo SAEExpansion factor
×1×2×4×8×16×64
BatchTopK110.01 / 0.010.61 / -0.020.73 / -0.080.71 / -0.060.87 / -0.070.90 / -0.101.00 / -0.11
170.01 / 0.010.65 / 0.010.79 / -0.020.86 / -0.070.86 / -0.080.93 / -0.081.00 / -0.12
220.01 / 0.010.66 / 0.010.79 / 0.010.80 / 0.010.88 / -0.080.92 / -0.061.00 / -0.11
230.01 / 0.010.73 / 0.010.72 / 0.010.83 / 0.010.89 / -0.020.93 / -0.061.00 / -0.10
last0.01 / 0.010.57 / 0.010.78 / 0.010.78 / 0.010.81 / -0.010.85 / -0.041.00 / -0.10
Matryoshka110.01 / 0.010.84 / -0.060.90 / -0.070.95 / -0.081.00 / -0.110.89 / -0.101.00 / -0.10
170.01 / 0.010.86 / -0.040.84 / -0.050.93 / -0.070.94 / -0.090.96 / -0.081.00 / -0.14
220.01 / 0.010.83 / 0.010.83 / 0.010.87 / -0.020.94 / -0.061.00 / -0.111.00 / -0.11
230.01 / 0.010.82 / 0.010.84 / 0.010.89 / -0.040.93 / -0.040.96 / -0.061.00 / -0.11
last0.01 / 0.010.82 / 0.010.91 / 0.010.89 / -0.030.93 / -0.050.91 / -0.071.00 / -0.12
+ +Table A6: Comparison of the best / worst MS of neurons in a CLIP ViT-Large model. CLIP ViT-B is used as the image encoder $E$ . + +
SAE typeLayerNo SAEExpansion factor
×1×2×4×8×16×64
BatchTopK110.50 / 0.470.80 / 0.410.87 / 0.380.90 / 0.280.91 / 0.270.95 / 0.241.00 / 0.20
170.50 / 0.470.84 / 0.370.87 / 0.330.94 / 0.350.94 / 0.280.96 / 0.241.00 / 0.14
220.50 / 0.470.82 / 0.390.85 / 0.380.89 / 0.370.93 / 0.290.93 / 0.151.00 / 0.15
230.50 / 0.470.81 / 0.410.84 / 0.400.89 / 0.350.91 / 0.270.93 / 0.241.00 / 0.08
last0.50 / 0.470.80 / 0.400.84 / 0.400.87 / 0.360.87 / 0.310.89 / 0.251.00 / 0.17
Matryoshka110.50 / 0.470.90 / 0.390.95 / 0.310.97 / 0.231.00 / 0.220.94 / 0.181.00 / 0.19
170.50 / 0.470.94 / 0.330.93 / 0.350.96 / 0.290.96 / 0.220.97 / 0.141.00 / 0.11
220.50 / 0.470.88 / 0.400.87 / 0.330.89 / 0.290.94 / 0.231.00 / 0.151.00 / 0.06
230.50 / 0.470.85 / 0.400.86 / 0.350.90 / 0.350.91 / 0.190.93 / 0.171.00 / 0.14
last0.50 / 0.470.85 / 0.410.88 / 0.400.89 / 0.310.91 / 0.260.92 / 0.171.00 / 0.09
+ +Table A7: Comparison of the best / worst MS of neurons in a SigLIP SoViT-400m model. CLIP ViT-B is used as the image encoder $E$ . + +
SAE typeLayerNo SAEExpansion factor
×1×2×4×8×16×64
BatchTopK110.49 / 0.480.61 / 0.410.83 / 0.290.88 / 0.270.90 / 0.231.00 / 0.121.00 / 0.15
160.53 / 0.470.74 / 0.380.75 / 0.340.93 / 0.250.94 / 0.200.93 / 0.221.00 / 0.18
210.54 / 0.470.76 / 0.380.77 / 0.350.83 / 0.250.89 / 0.170.95 / 0.201.00 / 0.11
last0.50 / 0.470.83 / 0.410.86 / 0.400.88 / 0.370.92 / 0.330.93 / 0.201.00 / 0.11
Matryoshka110.49 / 0.480.70 / 0.400.93 / 0.290.77 / 0.270.93 / 0.180.91 / 0.221.00 / 0.16
160.53 / 0.470.78 / 0.400.84 / 0.290.91 / 0.190.93 / 0.181.00 / 0.191.00 / 0.16
210.54 / 0.470.85 / 0.390.81 / 0.370.83 / 0.250.93 / 0.240.94 / 0.211.00 / 0.15
last0.50 / 0.470.87 / 0.400.87 / 0.380.89 / 0.300.91 / 0.250.94 / 0.151.00 / 0.15
+ +In Figure A7 we plot MS across single neurons. We consider setups in which (a) neurons of CLIP ViT-L are evaluated with DINOv2 as the image encoder $E$ , (b) neurons of CLIP ViT-L are evaluated with CLIP ViT-B as $E$ , and (c) neurons of SigLIP SoViT-400m are evaluated with CLIP ViT-B as $E$ . In all three cases SAE neurons are more monosemantic compared to the original neurons of the models. It shows that MS results are consistent across different architectures being both explained and used as $E$ . + +![](images/c2a0948d34409e4ea4a1d57f24fdb123d2df613712f552d9c4aba61fbd69fd93.jpg) +(a) Neurons of CLIP ViT-L evaluated with DINOv2 ViT-B as the image encoder $E$ + +![](images/a6e46bee45704b0ddb7e101340ca4cd4d17b3f940e733734eabe17842317302f.jpg) +Figure A7: MS in decreasing order across neurons. Results are shown for the last layers of two different models, without SAE (black dashed line), and with SAE being trained with expansion factor 1 (green solid line). MS is computed with distinct image encoders $E$ . + +![](images/762e50f7e01b23f3b6b324a9d9e1830377d2f68b1bcbee5b17ca1ab6c5408041.jpg) +(b) Neurons of CLIP ViT-L evaluated with CLIP ViT-B as the image encoder $E$ +(c) Neurons of SigLIP SoViT 400m evaluated with CLIP ViT-B as the image encoder $E$ + +In Figures A8 and A9, we plot again MS scores across neurons for SAEs trained with different expansion factors and sparsity levels, but using CLIP ViT-B as the image encoder $E$ . We observe very similar patterns when compared to the MS computed using DINOv2 ViT-B. Both higher expansion factor and lower sparsity helps find more of the monosemantic units. + +![](images/f42ef9df60431fdffb4437ff24cb3028a52938d01d1eb4ea5435f6ab8cd6018f.jpg) +(a) Impact of expansion factor $\varepsilon$ +Figure A8: Monosemanticity Scores (computed using CLIP ViT-B) in decreasing order across neurons, normalized by width. Results are shown for the last layer of the model, without SAE ("No SAE", in black dashed line), and with SAE using either (a) different expansion factors (in straight lines, for $\varepsilon = 1$ , for $\varepsilon = 4$ and for $\varepsilon = 16$ ) or (b) different sparsity levels, with straight lines for $K = 1$ , for $K = 10$ , for $K = 20$ , and for $K = 50$ . + +![](images/54748fc2fd50550cef59239709f8e881aedf810e0d130387391412422b348681.jpg) +(b) Impact of sparsity factor $K$ + +![](images/570ee02cdb05ae358eaba63e26696e64f0c44200d92fbeaa99a3369548fa164f.jpg) +Figure A9: Monoseismicity Scores (computed using CLIP ViT-B) in decreasing order across neurons without normalizing by width. Results are shown for a layer without SAE ("No SAE"), and with SAE using different expansion factors $(\times 1, \times 4$ and $\times 16)$ . + +# E.3 Matryoshka hierarchies + +We train and evaluate the SAE on embeddings extracted from iNaturalist [55] dataset using an expansion factor $\varepsilon = 2$ and groups of size $\mathcal{M} = \{3,16,69,359,1536\}$ . These group sizes correspond to the numbers of nodes of the first 5 levels of the species taxonomy tree of the dataset, i.e. the respective number of "kingdoms", "phylums", "classes", "orders", and "families". + +To measure the granularity of the concepts, we map each neuron to the most fitting depth in the iNaturalist taxonomy tree to compare the hierarchy of concepts within the Matryoshka SAE with human-defined ones. To obtain this neuron-to-depth mapping, we select the top-16 activating images per neuron, and compute the average depth of the Lowest Common Ancestors (LCA) in the taxonomy tree for each pair of images. For instance, given a neuron with an average LCA depth of 2, we can assume that images activating this neuron are associated to species from multiple "classes" of the same "phylum". We report the average assigned LCA depth of neurons across the Matryoshka group level in Table A8. We notice that average LCA depths are correlated with the level, suggesting that the Matryoshka hierarchy can be aligned with human-defined hierarchy. We additionally aggregate statistics of MS of neurons for each level. Average and maximum MS also correlates with the level, confirming that the most specialized neurons are found in the lowest levels. + +Table A8: Average LCA depth and monoseismicity (MS) scores across neurons at each level in the Matryoshka nested dictionary. + +
Level01234
Depth3.332.923.853.864.06
MSAvg.0.060.080.090.160.24
Max.0.110.300.290.690.76
Min.0.040.030.030.03-0.05
+ +# F Reconstruction of SAEs + +In Table A9 and Table A10, we report respectively $R^2$ and sparsity $(L_0)$ , for the two SAE variants we compare in Section 4.2. As BatchTopK activation enforces sparsity on batch-level, during test-time it is replaced with $\mathrm{ReLU}(\mathbf{x} - \gamma)$ , with $\mathbf{x}$ is the input and $\gamma$ is a vector of thresholds estimated for each neuron, as the average of the minimum positive activation values across a number of batches. For this reason the test-time sparsity may slightly differ from $K$ fixed at the value of 20 in our case. + +We report in Table A11 the detailed metrics $(R^2, L_0$ and statistics of MS) obtained for SAEs trained with different $K$ values considered in Section 4.2. + +Table A9: Comparison of $R^2$ (in %) by different SAEs trained with $K = 20$ for a CLIP ViT-L model. + +
SAE typeLayerNo SAEExpansion factor
x1x2x4x8x16x64
BatchTopK1110074.775.075.175.074.773.5
1710070.471.972.672.972.972.5
2210068.772.674.976.076.877.4
2310067.271.574.075.376.076.8
last10070.174.677.178.278.679.1
Matryoshka1110072.873.974.575.175.274.5
1710067.369.570.771.872.672.7
2210065.569.671.574.075.476.6
2310063.968.571.073.174.874.6
last10066.871.674.176.077.678.2
+ +Table A10: Comparison of true sparsity measured by $L_{0}$ -norm for different SAEs trained with $K = 20$ for a CLIP ViT-L model. + +
SAE typeLayerNo SAEExpansion factor
x1x2x4x8x16x64
BatchTopK11102419.719.519.419.620.022.9
17102419.419.419.219.619.522.3
22102419.619.719.719.820.323.0
23102419.819.819.920.120.322.2
last76819.919.919.920.120.222.2
Matryoshka11102419.419.519.419.619.821.3
17102419.319.319.319.419.520.5
22102419.719.719.619.819.922.0
23102419.719.819.819.920.625.1
last76820.019.919.819.920.222.5
+ +Table A11: Statistics for SAEs trained with different sparsity constraint $K$ on activations of the last layer with expansion factor 16. "No SAE" row contains results for raw activations before attaching the SAE. + +
KL0R2(%)MS
MinMaxMean
10.931.3-0.030.900.37 ± 0.20
109.960.60.010.790.19 ± 0.16
2020.066.80.010.820.16 ± 0.17
5050.174.90.010.690.07 ± 0.08
No SAE--0.010.010.01 ± 0.00
+ +![](images/c9f56a8e573557130e8ce147f258e003ffa739fd8581f157375bf3f38bd83347.jpg) +Figure A10: Images highly activating the neuron we intervene on in Figure 6, which we manually labeled as "Pencil Neuron". + +# G Uniqueness of concepts + +The sparse reconstruction objective regularizes the SAE activations to focus on different concepts. To confirm it in practice, we collect top-16 highest activating images for each neuron of SAE and compute Jaccard Index $J$ between every pair of neurons. The images come from training set. We exclude 10 out of 12288 neurons for which we found less than 16 activating images and use Matryoshka SAE trained on the last layer with expansion factor of 16. We find that $J > 0$ for 16000 out of 75368503 pairs $(>0.03\%)$ and $J > 0.5$ for only 20 pairs, which shows very high uniqueness of learned concepts. + +# H Additional qualitative results + +We illustrate in Figure A10 the highly activating images for the "Pencil" neuron, which we used for steering in Figure 6. In Figures A11 and A12 we provide more randomly selected examples of neurons for which we computed MS using two different image encoders. In both cases we see a clear correlation between score and similarity of images in a grid. + +![](images/ef2326c50a5422e8116b0f414e7a74af93915a3e96833a0ca62cfd606691c708.jpg) +Figure A11: Qualitative examples of highest activating images for different neurons from high (left) to low (right) MS score. As the metric gets higher, highest activating images are more similar, illustrating the correlation with monoseismicity. DINOv2 ViT-B is used as the image encoder $E$ . + +![](images/1d44efecbd1558daa05751ac640235caa1f6094b7e51948bb1f57c12e83f49fa.jpg) +Figure A12: Qualitative examples of highest activating images for different neurons from high (left) to low (right) MS score. As the metric gets higher, highest activating images are more similar, illustrating the correlation with monoseismicity. 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Existing methods encode the intrinsic dynamics of high-dimensional observations through latent representations and predict autoregressively. However, these latent representations lose the inherent spatial structure of spatiotemporal dynamics, leading to the predictor's inability to effectively model spatial interactions and neglect emerging dynamics during long-term prediction. In this work, we propose SparseDiff, introducing a test-time adaptation strategy to dynamically update the encoding scheme to accommodate emergent spatiotemporal structures during the long-term evolution of the system. Specifically, we first design a codebook-based sparse encoder, which coarsens the continuous spatial domain into a sparse graph topology. Then, we employ a graph neural ordinary differential equation to model the dynamics and guide a diffusion decoder for reconstruction. SparseDiff autoregressively predicts the spatiotemporal evolution and adjust the sparse topological structure to adapt to emergent spatiotemporal patterns by adaptive re-encoding. Extensive evaluations on representative systems demonstrate that SparseDiff achieves an average prediction error reduction of $49.99\%$ compared to baselines, requiring only $1\%$ of the spatial resolution. + +# 1 Introduction + +The dynamics of complex systems are driven by the nonlinear interactions and co-evolution of numerous components, giving rise to rich emergent spatiotemporal structures, as seen in fluid dynamics [1], climate science [2], and molecular dynamics [3]. Accurate long-term prediction of these systems is crucial for many real-world applications [4, 5]. However, complex systems are typically observed in high-dimensional spaces with unknown intrinsic dynam- + +![](images/a3b63bbc442d0a1a9e058801db115cc3e63ef899444a1a79030a60bdd54f3300.jpg) +Figure 1: Adapting probe topologies. + +ics, making high-fidelity, long-term prediction a challenging and computationally expensive problem [6, 7, 8]. + +Data-driven methods offer an equation-free and computationally friendly approach. The core idea is to build reduced-order models of high-dimensional observations using encoder-decoders, which project observed states to a low-dimensional latent space for prediction, thereby improving computational efficiency [6, 9, 10, 11, 12]. However, most existing work employs parameterized neural network encoders that implicitly encode spatial structure by compressing observational spatial information into a latent representation vector. This conversion of spatially-correlated data into latent vectors hinders predictors from effectively understanding and modeling the system's inherent spatial interactions and structural relationships [5]. For instance, in fluid convection, heat drives fluid to form rising plumes and sinking cold flows. These specific spatial structures, their relative positions, and interactions are critical for understanding and predicting convection patterns [13]. Considering that spatial interaction patterns and emergent structures continuously change during long-term evolution, losing such information progressively amplifies the misalignment between the encoder-decoder and predictor in long-term predictions. This leads to a core question: Can we construct a reduced-order model that preserves crucial spatial structure and continuously adapts to newly emerging dynamics patterns during prediction? + +This work is inspired by a recent finding that the system state in the entire spatial domain can be effectively reconstructed from sparse observation points [14, 15, 16]. The data disparity between sparse points and the full spatial grid strongly suggests a promising encoding paradigm: aggregating the full-space dynamic information onto a much smaller set of sparse probes [17]. These probes and their spatial topology constitute the dynamical skeleton of the spatiotemporal dynamics, as shown in Figure 1. In this paradigm, the model is capable of adapting to the system's latest spatiotemporal patterns during long-term predictions by dynamically adjusting probe positions and topology based on the re-encoded predicted states [6, 18]. Nevertheless, achieving effective probe-based aggregation and adaptive updating in practice faces two key challenges: 1) A lack of theoretical understanding for aggregating rich spatiotemporal dynamics from a continuous domain into a small set of discrete probes; 2) Frequent re-encoding during long-term prediction introduces significant computational overhead and potential accumulation of prediction errors. + +To address these challenges, we propose a novel Sparse Diffusion Autoencoder, SparseDiff, consisting of a codebook-based sparse encoder and an unconditional diffusion decoder. The sparse encoder learns a representative pattern codebook from historical spatiotemporal trajectories, enabling it to dynamically aggregate and project full-space observational data onto a spatial topology formed by sparse probes, thus constructing a reduced-order model. On this probe topology, we model spatiotemporal dynamics using a graph neural ordinary differential equation and explicitly introduce a diffusion term to capture spatial interactions and information propagation among probes. Finally, we pad the probe predictions to the full spatial domain to serve as the initial state for the diffusion process, enabling rapid reconstruction of the full spatiotemporal field. SparseDiff's key innovation and strength lie in its test-time adaptation: during long-term prediction, it utilizes the learned codebook to re-encode the latest prediction, dynamically adjusting and constructing a probe topology better matched to the current system state to continuously adapt to emerging spatiotemporal patterns. Experimental evaluation on simulated and real-world systems demonstrates that SparseDiff outperforms baselines by over $49.99\%$ in long-term predictions, and reliably predicts full-space spatiotemporal dynamics using less than $1\%$ of grid points as probes. + +The highlights of this work are summarized as follows: + +- We propose SparseDiff, a novel autoencoder that uses sparse probes to construct a reduced-order model that effectively preserves the spatial structure of spatiotemporal dynamics. +- We introduce a test-time adaptation strategy via re-encoding, allowing the model to dynamically adapt its probe-based representation to emerging dynamics patterns, significantly improving long-term prediction accuracy and computational efficiency. +- Experiments demonstrate that SparseDiff significantly outperforms baselines in long-term prediction accuracy and achieves high efficiency by reliably predicting full-space dynamics using only approximately $1\%$ of grid points as probes. Our code is open-source: https://github.com/ tsinghua-fib-lab/SparseDiff . + +# 2 Preliminary + +# 2.1 Problem Definition + +In this work, we focus on spatiotemporal dynamics $\frac{du}{dt} = f(u,x,t,\frac{\partial u}{\partial x},\frac{\partial^2u}{\partial x^2},\ldots)$ on a 2D regular spatial domain $x\subset \mathbb{R}^2$ . Such systems can often be formalized as time-dependent partial differential equations, for example, the Navier-Stokes equations, which include a Laplacian operator term acting on the spatial domain [19]. Without loss of generality, we address the problem of data-driven modeling of system dynamics from historical evolving trajectories $U = \{u(x,t)\} ^T$ , thereby extending our focus to real-world systems like climate dynamics where explicit equations are unknown. In practice, the spatial domain is discretized into an $h * w$ grid. Consequently, the full-domain observation state at each timestamp is represented as a tensor $u\in \mathbb{R}^{h\times w}$ . Given a historical observation trajectory of lookback steps, we predict multiple future steps in an autoregressive manner, forecasting one step at a time. + +# 2.2 Guided Diffusion for Sparse Reconstruction + +Diffusion methods have recently been shown to reliably reconstruct the full spatial domain's system state, guided by sparse observations [16, 15, 14]. Diffusion models aim to learn a probabilistic mapping from a simple prior distribution, such as a standard Gaussian to a complex target data distribution [20, 21, 22]. This is achieved by defining a forward diffusion process that gradually adds noise to data instances and training a reverse process to sequentially denoise them. We represent the original data point as $x_0$ . The forward stage transforms $x_0$ into a noisy version $x_n$ over $n$ steps, governed by the equation $x_n = \sqrt{\overline{a}_n} x_0 + \sqrt{1 - \overline{a}_n} \epsilon_n$ , $\epsilon_n$ and $\{\overline{a}_n\}$ represent the Gaussian noise and noise schedule [23], respectively. The learned reverse diffusion process models the transition from noise back to data through a sequence of conditional distributions given by + +$$ +p _ {\theta} \left(x _ {n - 1} \mid x _ {n}\right) := \mathcal {N} \left(x _ {n - 1}; \mu_ {\theta} \left(x _ {n}, n\right), \sigma_ {n} ^ {2} \mathbf {I}\right), \tag {1} +$$ + +where $\mu_{\theta} = \frac{1}{\sqrt{\alpha_n}} (x_n - \frac{1 - \alpha_n}{\sqrt{1 - \overline{\alpha}_n}}\epsilon_\theta (x_n,n))$ and $\{\sigma_n\}$ are step dependent constants. The term $\epsilon_{\theta}$ represents the model's prediction of the added noise, typically implemented using a parameterized neural network architecture like a UNet or Transformer. The optimization of this network's parameters is performed by minimizing the objective function [23] + +$$ +L _ {n} = \mathbb {E} _ {n, \epsilon_ {n}, x _ {0}} | | \epsilon_ {n} - \epsilon_ {\theta} (\sqrt {\bar {\alpha} _ {n}} x _ {0} + \sqrt {1 - \bar {\alpha} _ {n}} \epsilon_ {n}, n) | | ^ {2}. \tag {2} +$$ + +This loss function is derived from the negative log-likelihood $\mathbb{E}_{x_0\sim q(x_0)}[-p_\theta (x_0)]$ . Once trained, the diffusion model progressively denoises from Gaussian noise to yield high-fidelity data samples. To guide diffusion in reconstructing the full spatial domain state from $K$ sparse observations $\mathcal{M}(x_K)$ , Bayes' rule guides the diffusion gradient direction [24] to be + +$$ +\nabla_ {x _ {n}} \log p (x _ {n} | \mathcal {M}) \approx - \frac {1}{\sqrt {1 - \bar {\alpha} _ {n}}} \epsilon_ {\theta} - \zeta \nabla_ {x _ {n}} \| y - \mathcal {M} (x _ {K}) \| _ {2} ^ {2}, \tag {3} +$$ + +where $y$ represents the noise values at the sparse observation points and $\zeta = 1 / \sigma^2$ . + +# 3 Methodology + +In this section, we first introduce the proposed sparse diffusion autoencoder to discover the skeleton of spatiotemporal dynamics, namely the probe topology. Subsequently, we design a diffusion graph neural ordinary differential equation on the probe topology to model spatiotemporal dynamics. Finally, we propose the test-time adaptation strategy to dynamically sense emerging spatiotemporal patterns in long-term prediction. The overall framework is illustrated in Figure 2. + +# 3.1 Sparse Diffusion Autoencoder + +We introduce our approach for discovering the dynamical skeleton of complex systems, utilizing a codebook-based sparse encoder and a guided diffusion decoder. + +![](images/00017918bb7f9a2a93d344933d534b25674d2886a581f7466634c62e6b104af5.jpg) +Figure 2: Overall framework of proposed Sparse Diffusion Autoencoder. + +# 3.1.1 Codebook-based Sparse Encoder + +To discover the skeleton of spatiotemporal dynamics on the spatial domain, we first identify and record the rich dynamical patterns inherent in complex systems. Specifically, we maintain a codebook $C$ containing $K$ codewords for spatiotemporal dynamics, where each codeword $c \in \mathbb{R}^d$ is a $d$ -dimensional learnable vector. For a given historical observation $\mathbf{u} \in \mathbb{R}^{t \times h \times w}$ , we employ an encoder $\mathcal{E}$ to encode along the time dimension, yielding a $d$ -dimensional representation vector $z \in \mathbb{R}^{d \times h \times w}$ for the full-domain states. We replace the representation vector $z$ with the most similar codeword $c$ , which serves as the decoder's input. The decoder then reconstructs the original observation $\hat{\mathbf{u}}$ . The encoder-decoder and the codebook are trained by minimizing the objective function [25] + +$$ +L = \log p (\mathbf {u} | c) + \left\| \operatorname {s g} [ \mathcal {E} (\mathbf {u}) ] - c \right\| _ {2} ^ {2} + \beta \left\| \mathcal {E} (\mathbf {u}) - \operatorname {s g} [ c ] \right\| _ {2} ^ {2}, \tag {4} +$$ + +where sg indicates gradient detachment. Through this process, we effectively record the rich spatiotemporal dynamical patterns from the observation trajectories within the pretrained codebook. + +In the inference stage, we first discretize the historical observation $u$ into a set of $k$ hit codewords $\{c_i\}_{i=1}^k$ using pretrained encoder $\mathcal{E}$ and codebook $C$ . We define the governing region of each hit codeword $c_i$ as the set of spatial grid points that are mapped to it, denoted as $u_{c_i}$ . Consequently, the full spatial domain is divided into $k$ region types by the $k$ hit codewords (a single region type might consist of dispersed patches). These $k$ regions represent $k$ local-scale spatiotemporal units. Therefore, we accordingly construct probe set $\mathcal{V} = \{v_i \in \mathbb{R}^l\}_{i=1}^k$ to represent them, where $l$ is the lookback window size. + +For probe $v_{i}$ , we select one spatial grid point from its corresponding codeword's governing region $u_{c_i}$ as its coordinate. We use random selection here instead of averaging, ensuring that the probe falls within patches even if the patches of its region are spatially dispersed. Subsequently, we average the historical observation sequences of all grid points within $u_{c_i}$ to aggregate them as probe states $\mathbf{v}_{\mathbf{i}} \in \mathbb{R}^{l}$ . Finally, we connect all probes to construct the topological structure $\mathcal{G} = \{\mathcal{V}, E\}$ , and assign different weights $e_{ij} \in E$ to each edge to characterize the spatial association strength. Specifically, edge weights $e_{ij}$ quantify the spatial association strength from probe $v_{i}$ to probe $v_{j}$ . We consider the governing region $u_{c_i}$ associated with $v_{i}$ and spatial neighborhood of every grid point within $u_{c_i}$ . We count how many grid points belonging to $v_{j}$ 's governing region $u_{c_j}$ appear within the combined neighborhoods of all points in $u_{c_i}$ . This count is normalized along all probes to determine the edge weight $e_{ij}$ . Thereby, we obtain the probe topology $\mathcal{G}$ of the spatiotemporal dynamics. + +# 3.1.2 Guided Diffusion Decoder + +We pretrain an unconditional diffusion model to capture the spatiotemporal patterns of the system state $u$ . Given the set of probes $\mathcal{V}$ , we use these probe values $v_{i}$ as sparse observations to guide the reconstruction by the diffusion decoder. Existing works [16, 15, 14] typically directly substitute probe values as the sparse observation $\mathcal{M}(x_K)$ in Equation 3 to guide reconstruction. However, our approach differs in that we guide the governing regions $u_{c_i}$ corresponding to the probes. That is, our probes represent the dynamical information of the entire governing region, not just single grid points. Therefore, we use probe value $v_{i}$ to guide diffusion in reconstructing its governing region. + +Specifically, we fill the probe value $v_{i}$ to each grid point within its governing region $u_{c_i}$ , serving as $\mathcal{M}(x_K)$ in Equation 3 to guide reconstruction. Furthermore, considering the computationally intensive of the denoising process, we use the filled full-space state instead of Gaussian noise as the starting point for denoising to accelerate the diffusion sampling process. + +# 3.2 Probe-graph Diffusive Predictor + +Given the probe topology $\mathcal{G}$ and probe states $\mathbf{v_i}$ , we now predict the spatiotemporal dynamics of the system. Specifically, we model this process as a neural diffusion process on the graph [26] + +$$ +h _ {t} = h _ {0} + \int_ {0} ^ {t} (\mathrm {A - I}) * h _ {\tau} d \tau , +$$ + +$$ +h _ {0} = \phi (\mathbf {v}), +$$ + +$$ +v _ {t} = \psi (h _ {t}), +$$ + +where $h_t$ refers to the representation at time $t$ , $\phi$ and $\psi$ are the parameterized encoder and decoders, and $A$ is the time-varying diffusion coefficient between probes, calculated by a parameterized attention network. To explicitly model the spatial association strength between probes, we introduce the edge weights into the calculation of the diffusion coefficient as + +$$ +a (h _ {i}, h _ {j}) = \mathrm {s o f t m a x} (\frac {(\mathbf {W} _ {K} h _ {i}) ^ {\top} \mathbf {W} _ {Q} h _ {j} + \mathbf {W} _ {E} e _ {i j}}{d _ {k}}), +$$ + +where $h_i$ denotes the representation of the $i$ th probe, $\mathrm{W}_Q$ , $\mathrm{W}_K$ , and $\mathrm{W}_E$ are learnable matrices. We employ a multi-head attention mechanism $\mathrm{A} = \frac{1}{k}\sum_k\mathrm{A}^k$ to capture complex dynamical mechanisms. + +# 3.3 Test-time Adapting Prediction + +During long-term prediction, complex system dynamics exhibit continuous emergence of spatiotemporal patterns, implying time-varying dynamical spatial interactions. Therefore, we continuously update the predictor during testing to adapt to the latest spatiotemporal dynamics patterns. For a $T$ -step prediction, the prediction window is divided into $N$ sub-windows $\{w_{n}\}^{N}$ . We assume each window possesses specific spatial interactions, with a corresponding probe topology sequence $\{\mathcal{G}_{w_n}\}^N$ . We update the probe topology at the beginning of each window through re-encoding. Specifically, at a window transition time, we decode the probe states back to full space. Following the method in Section 3.1.1, we then re-encode the probe topology $\mathcal{G}$ for the latest dynamics within the current window using encoder $\mathcal{E}$ and codebook $C$ . This topology is updated again at the next transition time. + +We design a dynamic update strategy to dynamically determine the window transition time. Instead of using fixed-length sub-windows, we continuously monitor whether the recent evolution of each probe remains well-aligned with its assigned codeword in the latent space of the encoder $\mathcal{E}$ . Specifically, after each re-encoding, we obtain $k$ hit codewords $\{c_i\}_{i=1}^k$ , where each $c_i \in \mathbb{R}^d$ corresponds to a governing region. For each codeword, a probe $v_i$ is randomly selected to represent its region and forms a probe graph $\mathcal{G}$ for prediction. At each prediction step, we compute the latent consistency score $\chi_t$ by encoding the past $T$ -step trajectory of each probe $v_i$ and evaluating its cosine similarity with the associated codeword $c_i$ : + +$$ +\chi_ {t} = \frac {1}{k} \sum_ {i = 1} ^ {k} \cos \left(\mathcal {E} (v _ {i} ^ {t - T: t}), c _ {i}\right). +$$ + +When $\chi_{t}$ drops below a predefined threshold $\tau$ , it indicates that many probe representations have drifted away from their original latent codewords, suggesting a mismatch between the current probe partition and the evolving system dynamics. In response, we decode the probe graph of the past $T$ steps back to the full space using the diffusion decoder and recompute the probe topology via re-encoding. This strategy enables the model to adaptively reallocate codewords and update the graph structure in accordance with the varying speed and patterns of spatiotemporal evolution. + +Table 1: Average performance of the trajectories predicted from different initial conditions with standard deviation from 10 runs. The best results are highlighted in bold. + +
SystemsLambda-OmegaNavier-StokesSwift-HohenbergCylinder-FlowReal-world
RMSE ×10-2SSIM ×10-1RMSE ×10-2SSIM ×10-1RMSE ×10-2SSIM ×10-1RMSE ×10-2SSIM ×10-1RMSE ×10-2SSIM ×10-1
ConvLSTM5.613 ± 0.5579.224 ± 0.12530.674 ± 2.7044.696 ± 0.0839.556 ± 0.0599.399 ± 0.00611.098 ± 0.3305.265 ± 0.14413.682 ± 2.7777.288 ± 1.195
FNO5.138 ± 0.5579.170 ± 0.51415.992 ± 1.0627.262 ± 1.64219.693 ± 5.5919.614 ± 0.00919.854 ± 25.2487.583 ± 0.52512.639 ± 3.8786.775 ± 1.368
UNet8.324 ± 0.4978.014 ± 0.39316.277 ± 1.5827.092 ± 0.83714.881 ± 2.7899.206 ± 0.22521.349 ± 4.9036.203 ± 0.19613.928 ± 2.8556.328 ± 0.974
G-LED6.506 ± 0.3958.992 ± 0.39612.334 ± 0.4858.095 ± 0.9638.214 ± 0.9379.572 ± 0.21110.021 ± 0.8737.059 ± 0.29110.304 ± 2.0096.768 ± 0.539
Ours2.912 ± 0.1879.601 ± 0.21811.130 ± 3.2418.492 ± 0.7937.628 ± 3.1029.675 ± 0.1928.544 ± 0.2397.392 ± 0.1717.957 ± 1.2077.781 ± 1.034
53.56%9.05%38.64%28.63%57.42%1.89%62.67%17.39%37.86%15.51%
+ +# 4 Experiments + +In this section, we validate the accuracy and efficiency of SparseDiff on simulated PDE systems and real-world datasets. Furthermore, we evaluate its robustness and generalization ability and examine the specific contribution of its components through ablation studies. + +# 4.1 Experimental Setup + +We conduct experimental validation on four PDE systems, including: 1) Lambda-Omega; 2) Navier-Stokes; 3) Cylinder Flow; and 4) Swift-Hohenberg systems. They encompass complex diffusion effects, convection terms, and higher-order spatial interaction terms, among other nonlinear dynamic components. Additionally, we also evaluate SparseDiff's performance in real-world applications on an open-source climate record dataset [27]. All models are trained on trajectories with different initial conditions and predict long-term (more than 100 steps) on new trajectories. Details on the equations, data generation, and training settings are in Appendix A. + +Baselines We compare our method against a set of state-of-the-art spatiotemporal forecasting models, including operator learning methods, recurrent architectures, and generative frameworks. Specifically, we consider Fourier Neural Operator (FNO) [28], ConvLSTM [29], UNet [30], and G-LED [31]. A detailed description of each baseline is provided in Appendix C. + +# 4.2 Main Results + +PDE systems Table 1 shows the long-term prediction performance of all models. SparseDiff achieves the optimal prediction quality in almost all scenarios. This indicates that SparseDiff's test-time adaptation enhances long-term prediction by timely sensing of new spatio-temporal dynamics. Furthermore, compared to other baselines, another characteristic of SparseDiff is that it performs dynamic prediction on the probe topology. Other methods predict in the original or uniformly downsampled grid space and, unlike SparseDiff, fail to discover the spatial interactions of complex systems and structure them into an explicit topology to enhance the predictor's representational capacity. + +Real-world dataset We use the SEVIR dataset [27] for real-world weather forecasting. Specifically, we select the GOES-16 Channel 09 $(6.9\mu m)$ infrared imagery ir069, which captures mid-level water vapor and is widely used for storm tracking. Each frame covers a $384\times 384\mathrm{km}$ area at $2\mathrm{km}$ resolution, yielding a $192\times 192$ grid, with a temporal resolution of 5 minutes. Experimental results are shown in Figure 3, where SparseDiff achieves the most faithful prediction of complex climate patterns with the lowest error. This indicates that the proposed probe-based encoding scheme is of great value for applications in real-world com + +![](images/d6374288036e2643f16435a74b5d6fead4728d09197c38cfdf7e0a12a8f3f071.jpg) +a + +![](images/ca80f5e5d00156ee7398323b86de2f6a28f06eaa89aeb5152637fbdc615a21c4.jpg) +SparseDiff + +![](images/d54c9bba9972f2708e90bd15ea3cceb3f1fada29a2369ef1cb600038f5c44d58.jpg) +Figure 3: Real-world dataset. (a) Prediction visualization of different models. (b) RMSE comparison of different models. + +![](images/bbf4d6b7f720d1be262d9bab917a40129f1290b9f0a98641f1ea016a0f29aac0.jpg) +G-LED + +![](images/574aeb5dfe0202b84b0fa4242806a1cd3ea3b35f9186c31a7f0b042269df8089.jpg) +b + +plex systems, especially in large-scale observational data scenarios in geoscience. SparseDiff encodes + +![](images/6e81f38d53f9e5dd8be158b1be7553b2e985db8a1e83de3a1b612fcae16379d7.jpg) +Figure 4: Ablation studies. Prediction performance with (a) uniform and (b) random probe selection. (c) Impact of probe topology edge weights on prediction. + +![](images/9a58ca054bd287e17539ea3d0a886ff5b0af2950ffec0800e309bb681757aece.jpg) + +![](images/912c3916766fd7aa0b477a3b1813747f58def8552c03654155d83775a253218e.jpg) + +high-dimensional observations into a sparse probe topology, thus reducing computational overhead while ensuring accuracy. + +# 4.3 Ablation Study + +We capture a compact low-dimensional skeleton of spatiotemporal dynamics through codebook learning, thereby improving prediction performance. The contrasting version is uniform downsampling or random selection of probes. We also establish heuristic calculation rules for the edge weights of the probe topology. The contrasting version is an unweighted graph (where all weights are set to 1). In the following, we ablate these two key components. + +Selection of Probes Probes are the most critical components in the SparseDiff framework, as they serve as the information carriers that represent the coarse-grained structure of the spatiotemporal dynamics. Their selection fundamentally determines the prediction quality of the entire model. Figures 4a and b illustrate the prediction performance of SparseDiff on the Swift-Hohenberg system when the probes are selected via spatially uniform and random sampling, respectively. We observe that even when the number of probes is increased to 1024, these two variants still perform significantly worse than our method. In contrast, the original version achieves high-quality predictions with as few as 150 probes. This highlights that our proposed codebook-based sparse encoder not only compresses the spatial domain effectively but also selects informative and dynamically representative probes. It demonstrates a strong ability to extract the low-dimensional spatiotemporal skeleton where the intrinsic dynamics reside. + +Probe-Graph Edge Weights When the edge weight feature is disabled, the edge-feature-aware attention mechanism described in Section 3.2 degenerates into standard inter-node attention that treats all probe connections equally. Without incorporating the spatial association strength $e_{ij}$ , the model loses the ability to capture relative distances and directional relationships between probes, which are essential in spatiotemporal systems. As shown in Figure 4c, this leads to a notable performance drop on the Navier-Stokes system, where accurate modeling of spatial interactions is critical for capturing fine-grained dynamics. + +# 4.4 Robustness + +Here, we evaluate the robustness of SparseDiff to data observation noise and codebook size settings. Specifically, we first test the prediction performance with different preset codebook sizes on the Navier-Stokes system. The codebook size is the upper limit on the number of activated codewords. It determines the maximum number of probes SparseDiff can select to perceive spatiotemporal dynamics. Experimental results are shown in Figure 5a, where the prediction accuracy (SSIM) converges after the codebook size reaches 150. For the Navier-Stokes system with $128 \times 128$ resolution, this is + +![](images/b4d7bdf830171f6d451852d5c4139af467c42bcfc5f4ebdb4aae57eb80b42cfc.jpg) +Figure 5: Robustness experiments. (a) Impact of codebook size on SparseDiff's performance on the Navier-Stokes system. (b) Impact of the noise on SparseDiff's performance on the real climate dataset. + +![](images/afe96d62e859cbc0b6c8d9549f5822aa78231346dfb2b62f506c5d4aca9591f0.jpg) + +quite a small number. This indicates that SparseDiff utilizes probes efficiently and achieves superior prediction with very low computational overhead. + +Next, we examine SparseDiff's robustness to real-world noise. We apply Gaussian noise with varying percentage intensity to a real-world climate dataset (relative to the original data amplitude). Experimental results are shown in Figure 5b, where SparseDiff's performance deteriorates as the noise increases, but at moderate noise levels, it still outperforms the baseline without noise. + +# 4.5 Generalization + +We examine SparseDiff's out-of-distribution generalization ability. Specifically, using the Cylinder Flow system as the experimental subject, we evaluate SparseDiff's ability to predict flows with Reynolds numbers not included in the training set. For turbulent systems, the Reynolds number influences the viscous coefficient, thereby affecting collision frequency. For fluids with high Reynolds numbers, the flow exhibits chaotic tendencies and is thus difficult to predict over long periods. We train SparseDiff on Reynolds numbers less than 500 and predict on larger Reynolds numbers. Experimental results are shown in Figure 6, where the baseline shows rapid performance degradation in out-of + +distribution scenarios, whereas SparseDiff's performance consistently outperforms the baseline and exhibits fluctuations. The reason may be that a rich set of spatiotemporal dynamic patterns is recorded in SparseDiff's pretrained codebook. Although there are differences in the turbulent dynamic behaviors at different Reynolds numbers, local small-scale dynamic patterns share similarities and are thus recognized by SparseDiff and accurately generalized. + +![](images/6040dc42d10508e6a664cf5edf46fc2f3dba4f7cfc8c56998428a4fce0890ac9.jpg) + +![](images/bfee07eee83b5886f9272829b98566605bb6ebda354e5e13b886b1b981664fe5.jpg) +Figure 6: Generalization experiments. Visualized (a) ground truth and (b) SparseDiff prediction of turbulent at $Re = 660$ . (c) SSIM as a function of Reynolds number. + +![](images/11bae675dd51a353f42bf516740d0ff7a302fceb973840b4c0056c3463135620.jpg) + +# 4.6 Trade-off between Accuracy & Efficiency + +Here, we analyze how SparseDiff trades off between accuracy and efficiency. Specifically, during testing, SparseDiff can adjust the probe topology through re-encoding, thus constructing the most suitable form for newly emerging spatiotemporal dynamics. Frequent adjustments allow for sufficient perception of real-time dynamics, but the corresponding computational overhead increases. We conducted tests at different update intervals on the Navier-Stokes system, and the results are shown in Figure 7, where Rollout Step refers to the time index in the autoregressive prediction sequence, and each step + +![](images/4b67ff2767d33c8410917e9aa1729d9bbb2f5205238641323ca6a45b739fb939.jpg) +Figure 7: Trade-off of accuracy & efficiency. Circles represent SparseDiff, and crosses represent GLED. + +![](images/902aa3b8264e2517ec6fe20f8d79ae2c9257d123caf2bb92824907e6f9d032f8.jpg) + +corresponds to one forward prediction in the rollout process. According to Figure 7a, accuracy and time overhead exhibit a negative correlation. + +We also compare with the best-performing baseline, G-LED. It can enhance long-term prediction accuracy by decoding and re-encoding at intervals, but its accuracy ceiling is lower than SparseDiff, and its time overhead is much greater than SparseDiff. Our adaptive re-encoding strategy proposed in Section 3.3 achieves a balance between accuracy and efficiency, as shown by the red star in Figure 7a. + +# 4.7 Comparison of Graph Construction Schemes + +To validate our proposed region-aware edge weighting scheme, we compare it against two primary baseline categories: (a) Learnable Graph Construction: The Graph Kernel Network (GKN) [32], a method designed for learnable graph construction; and (b) Heuristic Alternatives: Using the same GRAND predictor but replacing our learned $e_{ij}$ with static graph structures, including k-nearest + +neighbors (kNN) based on Euclidean coordinates ( $k = 20\%$ of probe count), and spectral clustering followed by kNN in the embedding space. Table 2: Comparison of different graph + +We report the average RMSE over a 100-step autoregressive prediction on the LO system. As shown in Table 2, our region-aware edge scheme yields significantly better performance than both the learnable GKN and the heuristic variants. This demonstrates that explicitly encoding region-level adjacency improves the expressiveness of the model for probe-based dynamics. + +Table 2: Comparison of different graph construction methods. + +
MethodRMSE ×10-2
Ours2.873
GKN4.210
GRAND + kNN3.287
GRAND + spectral3.019
+ +# 5 Related Work + +# 5.1 Encoding Dynamics of Complex Systems + +Discovering the low-dimensional latent space containing the intrinsic dynamics of complex systems represents a core challenge in long-term prediction. To accelerate spatiotemporal dynamical system prediction, some earlier studies focus on approximating solutions on coarse grids. Lee et al. [7] utilize Gaussian processes and diffusion maps to coarse-grain high-resolution microscopic observations into coarse-scale PDEs. Bar-Sinai et al. [33] employ uniform grid downsampling of the continuous spatial domain for specific nonlinear partial differential equations. This coarse-grid approach has inspired several recent works [31, 34]. With advancements in representation learning, autoencoders have been employed to data-drivenly uncover the low-dimensional latent space of high-dimensional observations [6, 35]. Prediction models such as recurrent neural networks [6] and neural operators [9] operate directly within the low-dimensional space constructed by autoencoders. To ensure the latent space aligns with physical intuition, existing research suggests integrating autoencoders with physical priors [36]. Li et al. [11] restrict the output semantics of the encoder by specifying timescales and intrinsic dimensions. Wu et al. [10] combine delay embeddings with feature embeddings to discover the low-dimensional manifold of high-dimensional observations. Additionally, some work [37] learns the latent coordinates of partial differential equations through autoencoders to reveal governing equations. In contrast to these methods, our proposed probe topology adaptively senses spatial structure, without enforcing a regular grid. This explicitly preserves the tight spatial correlations of spatiotemporal dynamics with a high compression ratio. + +# 5.2 Generative Modeling for Complex Systems + +Diffusion models have performed remarkably well in generative tasks, with their significant achievements in video synthesis [38, 39] and time series modeling [40, 41] inspiring numerous studies in complex system dynamics prediction. Some works [42, 15] have utilized large-scale pre-trained diffusion models to reconstruct high-fidelity data from lower-fidelity samples or sparse measurement data. Physics knowledge is also integrated. For example, known partial differential equations provide physical constraints (PDE loss) for the denoising step, improving accuracy [43]. Beyond reconstruction, diffusion models have also been applied to generate specific dynamical data. Li et al. [44] propose a machine learning approach for generating single-particle trajectory data in high Reynolds number three-dimensional turbulence. Lienen et al. [13] treat turbulence simulation itself as a generative task, employing diffusion models to capture the distribution of turbulence induced by unseen objects and generate high-quality samples for downstream applications. Other strategies focus on directly embedding dynamics or prediction processes into the diffusion mechanism. Cachay et al. [45] align the temporal evolution axis of spatiotemporal dynamics with diffusion process steps, replacing noise injection with temporal interpolation and the denoising operation with prediction. G-LED [31], a more direct approach, incorporates system states as prediction targets and utilizes predicted future frames as conditions to guide the diffusion model in reconstructing the original high-fidelity states. Li et al. [5] embed multiscale features as conditioning for Diffusion modeling, while [46] dynamically adjust the number of denoising steps based on dynamical timestamps. Compared to these methods, we introduce a novel sparse encoder working in tandem with a diffusion decoder. This combination not only reveals the low-dimensional spatial structure of long-term dynamics but also enables test-time adaptation to emerging spatiotemporal dynamics. + +# 6 Conclusion + +Modeling the long-term dynamics of complex systems is a challenging problem. Data-driven methods are often constrained by the difficulty in explicitly preserving the inherent spatial interactions of spatiotemporal dynamics in latent vectors. This information loss is exacerbated by new patterns that continuously emerge during the system's long-term evolution. Therefore, we propose a novel Sparse Diffusion Autoencoder, SparseDiff, a prediction framework that preserves the system's spatial structure and adapts at test-time. SparseDiff coarsens the full spatial domain state into a probe topology by means of discrete codebook learning to construct a compact skeleton of spatiotemporal dynamics. The pretrained codebook is able to capture the rich dynamic patterns of complex systems. Once training is complete, SparseDiff can adjust the probe topology to adapt to emerging dynamic patterns during testing and without updating weights. On the probe topology, we design an edge-weight-aware graph neural diffusion ordinary differential equation to model spatiotemporal dynamics, thereby predicting the future states of the probes and guiding the diffusion model to efficiently reconstruct back to the full spatial domain. Experiments on simulated and real-world systems show that SparseDiff can achieve optimal long-term prediction and exhibits excellent robustness and generalization ability. + +Limitation & Future Work SparseDiff's spatiotemporal dynamics module relies on the perception of edge weights, therefore the rules for calculating weights will affect its long-term prediction quality. Furthermore, the state initialization of probes is achieved through average aggregation, which may lose high-frequency information at a few grid points. Future research will focus on end-to-end graph structure learning and probe initialization strategies. + +# 7 Author Contributions + +Ruikun Li, Huandong Wang and Yong Li launched this research and provided the research outline. Ruikun Li, Jingwen Cheng and Huandong Wang designed the research methods. Jingwen Cheng performed the experiments and prepared the figures & technical appendix. Ruikun Li wrote the original manuscript. Huandong Wang and Yong Li provided critical revisions. + +# Acknowledgements + +This work was supported in part by the National Natural Science Foundation of China under grant 62171260 and 92270114. + +# References + +[1] Dmitrii Kochkov, Jamie A Smith, Ayya Alieva, Qing Wang, Michael P Brenner, and Stephan Hoyer. Machine learning-accelerated computational fluid dynamics. 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Dynamical diffusion: Learning temporal dynamics with diffusion models. arXiv preprint arXiv:2503.00951, 2025. +[47] Qian Zhao, David B. Lindell, and Gordon Wetzstein. Learning to solve PDE-constrained inverse problems with graph networks. arXiv preprint, 2022. + +# Impact Statement + +This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. + +# A Data Generation + +Below, we provide an overview of the dynamics and data generation process for each complex system. Lambda-Omega (LO) system is governed by + +$$ +\left\{ \begin{array}{l} \dot {u} _ {t} = \mu_ {u} \Delta u + (1 - u ^ {2} - v ^ {2}) u + \beta \left(u ^ {2} + v ^ {2}\right) v \\ \dot {v} _ {t} = \mu_ {v} \Delta v + (1 - u ^ {2} - v ^ {2}) v - \beta \left(u ^ {2} + v ^ {2}\right) u, \end{array} \right. \tag {5} +$$ + +where $\Delta$ is the Laplacian operator. Here, we set $\mu_v$ and $\mu_v$ to 0.5, while $\beta$ is 1.0. + +Navier-Stokes (NS) system is governed by: + +$$ +\dot {\omega} _ {t} + (\mathbf {u} \cdot \nabla) \omega = \nu \Delta \omega + f, \tag {6} +$$ + +$$ +\nabla^ {2} \psi = - \omega , \tag {7} +$$ + +$$ +\mathbf {u} = (u, v) = \left(\frac {\partial \psi}{\partial y}, - \frac {\partial \psi}{\partial x}\right), \tag {8} +$$ + +$$ +\omega = (\nabla \times \mathbf {u}) \cdot \hat {\mathbf {z}} = \frac {\partial v}{\partial x} - \frac {\partial u}{\partial y}, \tag {9} +$$ + +where $f = A\left(\sin \left(2\pi (x + y + s)\right) + \cos \left(2\pi (x + y + s)\right)\right)$ is the driving force, and $\nu$ is the viscosity coefficient. Parameter values used in this simulation are set as follows: $\nu = 1.0$ , forcing amplitude $A = 0.1$ , and phase shift $s = 0$ . + +Swift-Hohenberg (SH) system is simulated as: + +$$ +\frac {\partial u}{\partial t} = r u - 2 \Delta u - \Delta^ {2} u + g u ^ {2} - u ^ {3}, \tag {10} +$$ + +where $r$ is the linear instability parameter, $g$ controls the strength of the quadratic nonlinearity, and $\Delta^2$ denotes the biharmonic operator. In this system we set $r$ to 0.7 and $g$ 1.0 separately. + +Cylinder flow (CY) system is governed by: + +$$ +\left\{ \begin{array}{l} \dot {u} _ {t} = - u \cdot \nabla u - \frac {1}{\alpha} \nabla p + \frac {\beta}{\alpha} \Delta u, \\ \dot {v} _ {t} = - v \cdot \nabla v + \frac {1}{\alpha} \nabla p - \frac {\beta}{\alpha} \Delta v. \end{array} \right. \tag {11} +$$ + +where $\alpha$ is set to 1.0, and $\beta$ corresponds to the dynamic viscosity $\mu$ as defined in Equation 12. + +The systems under study are simulated across a diverse range of initial conditions. Temporal downsampling by a factor of 10 and 25 are applied to the LO and NS systems separately to reduce redundancy, while all the systems are spatially rescaled to a uniform $128 \times 128$ resolution. + +The Cylinder flow system is modeled via the lattice Boltzmann method (LBM) [6], capturing complex vortex shedding phenomena governed by the Navier-Stokes equations in the presence of a cylindrical obstacle. The simulation operates on a lattice velocity grid, where relaxation dynamics are dictated by the kinematic viscosity and the Reynolds number. To ensure data quality, we begin recording after the flow reaches a statistically steady turbulent regime. The outputs are resampled in time by a factor of 300 and spatially interpolated to a $128 \times 64$ grid. We generate 50 training and 20 testing trajectories across varying flow conditions, with Reynolds numbers uniformly sampled: 10 training samples in the range $\mathrm{Re} \in [100, 500]$ and 10 out-of-distribution (OOD) samples in $\mathrm{Re} \in [500, 1000]$ . Based on the relation + +$$ +\mu = \frac {\rho U _ {m} D}{\mathrm {R e}} \tag {12} +$$ + +where $\rho = 1$ , $U_{m} = 0.08$ , and $D = 0.2$ , the corresponding dynamic viscosity spans $\mu \in [3.2 \times 10^{-5}, 1.6 \times 10^{-4}]$ for training and $\mu \in [1.6 \times 10^{-5}, 3.2 \times 10^{-5}]$ for OOD cases. + +To standardize input scales and facilitate stable training, we apply min-max normalization independently across all channels. + +# B Model Architecture + +Our model is composed of three key modules: a Codebook-based Sparse Encoder for spatiotemporal discretization, a Probe-graph Diffusive Predictor for latent dynamics modeling, and a Guided Diffusion Decoder for reconstructing full-resolution system states. Below, we detail the architectural settings used in each component. + +```ini +# --- Sparse Encoder --- +hidden_dim = 1024 # hidden dimension of MLP +embedding_dim = 512 # Latent dimension (d) +num_embeddingings = M # Codebook size, hyperparameter +# --- Diffusive Predictor --- +input_steps = 10 # number of lookback steps +feature_dim = 256 # Feature dimension +num_heads = 8 # Attention heads for calculating Matrix A +ODE_method = 'rk4' # Numerical solver for ODE integration +# --- Unconditioned Diffusion --- +n_channels = 128 # Base number of channels +chMULTS = [1, 2, 2] # Channel multiplier for each resolution level +is_attn = [False, False, True] # Whether to apply self-attention +dropout = 0.1 # Dropout rate in residual blocks +n_blocks = 2 # Number of residual blocks per resolution level +``` + +Table 3: Trainable parameter counts (in millions). + +
ModelSparseDiffFNOConvLSTMUNetG-LED
EncoderPredictorDiffusion
Params (M)2.3 × 10-21.3225.823.905.4710.9126.3
+ +# C Baseline Implementation + +We provide a brief description of the baseline methods used for comparison in our experiments. These methods represent a diverse set of state-of-the-art approaches for modeling spatiotemporal dynamics. Their corresponding trainable parameter counts are summarized in Table 3. + +- FNO [28]: Fourier Neural Operator leverages fast Fourier transforms to model spatially continuous operators, enabling efficient learning of solution mappings for partial differential equations. It is widely adopted for learning complex physical dynamics. +- ConvLSTM [29]: ConvLSTM integrates convolutional structures into recurrent networks, allowing spatial correlations to be preserved while capturing temporal dependencies. It is a standard baseline for video and sequence-based spatial forecasting tasks. +- UNet [30]: UNet employs an encoder-decoder structure with skip connections to effectively combine global context and local details. It is particularly suitable for dense prediction tasks involving structured outputs. +- G-LED [31]: G-LED is a generative latent evolution model that models temporal dynamics using autoregressive attention in latent space, and reconstructs full-resolution outputs using a Bayesian diffusion model. It achieves strong performance on high-dimensional physical systems. + +# D Additional Results + +# D.1 Application to Irregular or Sparse Spatial Domain + +While our current implementation operates on 2D regular grids, SparseDiff can be applied to irregular or sparse spatial data by first transforming the inputs into a regular grid through interpolation or zero-filling, enabling processing within the same framework. + +To validate this, we conduct two experiments: + +a. Sparse sampling on Navier-Stokes system: On the Navier-Stokes system, we randomly sample only $10\%$ of grid points. These sparse samples are then interpolated onto a $128 \times 128$ regular grid before being fed into SparseDiff. Despite the sparsity, SparseDiff outperforms baselines that are given full-resolution inputs, as shown in Table 4. + +Table 4: Sparse sampling on the Navier-Stokes system. We report average metrics of 100 prediction steps. + +
MethodRMSE ×10-2↓SSIM ×10-1↑
FNO (full grid)15.997.26
G-LED (full grid)12.338.10
SparseDiff (full grid)11.398.36
SparseDiff (10% observations)12.278.19
+ +b. Irregular 2D wave equation dataset [47]: We also evaluate SparseDiff on the 2D wave equation with observations sampled on irregular meshes. These inputs are first completed onto regular grids using interpolation within observed regions and zero-filling in excluded regions before being processed by the model. Compared to FNO, SparseDiff maintains stable prediction accuracy (Table 5). + +Table 5: Results on the irregular 2D wave equation dataset [47] of 100 prediction steps. + +
MethodRMSE ×10-2↓SSIM ×10-1↑
FNO23.767.81
SparseDiff15.588.25
+ +While these results demonstrate practical applicability to irregular or sparse inputs, we acknowledge certain limitations of this approach: + +- For highly non-uniform spatial distributions, interpolating to a uniform grid can be problematic. Also, the sparse observation may be too under-sampled to allow reliable interpolation, leading to unnecessary computational cost or significant reconstruction errors. +- If the distribution of spatial inputs shifts between training and testing, performance may degrade, as our current model lacks coordinate-querying capabilities (unlike operator-learning models such as neural operators), and cannot generalize across arbitrary spatial layouts. + +Nevertheless, the core design of SparseDiff is not inherently tied to regular grids. The key innovation of SparseDiff lies in its latent probe representation, which aggregates and models regional dynamics through sparse, learnable units. This representation is inherently flexible and does not require regular spatial structures. + +The main constraint stems from the UNet-based diffusion decoder, which requires regular-grid input. However, this is a design choice rather than a fundamental limitation of the method. In future work, the diffusion decoder could be replaced with architectures that naturally support irregular domains, such as Graph Neural Networks. Alternatively, it could be substituted with other super-resolution models, including transformer-based approaches and operator-learning models, to further extend SparseDiff to irregular spatial settings. + +# D.2 Visualization + +We provide long-term prediction visualizations on three representative systems: NS, SH, and CY. For each case, the left column shows the ground truth, the middle column presents the prediction reconstructed via the diffusion decoder, and the right column shows the vanilla reconstruction by directly filling each codeword region with its corresponding probe value. + +![](images/171af13c0beab593f0872bde478067943c6a985a298c2b9813a4c771ddf3ce2a.jpg) +(a) CY system + +![](images/5fb6e06aad84dcd39e70089b8cf077299be73dd87cc98ff78b24a800d5b9913a.jpg) +(b) NS system + +![](images/8be877251a6480a04893ea86f0731a7fa9426c3cb00091236e8a025432786285.jpg) +(c) SH system +Figure 8: Long-term prediction results on three representative systems. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Can be found at the last paragraph of the introduction section. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: Can be found at the limitation paragraph of the conclusion section. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory Assumptions and Proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [NA] + +Justification: This paper does not include theoretical results. + +# Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental Result Reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +# Answer: [Yes] + +Justification: Can be found in the data generation and supplementary materials sections of the appendix. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. +- Depending on the contribution, reproducibility can be accomplished in various ways. 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Bui + +Australian National University + +thang.bui@anu.edu.au + +Michalis K. Titsias + +Google DeepMind + +mtitsias@google.com + +# Abstract + +Inducing-point-based sparse variational Gaussian processes have become the standard workhorse for scaling up GP models. Recent advances show that these methods can be improved by introducing a diagonal scaling matrix to the conditional posterior density given the inducing points. This paper first considers an extension that employs a block-diagonal structure for the scaling matrix, provably tightening the variational lower bound. We then revisit the unifying framework of sparse GPs based on Power Expectation Propagation (PEP) and show that it can leverage and benefit from the new structured approximate posteriors. Through extensive regression experiments, we show that the proposed block-diagonal approximation consistently performs similarly to or better than existing diagonal approximations while maintaining comparable computational costs. Furthermore, the new PEP framework with structured posteriors provides competitive performance across various power hyperparameter settings, offering practitioners flexible alternatives to standard variational approaches. + +# 1 Introduction + +Gaussian processes (GPs) provide a principled framework for modelling functions that offer calibrated uncertainty and safeguard against overfitting, among many other benefits (see e.g., Rasmussen & Williams, 2006). However, their computational requirement, cubic in the number of training data $N$ , is prohibitive for many practical applications. This bottleneck motivates the development of a plethora of scalable approximation methods (Quinonero-Candela & Rasmussen, 2005; Liu et al., 2020), with sparse variational methods using inducing points arguably the most popular (Titsias, 2009; Hensman et al., 2013). + +The key idea behind sparse variational GPs (SVGPs) is to approximate the posterior process using a small set of $M \ll N$ inducing points, reducing the computational complexity to $\mathcal{O}(NM^2)$ or $\mathcal{O}(M^3)$ in the batch and stochastic settings, respectively. A key assumption in the standard SVGP approximation is the prior distribution of the non-inducing function values conditioned on the inducing points remains unchanged in the approximate posterior, that is, $q(f_{\neq u}|\boldsymbol{u}) = p(f_{\neq u}|\boldsymbol{u})$ . Titsias (2025); Bui et al. (2025) recently showed that relaxing this assumption yields provably tighter variational bounds. In particular, the key innovation is slightly adjusting covariance of $q(f_{\neq u}|\boldsymbol{u})$ by a diagonal scaling matrix $\mathbf{M}$ , leading to improved predictive performance while maintaining computational tractability. This approach has the original SVGP approach as a special case when $\mathbf{M} = \mathbf{I}$ . Such improvement begs the question: can we achieve even better approximations by considering more expressive structures for $\mathbf{M}$ while preserving efficient computation? + +To this end, we propose using block-diagonal structures for $\mathbf{M}$ and show that this choice leads to provably tighter variational bounds compared to existing diagonal approximations while maintaining the same computational complexity and ease of implementation. We then show that these structured approximations can also help with other inference schemes beyond variational inference. Specifically, + +certain structural choices for M lead to tractable Power Expectation Propagation (PEP) updates and approximate log marginal likelihood. This greatly extends and improves over the unifying framework of Bui et al. (2017). + +The remainder of this paper is organised as follows. Section 2 reviews sparse variational GPs, recent advances in structured approximations, and the PEP framework for sparse GPs. Section 3 presents the proposed block-diagonal variational approximation. Section 4 extends the existing PEP framework with various structured posteriors. Section 5 evaluates the proposed methods on a suite of tasks. We then discuss related work in section 6 and conclude with a discussion of future directions in section 7. + +# 2 Background + +We first provide a summary of inducing-point sparse variational Gaussian processes (SVGP; Titsias, 2009; Hensman et al., 2013, 2015; Matthews et al., 2016), a recently proposed tighter bound (Bui et al., 2025; Titsias, 2025), and a power-EP based approach (Bui et al., 2017). Consider the supervised learning setting with an unknown input-output mapping $f$ , a GP prior over this function $p(f|\gamma) = \mathcal{GP}(f;0,k_{\gamma})$ , and a pointwise likelihood $p(\pmb{y}|f,\mathbf{X},\omega) = \prod_{n}p(y_{n}|f(\pmb{x}_{n}),\omega)$ , where $\mathbf{X}\in \mathbb{R}^{N\times D}$ and $\pmb{y}\in \mathbb{R}^{N}$ are the training inputs and outputs, $k_{\gamma}$ is the covariance function governed by hyperparameters $\gamma$ , and $\omega$ is the likelihood hyperparameters. In what follows, we will use $\theta$ to denote these hyperparameters and, when clear, drop the dependence on $\theta$ for brevity. Inference and learning in this model are computationally challenging for large-scale datasets due to the $\mathcal{O}(N^3)$ complexity; thus, efficient approximations are required. Sparse variational methods parameterise an approximate posterior based on $M$ inducing points, $\{\pmb{z}\in \mathbb{R}^{M\times D},\pmb{u}\in \mathbb{R}^{M}\}$ , with $M\ll N$ , as follows, + +$$ +q (f) = p \left(f _ {\neq} f, u \mid f, u\right) q \left(f \mid u\right) q (u), \tag {1} +$$ + +where $\pmb{f} = [f(\pmb{x}_1),\dots ,f(\pmb{x}_N)]$ . Note that the factorisation here mirrors that in the prior, $p(f) = p(f_{\neq f,u}|f,u)p(f|u)p(u)$ , where $p(\pmb {u}) = \mathcal{N}(\pmb {u};\mathbf{0},\mathbf{K}_{\mathrm{uu}})$ $p(\pmb {f}|\pmb {u}) = \mathcal{N}(\pmb {f};\mathbf{K}_{\mathrm{fu}}\mathbf{K}_{\mathrm{uu}}^{-1}\pmb {u},\mathbf{D}_{\mathrm{ff}})$ $\mathbf{D}_{\mathrm{ff}} = \mathbf{K}_{\mathrm{ff}} - \mathbf{Q}_{\mathrm{ff}},\mathbf{Q}_{\mathrm{ff}} = \mathbf{K}_{\mathrm{fu}}\mathbf{K}_{\mathrm{uu}}^{-1}\mathbf{K}_{\mathrm{uf}},\mathbf{K}_{\mathrm{ff}} = k(\mathbf{X},\mathbf{X}),\mathbf{K}_{\mathrm{fu}} = k(\mathbf{X},\mathbf{z})$ , and $\mathbf{K}_{\mathrm{uu}} = k(\mathbf{z},\mathbf{z})$ Note that we use $f$ to denote the function and $\pmb{f}$ to denote the function values at the training inputs. The resulting variational lower bound to the log marginal likelihood is + +$$ +\mathcal {F} _ {0} = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] - \int q (\boldsymbol {u}) \mathrm {K L} [ q (\boldsymbol {f} | \boldsymbol {u}) | | p (\boldsymbol {f} | \boldsymbol {u}) ] + \sum_ {n} \int q (\boldsymbol {u}) q (f (\boldsymbol {x} _ {n}) | \boldsymbol {u}) \log p (y _ {n} | f (\boldsymbol {x} _ {n})). +$$ + +When $q(\pmb {f}|\pmb {u}) = p(\pmb {f}|\pmb {u}) = \mathcal{N}(\pmb {f};\mathbf{K}_{\mathrm{fu}}\mathbf{K}_{\mathbf{uu}}^{-1}\pmb {u},\mathbf{D}_{\mathbf{ff}})$ , the bound above becomes, + +$$ +\mathcal {F} _ {1} (q (\boldsymbol {u}), \theta) = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] + \sum_ {n} \int q (\boldsymbol {u}) p (f (x _ {n}) | \boldsymbol {u}) \log p (y _ {n} | f (x _ {n})), \tag {2} +$$ + +commonly known as the uncollapsed SVGP bound (Hensman et al., 2015; Titsias, 2009). This bound conveniently allows both (i) tractable computation $\left[\mathcal{O}(NM^2)\right.$ in the batch setting or $\mathcal{O}(BM^2 + M^3)$ where $B$ is the batch size in the mini-batch setting] and (ii) tractably handling of non-Gaussian likelihoods using quadrature or Monte Carlo estimation for the expected log-likelihood terms. For the Gaussian likelihood, the bound can be simplified to + +$$ +\mathcal {F} _ {1, r} (q (\boldsymbol {u}), \theta) = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] + \sum_ {n} \left[ \int q (\boldsymbol {u}) \log \mathcal {N} \left(y _ {n}; \mathbf {k} _ {f _ {n} \mathbf {u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \sigma^ {2}\right) - \frac {d _ {n n}}{2 \sigma^ {2}} \right], \tag {3} +$$ + +where $\sigma^2$ is the observation noise and $d_{nn} = [\mathbf{D}_{\mathbf{ff}}]_{nn}$ . Furthermore, an optimal form for $q(\pmb{u})$ can be found, $q(\pmb{u}) \propto p(\pmb{u}) \mathcal{N}(\pmb{y}; \mathbf{K}_{\mathbf{fu}} \mathbf{K}_{\mathbf{uu}}^{-1} \pmb{u}, \sigma^2 \mathbf{I}_N)$ , yielding the following analytic collapsed bound (Titsias, 2009), + +$$ +\mathcal {F} _ {1, r c} (\theta) = \log \mathcal {N} (\boldsymbol {y}; \mathbf {0}, \mathbf {Q} _ {\mathbf {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2 \sigma^ {2}} \sum_ {n} d _ {n n}. \tag {4} +$$ + +The SVGP approach above has arguably been the most popular scalable GP approach in the literature. More recently, Bui et al. (2025); Titsias (2025) show that this approach can be improved by relaxing the $q(\pmb{f}|\pmb{u}) = p(\pmb{f}|\pmb{u})$ assumption. Specifically, when $q(\pmb{f}|\pmb{u}) = \mathcal{N}(\pmb{f};\mathbf{K}_{\mathrm{fu}}\mathbf{K}_{\mathrm{uu}}^{-1}\pmb{u},\mathbf{D}_{\mathrm{ff}}^{1/2}\mathbf{M}\mathbf{D}_{\mathrm{ff}}^{1/2})$ , + +where $\mathbf{M} = \mathrm{diag}([m_1,\dots,m_N])$ , the uncollapsed and collapsed bounds in the regression case are: + +$$ +\mathcal {F} _ {2, r} (q (\boldsymbol {u}), \theta) = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] + \sum_ {n} \left[ \int q (\boldsymbol {u}) \log \mathcal {N} \left(y _ {n}; \mathbf {k} _ {f _ {n} \mathbf {u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \sigma^ {2}\right) - \frac {1}{2} \log \left(1 + \frac {d _ {n n}}{\sigma^ {2}}\right) \right] \tag {5} +$$ + +$$ +\mathcal {F} _ {2, r c} (\theta) = \log \mathcal {N} (\boldsymbol {y}; \mathbf {0}, \mathbf {Q} _ {\mathrm {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2} \sum_ {n} \log \left(1 + \frac {d _ {n n}}{\sigma^ {2}}\right). \tag {6} +$$ + +Note that the optimal form for $m_{n}$ is $m_{n} = \sigma^{2} / (\sigma^{2} + d_{nn}) < 1$ ; and eqs. (5) and (6) are tighter than eqs. (3) and (4) for fixed $\theta$ and $q(\pmb{u})$ since $\log \left(1 + d_{nn} / \sigma^2\right) \leq d_{nn} / \sigma^2$ . + +The posterior approximation in eq. (1) can also be used in other deterministic inference strategies. For example, in the regression case, for $q(\pmb{f}|\pmb{u}) = p(\pmb{f}|\pmb{u})$ , Bui et al. (2017) showed that Power-Expectation Propagation (PEP) yields an analytic collapsed approximate marginal likelihood, + +$$ +\mathcal {F} _ {3, r c} (\theta) = \log \mathcal {N} (\boldsymbol {y}; \mathbf {0}, \mathbf {Q} _ {\mathbf {f f}} + \alpha \mathbf {D} _ {\mathbf {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1 - \alpha}{2 \alpha} \sum_ {n} \log \left(1 + \alpha \frac {d _ {n n}}{\sigma^ {2}}\right), \tag {7} +$$ + +and a closed form $q(\pmb{u})$ , $q(\pmb{u}) \propto p(\pmb{u})\mathcal{N}(\pmb{y};\mathbf{K}_{\mathbf{fu}}\mathbf{K}_{\mathbf{uu}}^{-1}\pmb{u},\alpha \mathbf{D}_{\mathbf{ff}} + \sigma^2\mathbf{I}_N)$ , where $\alpha$ is the power hyperparameter in PEP. This framework encompasses a multitude of approximations, such as the SVGP approximation (as $\alpha \to 0$ ) and FITC (Snelson & Ghahramani, 2005; Qi et al., 2010) ( $\alpha = 1$ ). + +# 3 A block-diagonal structured variational approximation + +We first consider the following posterior approximation: + +$$ +q (f) = p \left(f _ {\neq} f, u | f, u\right) q (f | u) q (u), \quad q (f | u) = \mathcal {N} \left(f; \mathrm {K} _ {\mathrm {f u}} \mathrm {K} _ {\mathrm {u u}} ^ {- 1} u, \mathrm {C}\right), +$$ + +where we have not posited a form for the covariance $\mathbf{C}$ . Interestingly, this leads to the familiar optimal form for $q(\pmb{u})$ , $q(\pmb{u}) \propto p(\pmb{u}) \mathcal{N}(\pmb{y}; \mathbf{K}_{\mathbf{fu}} \mathbf{K}_{\mathbf{uu}}^{-1} \pmb{u}, \sigma^2 \mathbf{I}_N)$ . The resulting collapsed bound is, + +$$ +\mathcal {F} (\theta) = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q _ {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2} \mathrm {t r a c e} [ (\sigma^ {- 2} \mathbf {I} _ {N} + \mathbf {D _ {f f} ^ {- 1}}) \mathbf {C} ] - \frac {1}{2} \log | \mathbf {C} ^ {- 1} \mathbf {D _ {f f}} | + \frac {N}{2}. +$$ + +Except for some special cases, the bound above is as expensive as the original log marginal likelihood to compute. Specifically, as shown in the background, $\mathbf{C} = \mathbf{D}_{\mathbf{ff}}^{1/2}\mathbf{M}\mathbf{D}_{\mathbf{ff}}^{1/2}$ with $\mathbf{M} = \mathbf{I}_N$ (Titsias, 2009) or $\mathbf{M} = m\mathbf{I}_N$ (Artemev et al., 2021) or $\mathbf{M} = \mathrm{diag}(\{m_n\}_{n=1}^N)$ (Titsias, 2025; Bui et al., 2025) admit tractability, and each move (from $\mathbf{I}_N$ to $m\mathbf{I}_N$ , and from $m\mathbf{I}_N$ to $\mathrm{diag}(\{m_n\}_{n=1}^N)$ ) makes the bound tighter. It is thus natural to enquire what structure to encode in $\mathbf{M}$ to further improve the bound, retain tractable computation, and potentially improve predictive performance. + +We now consider one such structure, a block-diagonal $\mathbf{M}$ , $\mathbf{M} = \mathrm{blkdiag}(\{\mathbf{m}_b\}_{b=1}^B)$ , where $B$ is the number of blocks and $\mathbf{m}_b \in \mathbb{R}^{N_b \times N_b}$ . Substituting this into the bound above gives + +$$ +\mathcal {F} _ {4} (\theta) = \log \mathcal {N} (\pmb {y}; 0, \mathbf {Q _ {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2} \sum_ {b} \left[ \frac {1}{\sigma^ {2}} \mathrm {t r a c e} [ \mathbf {m} _ {b} \mathbf {D _ {f _ {b} f _ {b}}} ] + \mathrm {t r a c e} [ \mathbf {m} _ {b} ] - \log | \mathbf {m} _ {b} | - N _ {b} \right]. +$$ + +We can obtain the optimal $\mathbf{m}_b$ , $\mathbf{m}_b = (\mathbf{I}_b + \sigma^{-2}\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b})^{-1}$ , leading to the following collapsed bound, + +$$ +\mathcal {F} _ {4, r c} (\theta) = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q} _ {\mathbf {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2} \sum_ {b} \log | \mathbf {I} _ {b} + \sigma^ {- 2} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} |. \tag {8} +$$ + +Due to the Hadamard's inequality, $|\mathbf{I}_b + \sigma^{-2}\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b}| < \prod_i(1 + \sigma^{-2}[\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b}]_{ii})$ , and thus $\log |\mathbf{I}_b + \sigma^{-2}\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b}| < \sum_i\log (1 + \sigma^{-2}[\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b}]_{ii})$ . In other words, the bound in eq. (8) [M is block-diagonal] is provably tighter than the bound in eq. (6) [M is diagonal]. + +Similar to the standard SVGP approach, for large datasets, it is more convenient to work with the following uncollapsed bound that supports stochastic optimisation, + +$$ +\mathcal {F} _ {4, r} (\cdot) = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] + \sum_ {b = 1} ^ {B} \left[ \int q (\boldsymbol {u}) \log \mathcal {N} \left(\boldsymbol {y} _ {b}; \mathbf {K} _ {\mathbf {f} _ {b} \mathbf {u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \sigma^ {2} \mathbf {I} _ {b}\right) - \frac {1}{2} \log | \mathbf {I} _ {b} + \sigma^ {- 2} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} | \right]. \tag {9} +$$ + +If the B blocks are of roughly equal size, computing the bound in eq. (9) using the entire training set takes $\mathcal{O}(M^3 + N M^2 + B[N / B]^3)$ . However, in practice, we perform stochastic optimisation, + +![](images/589c69870288fbca5bf596a5ba7c6cdbb5d7ab7f355c24e38dd42421d74e6665.jpg) +Figure 1: Connections between the sparse GP regression methods from the Power-EP perspective. Green means computationally tractable, red means intractable, and blue represents the new methods presented in this paper. $B < N$ means the training points are partitioned into B disjoint blocks. $B = N$ means having the same number of blocks as training points, i.e., block size equal to 1. + +where we unbiasedly approximate the sum over blocks in eq. (9) using one random block to obtain the stochastic bound + +$$ +\widetilde {\mathcal {F}} _ {4, r} (\cdot) = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] + B \left[ \int q (\boldsymbol {u}) \log \mathcal {N} \left(\boldsymbol {y} _ {b}; \mathbf {K} _ {\mathbf {f} _ {b} \mathbf {u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \sigma^ {2} \mathbf {I} _ {b}\right) - \frac {1}{2} \log \left| \mathbf {I} _ {b} + \sigma^ {- 2} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} \right| \right], \tag {10} +$$ + +based on which we perform stochastic gradient updates by cycling over the $B$ blocks. If we judiciously choose the block size $\frac{N}{B}$ to be $M$ (i.e., block size equals to the number of inducing points), the computational requirement per iteration is only $\mathcal{O}(M^3)$ . Therefore, eq. (10) has a small implementation overhead compared to standard stochastic sparse GP objectives. The precise extra overhead involves taking the Cholesky decomposition of $\mathbf{I}_b + \sigma^{-2}\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b}$ , needed when computing the log-determinant regularisation term. + +We will next consider a special case. When we let all $\mathbf{m}_b$ matrices to be the same, $\mathbf{m}_b = \mathbf{m}$ , we arrive at the optimal $\mathbf{m}$ , $\mathbf{m} = (\mathbf{I}_b + B^{-1}\sigma^{-2}\sum_b\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b})^{-1}$ , and the resulting collapsed bound, + +$$ +\mathcal {F} _ {5} (\theta) = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q} _ {\mathbf {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {B}{2} \log | \mathbf {I} _ {b} + \frac {1}{B \sigma^ {2}} \sum_ {b} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} |. \tag {11} +$$ + +Since the log-determinant is a concave function on the cone of positive definite matrices, we can apply Jensen's inequality to show that the bound above is less tight compared to eq. (8). As the block size equals one, this becomes the spherical bound in Titsias (2025); Artemev et al. (2021). + +A disadvantage of diagonal and block diagonal structures in $\mathbf{M}$ is the expensive predictive covariance. However, we can approximate it by reverting to using $q(\pmb {f}|\pmb {u})\approx p(\pmb {f}|\pmb {u})$ at test time. Bui et al. (2025) noted that this approximation does not degrade the performance compared to the expensive exact predictive distribution. In other words, in practice, we only use the new structured posterior in training, and therefore, any improvement in predictive performance at test time will come from better $q(u)$ and hyperparameters. + +# 4 A more general approximation based on Power Expectation Propagation + +Although the variational sparse GP approach has captured the spotlight in the sparse GP literature, Bui et al. (2017) showed various variants of PEP can be as competitive or better. We will now revisit the framework of Bui et al. (2017) and explore how it can be improved by leveraging the recent innovation in structured posterior approximations (Titsias (2025); Bui et al. (2025) and section 3) originally developed in the variational inference setting. We first write down the joint density of the exact model and the approximate posterior, + +$$ +p (f, \boldsymbol {y}) = p \left(f _ {\neq} \boldsymbol {f}, \boldsymbol {u} \mid \boldsymbol {f}, \boldsymbol {u}\right) p (\boldsymbol {f} \mid \boldsymbol {u}) p (\boldsymbol {u}) \prod_ {n = 1} ^ {N} p \left(y _ {n} \mid \boldsymbol {f} _ {n}\right) \tag {12} +$$ + +$$ +q (f) \propto p \left(f _ {\neq} f, u | f, u\right) q (f | u) p (u) \prod_ {b = 1} ^ {B} t _ {b} (u) \tag {13} +$$ + +where the $N$ training points are partitioned into $B$ disjoint blocks, and the factors $t_b(\pmb{u})$ are assumed to be Gaussian. Instead of using $q(\pmb{f}|\pmb{u}) = p(\pmb{f}|\pmb{u})$ as in Bui et al. (2017), we consider $q(\pmb{f}|\pmb{u}) = \mathcal{N}(\pmb{f};\mathbf{K}_{\mathrm{fu}}\mathbf{K}_{\mathrm{uu}}^{-1}\pmb{u};\mathbf{D}_{\mathrm{ff}}^{1/2}\mathbf{M}\mathbf{D}_{\mathrm{ff}}^{1/2})$ where $\mathbf{M} = \mathrm{blkdiag}(\{\mathbf{m}_b\}_{b=1}^B)$ , that is, the blocks in $\mathbf{M}$ match that of the likelihood partitions. + +The PEP procedure (Minka, 2004) iteratively updates $t_b(\pmb{u})$ by (i) first remove an $\alpha$ -fraction of $t_b(\pmb{u})$ from $q(f)$ to form the cavity distribution, $q^{\backslash b}(f) = q(f) / t_b^\alpha(\pmb{u})$ , (ii) incorporate an $\alpha$ -fraction of the likelihood for the $b$ -th block $p(\pmb{y}_b|\pmb{f}_b) = \prod_{n=1}^{N_b} p(y_n|f_n)$ to form the tilted distribution, $\tilde{q}(f) = q^{\backslash b}(f)p^\alpha(\pmb{y}_b|\pmb{f}_b)$ , (iii) find a new approximation $q(f)$ that minimises KL[ $\tilde{q}(f)||q(f)|$ , and (iv) adjust $t_b(\pmb{u})$ based on the new posterior using $t_b(\pmb{u}) = [q(f) / q^{\backslash b}(f)]^{1/\alpha}$ or $t_b(\pmb{u}) \gets t_b^{1-\alpha}(\pmb{u})[q(f) / q^b(f)]$ . These steps are repeated for all blocks until convergence. Readers might have noticed that step (iii) is a daunting task as it involves moment matching for the entire Gaussian processes; however, due to the structure of the approximate posterior $q(f)$ , it is sufficient to perform moment matching for the finite function values $\pmb{u}$ (Bui et al., 2017). In addition, this procedure returns an estimate of the log marginal likelihood that can be used for hyperparameter optimisation. + +Mirroring the derivation in Bui et al. (2017), we can show the optimal form for $t_b(\pmb{u})$ has rank $N_b = |\pmb{f}_b|, t_b(\pmb{u}) = \mathcal{N}(\mathbf{K}_{\mathbf{f}_b\mathbf{u}}\mathbf{K}_{\mathbf{uu}}^{-1}\pmb{u}; \pmb{g}_b, \pmb{v}_b)$ . In the regression case, $\pmb{g}_b = \pmb{y}_b$ and $\pmb{v}_b = \alpha [\mathbf{D}_{\mathbf{ff}}^{1/2}\mathbf{M}\mathbf{D}_{\mathbf{ff}}^{1/2}]_{bb} + \sigma^2\mathbf{I}_b$ . The full derivation is rather lengthy and is included in the appendix; however, one can verify that this is a stable fixed point of the procedure by noting that the $\alpha$ -fraction of $t_b(\pmb{u})$ is identical to the contribution of $p^\alpha(y_b|\pmb{f}_b)$ to the posterior at $\pmb{u}$ , $\int d\pmb{f}_b q(\pmb{f}_b|\pmb{u}) p^\alpha(\pmb{y}_b|\pmb{f}_b)$ . The optimal $q(\pmb{u})$ is thus $q(\pmb{u}) \propto p(\pmb{u})\mathcal{N}(\pmb{y}; \mathbf{K}_{\mathbf{f}\mathbf{u}}\mathbf{K}_{\mathbf{uu}}^{-1}\pmb{u}, \alpha \mathrm{blkdiag}([\mathbf{D}_{\mathbf{ff}}^{1/2}\mathbf{M}\mathbf{D}_{\mathbf{ff}}^{1/2}]_{bb}]_{b=1}^B) + \sigma^2\mathbf{I}_N)$ . Furthermore, for the regression case, we can further derive the approximate marginal likelihood, + +$$ +\begin{array}{l} \mathcal {F} _ {5, r c} (\theta , \mathbf {M}) = \log \mathcal {N} (\boldsymbol {y}; \mathbf {0}, \mathbf {Q} _ {\mathbf {f f}} + \alpha \operatorname {b l k d i a g} \left(\left\{\left[ \mathbf {D} _ {\mathbf {f f}} ^ {1 / 2} \mathbf {M D} _ {\mathbf {f f}} ^ {1 / 2} \right] _ {b b} \right\} _ {b = 1} ^ {B}\right) + \sigma^ {2} \mathbf {I} _ {N}) \\ + \sum_ {b} \left[ - \frac {1 - \alpha}{2 \alpha} \log \left| \mathbf {I} _ {b} + \alpha \frac {[ \mathbf {D} _ {\mathbf {f f}} ^ {1 / 2} \mathbf {M D} _ {\mathbf {f f}} ^ {1 / 2} ] _ {b b}}{\sigma^ {2}} \right| - \frac {1}{2 \alpha} \log | \mathbf {I} _ {b} + \alpha (\mathbf {m} _ {b} - \mathbf {I} _ {b}) | + \frac {1}{2} \log | \mathbf {m} _ {b} | \right]. \\ \end{array} +$$ + +We note that, for a general $\alpha$ and $\mathbf{M}$ , including diagonal and block-diagonal cases, the PEP procedure above as well the approximate marginal likelihood for the regression case is computationally intractable (i.e., cubic in $N$ ) due to the need to find the (block-)diagonal of $\mathbf{D}_{\mathbf{ff}}^{1/2} \mathbf{M} \mathbf{D}_{\mathbf{ff}}^{1/2}$ . We will now discuss the tractable special cases. + +Remark 1 When $\mathbf{M}$ is diagonal or block-diagonal, the approximate marginal likelihood and posterior approximation above are only tractable as $\alpha \rightarrow 0$ . Specifically, when $\mathbf{M} = \mathrm{diag}(\{m_n\}_{n=1}^N)$ , the objective becomes the variational bound of Titsias (2025); Bui et al. (2025), and when $\mathbf{M} = \mathrm{blkdiag}(\{\mathbf{m}_b\}_{b=1}^B)$ , the objective matches the variational bound in eq. (8). + +Remark 2 When $\mathbf{M} = m\mathbf{I}_N$ , the approximate marginal likelihood and posterior approximation are computationally tractable for all $\alpha$ 's. In particular, the optimal $q(\pmb{u})$ is $q(\pmb{u}) \propto p(\pmb{u})\mathcal{N}(\pmb{y}; \mathbf{K}_{\mathbf{fu}}\mathbf{K}_{\mathbf{uu}}^{-1}\pmb{u}, m\alpha \mathrm{blkdiag}(\{\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b}\}_{b=1}^B) + \sigma^2\mathbf{I}_N)$ , and the approximate marginal likelihood becomes, + +$$ +\begin{array}{l} \mathcal {F} _ {6} (\theta , m) = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q} _ {\mathbf {f f}} + m \alpha \operatorname {b l k d i a g} \left(\left\{\mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} \right\} _ {b = 1} ^ {B}\right) + \sigma^ {2} \mathbf {I} _ {N}) \\ - \frac {1 - \alpha}{2 \alpha} \sum_ {b} \left[ \log \left| \mathbf {I} _ {b} + \frac {\alpha m \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}}}{\sigma^ {2}} \right| \right] - \frac {N}{2 \alpha} \log (1 + \alpha (m - 1)) + \frac {N}{2} \log (m). \tag {14} \\ \end{array} +$$ + +In this special case, we note the following. First, as a sanity check, we can see that when $m = 1$ , we recover the Power-EP approximate marginal likelihood of Bui et al. (2017): + +$$ +\mathcal {F} _ {6, m = 1} (\theta) = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q} _ {\mathbf {f f}} + \alpha \mathrm {b l k d i a g} (\mathbf {D} _ {\mathbf {f f}}) + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1 - \alpha}{2 \alpha} \sum_ {n} \log \left| \mathbf {I} _ {b} + \alpha \frac {\mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}}}{\sigma^ {2}} \right|. +$$ + +Second, when $\alpha = 1$ , the objective in eq. (14) becomes $\mathcal{F}_{6,\alpha=1}(\theta,m) = \log \mathcal{N}(\pmb{y};0,\mathbf{Q}_{\mathbf{ff}} + m\mathrm{blkdiag}(\mathbf{D}_{\mathbf{ff}}) + \sigma^2\mathbf{I}_N)$ . When $m = 1$ , this becomes the PITC marginal likelihood and for block size equal to one it further reduces to FITC (Quinonero-Candela & Rasmussen, 2005). + +Third, as $\alpha \to 0$ , we recover the spherical bound in (Bui et al., 2025; Titsias, 2025; Artemev et al., 2021). Only in this setting, we can derive the optimal $m = (1 + N^{-1}\sum_{n}d_{n} / \sigma^{2})^{-1}$ + +Finally, inspired by the uncollapsed variational bound, we can optimise an uncollapsed version of eq. (14) that supports stochastic optimisation as follows, + +$$ +\begin{array}{l} \mathcal {F} _ {6, r} (q (\boldsymbol {u}), \theta , \mathbf {M}) = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] + \sum_ {b} \left[ \int q (\boldsymbol {u}) \log \mathcal {N} (\boldsymbol {y} _ {b}; \mathbf {K} _ {\mathbf {f} _ {b} \mathbf {u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, m \alpha \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} + \sigma^ {2} \mathbf {I} _ {b}) \right] \\ - \frac {1 - \alpha}{2 \alpha} \sum_ {b} \left[ \log \left| {\bf I} _ {b} + \frac {\alpha m {\bf D} {\bf f} _ {b} {\bf f} _ {b}}{\sigma^ {2}} \right| \right] - \frac {N}{2 \alpha} \log \left(1 + \alpha (m - 1)\right) + \frac {N}{2} \log (m). \\ \end{array} +$$ + +That is, instead of running the PEP procedure, we can optimise the objective above to yield the same fixed point as PEP. We attempt to visualise the connections between the methods, the special cases and the broader literature in fig. 1. + +# 5 Experiments + +Having described the new block-diagonal structure in sparse variational GPs and revisited the unified work of Bui et al. (2017) in light of the new approximate posteriors, we will detail the experiments to qualitatively investigate (i) if the proposed block-diagonal approximation in section 3 yields better performance and, if yes, how, and (ii) whether having $m \neq 1$ benefits power expectation propagation in section 4 the same way it does to variational inference. All experiments were done on either a V100 GPU or a MacBook laptop. We provide an implementation here https://github.com/thangbui/tighter_sparse_gp. + +# 5.1 1-D regression and biases in hyperparameter estimation + +We first illustrate the difference between the proposed and existing methods on a simple 1D regression problem (Snelson & Ghahramani, 2005). In particular, we compare Titsias' collapsed bound in eq. (4) [SGPR], the bound of Titsias (2025); Bui et al. (2025) in eq. (6) [T-SGPR], the bound with block diagonal M in eq. (8) with 10 and 20 blocks [20 and 10 data points per block, respectively, BT-SGPR], the PEP approach of Bui et al. (2017) with $\alpha = 0.5$ [PEP], and the PEP approach in eq. (14) with $B = N$ and $\alpha = 0.5$ [T-PEP]. We used 5 inducing points in this experiment. The key results are summarised in fig. 2. It can be observed that (i) the block-diagonal approximation improves over the diagonal one in this example, (ii) increasing the number of training points in each block tightens the bound, (iii) the structured posterior approximation also helps in PEP, and (iv) hyperparameter optimisation using a more structured approximation tend to result in a smaller noise variance and a larger kernel variance. We note that the PEP approximate marginal likelihood is not guaranteed to be a lower bound and therefore optimising it can result in pathological behaviours, for example, when $\alpha = 1$ , the noise variance can be severely underestimated (Bauer et al., 2016). + +![](images/9c8f8b8c57cf6bb5e92721d199424dfdd0a9abc31e92ec1db240d8aa7b7b219c.jpg) +Figure 2: Left: Variational bounds during training on the Snelson dataset. Middle and right: Predictive mean and intervals using various methods and the final hyperparameter values. + +![](images/4164f226fe4d45059d5794f4c3fb8a25b86850b31b7fecee914950052d5b19a3.jpg) + +![](images/2180083db5c4d96f16d65194a421170438956c4edd2618201ffc55fa629e034c.jpg) + +To further investigate point (iv), we picked a subset of the KIN40K dataset with 5,000 data points, and ran an experiment to compare the sparse approximations with exact GP. For each method, we recorded in table 1 the exact or approximate log marginal likelihood, the predictive performance measured by root mean squared error (RMSE) and log likelihood (LL), the noise standard deviation $\sigma$ and the kernel lengthscales. Similar to the observation in the Snelson dataset above, the noise estimate is smaller when moving from $\mathbf{M} = \mathbf{I}_N$ to increasingly more structured $\mathbf{M}$ , translating + +to better predictions. The trend seems to be consistent across two numbers of inducing points. In addition, there is no notable difference in the lengthscales between PEP, T-PEP, and the structured variational approximations; however, these methods tend to leverage more dimensions than SGPR for $M = 256$ . + +Table 1: Exact/approximate marginal likelihoods, predictive performance, and lengthscales given by various methods on 5,000 samples from the KIN40K dataset. + +
M = 256M = 512
MethodObj.RMSELLσlengthscalesObj.RMSELLσlengthscales
Exact-0.660.120.800.00-0.660.120.800.00
SGPR0.880.26-0.140.300.660.220.020.25
T-SGPR0.780.22-0.060.260.510.180.110.21
BT-SGPR [50]0.750.22-0.050.250.500.180.120.20
BT-SGPR [10]0.660.20-0.030.230.440.170.130.19
PEP [0.5]0.660.23-0.020.220.420.200.140.19
T-PEP [0.5]0.480.200.020.180.180.160.190.14
+ +# 5.2 Block-diagonal structured variational approximation + +We next ran an experiment to validate the utility of the proposed block-structured approximation in section 3 on four real-world regression datasets1. For each dataset and each inducing point configuration ( $M = 256$ or $M = 512$ ), we compare the uncollapsed variational bounds of Titsias (2009); Hensman et al. (2015) [eq. (3), SVGP], Titsias (2025); Bui et al. (2025) [eq. (5), T-SVGP], and the proposed bound in eq. (9) [BT-SVGP], corresponding to $\mathbf{M} = \mathbf{I}_N$ , $\mathbf{M} = \mathrm{diag}(\{m_n\}_{n=1}^N)$ , and $\mathbf{M} = \mathrm{blkdiag}(\{\mathbf{m}_b\}_{b=1}^B)$ , respectively. We repeated the experiment 10 times, each using a random train/test split, a batch size of 500 (also the block size), random partitioning of the training data into blocks, and 300 epochs for training. The average variational bound (ELBO) and test performance after training are shown in fig. 3. Similar to the earlier experiments, the benefit of the block-structured approximation is also clearly demonstrated here: it tightens the variational bound compared to that of the diagonal $\mathbf{M}$ and consistently yields comparable or better predictive performance. We note again that (i) the estimated observation noise tends to be smaller when employing the new bound (see the appendix), and (ii) there is a minimal implementation overhead compared to Titsias (2009, 2025); Bui et al. (2025) to result in these gains. + +![](images/eef9af4309a7c31e4d92e0936184ff130999b2989233223cfb8c7933b2a714a2.jpg) +Figure 3: Lower bounds (ELBO) and predictive performance of various variational methods with $\mathbf{M} = \mathbf{I}_N$ [SVGP], $\mathbf{M} = \mathrm{diag}\left(\left\{m_n\right\}_{n=1}^N\right)$ [T-SVGP], and $\mathbf{M} = \mathrm{blkdiag}\left(\left\{\mathbf{m}_b\right\}_{b=1}^B\right)$ [BT-SVGP]. + +# 5.3 Power-EP with a structured approximate posterior $[\mathbf{M} = m\mathbf{I}_N]$ + +As shown in section 4, the structured approximate posterior considered by Titsias (2025) can be utilised in PEP and in the regression case, the approximate posterior and marginal likelihood are analytically available. To evaluate its practical utility, we ran an experiment on five small regression datasets, comparing the PEP approach of Bui et al. (2017) $[\mathbf{M} = \mathbf{I}_N]$ to the proposed approach in section 4 $[\mathbf{M} = m\mathbf{I}_N]$ . The typical performance across various inducing point configurations is shown in fig. 4, with the full results included in the appendix. It is noticeable that the Power-EP scheme with $m \neq 1$ tends to outperform the corresponding setting when $m = 1$ . To elucidate the trend, we plot the difference between the performance of $\mathbf{M} = \mathbf{I}_N$ and $\mathbf{M} = m\mathbf{I}_N$ in fig. 5. We note that $m \neq 1$ outperforms $m = 1$ on all datasets in terms of RMSE, but log-likelihood performance degrades when $\alpha$ is closer to 1. These results suggest that for $m \neq 1$ , intermediate $\alpha$ values such as 0.5 are most competitive in terms of both RMSE and LL, in line with recommendations from Bui et al. (2017) when $m = 1$ . + +![](images/eed6c056cca18e1a15c56028be2ecfff5df638218e7b13df0e6a9f1cc3343677.jpg) +Figure 4: Predictive performance of power expectation propagation with $\mathbf{M} = \mathbf{I}_N$ and $\mathbf{M} = m\mathbf{I}_N$ on two UCI datasets. Results for other datasets are in the appendix. + +![](images/574d1097f123701e02dc023e3e5d0fbd84497bce19515a83f41d2d12205e7c90.jpg) +Figure 5: Difference in PEP performance between $\mathbf{M} = \mathbf{I}_N$ and $\mathbf{M} = m\mathbf{I}_N$ on five UCI datasets. + +# 6 Related work + +The use of inducing points for sparse approximations in Gaussian processes has a rich history, to name a few approaches, sparse online GPs (Csato & Opper, 2002), DTC (Seeger et al., 2003), FITC approximation (Snelson & Ghahramani, 2005), and PITC (Quinonero-Candela & Rasmussen, 2005). The most notable was the variational approach of Titsias (2009), who introduced a principled method for selecting inducing points by optimising a variational lower bound. Hensman et al. (2013, 2015) extended this approach to enable stochastic optimisation and non-Gaussian likelihoods, significantly broadening the applicability of sparse GPs to large datasets. Other work on inducing point methods have exploited Kronecker products (Wilson & Nickisch, 2015), nearest neighbour structures (Tran et al., 2021; Wu et al., 2022) and inter-domain inducing points (Lázaro-Gredilla & + +Figueiras-Vidal, 2009; Hensman et al., 2018). Also, recent theoretical work (Burt et al., 2020) studied the approximation convergence with respect to the number of inducing points. + +Our work is most closely related to the recent advances by Titsias (2025); Bui et al. (2025), who showed that relaxing the standard assumption with diagonal scaling matrices improves the variational bound. Our block-diagonal extension naturally builds upon this line of work, showing practical benefits. Similarly, our extension of the PEP framework builds directly on Bui et al. (2017), expanding their unifying perspective by incorporating structured posterior approximations. Note that our work is distinct from the PITC approximation. PITC is derived from the prior modification perspective, where the prior is modified so that the blocks of function values are conditionally independent given the inducing points. Bui et al. (2017) showed that this is equivalent to EP when retaining the prior conditional in the approximate posterior, which differs from our proposed structured conditional distribution. + +A key component in the sparse GP approximate posterior is $q(\pmb{u})$ , and imposing additional structures for this object will likely lead to improvement. For example, Shi et al. (2020) showed that $q(\pmb{u})$ can be parameterised by two sets of inducing points, orthogonal to each other, leading to better predictive performance at a much lower compute cost compared to doubling up the inducing points in the standard SVGP approximation. This line of work is complementary to our work here, as it focuses on a different aspect of the posterior, and thus, the two approaches can be combined. + +A well-known pathology of variational sparse GP regression is the large estimated observation noise variance (Bauer et al., 2016). It can be partially alleviated by changing the objective function (Jankowiak et al., 2019) or mixing separate schemes for learning and inference (Li et al., 2023). Our work shows that principled structured variational approximations can also partly address this issue. + +# 7 Summary + +Approximation schemes using inducing points are the method of choice for scaling GP models to large datasets. We show that (i) these methods can be improved by introducing additional structures in the approximate posterior and (ii) these new structures can be applied to various inference strategies, including PEP and variational inference. The resulting methods show comparable or better predictive performance and smaller hyperparameter estimation biases in many standard regression tasks. + +There are several potential future directions. First, we have assumed that the size of the data blocks in a dataset is the same and the data partitioning in the experiments was random, but these can be adjusted based on the data characteristics, potentially tightening the variational objective further. Second, the power hyperparameter $\alpha$ in PEP can be made private per block; this will require an understanding of when variational or EP might work best and how to dynamically select $\alpha$ . Third, a full discussion for non-Gaussian likelihoods and models beyond GP regression (e.g., deep GPs, GP latent variable models) and how they benefit from structured approximations is a promising exploratory direction. + +# Acknowledgments + +We would like to thank the anonymous reviewers for the feedback. TDB would like to thank the National Computational Infrastructure (NCI Australia), an NCRIS enabled capability supported by the Australian Government, for the computing resources. + +# References + +Artem Artemev, David R Burt, and Mark van der Wilk. Tighter bounds on the log marginal likelihood of Gaussian process regression using conjugate gradients. In International Conference on Machine Learning, pp. 362-372, 2021. +Matthias Bauer, Mark van der Wilk, and Carl Edward Rasmussen. Understanding probabilistic sparse Gaussian process approximations. In Advances in Neural Information Processing Systems, pp. 1533-1541, 2016. + +Thang D. Bui, Josiah Yan, and Richard E. Turner. A unifying framework for Gaussian process pseudo-point approximations using power expectation propagation. Journal of Machine Learning Research, 18(104):1-72, 2017. +Thang D. Bui, Matthew Ashman, and Richard E. Turner. Tighter sparse variational Gaussian processes, 2025. +David R. 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In International Conference on Machine Learning, pp. 1511-1520, 2016. +Martin Jankowiak, Geoff Pleiss, and Jacob R Gardner. Sparse Gaussian process regression beyond variational inference. 2019. +Miguel Lázaro-Gredilla and Aníbal Figueiras-Vidal. Inter-domain Gaussian processes for sparse inference using inducing features. In Advances in Neural Information Processing Systems, volume 22, 2009. +Rui Li, ST John, and Arno Solin. Improving hyperparameter learning under approximate inference in Gaussian process models. In International Conference on Machine Learning, 2023. +Yingzhen Li, Jose Miguel Hernández-Lobato, and Richard E. Turner. Stochastic expectation propagation. In Advances in Neural Information Processing Systems, pp. 2323-2331, 2015. +Haitao Liu, Yew Ong, Xiaobo Shen, and Jianfei Cai. When Gaussian process meets big data: A review of scalable GPs. IEEE Transactions on Neural Networks and Learning Systems, pp. 1-19, 01 2020. +Alexander G de G Matthews, James Hensman, Richard Turner, and Zoubin Ghahramani. On sparse variational methods and the Kullback-Leibler divergence between stochastic processes. In International Conference on Artificial Intelligence and Statistics, pp. 231-239, 2016. +Thomas Minka. Power EP. Technical report, Microsoft Research, 2004. +Yuan Qi, Ahmed H Abdel-Gawad, and Thomas P Minka. Sparse-posterior Gaussian processes for general likelihoods. In Conference on Uncertainty in Artificial Intelligence, pp. 450-457, 2010. +Joaquin Quinonero-Candela and Carl Edward Rasmussen. A unifying view of sparse approximate Gaussian process regression. Journal of Machine Learning Research, 6(65):1939-1959, 2005. +Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning. The MIT Press, 2006. +Matthias W. Seeger, Christopher K. I. Williams, and Neil D. Lawrence. Fast forward selection to speed up sparse Gaussian process regression. In International Workshop on Artificial Intelligence and Statistics, pp. 254-261, 2003. + +Jiaxin Shi, Michalis K. Titsias, and Andriy Mnih. Sparse orthogonal variational inference for Gaussian processes. In International Conference on Artificial Intelligence and Statistics, pp. 1932-1942, 2020. +Edward Snelson and Zoubin Ghahramani. Sparse Gaussian processes using pseudo-inputs. Advances in Neural Information Processing Systems, 18, 2005. +Michalis K. Titsias. Variational learning of inducing variables in sparse Gaussian processes. In International Conference on Artificial Intelligence and Statistics, pp. 567-574, 2009. +Michalis K. Titsias. New bounds for sparse variational Gaussian processes, 2025. +Gia-Lac Tran, Dimitrios Milios, Pietro Michiardi, and Maurizio Filippone. Sparse within sparse Gaussian processes using neighbor information. In International Conference on Machine Learning, pp. 10369-10378, 2021. +Andrew Wilson and Hannes Nickisch. Kernel interpolation for scalable structured Gaussian processes (KISS-GP). In International Conference on Machine Learning, pp. 1775-1784, 2015. +Luhuan Wu, Geoff Pleiss, and John P Cunningham. Variational nearest neighbor Gaussian process. In International Conference on Machine Learning, pp. 24114-24130, 2022. + +# A Full derivation of the block-diagonal variational bound + +We start with a general posterior approximation of the form: + +$$ +q (f) = p \left(f _ {\neq \mathbf {f}, \mathbf {u}} \mid \mathbf {f}, \mathbf {u}\right) q (\mathbf {f} \mid \mathbf {u}) q (\mathbf {u}) \tag {15} +$$ + +$$ +q (\boldsymbol {f} | \boldsymbol {u}) = \mathcal {N} (f; \mathbf {K} _ {\mathbf {f u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \mathbf {C}) \tag {16} +$$ + +where we have not specified the form of the covariance matrix $\mathbf{C}$ . The variational lower bound to the log marginal likelihood is + +$$ +\mathcal {F} (q, \theta) = - \mathrm {K L} [ q (\boldsymbol {u}) | | p (\boldsymbol {u}) ] - \int q (\boldsymbol {u}) \mathrm {K L} [ q (\boldsymbol {f} | \boldsymbol {u}) | | p (\boldsymbol {f} | \boldsymbol {u}) ] + \int q (\boldsymbol {u}) q (\boldsymbol {f} | \boldsymbol {u}) \log p (\boldsymbol {y} | \boldsymbol {f}) \tag {17} +$$ + +Setting the gradient wrt $q(\pmb{u})$ to zeros gives, $q(\pmb{u}) \propto p(\pmb{u}) \exp \left[ \int q(\pmb{f}|\pmb{u}) \log p(\pmb{y}|\pmb{f}) \right]$ . In the regression case, $p(\pmb{y}|\pmb{f}) = \mathcal{N}(\pmb{y}; \pmb{f}, \sigma^2\mathbf{I}_N)$ and thus, + +$$ +\begin{array}{l} \int q (\boldsymbol {f} | \boldsymbol {u}) \log p (\boldsymbol {y} | \boldsymbol {f}) = \int \mathcal {N} (\boldsymbol {f}; \mathbf {K} _ {\mathbf {f u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \mathbf {C}) \log \mathcal {N} (\boldsymbol {y}; \boldsymbol {f}, \sigma^ {2} \mathbf {I} _ {N}) (18) \\ = \log \mathcal {N} (\boldsymbol {y}; \mathbf {K} _ {\mathbf {f u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2 \sigma^ {2}} \operatorname {t r a c e} (\mathbf {C}). (19) \\ \end{array} +$$ + +The middle term in the bound can be simplified to, + +$$ +\int q (\boldsymbol {u}) \mathrm {K L} [ q (\boldsymbol {f} | \boldsymbol {u}) | | p (\boldsymbol {f} | \boldsymbol {u}) ] = \frac {1}{2} \operatorname {t r a c e} \left(\mathbf {D} _ {\mathbf {f f}} ^ {- 1} \mathbf {C}\right) + \frac {1}{2} \log | \mathbf {D} _ {\mathbf {f f}} | - \frac {1}{2} \log | \mathbf {C} | - \frac {N}{2}. \tag {20} +$$ + +Substituting this and the optimal $q(\pmb{u})$ back to the bound gives, + +$$ +\mathcal {F} = \log \mathcal {N} (\boldsymbol {y}; \mathbf {0}, \mathbf {Q} _ {\mathbf {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2} \operatorname {t r a c e} [ (\mathbf {D} _ {\mathbf {f f}} ^ {- 1} + \sigma^ {- 2} \mathbf {I} _ {N}) \mathbf {C} ] - \frac {1}{2} \log | \mathbf {D} _ {\mathbf {f f}} | + \frac {1}{2} \log | \mathbf {C} | + \frac {N}{2}. +$$ + +When $\mathbf{C} = \mathbf{D}_{\mathrm{ff}}^{1 / 2}\mathbf{M}\mathbf{D}_{\mathrm{ff}}^{1 / 2}$ , the collapsed bound above becomes, + +$$ +\mathcal {F} (\theta) = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q _ {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2} \sum_ {b} \left[ \frac {1}{\sigma^ {2}} \mathrm {t r a c e} [ \mathbf {m} _ {b} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} ] + \mathrm {t r a c e} [ \mathbf {m} _ {b} ] - \log | \mathbf {m} _ {b} | - N _ {b} \right]. +$$ + +We can find the gradient of the bound wrt $\mathbf{m}_b$ + +$$ +\frac {\partial}{\partial \mathbf {m} _ {b}} \mathcal {F} = \frac {1}{2} \left[ \sigma^ {- 2} \mathbf {D} _ {\mathbf {f} _ {b}, \mathbf {f} _ {b}} + \mathbf {I} _ {b} - \mathbf {m} _ {b} ^ {- 1} \right] \tag {21} +$$ + +Setting this to zero gives $\mathbf{m}_b = (\mathbf{I}_b + \sigma^{-2}\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b})^{-1}$ , and the resulting $\mathbf{m}$ -collapsed bound: + +$$ +\mathcal {F} = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q} _ {\mathbf {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {1}{2} \sum_ {b} \log | \mathbf {I} _ {b} + \sigma^ {- 2} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} |. \tag {22} +$$ + +When the block size is 1, the above bounds become the bounds presented in Titsias (2025); Bui et al. (2025). + +We now consider a special case when we let all $\mathbf{m}_b$ matrices to be the same, $\mathbf{m}_b = \mathbf{m}$ . The gradient wrt $\mathbf{m}$ in this case is, + +$$ +\frac {\partial}{\partial \mathbf {m}} \mathcal {F} = \frac {1}{2} \sum_ {b} \left[ \sigma^ {- 2} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} + \mathbf {I} _ {b} - \mathbf {m} ^ {- 1} \right]. \tag {23} +$$ + +This leads to the optimal $\mathbf{m}$ , $\mathbf{m} = (\mathbf{I}_b + B^{-1}\sigma^{-2}\sum_b\mathbf{D}_{\mathbf{f}_b\mathbf{f}_b})^{-1}$ , and the corresponding $\mathbf{m}$ -collapsed bound, + +$$ +\mathcal {F} = \log \mathcal {N} (\boldsymbol {y}; 0, \mathbf {Q} _ {\mathbf {f f}} + \sigma^ {2} \mathbf {I} _ {N}) - \frac {B}{2} \log | \mathbf {I} _ {b} + \frac {1}{B \sigma^ {2}} \sum_ {b} \mathbf {D} _ {\mathbf {f} _ {b} \mathbf {f} _ {b}} |. \tag {24} +$$ + +A special case is when the block size is only 1, we arrive at the spherical diagonal approximation $\mathbf{M} = m\mathbf{I}$ (Titsias, 2025; Artemev et al., 2021). Note that, since the log-determinant is a concave function on the cone of positive definite matrices, we can apply Jensen's inequality to show that the bound above (when all $\mathbf{m}$ blocks are the same) is less tight compared to the bound when all blocks are different. + +# B Power-EP posterior and approximate marginal likelihood + +# B.1 Power EP steps + +Given a data set of $N$ input-output pairs $\{\pmb{x}_n, y_n\}_{n=1}^N$ , we use $M$ pseudo-points $\pmb{y}$ at locations $\pmb{z}$ to approximate the exact posterior. Power-EP posits the following approximation to the joint: + +$$ +p (f, \boldsymbol {y}) = p (f _ {\neq \boldsymbol {f}, \boldsymbol {u}} | \boldsymbol {f}, \boldsymbol {u}) p (\boldsymbol {f} | \boldsymbol {u}) p (\boldsymbol {u}) \prod_ {b} p (\boldsymbol {y} _ {b} | \boldsymbol {f} _ {b}) \approx p (f _ {\neq \boldsymbol {f}, \boldsymbol {u}} | \boldsymbol {f}, \boldsymbol {u}) q (\boldsymbol {f} | \boldsymbol {u}) p (\boldsymbol {u}) \prod_ {b} t _ {b} (\boldsymbol {u}) = q (f) +$$ + +where we have partitioned the data into $B$ disjoint blocks, $b$ indexes blocks of data and $t_b(\pmb{u})$ are the approximate factors. Crucially, we employ a structured conditional approximate posterior $q(\pmb{f}|\pmb{u}) = \mathcal{N}(\pmb{f};\mathbf{K_{fu}}\mathbf{K}_{uu}^{-1}\pmb{u},\mathbf{D_{ff}}^{1 / 2}\mathbf{MD}_{\mathrm{ff}}^{1 / 2})$ . The Power-EP procedure with power $\alpha$ iteratively updates the factors $\{t_b\}_{b = 1}^B$ as follows: + +1. Deletion step: Compute the cavity distribution by removing a fraction $\alpha$ of one approximate factor: + +$$ +q ^ {\backslash i} (f) \propto \frac {q (f)}{t _ {i} ^ {\alpha} (\boldsymbol {u})} = p (f _ {\neq \boldsymbol {f}, \boldsymbol {u}} | \boldsymbol {f}, \boldsymbol {u}) q (\boldsymbol {f} | \boldsymbol {u}) \frac {q (\boldsymbol {u})}{t _ {i} ^ {\alpha} (\boldsymbol {u})} = p (f _ {\neq \boldsymbol {f}, \boldsymbol {u}} | \boldsymbol {f}, \boldsymbol {u}) q (\boldsymbol {f} | \boldsymbol {u}) q ^ {\backslash i} (\boldsymbol {u}), \tag {25} +$$ + +where $q(\pmb{u}) = p(\pmb{u})\prod_{b}t_{b}(\pmb{u})$ and $q^{\setminus i}(\pmb{u}) = q(\pmb{u}) / t_i^\alpha (\pmb{u})$ + +2. Projection step: First, compute the tilted distribution by incorporating a corresponding fraction of the true likelihood factor: + +$$ +\tilde {p} (f) = q ^ {\backslash i} (f) p ^ {\alpha} \left(\boldsymbol {y} _ {i} \mid \boldsymbol {f} _ {i}\right) = p \left(f _ {\neq \boldsymbol {f}, \boldsymbol {u}} \mid \boldsymbol {f}, \boldsymbol {u}\right) q (\boldsymbol {f} \mid \boldsymbol {u}) q ^ {\backslash i} (\boldsymbol {u}) p ^ {\alpha} \left(\boldsymbol {y} _ {i} \mid \boldsymbol {f} _ {i}\right) \tag {26} +$$ + +Second, project the tilted distribution onto the new approximate posterior using KL divergence: + +$$ +q (f) \leftarrow \underset {q (f)} {\arg \min } \mathrm {K L} [ \tilde {p} (f) | | q (f) ] \tag {27} +$$ + +Due to the structure of the approximate posterior, this minimisation is achieved when the moments at the pseudo-inputs are matched: $\mathbb{E}_{\tilde{p}(f)}[\phi(\boldsymbol{u})] = \mathbb{E}_{q(f)}[\phi(\boldsymbol{u})]$ , where $\phi(\boldsymbol{u}) = \{\boldsymbol{u}, \boldsymbol{u}\boldsymbol{u}^T\}$ are the sufficient statistics (Bui et al., 2017). In practice, this can be done by using the moment-matching shortcut involving the gradients of the log-normalising constant of the tilted distribution. + +3. Update step: Compute the new fraction by dividing the new approximate posterior by the cavity: + +$$ +t _ {i, \text {n e w}} ^ {\alpha} (\boldsymbol {u}) = \frac {q (f)}{q \backslash i (f)} \tag {28} +$$ + +The factor then is updated using $t_i(\boldsymbol{u}) = t_{i,\mathrm{new}}(\boldsymbol{u})$ or with damping, $t_i(\boldsymbol{u}) = t_{i,\mathrm{old}}^{1 - \alpha}(\boldsymbol{u}) \cdot t_{i,\mathrm{new}}^{\alpha}(\boldsymbol{u})$ . + +# B.2 Optimal factors + +The factors are parameterised as follows, + +$$ +t _ {b} (\boldsymbol {u}) = \mathcal {N} (\boldsymbol {u}; z _ {b}, \mathbf {T} _ {1, b}, \mathbf {T} _ {2, b}) = z _ {b} \exp \left(\boldsymbol {u} ^ {T} \mathbf {T} _ {1, b} - \frac {1}{2} \boldsymbol {u} ^ {T} \mathbf {T} _ {2, b} \boldsymbol {u}\right) \tag {29} +$$ + +The posterior distribution over $\pmb{u}$ is therefore $q(\pmb{u}) = \mathcal{N}(\pmb{u};\mathbf{m},\mathbf{S})$ , where + +$$ +\mathbf {S} ^ {- 1} = \mathbf {K} _ {\mathbf {u u}} ^ {- 1} + \sum_ {b} \mathbf {T} _ {2, b} \tag {30} +$$ + +$$ +\mathbf {S} ^ {- 1} \mathbf {m} = \sum_ {b} \mathbf {T} _ {1, b}. \tag {31} +$$ + +Similarly, the cavity distribution over $\pmb{u}$ is $q^{\backslash i}(\pmb {u}) = \mathcal{N}(\pmb {u};\mathbf{m}^{\backslash i},\mathbf{S}^{\backslash i})$ , where + +$$ +\mathbf {S} ^ {\backslash i, - 1} = \mathbf {K} _ {\mathbf {u u}} ^ {- 1} + \sum_ {b \neq i} \mathbf {T} _ {2, b} + (1 - \alpha) \mathbf {T} _ {2, i} = \mathbf {S} ^ {- 1} - \alpha \mathbf {T} _ {2, i} \tag {32} +$$ + +$$ +\mathbf {S} ^ {\backslash i, - 1} \mathbf {m} ^ {\backslash i} = \sum_ {b \neq i} \mathbf {T} _ {1, b} + (1 - \alpha) \mathbf {T} _ {1, i} = \mathbf {S} ^ {- 1} \mathbf {m} - \alpha \mathbf {T} _ {1, i}. \tag {33} +$$ + +The moments of the tilted distribution (and the new posterior) can be computed efficiently using the following shortcuts, + +$$ +\mathbf {m} = \mathbf {m} ^ {\backslash i} + \mathbf {V} _ {\mathbf {u} \mathbf {f} _ {i}} ^ {\backslash i} \frac {d \log \tilde {Z} _ {i}}{d \mathbf {m} _ {\mathbf {f} _ {i}} ^ {\backslash i}}, \tag {34} +$$ + +$$ +\mathbf {V} = \mathbf {V} ^ {\backslash i} + \mathbf {V} _ {\boldsymbol {u} \boldsymbol {f} _ {i}} ^ {\backslash i} \frac {d ^ {2} \log \tilde {Z} _ {i}}{d (\mathbf {m} _ {\boldsymbol {f} _ {i}} ^ {\backslash i}) ^ {2}} \mathbf {V} _ {\boldsymbol {f} _ {i} \boldsymbol {u}} ^ {\backslash i} \tag {35} +$$ + +where $\tilde{Z}_i = \int q^{\setminus i}(\pmb {f}_i)p^\alpha (\pmb {y}_i|\pmb {f}_i)d\pmb {f}_i$ is the normaliser of the tilted distribution. + +At convergence, the optimal form of $\mathbf{T}_{2,b}$ is rank- $N_{b}$ , $\mathbf{T}_{2,b} = \mathbf{w}_{b}\mathbf{v}_{b}^{-1}\mathbf{w}_{b}^{T}$ , where $\mathbf{w}_{b} = \mathbf{V}_{\mathbf{u}\mathbf{u}}^{\backslash b, - 1}\mathbf{V}_{\mathbf{u}\mathbf{b}}^{\backslash b} = \mathbf{K}_{\mathbf{u}\mathbf{u}}^{-1}\mathbf{K}_{\mathbf{u}\mathbf{f}_{b}}, \mathbf{v}_{b} = -d_{2}^{-1} - \mathbf{V}_{\mathbf{b}\mathbf{u}}^{\backslash b}\mathbf{V}_{\mathbf{u}\mathbf{u}}^{\backslash b, - 1}\mathbf{V}_{\mathbf{u}\mathbf{b}}^{\backslash b}$ , and $d_{2} = \frac{d^{2}\log Z_{b}}{d(m_{b}^{\backslash b})^{2}}$ . + +In the regression case, at convergence, $t_b(\pmb{u}) = \mathcal{N}(\mathbf{K}_{\mathbf{f}_b\mathbf{u}}\mathbf{K}_{\mathbf{uu}}^{-1}\pmb{u};\pmb{y}_b,\alpha [\mathbf{D}_{\mathbf{ff}}^{1 / 2}\mathbf{M}\mathbf{D}_{\mathbf{ff}}^{1 / 2}]_{bb} + \sigma^2\mathbf{I}_b)$ . We can check this by computing the contribution of an $\alpha$ fraction of the exact likelihood to the posterior $q(\pmb{u})$ , + +$$ +\begin{array}{l} \int q \left(\boldsymbol {f} _ {b} \mid \boldsymbol {u}\right) p ^ {\alpha} \left(\boldsymbol {y} _ {b} \mid \boldsymbol {f} _ {b}\right) d \boldsymbol {f} _ {b} = \int \mathcal {N} \left( \right.\boldsymbol {f} _ {b}; \mathbf {K} _ {\mathbf {f} _ {b} \mathbf {u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \left[ \mathbf {D} _ {\mathbf {f f}} ^ {1 / 2} \mathbf {M D} _ {\mathbf {f f}} ^ {1 / 2} \right] _ {b b} \mathcal {N} ^ {\alpha} \left(\boldsymbol {y} _ {b}; \boldsymbol {f} _ {b}, \sigma^ {2} \mathbf {I} _ {b}\right) d \boldsymbol {f} _ {b} (36) \\ \propto \mathcal {N} \left(\boldsymbol {y} _ {b}; \mathbf {K} _ {\mathbf {f} _ {b} \mathbf {u}} \mathbf {K} _ {\mathbf {u u}} ^ {- 1} \boldsymbol {u}, \left[ \mathbf {D} _ {\mathbf {f f}} ^ {1 / 2} \mathbf {M D} _ {\mathbf {f f}} ^ {1 / 2} \right] _ {b b} + \sigma^ {2} \mathbf {I} _ {b} / \alpha\right), (37) \\ \end{array} +$$ + +which is exactly an $\alpha$ -fraction of the optimal factor listed above. + +# B.3 Power-EP approximate marginal likelihood + +After convergence, Power EP provides an approximate log marginal likelihood: + +$$ +\begin{array}{l} \log Z _ {\text {P E P}} = \log \int p \left(f _ {\neq \boldsymbol {f}, \boldsymbol {u}} | \boldsymbol {f}, \boldsymbol {u}\right) q (\boldsymbol {f} | \boldsymbol {u}) p (\boldsymbol {u}) \prod_ {b} t _ {b} (\boldsymbol {u}) d f (38) \\ = \mathcal {G} (q (\boldsymbol {u})) - \mathcal {G} (p (\boldsymbol {u})) + \frac {1}{\alpha} \sum_ {b} \left[ \log \tilde {Z} _ {b} + \mathcal {G} \left(q ^ {\backslash b} (\boldsymbol {u})\right) - \mathcal {G} (q (\boldsymbol {u})) \right] + \frac {1}{\alpha} \log \tilde {Z} _ {q}, (39) \\ \end{array} +$$ + +where + +$$ +\log \tilde {Z} _ {b} = \log \int q (\boldsymbol {f} _ {b} | \boldsymbol {u}) q ^ {\backslash b} (\boldsymbol {u}) p ^ {\alpha} (\boldsymbol {y} _ {b} | \boldsymbol {f} _ {b}) d \boldsymbol {f} _ {b} d \boldsymbol {u} \tag {40} +$$ + +$$ +\log \tilde {Z} _ {q} = \log \int q ^ {1 - \alpha} (\boldsymbol {f} _ {b} | \boldsymbol {u}) q ^ {\backslash b} (\boldsymbol {u}) p ^ {\alpha} (\boldsymbol {f} _ {b} | \boldsymbol {u}) d \boldsymbol {f} _ {b} d \boldsymbol {u} \tag {41} +$$ + +$$ +\mathcal {G} (q (\boldsymbol {u})) = \frac {M}{2} \log (2 \pi) + \frac {1}{2} \log | \mathbf {V} | + \frac {1}{2} \mathbf {m} ^ {\intercal} \mathbf {V} ^ {- 1} \mathbf {m} \tag {42} +$$ + +$$ +\mathcal {G} (p (\boldsymbol {u})) = \frac {M}{2} \log (2 \pi) + \frac {1}{2} \log | \mathbf {K} _ {\mathbf {u u}} | \tag {43} +$$ + +$$ +\mathcal {G} \left(q ^ {\backslash b} (\boldsymbol {u})\right) = \frac {M}{2} \log (2 \pi) + \frac {1}{2} \log | \mathbf {V} ^ {\backslash b} | + \frac {1}{2} \mathbf {m} ^ {\backslash b, \top} \mathbf {V} ^ {\backslash b, - 1} \mathbf {m} ^ {\backslash b} \tag {44} +$$ + +In the regression case, following closely the steps in (Bui et al., 2017), we can derive the closed-form approximate log marginal likelihood + +$$ +\begin{array}{l} \log Z _ {\mathrm {P E P}} = \log \mathcal {N} (\boldsymbol {y}; \mathbf {0}, \mathbf {Q} _ {\mathbf {f f}} + \alpha \operatorname {b l k d i a g} \left(\left\{\left[ \mathbf {D} _ {\mathbf {f f}} ^ {1 / 2} \mathbf {M D} _ {\mathbf {f f}} ^ {1 / 2} \right] _ {b b} \right\} _ {b = 1} ^ {B}\right) + \sigma^ {2} \mathbf {I} _ {N}) \\ + \sum_ {b} \left[ - \frac {1 - \alpha}{2 \alpha} \log \left| \mathbf {I} _ {b} + \alpha \frac {\left[ \mathbf {D} _ {\mathbf {f f}} ^ {1 / 2} \mathbf {M D} _ {\mathbf {f f}} ^ {1 / 2} \right] _ {b b}}{\sigma^ {2}} \right| - \frac {1}{2 \alpha} \log | \mathbf {I} _ {b} + \alpha (\mathbf {m} _ {b} - \mathbf {I} _ {b}) | + \frac {1}{2} \log | \mathbf {m} _ {b} | \right] \tag {45} \\ \end{array} +$$ + +# B.4 Extension to classification + +Instead of working with individual factors, we can use the stochastic Power-EP parameterisation (Li et al., 2015), i.e., assuming contributions from all blocks to the posterior are equal $t_b(\pmb{u}) = t(\pmb{u})$ . In addition, instead of running stochastic Power-EP iteration, we can directly work with $q(\pmb{u}) \propto p(\pmb{u})t^{B}(\pmb{u})$ and optimise the Power-EP energy, also known as the black-box $\alpha$ -divergence objective (Hernández-Lobato et al., 2016). We will explore this direction in future work. + +# C Additional experimental results + +# C.1 Experimental set-up + +In addition to the details in the main text, we provide additional information here. For all experiments involving the block-diagonal matrix $M$ , we randomly partitioned the training data into $B$ blocks. In the Snelson, kin40k, and Power-EP experiments, we optimised the collapsed bound using the L-BFGS optimiser. In the block-diagonal experiments with medium-scale datasets, we used the Adam optimiser with a learning rate of 0.005. To initialise the inducing point locations, we picked $M$ random training inputs in the Snelson experiment, and employed k-means clustering for all other experiments. For the later datasets, we used the median distance between the data points to initialise the lengthscales and set the initial observation noise variance to 0.1. + +# C.2 Snelson dataset + +We compared several sparse variational GP variants, including SGPR, T-SGPR, and BT-SGPR, with $M = 10$ to exact GP regression, and the objective and hyperparameters collected during optimisation are included in fig. 6. We note that, by using structured approximations, (i) the variational bound that is provably tighter for fixed hyperparameters indeed is tighter in practice, and (ii) the observation noise variance (the kernel variance) is smaller (larger). + +![](images/ec6a9c796d85c40e238bf5f11004d8e862dd300493edb2f370fd454ec0841c50.jpg) + +![](images/10326c2e6095a4bf53c359db3fa4c56775a8b2aa8048d73d189fc1efeb7a2cda.jpg) + +![](images/7ffa446d05271bb2e6a99f9eee1e4ee1bba70bac10bc5d418db2f9b800f382de.jpg) +Figure 6: Objectives and hyperparameters provided by sparse variational and exact methods. + +![](images/7eb5fe88990852e8b02c9617fe1a3b63370f057c25e8b070310e020dfc9e6658.jpg) + +# C.3 KIN40K hyperparameters + +We include the full results, including the standard errors, for the KIN40K experiment in table 2. + +# C.4 Block-diagonal structured variational approximation + +In addition to the predictive performance metrics in the main text, we also recorded the estimated hyperparameters when using the new structured variational approximations. These results are included in fig. 7, and agree with observations in smaller datasets (Snelson and kin40k): kernel variance and observation noise variance tend to be larger and smaller, respectively, when using the improved bounds. + +# C.5 Power-EP + +We include the full results for all five datasets considered in the main text in fig. 8. + +Table 2: Exact/approximate marginal likelihoods, predictive performance, and lengthscales given by various methods on 5,000 samples from the KIN40K dataset, including the standard errors across three repeats + +
M = 256M = 512
MethodObj.RMSELLσlengthscalesObj.RMSELLσlengthscales
Exact-0.656 ± 0.0050.117 ± 0.0000.796 ± 0.0040.001 ± 0.000-0.656 ± 0.0050.117 ± 0.0000.796 ± 0.0040.001 ± 0.000
SGPR0.883 ± 0.0060.256 ± 0.001-0.136 ± 0.0020.299 ± 0.0010.660 ± 0.0060.215 ± 0.0010.022 ± 0.0020.252 ± 0.001
T-SGPR0.779 ± 0.0060.223 ± 0.001-0.057 ± 0.0020.259 ± 0.0010.515 ± 0.0050.184 ± 0.0000.115 ± 0.0010.206 ± 0.001
BT-SGPR B = 500.752 ± 0.0060.217 ± 0.000-0.045 ± 0.0020.250 ± 0.0010.499 ± 0.0060.181 ± 0.0000.120 ± 0.0020.201 ± 0.001
BT-SGPR B = 100.659 ± 0.0060.200 ± 0.001-0.032 ± 0.0020.227 ± 0.0010.437 ± 0.0060.173 ± 0.0000.133 ± 0.0020.186 ± 0.001
PEP α = 0.50.661 ± 0.0060.235 ± 0.001-0.015 ± 0.0020.225 ± 0.0010.422 ± 0.0050.200 ± 0.0010.140 ± 0.0020.187 ± 0.000
T-PEP α = 0.50.480 ± 0.0050.200 ± 0.0000.024 ± 0.0020.182 ± 0.0010.184 ± 0.0060.164 ± 0.0000.190 ± 0.0030.138 ± 0.001
+ +![](images/065b2db6d9ebe11da0f1c3fd58d882a6df23a7f0b779f9160372b45fb1581a50.jpg) +SVGP T-SVGP BT-SVGP M=256 M=512 Tighter gives higher values ---- Tighter gives lower values +Figure 7: Estimated hyperparameters by using SVGP, T-SVGP and BT-SVGP on four UCI datasets. + +![](images/58204dae8f342f5ce27a37a76c16b45eeab4e52281864e559b77f0532a9f237f.jpg) +Figure 8: A comparison between $\mathbf{M} = \mathbf{I}$ and $\mathbf{M} = m\mathbf{I}$ for Power Expectation Propagation. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. 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However, most generative models rely on dense grid-based pixels or latents, neglecting this inherent sparsity. In this paper, we explore modeling visual generation paradigm via sparse non-grid latent representations. Specifically, we design a sparse autoencoder that represents an image as a small number of latents with their positional properties (i.e., regions of interest, RoIs) with high reconstruction quality. We then explore training flow-matching transformers jointly on non-grid latents and RoI values. To the best knowledge, we are the first to address spatial sparsity using RoIs in generative process. Experimental results show that our sparse flow-based transformers have competitive performance compared with dense grid-based counterparts with significantly reduced lower compute, and reaches a competitive 2.76 FID with just 64 latents on class-conditional ImageNet $256 \times 256$ generation. + +# 1 Introduction + +Deep visual generative models have advanced significantly in recent years, achieving impressive visual quality on image [1, 2], video [3], and 3D domains [4]. Current visual generation pipelines typically start by encoding raw data (e.g., images) into compact latent representations with autoencoders, and then use diffusion, masking modeling, or autoregressive methods to generate such latents, as this pipeline exemplified by latent diffusion [1]. Flagship text-to-image models, e.g., Stable Diffusion [5] and FLUX [6], follow this line and compress spatial dimensions at typically $8 \times$ factor, significantly lowering computational costs and modeling complexity in generative training. + +Though being a core component of visual generation, autoencoders conventionally assume a grid-based space of latent structures with the uniform information density. However, natural images often exhibit highly non-uniform information density and require adaptive computation across spatial locations [7, 8]. For example, in a landscape image, the sky background occupies numerous pixels while being worth fewer latent units to reconstruct and generate. In contrast, intricate foreground objects may require more latents to capture their details. Existing visual generation pipelines fail to address this point, as they rely on dense uniform grid-based latent structures and cannot adaptively allocate more computation to intricate foregrounds. + +This paper aims to study this point. First, we propose a sparse visual autoencoder that learns to compress an image into a set of sparse non-grid latents along with their positional property, i.e., region of interests (RoIs), and then recover image pixels from them. The RoIs explicitly characterize the spatial locations of the latents in bounding box formulation, and can be learned jointly with latents in an end-to-end manner by the plain reconstruction loss. The resulting sparse visual autoencoder reaches high compression rates by prioritizing latents to detailed regions while maintaining high reconstruction fidelity. Then, we design sparse flow-based transformers to generate latents and RoIs by modeling the joint flow of them with the velocity prediction in the denoising process [9, 10]. At + +![](images/68dfd5d85bdfbc9630c4c45e40cab853e74a59394f6d8291f9b42a3794fc3b66.jpg) +Figure 1: Left: conventional autoencoders encode pixels into latent grid representations. Right: our method encodes them into fewer non-grid latents with region of interests (RoIs). + +every timestep, our model learns to estimate both latent and RoI velocity from initial noise to target samples. Divergent from prior grid-based latent approaches, our method dynamically adjusts latent spatial positions during sampling via ordinary differential equations (ODEs) at inference, allowing adaptive refinement of both content and spatial focus. + +We show the feasibility of representing and generating images with sparse latents and RoIs on the challenge ImageNet benchmark. Our proposed sparse flow autoencoder, SF-VAE, can represent $256 \times 256$ images with just 64 latents with 0.70 reconstruction FID, or even down to 32 latents with 1.70 rFID. Then, the presented sparse flow-based transformers, SF-SiTs, have competitive performance on par with diffusion/flow-based grid-based transformers [11, 10]. The largest SF-SiT, XL variant, can reach 2.76 FID with classifier-free guidance [12] on the class-conditional ImageNet generation benchmark with just 64 latents. + +# 2 Related Work + +Diffusion models. In recent years, generative models has been marked as a breakthrough in the field of visual synthesis [1, 13, 14]. Commercial systems like DALL-E [2] or FLUX [6] are typically rooted in denoising diffusion architectures. The seminal work, denoising diffusion probabilistic models [15], take the image generative process as a gradual denoising trajectory, iteratively refining pure noise into target images. Building on this foundation, subsequent advancements further accelerated and refined diffusion-based generation. Improved variants including [16, 17, 9] investigate the training and sampling trajectories, enabling high-quality results with fewer sampling steps. Latent diffusion models [1] democratize the high resolution image synthesis by operating in a compressed latent space with reduced computational costs. The following up work, including diffusion/flow-matching transformers [18, 11, 10], also follows this convention to speed up training. Although training diffusion models directly on raw pixels is technically feasible [19, 20], the preference for latent space modeling stems from practical challenges: raw pixel data often contains high-frequency details and perceptually complex patterns that are computationally intensive and difficult for diffusion processes to model effectively. + +Latent space for diffusion models. The compact latent space is crucial for diffusion models to achieve high-quality image synthesis. Latent diffusion models [1] propose to train an autoencoder to map raw pixels to a latent space first, where the latent space is typically $8 \times$ spatially downsampled and comes with 4 channels, reaching a compression rate of 48. The follow up work on autoencoders, including [5, 21, 22], mainly investigate the channel number and shows that increasing channel number can improve the quality of diffusion samples via larger transformer models. Recently proposed deep compression autoencoders (DC-AE) [23] compress the latent space at more aggressive spatial downsampling rates, e.g., 32 or 64, further reducing the training cost of diffusion models. + +However, there is a lack of exploration and discussion on the structure of latents for diffusion models. Most autoencoders for diffusion models encode pixels into dense 2D grid-based latents and ignore the underlying non-uniform and sparse structures in natural images, where a background region in an image might be worth less latents than foregrounds. Here in this paper, we study this sparsity as well as visual non-uniformity explicitly with region of interests (RoIs) along with latents for diffusion models, following the sparse visual generation research line [8]. + +# 3 Methods + +Our image synthesis pipeline follows the common practice of latent diffusion models: an autoencoder first that compresses images into a set of latents and RoIs, followed by a generative model that takes noised latents and RoIs as input and predicts diffusion targets. We first describe the design of our sparse flow autoencoders, and then introduce sparse flow-based generative transformers for modeling joint flow. + +# 3.1 Sparse Flow Autoencoders + +Given an image $\mathbf{I} \in \mathbb{R}^{H \times W \times 3}$ , conventional autoencoders encode it into grid-based latent representations and decode latents back into pixels by the encoder $\mathcal{E}$ and decoder $\bar{\mathcal{D}}$ : + +$$ +\mathbf {z} = \mathcal {E} (\mathbf {I}) \in \mathbb {R} ^ {H / f \times W / f \times d}, \tag {1} +$$ + +$$ +\hat {\mathbf {I}} = \mathcal {D} (\mathbf {z}) \in \mathbb {R} ^ {H \times W \times 3}, \tag {2} +$$ + +where $f$ is the downsampling factor, typically 8 in practice, and $d$ is the latent dimension. The training for $\mathcal{E}$ and $\mathcal{D}$ is done by minimizing the reconstruction loss $\ell_{\mathrm{rec}}(\mathbf{I},\hat{\mathbf{I}})$ . Variational autoencoders also impose a Kulback-Leibler divergence loss $\ell_{\mathrm{KL}}$ on $\mathbf{z}$ to regularize the latent distribution [24]. + +Latent and RoI representation. Different from grid-based latent representations, we propose sparse flow variational autoencoders, SF-VAE, to use a sparse set of latents $\mathbf{z} \in \mathbb{R}^{N \times d}$ with their corresponding regions of interests (RoIs) $\mathbf{r} \in \mathbb{R}^{N \times 4}$ to represent an image. The latent space of SF-VAE is simply structured as one flattened dimension space, and the spatial property of latents is characterized in RoIs. The RoIs are represented as bounding boxes in the format of $(x, y, h, w)$ , where $(x, y)$ and $(h, w)$ are center points and height and width. The encoder $\mathcal{E}$ now outputs both latents and RoIs, and the decoder $\mathcal{D}$ takes both latents and RoIs as input to reconstruct raw pixels: + +$$ +(\mathbf {z}, \mathbf {r}) = \mathcal {E} (\mathbf {I}), \quad \tilde {\mathbf {I}} = \mathcal {D} (\mathbf {z}, \mathbf {r}). \tag {3} +$$ + +By eliminating the grid-based spatial prior, the number of latents $N$ can be decoupled from image pixels $H\times W$ and can be arbitrarily chosen and further greatly reduced according to our experiments. + +Encoding pixels into latents and RoIs. Typical object detectors [25, 26] can encode image pixels into latent representations and RoIs. However, they often lack emphasis on backgrounds and need bounding box annotations as individual supervision. Here, we resort to the SparseFormer architecture [27] to build our encoder $\mathcal{E}$ from scratch, which can encode image pixels into latents and RoIs in an end-to-end manner without bounding box supervision. SparseFormer takes early image features $\tilde{\mathbf{I}}$ as input and gradually refines latents $\mathbf{z}$ and RoIs $\mathbf{r}$ via local image features within RoIs $\mathbf{r}$ by several SparseFormer transformer layers, where refinement on $\mathbf{z}$ and $\mathbf{r}$ are both differentiable: + +$$ +\left(\mathbf {z} ^ {t}, \mathbf {r} ^ {t}\right) = \operatorname {S P A R S E F O R M E L A Y E r} _ {t} \left(\tilde {\mathbf {I}}, \mathbf {z} ^ {t - 1}, \mathbf {r} ^ {t - 1}\right), \tag {4} +$$ + +where $\mathbf{z}^0$ and $\mathbf{r}^0$ are parameters of the model. The RoIs $\mathbf{r}$ are updated using delta-formulation [25] on $(x,y,h,w)$ . + +Decoding latents and RoIs back into pixels. To learn the distribution of latents and RoIs given an image by just pixel reconstruction, we need to decode them back into raw pixels in a fully differentiable way, that is, the reconstruction loss needs to be both differentiable to latents and RoIs. Unfortunately, to our best knowledge, mapping latents and RoIs back into raw pixels in a differentiable way has not been deeply investigated in previous works. We have first tried a simple idea, RoI-aware cross attention, but they struggle to recover high frequency details even in a long training schedule, and cannot learn compact latent RoIs with pixel reconstruction. + +Inspired by advance in neural rendering [28, 29], we design our decoder $\mathcal{D}$ with the neural field approach and divide-and-conquer strategy. Specifically, we consider a latent and its RoI indexed by $i$ as a neural field function $\mathcal{F}_{(\mathbf{z}_i,\mathbf{r}_i)}:\mathbb{R}^2\to \mathbb{R}^3$ whose input is the pixel coordinate and output is the RGB tuple. In other words, a latent and its RoI can be decoded into a pixel image individually. Considering rich high frequency details in natural images [30], we design a neural field based on cosine transform bases similar to 2D discrete cosine transform (DCT) [31] but in a continuous + +![](images/a08d11efe3e007df01fd1883cabcb3056b4efbc24ab935bd7bf34a501053e5ab.jpg) +Figure 2: Structure of our SF-VAE and neural field-based decoder. For a latent tuple $(\mathbf{z}_i, \mathbf{r}_i)$ , the neural field function $\mathcal{F}_{(\mathbf{z}_i, \mathbf{r}_i)}$ first transforms pixel coordinates into 2D cosine bases conditioned on $\mathbf{r}_i$ and then feed these bases into an MLP whose parameters are conditioned on $\mathbf{z}_i$ to get downsampled intermediate features. The pixel decoder then decode these features into raw image pixels. For clarity, we show only a single latent case. + +coordinate space. Given a raw pixel's coordinate $(x,y)$ in the image, we first compute relative coordinate $(x',y')$ with regard to the RoI $\mathbf{r}_i$ for a single pixel output: + +$$ +x ^ {\prime} = \left(x - x _ {i}\right) / w _ {i}, \quad y ^ {\prime} = \left(y - y _ {i}\right) / h _ {i}, \tag {5} +$$ + +then compute our customized cosine transform bases $\mathbf{X} \in \mathbb{R}^{C}$ : + +$$ +\mathbf {X} _ {c} = \cos [ \mathbf {f} _ {y} (c) (y ^ {\prime} + 0. 5) \pi ] \cos [ \mathbf {f} _ {x} (c) (x ^ {\prime} + 0. 5) \pi ], \tag {6} +$$ + +where the number of bases $C$ needs to be a squared number, and $\mathbf{f}_y(c)$ and $\mathbf{f}_x(c)$ are the frequency of the $c$ -th channel in $y$ and $x$ directions, linearly increasing from 0 to $\sqrt{C} - 1$ . + +We use a two-layered MLP to transform these cosine bases into final output values, where MLP's weights are conditioned by the latent $\mathbf{z}_i$ through a shared feed forward layer. Note that all these computation can be done with matrix multiplication efficiently. However, we find that directly recovering raw RGB pixels from $\mathcal{F}_{(\mathbf{z}_i,\mathbf{r}_i)}$ is not memory friendly despite being efficient, since we need to "trace" 65536 pixels with their bases and MLP activations for a single latent in a $256\times 256$ image, which is unrealistic. Therefore, we retarget the neural field to output a downsampled feature map $\mathbf{I}'\in \mathbb{R}^{H / 8\times W / 8\times C}$ , use the softmax function to blend different feature maps produced by different latents, and use a upsampling decoder, $\mathcal{D}_{\mathrm{pix}}$ , to upsample $\mathbf{I}'$ to the final image $\hat{\mathbf{I}}$ of the input size. + +Overall autoencoder architecture. In the resulting SF-VAE, a latent does not need to correspond to a fixed spatial region across different images, and we can decouple the number of latents $N$ from the image size and further reduce $N$ . Our default number of latents and RoIs is $N = 64$ for a $256 \times 256$ image, where the latent dimension $d$ is 32, reaching a high $96 \times$ compression rate2. We train SF-VAE in an end-to-end manner using regular VAE loss with a minor difference: we only apply the KL loss to latent variables $\mathbf{z}$ since RoIs $\mathbf{r}$ do not necessarily follow a standard Gaussian distribution. We also design our SF-VAE to be as parameter lightweight as possible. Visualizations in Figure 3 show that SF-VAE can learn semantic and compact latent RoIs and achieve high reconstruction quality, allowing latents to focus on intricate foreground objects. + +# 3.2 Sparse Flow Transformers + +Now we have compact latent space defined by latent variables $\mathbf{z}$ and RoIs $\mathbf{r}$ for an image, we describe study how to generate them jointly. Here we first describe the flow matching framework [9, 32] for continuous data modeling, and then delve into our flow formulation and the design of our flow-based generative transformers. + +Flow matching. Assume we want to model a target data distribution $p(\mathbf{x})$ , flow matching formulates a process starting from a sample drawn from the starting distribution, typically $\mathbf{x}_0 \sim \mathcal{N}(0,1)$ , to a + +![](images/d9d059093e7b69d976ec182af4d3258687e57fcfc6ae71ae3179735044318216.jpg) +Figure 3: Visualizations of SF-VAE, and learned latent RoIs $(N = 64)$ . Note different colors are only used for distinguishing different RoIs. + +![](images/7697c5887cff7c90f164843ac0416b13c0ad8371926080a301aca43e09a16049.jpg) + +![](images/a724af12578f7e036618ae2cb9a8a7de0b335ef392b4f4f194e22f4aa8a13609.jpg) +Figure 4: Asynchronous interpolating schedule for latents and RoIs. RoIs approach target RoIs faster than latents as time $t$ increases using $\beta = 2$ polynomial schedule. + +![](images/4585f9cd1192dc52aa374e153d0c47fa1fb0f65646a2f46d219df47ada781c5c.jpg) + +sample from the target distribution $\mathbf{x}_1\sim p(\mathbf{x})$ by continuous time process: + +$$ +\mathbf {x} _ {t} = \sigma_ {t} \mathbf {x} _ {0} + \alpha_ {t} \mathbf {x} _ {1}, \tag {7} +$$ + +where $t \in [0,1]$ , $\sigma_t$ and $\alpha_t$ are interpolating coefficients characterizing an interpolating flow. The flow matching requires $\sigma_0 = 1$ , $\alpha_0 = 0$ , $\sigma_1 = 0$ , and $\alpha_1 = 1$ . A popular choice is the linear interpolation, $\sigma_t = 1 - t$ and $\alpha_t = t$ . The velocity, or the derivation on $\mathbf{x}_t$ in this process is + +$$ +\mathbf {v} _ {t} = \frac {d \mathbf {x} _ {t}}{d t} = \dot {\sigma} _ {t} \mathbf {x} _ {0} + \dot {\alpha} _ {t} \mathbf {x} _ {1}. \tag {8} +$$ + +The model $\mathcal{F}$ is trained by minimizing the $L_{2}$ loss between the predicted velocity and the true velocity over sampled points and sampled paths: + +$$ +\mathbb {E} _ {t, \mathbf {x} _ {0}, \mathbf {x} _ {1}} \| \mathcal {F} (t, \mathbf {x} _ {t}) - \mathbf {v} _ {t} \| ^ {2}. \tag {9} +$$ + +During inference, flow matching draws an initial sample $\mathbf{x}_0\sim \mathcal{N}(0,1)$ and solve the ordinary differential equation (ODE) to get the target sample $\mathbf{x}_1$ : + +$$ +\frac {d \mathbf {x} _ {t}}{d t} = \mathcal {F} (t, \mathbf {x} _ {t}). \tag {10} +$$ + +Flow matching eliminates the noise introduction in the sampling stage [15] and can model most arbitrary distribution, which is suitable for our RoI distribution modeling. Recent SiT models [10] also shows flow matching as the prediction target with transformers [33] can surpass ones with the diffusion target. + +Joint latent and RoI flow. Different from SiT models only to model grid-based latents, we need to model both the joint distribution of latents $\mathbf{z}$ with their RoIs $\mathbf{r}$ by SF-VAE in our flow-based generative transformers. The latents $\mathbf{z}$ in SF-VAE are not structured as a grid, and their positional properties are encoded in RoIs $\mathbf{r}$ . Here, we keep most of the transformer architecture in SiT unchanged but retarget it to take both latents and RoIs as input and predict the velocity of both them: + +$$ +\left(\hat {\mathbf {v}} _ {z, t}, \hat {\mathbf {v}} _ {r, t}\right) = \mathcal {F} \left(t, \mathbf {z} _ {t}, \mathbf {r} _ {t}\right), \tag {11} +$$ + +where $\mathbf{z}_t$ and $\mathbf{r}_t$ are interpolated latents and RoIs following Equ 7 at time $t$ , $\hat{\mathbf{v}}_{z,t}$ and $\hat{\mathbf{v}}_{r,t}$ are estimated velocity of latents and RoIs. We name it as SF-SiT for modeling joint latent and RoI flow. We choose the initial distribution of latents and RoIs both to be standard Gaussian distributions for simplicity. + +Given the number of latents $N$ already being highly reduced (e.g., 64), we do not perform any latent "grouping" and "ungrouping" operations in SiT to reduce computation. In other words, a latent from SF-VAE directly corresponds to a token in the transformer in SF-SiT. + +RoI-based positional encoding. Since we get rid of the grid-based latents, it is crucial to inject positional information dependent on RoIs $\mathbf{r}_t$ into the transformer. Otherwise, the latents $\mathbf{z}$ becomes fully permutation invariant due to the transformer's architecture. We use the sinusoidal positional encoding on four values of RoIs, $(x,y,h,w)$ , and concatenate them together. We add them as position encoding to a corresponding token in the transformer. It is worth noting that this positional encoding is totally based on RoIs and therefore changes for the same latent across different training and inference timesteps as RoIs move across different timesteps. + +Asynchronous interpolation schedule. The time $t$ indicates the signal-to-noise ratio (SNR) in the interpolation schedule, where $t = 0$ means pure noise and $t = 1$ means target data. Recall that we need to model both $\mathbf{z}$ and $\mathbf{r}$ and therefore SF-SiT are required to handle pure noise RoIs at time $t = 0$ as well as not-so-clean RoIs in early $t$ period. This differs from the regular SiT where the position information is always clean, and the model have strong position information given at any $t$ to predict the latent velocity to denoise them. Therefore, we design an asynchronous interpolation schedule for RoIs compared with latents, where we interpolate RoIs to approach the target RoIs faster than latents in a polynomial way: + +$$ +\sigma_ {\mathbf {r}, t} = (1 - t) ^ {\beta}, \quad \alpha_ {\mathbf {r}, t} = 1 - (1 - t) ^ {\beta}, \tag {12} +$$ + +where $\beta$ is a control parameter, and reduces to the synchronous linear interpolation when $\beta = 1$ . We keep the linear interpolation schedule untouched for latents: + +$$ +\sigma_ {\mathbf {z}, t} = 1 - t, \quad \alpha_ {\mathbf {z}, t} = t. \tag {13} +$$ + +This asynchronous interpolation schedule for latents and RoIs allows the velocity prediction for $\mathbf{z}_t$ to be more position-aware with cleaner RoIs as $\mathbf{r}_t$ approaches the target RoIs faster, as shown in the Fig 4. It is worth noting that the asynchronous interpolation schedule only affects the training phase, i.e., the velocity computation of latents and RoIs. The inference logic is the same as regular SiT, where we solve the ODE to sample the target latents and RoIs. To force the model to learn the accurate RoI distribution, we also impose an additional $L_{1}$ between the predicted RoI velocity and the ground-truth, with a balancing weight $w_{L_1}$ , as proposed by LayoutFlow [34] to model bounding box flow more accurately in layout generation. + +# 4 Experiments + +To verify the feasibility of synthesizing images with sparse non-grid latents, we conduct experiments on the standard ImageNet benchmark [35], mainly on $256 \times 256$ images. We first discuss our autoencoders, SF-VAE, and then present the results of our generative model, SF-SiT. Note that our aim is not to achieve new state-of-the-art results on a benchmark, but to explore sparsity in image generation pipeline. + +# 4.1 Sparse Flow Autoencoders + +Setup. As discussed in previous section, our SF-VAE consists up of a SparseFormer encoder $\mathcal{E}$ and a neural field-based decoder $\mathcal{D}$ , where the latent space defined by latent variables and RoIs. To keep as lightweight as possible, we design a SparseFormer encoder of 8 blocks with transformer dimension 512 to extract latents and RoIs, where the leading 4 blocks extract RoI regional features from the image. The neural field decoder also consists of 8 transformer blocks of $512\dim$ , and then + +Table 1: Reconstruction results on 50K ImageNet validation samples. + +
resolutionmethodlatent shapeparamsrFID↓PSNR↑SSIM↑LPIPS↓
256 × 256SD-VAE-ema-f832 × 32 × 484M0.6324.980.8040.062
DC-AE-in-f32c328 × 8 × 32323M0.7723.920.7650.086
DC-AE-mix-f32c328 × 8 × 32323M0.9623.750.7630.088
SF-VAE64 × 32133M0.7023.240.7430.085
512 × 512SD-VAE-ema-f864 × 64 × 484M0.1927.360.8490.061
DC-AE-in-f32c3216 × 16 × 32323M0.2026.230.8150.078
SF-VAE256 × 32133M0.2925.030.7870.088
+ +produce dynamic MLP's parameters, where the hidden dimension of two-layer MLP is 64. The output of the neural field decoder is then decoded into raw pixels by $\mathcal{D}_{\mathrm{pixel}}$ of conventional VAE decoder architectures [1] but we reduce the dimension of most convolutional layers. The computational cost of the resulting SF-VAE decoder is 153GFLOPs for a $256 \times 256$ image, where the SD-VAE-ema-f8 decoder needs 311GFLOPs. The latent $\mathbf{z}$ is 32-dim, and the default number of latents is 64. The loss follows the convention of VAE in latent diffusion models, with the $L_{2}$ loss for pixel reconstruction, perceptual loss [36] and GAN adversarial loss [37] for visual perceptual details, as well as KL divergence [24] for the latent regularization on $\mathbf{z}$ . We train the SF-VAE on the ImageNet training set with a batch size of 128 for 320K iterations (equivalent to 32 epochs) with a learning rate of $10^{-4}$ . We perform reconstruction FID evaluation [38] on the ImageNet validation 50K samples. + +Reconstruction results. We compare our SF-VAE with the popular SD-VAE-f8 [1] that comes with a downsampling rate of 8, resulting in a latent shape of $32 \times 32 \times 4$ . We also include the recently proposed deep compression autoencoder (DC-AE) [23] with a latent shape of $8 \times 8 \times 32$ for comparison, which is exactly the same as our default latent shape. The results are shown in Table 1. + +The reconstruction FID of our SF-VAE is 0.70, which is competitive with the SD-VAE (rFID 0.63) with 1024 latents. Also, if counting the latent dimension together, the SF-VAE comes with $64 \times 32 = 2048$ units, much less than SD-VAE's $32 \times 32 \times 4 = 4096$ units. Compared with DC-AE also trained on ImageNet samples and with the same latent shape, our SF-VAE achieves a lower rFID score while maintaining significantly fewer parameters. + +More importantly, the reconstruction results further demonstrate that non-grid latents are also effective for image reconstruction. These findings align with recent studies, which show that explicit 2D structured latents are not necessary for high-quality reconstructions and token numbers can be extremely compressed [39-41]. However, our methods does not necessarily belong to the 1D tokenization family, since there are explicit geometric properties, i.e., RoIs, associated with each latent. These 2D RoI representations also maintain the translation equivariance property of this pipeline, while 1D tokenization methods typically fail to address this point. + +Varying number of latents and higher image resolution. As discussed before, SF-VAE enjoys the flexibility of decoupling the number of latents from a specified image resolution. Here, we investigate our SF-VAE with more aggressive latent reduction for a fixed resolution of $256 \times 256$ in Table 2. We do not change the architecture of our SF-VAE, but only changes the number of latents from 64 to 48 and 32 and maintains the latent dimension unchanged. To reduce training costs, we initialize SF-VAE with new latent configurations from pre-trained weights of the 64-latents model, and finetune them for 20K iterations. Here, we find 32 tokens have already been able to reconstruct pixels with an rFID of 1.70, and 48 tokens give a competitive 0.99 rFID. + +![](images/40881d4ec310832530534f4f2b3aa0b9017b5bdba042ba7daa58f6067e5d07e6.jpg) + +![](images/99662149c0bb315f3b9ea63584128aaae0e52041e63074af20f22af8c88ae540.jpg) + +![](images/ee92893472f04611652c2171ba300fc5341685a731114e920ca121cf9abd983e.jpg) + +![](images/31a83758e3903d78f6e84059c4b32dbc251292b00ff40138c3ed94dd1ced0e8a.jpg) + +![](images/1c2ea61d5a3cc73e979c94b2e72ace06eb8cb3e50e8f1fe138f690ce1658ae88.jpg) +512x512 input + +![](images/e6352666e2458521ed67e2d56fde5b6ab4cff7e2f77d1e298b18f4421ca57bdd.jpg) +128 latents + +![](images/084673ea9bb7478f8cb02b7eb584d73bc0a98164d08e72a477f5d40a5ca2e3f4.jpg) +256 latents +Figure 5: SF-VAE reconstructions with different latent numbers. Better zoom in. + +Table 2: Token numbers for SF-VAE reconstructions. + +
resolution#latentrFID
256 × 2561280.35
640.70
480.99
321.70
512 × 5122560.66
1281.29
+ +Table 3: Effects of asynchronous interpolating schedule. + +
interpolating scheduleFID↓
linear, β = 144.4
async, β = 242.0
async, β = 444.6
async, min(2 × t, 1)54.0
async, min(4 × t, 1)55.9
decoupled, RoIs first63.3
+ +Table 4: Effect of $L_{1}$ loss. + +
wL1FID
0.043.9
0.242.0
0.542.6
1.045.1
+ +Table 5: Comparison with diffusion/flow-based transformers with grid-based latents under 400K training iterations without CFG. *SiT reproduced using official code. † sampled with ODE Euler sampler with NFE=250 for fair comparison. Flops are calculated with one single NFE. + +
method#tokens#paramsflops (G)FID↓IS↑
DiT-B/2256130M21.843.5-
SiT-B/2256130M21.833.0-
SiT-B/2*†256130M21.837.039.6
SF-SiT-B†64138M5.933.641.4
DiT-XL/2256675M114.419.5-
SiT-XL/2256675M114.417.2-
SiT-XL/2*†256675M114.418.870.9
SF-SiT-XL†64685M29.318.671.6
+ +We also study SF-VAE on higher image resolution of $512 \times 512$ in Table 2. We first scale up the number of latents to 256 similar to dense grid-based VAEs, where latent numbers are increasing quadratically regarding input image size. Then we reduce the number of latents to 128. We find that our SF-VAE can still achieve competitive reconstructions for higher images in Figure 5. + +# 4.2 Sparse Flow Transformers + +Setup. Our SF-SiT generally follows the design of original SiT transformers [10], where the input and output are retargeted to latents $\mathbf{z}$ and RoIs $\mathbf{r}$ , and the velocity of them, $\hat{\mathbf{v}}_z$ and $\hat{\mathbf{v}}_r$ . We mainly experiment SF-SiT-B and SF-SiT-XL variants, whose configurations strictly follow the original SiT-B and SiT-XL. Since SF-SiT are now required to output RoI velocity besides latent velocity, we add an additional AdaLN head [42] to predict RoI velocity, where the output is also initialized to zeros similar to latent prediction. + +We train SF-SiT variants from scratch on ImageNet training set with 64 latents and RoIs from SF-VAE, unless otherwise specified. In SF-SiT, each latent directly corresponds to a token in the transformer blocks, as our architecture eliminates extra patchifying operations. The training configurations are the same as the original SiT models, i.e., the global batch is 256 and the Adam optimizer [43] with constant learning rate $10^{-4}$ . We set the default $\beta$ to 2 in the asynchronous interpolating schedule and the weight $w_{L_1}$ to 0.2 for the additional L1 loss on RoI velocity prediction. We evaluate our SF-SiT using ADM evaluation pipeline [44] and report FID with 50K samples. + +Comparison with grid-based diffusion/flow transformers. We first compare our methods under the 400K iteration budget with grid-based DiT/SiT baseline [11, 10] in Table 5 without classifier-free guidance (CFG). It is worth noting that original SiT models adopt the SDE sampler during inference and report their results. This differs from SF-SiT models, which use simple ODE sampler for latent and RoI inference. Due to the lack of publicly available intermediate SiT checkpoints, we reproduce SiT-B and SiT-XL variants using the official code, and inference samples with ODE Euler sampler. We also align the number of function evaluations (NFE) in SF-SiT and SiT models to 250 to make fair comparison. + +As shown in Table 5, SF-SiT models, despite operating on significantly reduced latents, achieve performance on par with SiT models using the Euler ODE sampler. We experimented SF-SiT models with the original SiT's SDE sampler but observed degraded RoI quality and visual distortion in decoded images. We hypothesize this sensitivity arises from that SF-VAE encoded RoIs being deviating from standard Gaussian distributions, making them incompatible with the noise introduced by SDE sampling. Overall, these findings suggest that jointly modeling the flow of latents and RoIs have competitive performance paired with ODE-based samplers. + +Ablation studies. For all ablation studies, we resort to SF-SiT-B variant with 200K budget due to the limited computational resource. We also inference samples in ablations with the Heun ODE solver with 50 NFE for efficiency. + +Asynchronous interpolation schedule. Our proposed SF-SiT introduces additional RoIs $\mathbf{r}$ as additional modeling targets. As discussed before, we propose the asynchronous interpolation schedule + +![](images/aafc224ffd8a631dd7f81d017ce9d485d4db4ca9cc69b8b82f79ca7abf905e7e.jpg) +Figure 6: Visualization of RoI flows and decoded images with SF-SiT-XL and CFG=4.0. + +for $\mathbf{r}$ to be more compatible in the joint modeling with the latents $\mathbf{z}$ . Here we investigate the effect of the asynchronous interpolation schedule in Table 3. We can find that the asynchronous interpolation schedule $\beta = 2$ for $\mathbf{r}$ gives the best result. In contrast, synchronous interpolation $(\beta = 1)$ between $\mathbf{r}$ and $\mathbf{z}$ , or purely linear interpolation for both, results in the inferior FID. We also try more aggressive asynchronous schedule, where the interpolated RoI $\mathbf{r}$ reaches the target RoI linearly but at the earlier timestep, e.g., $t = 0.25$ for $\min(4 \times t, 1)$ . These clamped linear schedules are also asynchronous, where we suppose to sample RoIs first and then sample latents in a hard way, leading to significant degradation in FID. Finally, the decoupled interpolation schedule indicates that we first schedule RoI denoising separately and individually when $t$ is in $[0, 0.5]$ and then schedule latent denoising when $t$ is in $[0.5, 1]$ , where the denoising processes for $\mathbf{r}$ and $\mathbf{z}$ are fully decoupled and disjoint, leading to the worst FID. These findings show that the joint flow of latents and RoIs is crucial to good performance in SF-SiT framework but also needs soft asynchronous interpolation between $\mathbf{z}$ and $\mathbf{r}$ to have better performance. + +Roi velocity $L_{1}$ loss. We also investigate the effect of $L_{1}$ loss on the RoI velocity prediction, as proposed by. In line with [34], the $L_{1}$ loss weight 0.2 gives the best result. + +Longer training schedule. We further train the largest SF-SiT variant, SF-SiT-XL, with longer training schedule. To accelerate convergence on limited hardware resource, we use REPA [45] for our SF-SiT-XL training. We use DINOv2-B/14 [46] as our REPA align target. Since our SF-SiT + +Table 6: Class conditional generation results on ${256} \times {256}$ ImageNet. $P$ refers to precision and $R$ refers to recall following [44]. + +
method#tokenstrain stepsFID↓IS↑P↑R↑
LDM-4 [1]--10.56103.50.710.62
DiT-XL/2 [11]2567M9.62121.50.670.67
SiT-XL/2 [10]2567M8.26131.60.680.67
SiT-XL/2 w/ REPA [45]2561M6.4---
SF-SiT-XL w/ REPA641.4M7.35131.90.700.66
DiT-XL/2 (CFG=1.5)2567M2.27278.20.830.57
SiT-XL/2 (CFG=1.5)2567M2.06277.50.830.59
SF-SiT-XL w/ REPA (CFG=1.5)641.4M2.99279.20.820.54
SF-SiT-XL w/ REPA (CFG=1.375)641.4M2.76247.70.810.58
+ +does not form grid-based feature maps, we need to transform intermediate representations in SF-SiT transformer blocks into a grid using the time-dependent interpolated $\mathbf{r}_t$ as positional properties. Here, we use a simple cross attention from dense queries on grids to SF-SiT tokens with the positional embedding based on $\mathbf{r}_t$ added into keys and values. Dense grid queries are then passed through an MLP to align with DINOv2. + +Table 6 shows SF-SiT-XL (w/ 64 latents) results with longer training budget. Note that here we only want to show the potential of SF-SiT-XL on our limited budget with the fast convergence of REPA (it have already took 7 days on 8 A100s to complete 1.4M steps). SF-SiT-XL reaches the competitive performance (7.35 FID w/o CFG and 2.76 w/ CFG) with significantly reduced tokens and computation used. It is expected to have better results when training for more iterations and we will investigate them further when computational resources are available. + +# 5 Conclusion + +In this paper, we proposed a paradigm for visual generative modeling by using non-grid sparse latents with their positional properties, challenging the dense grid convention for image synthesis. We first designed sparse flow variational autoencoder, SF-VAE, which encodes raw image pixels into latents together with their RoIs. With SF-VAE, we can compress $256 \times 256$ images with down to 32 latents for good reconstruction fidelity. We then propose sparse flow-based generative transformers, SF-SiT, to model the joint flow of latents and RoIs with the highly reduced latent space. With carefully-designed asynchronous interpolation schedule and loss for RoIs, SF-SiT models have competitive performance compared with diffusion/flow transformers. Although not targeting new state-of-the-art results, SF-SiT-XL with just 64 tokens still reaches a competitive 2.76 FID on ImageNet with longer training schedule. We hope that our work can facilitate further research directions on non-grid and sparse generative methods, as well as sparse approaches for generation in other domain, e.g., audio and video. + +Limitations. Due to the lack of computational resources, we only train our SF-SiT-XL variant with the help of representation alignment (REPA) for fast convergence, and under fewer tokens settings, it might require more training iterations to converge with RoI modeling compared with grid-based SiT/DiTs for large parameter models. Also, the large scale text-to-image experiments remain exploration in the future. As this paper mainly aims for visual generative modeling, it inherits safety risks regarding its generated content. As the standard ImageNet being the training dataset, which is known to be a relatively safe for academic purposes, these concerns might not be fully addressed in our current framework. + +Acknowledgement. This project is supported by the National Research Foundation, Singapore under its NRFF Award NRF-NRFF13-2021-0008. + +# References + +[1] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. In CVPR, pages 10674-10685. IEEE, 2022. +[2] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In ICML, volume 139 of Proceedings of Machine Learning Research, pages 8821-8831. 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Vo, Marc Szafraniec, Vasil Khalidov, Pierre Fernandez, Daniel Haziza, Francisco Massa, Alaaeldin El-Nouby, Mido Assran, Nicolas Ballas, Wojciech Galuba, Russell Howes, Po-Yao Huang, Shang-Wen Li, Ishan Misra, Michael Rabbat, Vasu Sharma, Gabriel Synnaeve, Hu Xu, Hervé Jégou, Julien Mairal, Patrick Labatut, Armand Joulin, and Piotr Bojanowski. Dinov2: Learning robust visual features without supervision. Trans. Mach. Learn. Res., 2024, 2024. + +# NeurIPS Paper Checklist + +# 1. 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Memory-efficient Zeroth-order (MeZO) optimizers, recently proposed to address this issue, only require forward passes during training, making them more memory-friendly. However, compared with exact gradients, ZO-based gradients usually exhibit an estimation error, which can significantly hurt the optimization process, leading to slower convergence and suboptimal solutions. In addition, we find that the estimation error will hurt more when adding to large weights instead of small weights. Based on this observation, this paper introduces Sparse MeZO, a novel memory-efficient zeroth-order optimization approach that applies ZO only to a carefully chosen subset of parameters. We propose a simple yet effective parameter selection scheme that yields significant performance gains with Sparse-MeZO. Additionally, we develop a memory-optimized implementation for sparse masking, ensuring the algorithm requires only inference-level memory consumption, allowing Sparse-MeZO to fine-tune LLaMA-30b on a single A100 GPU. Experimental results illustrate that Sparse-MeZO consistently improves both performance and convergence speed over MeZO without any overhead. For example, it achieves a $9\%$ absolute accuracy improvement and 3.5x speedup over MeZO on the RTE task. Code is available at https://github.com/NUS-HPC-AI-Lab/SparseMeZO. + +# 1 Introduction + +Fine-tuning large language models for specific tasks or datasets has become a prevalent practice in machine learning. However, a major obstacle in fine-tuning is the substantial memory requirements, which escalate as models increase in size and complexity, thereby limiting the scalability and accessibility for those with limited computational resources. + +To mitigate the memory constraints, Parameter Efficient Fine-Tuning (PEFT) has been developed, allowing for the modification of only a subset of parameters and achieving comparable results to full model tuning (Hu et al., 2021; Lester et al., 2021; Li & Liang, 2021; Zaken et al., 2021; Zhang et al., 2023). However, PEFT methods still necessitate the calculation of gradients for backpropagation and caching of numerous activations during training, which introduces additional memory overhead. For instance, Malladi et al. demonstrates that, even with PEFT, training still requires approximately 6 times more memory than the memory cost for inference. This discrepancy raises a critical question: Can large language models be fine-tuned solely with the cost of inference? + +In response to these challenges, zeroth-order (ZO) optimization presents a promising solution (Spall, 1992). ZO optimization is a gradient-free method that estimates gradients using only the forward pass of the model, eliminating the need for backpropagation and, consequently, reducing memory usage. MeZO (Malladi et al., 2023) is a recently proposed zeroth-order method for fine-tuning LLMs that has demonstrated impressive performance. + +However, compared to exact gradients, ZO-based gradients usually exhibit an estimation error, which can be defined as noise. This noise can significantly hurt the optimization process, leading to slower convergence and suboptimal solutions. Moreover, we find that the estimated ZO gradient is difficult to generalize across batches. Specifically, while it can successfully reduce the training loss on the sampled batch with a high probability, it is more likely to increase the loss on other batches. + +To address this challenge, we investigate the impact of gradient noise in zeroth-order optimization for LLM fine-tuning. We measure how the noise affects optimization by evaluating its effect on generalization performance across different data batches. Interestingly, our experiments reveal that the noise has a more significant impact when added to large weights compared to + +![](images/ba9849cff2fb90886e84a9c1df367105d6ff2d09b1b87f056b630f66acad94b4.jpg) +Figure 1: Performance of MeZO and Sparse-MeZO (S-MeZO) on RTE task. S-MeZO can achieve $3.5\mathrm{x}$ speedup compared with MeZO. + +small weights. Based on this finding, we propose a novel sparse memory efficient zeroth-order method (Sparse-MeZO) to selectively optimize small weights, which are more resilient to noise perturbation. By focusing on these noise-resistant weights, we demonstrate that our method enables the use of larger learning rates, leading to improved performance and faster convergence. Our contributions can be summarized as follows: + +- In this paper, we investigate the impact of gradient noise in zeroth-order optimization for LLM fine-tuning. Our evaluations show that the gradient noise can make the estimated ZO gradient difficult to generalize across batches and the noise will hurt more when adding to large weights instead of small weights. +- Based on the above finding, we propose a sparse Memory-Efficient Zeroth-Order optimization method Sparse-MeZO (S-MeZO) for large language model fine-tuning. We also provide theoretical analysis to show the convergence of Sparse-MeZO. +- Different from the efficient implementation with random seed in MeZO, we propose a novel memory-efficient implementation of Sparse-MeZO, which can compute the sparse mask and perturb parameters in the forward pass. The technique enables fine-tuning LLaMA-30b with Sparse-MeZO on a single A100 GPU. +- We conduct empirical studies on LLaMA, OPT, and Mistral. The experimental results demonstrate that Sparse-MeZO can improve the fine-tuning performance and yield a faster convergence rate compared with vanilla MeZO across a wide range of natural language processing tasks. For example, it achieves a $9\%$ absolute accuracy improvement and 3.5x speedup over MeZO on the RTE task, as shown in Figure 1. + +# 2 Preliminaries + +# 2.1 Parameter-Efficient Fine-Tuning + +Parameter-Efficient Fine-Tuning (PEFT) is designed to facilitate efficient adaptation by updating only a subset of the model's parameters, rather than fine-tuning the entire model (Hu et al., 2021; Zaken et al., 2021; Pan et al., 2024; Liu et al., 2025). These PEFT approaches can be categorized into selective and additive methods. + +Selective Methods. Selective Methods try to selectively fine-tune a portion of a model and these methods have been explored in various studies. For example, Zaken et al.; Cai et al. focused on the + +model's bias terms, finding that fine-tuning these terms alone could rival the results of fine-tuning the entire model. However, the effectiveness of this approach diminishes with larger datasets, as shown in further analysis by Zaken et al.. Beyond static parameter adjustments, there has been an exploration into dynamically modifying parts of the model (Brock et al., 2017). This concept was later applied to language models, with AutoFreeze (Liu et al., 2021b) confirming its viability. + +Additive Methods. Additive methods, as an alternative to updating existing parameters, involve incorporating new layers into models, with the fine-tuning process focusing solely on these added layers (Houlsby et al., 2019; Hu et al., 2021; Lin et al., 2020; Rebuffi et al., 2017). Traditional techniques in this category, such as adapters (Houlsby et al., 2019), implemented layer additions in a sequential manner, which unfortunately led to increased inference latency. LoRA (Hu et al., 2021) has been proposed to mitigate this issue, which freezes the weights of the pre-trained model and introduces trainable matrices based on rank decomposition into each layer. IA3 (Liu et al., 2022) introduced innovative methods for adding parameters, balancing parameter count with accuracy, while LST (Sung et al., 2022) introduced a highway structure that learns only small, auxiliary channels. + +# 2.2 Zeroth-Order Optimization + +Unlike traditional gradient-based optimization methods that rely on derivatives to guide the search for optimal solutions, Zeroth-Order (ZO) optimization techniques do not require derivatives for optimization Spall (1992); Liu et al. (2018, 2019); Shu et al. (2023). These methods utilize only the value of the objective function, denoted as $f(x)$ , at any chosen point $x$ . To estimate the gradient in the direction of vector $z$ , the objective function is assessed at two points in close proximity, $f(x + \epsilon z)$ and $f(x - \epsilon z)$ , with $\epsilon$ being a minimal value. Following this, conventional optimization algorithms, such as gradient descent or coordinate descent, are implemented using these approximated gradient values. Currently, ZO methods have been widely used in various applications, such as adversarial attack and defense (Chen et al., 2017; Ilyas et al., 2018; Tu et al., 2019; Ye et al., 2018), Auto-ML (Ruan et al., 2019; Wang et al., 2022), natural language processing (Sun et al., 2022a,b), reinforcement learning (Vemula et al., 2019), Signal Processing (Liu et al., 2020), and on-chip training (Gu et al., 2021). + +# 2.2.1 MeZO + +ZO-SGD employs SPSA (Spall, 1992) to estimate the gradient. In general, conventional ZO-SGD algorithms utilizing SPSA consume twice the inference memory. MeZO (Malladi et al., 2023) is a memory-efficient variant of ZO-SGD. It circumvents the storage of gradients by saving the random seed and resampling the same random noise $\mathbf{z}$ with the seed during forward process. More specifically, to calculate $\mathcal{L}(\theta + \epsilon z) - \mathcal{L}(\theta - \epsilon z)$ , MeZO will sample a noise $\mathbf{z}$ to perturb $\theta$ to $\theta + \epsilon z$ and then calculate $\mathcal{L}(\theta + \epsilon z)$ . Then it resamples the same noise $\mathbf{z}$ with the same seed and move the parameter back $\theta - \epsilon z$ and calculates the loss. As a result, the zeroth-order gradient estimator can be computed without any memory overhead, leading to numerous variants of MeZO being proposed in the literature (Zhang et al. (2024); Yang et al. (2024); Jiang et al. (2024); Chen et al. (2024); Wang et al. (2024); Yu et al. (2025); Tan et al. (2025); Sun et al. (2025); Zhang et al. (2025); Zhou et al. (2025). + +# 2.2.2 Sparsity for Zeroth-order Optimization + +The hypothesis proposed by Frankle & Carbin, known as the lottery ticket hypothesis, showed that within a densely connected neural network that is randomly initialized, there exists a subnetwork of sparse yet high-quality connections. Based on the hypothesis, model pruning aims to identify and preserve the crucial 'winning tickets' - sparse subnetworks within the larger neural network that can achieve comparable or even superior performance, such as Wanda (Sun et al., 2023) and SparseGPT (Frantar & Alistarh, 2023). In addition, Dynamic Sparse Training (DST) has been proposed to reduce the training and inference cost in first-order optimization (Liu et al., 2021a; Evci et al., 2020). Recently, several related works have tried to apply the sparsity to zeroth-order optimization (Balasubramanian & Ghadimi, 2018; Cai et al., 2021, 2022; Chen et al., 2023; Gu et al., 2021; Ohta et al., 2020; Wang et al., 2018). For example, DeepZero (Chen et al., 2023) proposes a novel ZO training protocol with model pruning guided sparsity. + +![](images/e83c53f587f948e6ecd5ba008f982735d29ceb74f146ed9a7d56b855ff9d72a9.jpg) +(a) RTE-LR + +![](images/bfa4f097aac124ea96cf6b0f1a05b6c1aa242cf76aba2c4514519b793eaea85e.jpg) +(b) RTE-Batch +Figure 2: (a) Test Accuracy with Different Learning Rates on RTE Task. We find MeZO is very sensitive to the selection of learning rate. Even a small increase from $1 \times 10^{-6}$ to $2 \times 10^{-6}$ causes divergence and instability. (b) Probability of Loss Increase on Different Batch. We find the estimated ZO gradient can successfully reduce the loss on the same batch but may be difficult to decrease the loss on the new held-out batch. (c) Continuing training from the drop point with small and large weights. We find that optimizing only small weights can recover and further improve test accuracy. + +![](images/73b90c93b15747171d9ab399793f1251af90c1b632b0d2c24b697bb17cf5ea3e.jpg) +(c) RTE-Magnitude + +# 3 Proposed Method + +# 3.1 Empirical Observation on MeZO + +For large language models, zeroth-order optimization algorithms like MeZO are often necessary when exact gradients are unavailable or prohibitively expensive to compute. However, compared with exact gradients, these methods inherently introduce noise in the gradient estimates used for optimization. Specifically, the zeroth-order gradient $g_{z}(\theta)$ is approximated as $g_{z}(\theta) = \frac{\mathcal{L}(\theta + \epsilon z) - \mathcal{L}(\theta - \epsilon z)}{2\epsilon} z$ , where $\mathcal{L}$ is the loss function. As shown in Figure 2(a), MeZO exhibits extreme sensitivity to the choice of learning rate. Even a small increase from $1 \times 10^{-6}$ to $2 \times 10^{-6}$ causes divergence and instability, while this larger learning rate is totally fine when fintuning with first-order methods. This suggests that the gradient noise introduced by the zeroth-order approximation, defined as $\delta = g(\theta) - g_{z}(\theta)$ where $g(\theta)$ is the exact gradient, significantly hinders the optimization process when large step sizes are used. This motivates us to analyze the effects of this gradient noise $\delta$ and understand how it impacts optimization performance. + +To quantify how the gradient noise $\delta$ hurts the optimization process, we evaluate its effect on the generalization performance of the estimated gradients. Specifically, we measure whether the zeroth-order gradient estimate computed on one batch can effectively reduce the loss on other held-out batches. For a batch $\mathcal{B}_t = \{\mathcal{B}_t^1,\mathcal{B}_t^2\}$ with 32 data points, we use 16 samples to estimate the zeroth-order gradient $g_{z}(\theta ;\mathcal{B}_{t}^{1})$ on batch $\mathcal{B}_t^1$ , and evaluate it on the remaining 16 held-out samples $\mathcal{B}_t^2$ . The results are shown in Figure 2. Interestingly, we find a stark contrast in performance - while the estimated gradient $g_{z}(\theta ;\mathcal{B}_{t}^{1})$ can reliably reduce the loss on the same batch $\mathcal{B}_t^1$ it was computed on (90% success rate), it only manages to decrease the loss on the new held-out batch $\mathcal{B}_t^2$ around 50% of the time. This suggests that the zeroth-order gradient estimates suffer from overfitting or noise that makes them less generalizable to unseen data samples. The gradient noise $\delta$ , while allowing descent on the current batch, appears to introduce errors that prevent reliable descent directions for unseen batches. Therefore, the noise $\delta$ can be seen as hurting the optimization process by degrading the generalization performance of the parameter updates. + +Next, we aim to understand if this effect is uniform across all model parameters or if certain parameter groups are more vulnerable to noise corruption. We notice the nature of vanilla MeZO, where $\frac{\mathcal{L}(\theta + \epsilon z) - \mathcal{L}(\theta - \epsilon z)}{2\epsilon}$ is used to estimate the gradient, and all parameters share the same value of $\frac{\mathcal{L}(\theta + \epsilon z) - \mathcal{L}(\theta - \epsilon z)}{2\epsilon}$ . This means not all parameters are optimized in the true gradient direction, which could be a limitation. To analyze this, we divide the parameters into different groups based on their magnitude - the top $20\%$ largest weights are considered "large", while the bottom $20\%$ are "small". Interestingly, our experiments reveal that the gradient noise $\delta$ hurts optimization more when added to large weights compared to small weights. As shown in Figure 2(c), when continuing training from the point where test accuracy drops (due to noise), we find that optimizing only the small weights can recover and further improve test accuracy. Therefore, the experimental illustrate that noise $\delta$ does + +not impact all parameters equally - it disproportionately disrupts the optimization of larger weights. Selectively optimizing smaller, noise-resilient weights may be a promising direction to mitigate the effects of gradient noise in zeroth-order optimization. In the next section, we will introduce the proposed Spare-MeZO algorithm, which can only select small weights to perturb and update weights. + +# 3.2 Sparse-MeZO + +Consider a labelled dataset $\mathcal{D} = \{(x_i, y_i)\}_{i \in [|\mathcal{D}|]}$ and let $\mathcal{L}(\theta; \mathcal{B})$ denotes the loss on a minibatch $\mathcal{B}$ . We can define a sparse mask $m \in \{0, 1\}^d$ to selectively sample the random noise $z \in \mathbb{R}^d$ with $z \sim \mathcal{N}(0, I_d)$ on the sub-net of pre-trained model. A sparsified version of random perturbation can be defined as $\hat{z} \in \mathbb{R}^d$ : + +$$ +\hat {z} = \boldsymbol {m} \odot z. \tag {1} +$$ + +Based on this sparse perturbation $\hat{z}$ , we can redefine MeZO algorithm on Section 2.2.1 as Sparse-MeZO. The main difference is from the estimated gradient $g_{\hat{z}}(\theta)$ , which can be defined as: + +$$ +\begin{array}{l} \boldsymbol {g} _ {\hat {z}} (\boldsymbol {\theta}) = \frac {\mathcal {L} (\boldsymbol {\theta} + \epsilon \hat {z} ; \mathcal {B}) - \mathcal {L} (\boldsymbol {\theta} - \epsilon \hat {z} ; \mathcal {B})}{2 \epsilon} \hat {z} \\ = \frac {\mathcal {L} (\theta + \epsilon m \odot z ; \mathcal {B}) - \mathcal {L} (\theta - \epsilon m \odot z ; \mathcal {B})}{2 \epsilon} \hat {z}, \tag {2} \\ \end{array} +$$ + +where $\epsilon$ represents the perturbation scale. Based on the observations from our motivation, we can create a sparse mask, $m$ , determined by parameter magnitudes. Specifically, we only update parameters of smaller magnitude. These targeted parameters are defined as $\hat{\theta} = m \odot \theta$ . It's important to note that we still preserve the complete set of parameters, but we apply sparse perturbations and gradient estimations only to the selected ones. This approach allows us to integrate the sparse mask into the standard MeZO method as a straightforward, adaptable tool. Then, we will introduce when and how to calculate the mask. + +- Constant Mask: Setting the Mask Before Training. We compare the parameter values to a threshold for each layer to set the mask before training begins. However, a significant downside of this approach is the extra memory required to store a sparse mask, which is as large as the pre-trained model itself. Our goal is for our method to enhance performance without using more GPU memory or causing extra overhead. +- Dynamic Mask: Determining Mask at Each Iteration. We can establish a threshold for each layer before training and then generate the mask by comparing parameter values to this threshold during each iteration. This method avoids the necessity of storing a large mask, $m$ . + +In this paper, we'll employ a dynamic mask to choose which parameters to perturb and update, addressing the issue of memory constraints. In addition, we determine thresholds using a principled sparsity-based approach. Specifically, we use a percentile-based method where the threshold is set based on a target sparsity level. + +The pseudo-code is provided in Algorithm 1. This algorithm outlines that we first establish the threshold $h_i$ for each layer before beginning training. We then use GetMask (Algorithm 3) to compare each parameter against its threshold $h_i$ and create the mask $m$ . Following this, we introduce the function PerturbParameters (AlGORITHM 2) to generate a Gaussian noise sample $z \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_d)$ and apply the mask $m$ to produce a sparse perturbation $\hat{z} = m \odot z$ . With $\hat{z}$ , we perturb the current parameters $\theta_t$ to get new parameters $\theta_t + \epsilon \hat{z}$ and $\theta_t - \epsilon \hat{z}$ . This allows us to compute two distinct loss values: $l^+ = \mathcal{L}(\theta_t + \epsilon \hat{z})$ and $l^- = \mathcal{L}(\theta_t - \epsilon \hat{z})$ . From these losses, we calculate the estimated sparse gradient $g_m(\theta_t) = \text{proj\_grad} * \hat{z}$ , where $\text{proj\_grad} = \frac{l^+ - l^-}{2\epsilon}$ . Finally, this gradient can be used with a learning rate $\eta_t$ to update $\theta_t$ . + +# 3.3 Memory-Efficient Implementation of Sparse-MeZO + +In this paper, our primary aim is to introduce an efficient method for fine-tuning language models using zeroth-order optimization, enhancing performance on downstream tasks. As outlined in Algorithm 1, our approach involves perturbing the parameters $\theta_{t}$ twice to generate two distinct sets of parameters, $\theta_{t}^{\prime} = \theta_{t} + \epsilon z$ and $\theta_{t}^{\prime \prime} = \theta_{t} - \epsilon z$ . We then use the estimated gradient to update the + +Algorithm 1 Sparse-MeZO (S-MeZO) +Require: $\theta$ represents pre-trained LLM weight, $N$ is the number of layers in model, learning rate $\eta_t$ , $s$ represents sparsification interval. Initialize random seed $s$ +Determine threshold $h = h_1, \ldots, h_N$ , of each layer with the sparsification interval +for $t \gets 1$ to $T$ do +Sample Minibatch $\mathcal{B}$ from $X$ and random seed $s$ . + $m \gets \operatorname{GetMask}(\theta_t, h)$ $\theta_t \gets \operatorname{PerturbParameters}(\theta_t, \epsilon, s, m)$ $l^+ = \mathcal{L}(\theta_t; \mathcal{B})$ $\theta_t \gets \operatorname{PerturbParameters}(\theta_t, -2\epsilon, s, m)$ $l^- = \mathcal{L}(\theta_t; \mathcal{B})$ $\theta_t \gets \operatorname{PerturbParameters}(\theta, \epsilon, s, m)$ +proj_grad $\leftarrow (l^+ - l^-) / (2\epsilon)$ +Reset random seed $s$ +for $\theta_i \in \theta$ do + $z_i \sim \mathcal{N}(0, 1)$ $\theta_i \gets \theta_i - \eta_t * \operatorname{proj\_grad} * m_i * z$ +end for +end for + +original parameters $\theta_{t}$ . This step typically requires storing two separate sets of parameters, leading to increased memory usage during fine-tuning. + +Recently proposed MeZO, conserves memory by saving random seeds $s$ and using it to resample $z$ for calculating $\theta_t^\prime$ , $\theta_t^{\prime \prime}$ , and reconstructing $\theta_{t}$ without needing extra memory. However, applying a sparse mask $m$ for calculating sparse perturbation $\hat{z}$ in MeZO poses a memory issue. We cannot simply reconstruct $\hat{z}$ by saving the random seed because the sparse mask, determined by parameter magnitudes, changes when parameters are altered by the perturbation. To address this, we propose potential solutions for the memory issue. + +1-bit Quantization: We can apply 1-bit quantization to store the mask $m$ , as it consists solely of 0s and 1s. However, this method still increases memory usage, which isn't our goal. As a solution, we introduce a novel, memory-saving approach for zeroth-order optimization that calculates the mask $m$ on the fly during the forward pass. + +Calculating the Mask During the Forward Pass: By computing the mask and perturb parameters in the forward pass, we eliminate the need to store perturbed parameters $\theta_t'$ and $\theta_t''$ . This means we only have to keep the original parameters $\theta_t$ throughout training. For vanilla implementation, we first need to calculate the perturbed parameters with mask $m$ : $\theta_t' = \theta_t + \epsilon m \odot z$ . After that, we can use perturbed parameters $\theta_t'$ to calculate the loss value $l^+$ with the forward process. For example, the output of layer $i$ can be defined as $y^{(i)} = \theta_t'^{(i)} x^{(i)} + b^{(i)}$ . Noted that we need to save the vanilla parameters $\theta_t$ and mask $m$ for vanilla implementation. However, for our proposed efficient implementation, we only need to save vanilla parameters $\theta_t$ . More specially, we can calculate the mask $m^{(i)}$ of layer $i$ during the forward process and then obtain the output of this layer: $y^{(i)} = (\theta_t^{(i)} + \epsilon m(\theta_t) z^{(i)}) x^{(i)} + b^{(i)}$ , where $m(\cdot)$ represents GetMask to calculate mask $m^{(i)}$ . Then, we can release the memory of mask $m^{(i)}$ and calculate the output and mask of the next layer. + +# 4 Experiments + +Following a similar setting to MeZO, we evaluate the performance of our proposed method on SuperGLUE Wang et al. (2019). + +# 4.1 Experimental Setting + +Datasets. To verify the performance gain of our proposed method, we conduct experiments on various fine-tuning tasks include SST-2 (Socher et al., 2013), RTE (Bentivogli et al., 2009; Dagan et al., 2005; + +![](images/35455d5e062bdd194e586b48106429cd8c3d25f1800692d6d5d9ad3c04759baa.jpg) +(a) RTE + +![](images/a19c2e42a4fcd5ed5fbbfc5a3712047bcd1c0f65b4953766456e73d470c5cb94.jpg) +(b)BoolQ +Figure 3: Convergence Curves of Fine-Tuning LLaMA-7b with MeZO and Sparse-MeZO (S-MeZO) on (a) RTE, (b) BoolQ, (c) WIC tasks. + +![](images/c0be2c9f242194cc981377988261c53b835110b44da54d6b585e0018b5e21cbe.jpg) +(c) WIC + +Giampiccolo et al., 2007), BoolQ (Clark et al., 2019), WIC (Pilehvar & Camacho-Collados, 2018), MultiRC (Khashabi et al., 2018)) and multi-class task COPA (Roemmele et al., 2011). + +Models. We primarily use pre-trained LLaMA-7b Touvron et al. (2023) to evaluate the performance of our proposed method on downstream tasks. To further demonstrate our method's versatility, we also test it with Mistral-7B-v0.1 Jiang et al. (2023) and OPT-13b Zhang et al. (2022). We provide more details about the results in the Appendix E. Additionally, to examine our method's scalability, we evaluate it on larger models, such as LLaMA-30b. + +Baselines. First, we compare our method to the vanilla MeZO to demonstrate how sparsification enhances MeZO's convergence speed and overall performance. Additionally, to show that our proposed S-MeZO effectively identifies and modifies crucial parameters, we contrast it with R-MeZO (a version of MeZO applying a random mask to select parameters for optimization). In addition, we also explore the impact of zero-shot optimization on improving a pre-trained language model's capabilities through experiments with zero-shot learning and in-context learning techniques (Brown et al., 2020). Lastly, to understand the performance gap between zeroth-order and first-order optimization in fine-tuning large language models, we present results from conventional full-parameter fine-tuning (FT) using the Adam optimizer, the most widely used method for such tasks. In addition, we also compare MeZO and its variants against LoRA, the most widely adopted PEFT method. + +Training Procedure. We adopt most of the training hyperparameters from the standard MeZO, including dataset configuration, batch size, training epochs, epsilon value, and task prompts, with the key difference being a higher learning rate for S-MeZO due to updating only a subset of the parameters (Zhou et al., 2023). We perform the experiments using three different seeds and report the average of the outcomes. + +# 4.2 Performance on SuperGLUE + +To evaluate the performance of our proposed method S-MeZO, we initially tested it on the SuperGLUE benchmark using the LLaMA-7b model. The fine-tuning results, presented in Table 12, reveal that our S-MeZO method outperforms other zero-order (ZO) techniques like MeZO and R-MeZO. For instance, S-MeZO boosts MeZO's accuracy from $71.7\%$ to $80.7\%$ on RTE $(\uparrow 9\%)$ and from $75.9\%$ to $80.9\%$ on BoolQ $(\uparrow 5\%)$ . Furthermore, all zeroth-order-based methods surpassed the performance of Zero-shot learning and in-context learning, demonstrating that zeroth-order optimization significantly enhances the pre-trained model's effectiveness on downstream tasks. Finally, we can find S-MeZO significantly bridges the performance gap between zero-order and first-order optimization methods. + +To further verify the generality of our proposed S-MeZO, we also evaluate it on Mistral-7B-v0.1. The performance is shown in Table 11. We can find that S-MeZO can consistently improve the performance of vanilla MeZO and narrow down the performance gap between zeroth-order optimization and first-order optimization. For example, S-MeZO can improve the accuracy of vanilla MeZO from 81.6 to 85.3 on BoolQ and then achieve a comparable performance with full fine-tuning. + +We conducted additional evaluations of S-MeZO on LLaMA2-7b and compared it against an expanded set of baseline methods. These include methods proposed by Zhang et al. (Zhang et al., 2024) such as ZO-SGD-Cons, ZO-SGD-Sign, and ZO-SGD-Adam, as well as other significant baselines including ZO-AdaMU (Jiang et al., 2024) and AdaZeta (Yang et al., 2024). The results are shown in the Table + +
MethodBoolQRTEWICMultiRCSST-2COPAAverage
Zero-Shot65.149.550.655.879.759.760.1
ICL67.454.552.758.781.284.466.5
LoRA84.582.367.678.395.086.082.3
FT84.583.668.480.295.785.082.9
MeZO75.971.761.469.894.686.376.6
MeZO-LoRA77.974.960.872.695.084.377.6
R-MeZO76.975.262.168.194.684.376.9
S-MeZO80.9 (↑5.0)80.7 (↑9.0)64.9 (↑3.5)73.3 (↑3.5)95.0 (↑0.4)86.7 (↑0.4)80.3 (↑3.7)
+ +Table 1: Accuracy of Fine-Tuning LLaMA-7b on SuperGLUE (1,000 examples). ICL: In-Context Learning, FT: full-parameter fine-tuning with Adam, R-MeZO: MeZO with Random Mask. + +2. We find that S-MeZO consistently outperforms other zeroth-order methods, improving MeZO's accuracy from $78.8\%$ to $82.2\%$ on $\mathrm{BoolQ}$ ( $\uparrow 3.4\%$ ) and from $70.2\%$ to $77.6\%$ on RTE ( $\uparrow 7.4\%$ ). While LoRA achieves the highest average accuracy of $83.0\%$ , S-MeZO significantly narrows this gap with an average of $80.0\%$ , surpassing both MeZO ( $76.3\%$ ) and R-MeZO ( $76.6\%$ ). + +
ModelMethodBoolQRTEWICSST-2Average
LLaMA2-7bLoRA84.083.268.896.083.0
LLaMA2-7bMeZO78.870.262.294.076.3
LLaMA2-7bMeZO-LoRA80.376.563.094.078.5
LLaMA2-7bZO-SGD-Cons77.168.663.094.075.7
LLaMA2-7bZO-SGD-Sign68.261.055.682.866.9
LLaMA2-7bZO-SGD-Adam78.973.662.193.877.1
LLaMA2-7bZO-AdaMU78.276.563.093.677.8
LLaMA2-7bAdaZeta79.875.862.094.077.9
LLaMA2-7bR-MeZO76.774.061.494.276.6
LLaMA2-7bS-MeZO82.2 (↑ 3.4)77.6 (↑ 7.4)65.3 (↑ 3.1)94.8 (↑ 0.8)80.0 (↑ 3.7)
+ +Table 2: Accuracy of Fine-Tuning LLaMA2-7b on SuperGLUE (1,000 examples). + +# 4.3 Performance on Commonsense Reasoning and Mathematics Tasks + +To further verify the performance of Sparse MeZO on more challenging tasks, we have conducted additional experiments on commonsense reasoning tasks (PIQA, SIQA, BoolQ) and a mathematics task (AQuA (Ling et al., 2017)), consistent with the evaluation protocols used in SensZOQ (Guo et al.). + +The results, presented in the Table 3, demonstrate that Sparse MeZO (S-MeZO) consistently + +outperforms MeZO across all these more complex tasks. In particular, we observe substantial improvements on the AQuA mathematics task $(+2.6\%)$ and the SIQA commonsense reasoning task $(+1.7\%)$ , further validating the effectiveness of our approach across diverse tasks. + +
MethodBoolQPIQASIQAAQuA
MeZO76.684.368.524.0
S-MeZO79.285.370.226.6
+ +Table 3: Accuracy of Fine-Tuning Mistral-7B on Challenging Tasks. + +# 4.4 Convergence Rate + +To verify that S-MeZO converges faster than MeZO, we carried out multiple experiments for comparison. The accuracy over steps is plotted in Figure 3, which shows that S-MeZO can use fewer steps to achieve a better performance than vanilla MeZO. For example, S-MeZO only needs about 5,000 + +steps to achieve $70\%$ accuracy but vanilla MeZO needs 17,500 steps. Finally, S-MeZO can achieve about 3.5x speedup on RTE and 3x speedup on BoolQ. + +# 4.5 Memory Usage + +Table 4 shows the memory consumption for MeZO, S-MeZO, and traditional full-parameter fin-tuning of LLaMA-7b. The data reveal that S-MeZO does not require more memory than MeZO and offers a substantial saving of roughly 12 times less GPU memory compared to full-parameter fin-tuning. For instance, S-MeZO with Efficient Implementation (S-MeZO-EI) cuts down the memory needed from 158.6 GB for full tuning to just 14.6 GB on MultiRC task. In addition, S-MeZO with efficient implementation can reduce the memory cost from 28.3 GB of vanilla S-MeZO to 14.6 GB across all five tasks, which also illustrates the efficiency of our proposed implementation method: Calculating the Mask During the Forward Pass. As a result, we can use only inference memory cost to fine-tune large language models. + +
MethodSST-2RTEBoolQWICMultiRCCOPAAverage
FT114.7123.7128.7115.3158.6119.1128.2
LoRA15.719.525.516.134.223.122.4
MeZO14.614.614.614.614.614.614.6
S-MeZO28.328.328.328.328.328.328.3
S-MeZO-EI14.614.614.614.614.614.614.6
+ +# 4.6 Sparse Rate + +For S-MeZO, we need to define the sparsity of the pre-trained model before starting to fine-tune it. To analyze the effects of sparsity value on the performance, we conduct experiments with various sparsity values (from 0.0 to 0.8). Table 10 summarizes these experimental results with different sparsity values. We can find that a significant performance gain can be obtained when we use the sparsity value from 0.5 to 0.8. In addition, for most tasks, a sparsity value of 0.8 usually means a better performance. For example, S-MeZO can improve the accuracy from $71.7\%$ to $82.3\%$ (when $r = 0.8$ ). It can also obtain a performance gain of $6.6\%$ for $\mathrm{BoolQ}$ (from $75.9\%$ to $82.5\%$ ). + +# 4.7 Scalability + +In Table 12, we mainly introduce the performance of our methods on LLaMA-7b. A direct question is whether our proposed method can scale to larger language models. Therefore, in this section, we further explore our proposed method S-MeZO on LLaMA-30b. As shown in Table 5, we can see that the a larger model usually can obtain a better fine-tuned performance. For example, the accuracy on RTE with MeZO can be improved from $71.1\%$ on LLaMA-7b to $76.9\%$ on LLaMA-30b. Our method S-MeZO can further improve the performance on RTE to + +Table 4: Memory Usage (batch size = 1) of Fine-Tuning LLaMA-7b on SuperGLUE (1,000 examples). EI represents the Efficient Implementation in section 3.3. + +
ModelMethodBoolQRTEWIC
LLaMA-7bMeZO75.971.761.4
LLaMA-7bS-MeZO80.980.764.9
LLaMA-30bMeZO83.876.963.3
LLaMA-30bS-MeZO85.782.167.3
+ +Table 5: Accuracy of Fine-Tuning LLaMA-7b and LLaMA-30b on SuperGLUE. + +82.1% on LLaMA-30b. In addition, S-MeZO can further improve the accuracy on BoolQ to 85.7% on LLaMA-30b. + +# 5 Conclusion + +In this paper, we propose a novel memory-efficient zeroth-order fine-tuning method Sparse-MeZO, which can use a similar memory cost to the inference process. We evaluate the performance of fine-tuning LLaMA and OPT with Sparse-MeZO on SuperGLUE benchmark and the experimental results illustrate that Sparse-MeZO can achieve a higher accuracy and faster convergence. Finally, we can fine-tune LLaMA-30b on a single A100 GPU. + +**Limitation:** There is still a performance gap between our proposed method Sparse-MeZO and first-order fine-tuning methods. We plan to address these limitations and enhance Sparse-MeZO's capabilities in our future research and conduct more experiments on state-of-the-art pre-trained language models. + +# 6 Acknowledgements + +Yang You's research group is being sponsored by NUS startup grant (Presidential Young Professorship), Singapore MOE Tier-1 grant, ByteDance grant, ARCTIC grant, SMI grant and Alibaba grant. This work is also supported by NSF 2048280, 2325121, 2244760, 2331966 and ONR N0001423-1-2300:P00001. + +# References + +Krishnakumar Balasubramanian and Saeed Ghadimi. Zeroth-order (non)-convex stochastic optimization via conditional gradient and gradient updates. Advances in Neural Information Processing Systems, 31, 2018. +Luisa Bentivogli, Peter Clark, Ido Dagan, and Danilo Giampiccolo. The fifth pascal recognizing textual entailment challenge. TAC, 7(8):1, 2009. +Andrew Brock, Theodore Lim, James M Ritchie, and Nick Weston. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The proposed method is developed by ourselves. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A The Prompts in LLaMA and OPT + +
DatasetTypePrompt
SST-2cls.{premise}Does this mean that “{hypothesis)” is true? Yes or No?Yes/No
RTEcls.Suppose “{premise}” Can we infer that “{hypothesis}?”? Yes, No, or Maybe?Yes/No/Maybe
BoolQcls.{passage} {question}?Yes/No
WICcls.Does the word “{word}” have the same meaning in these two sentences? Yes, No? {sent1} {sent2}Yes/No
MultiRCcls.{paragraph}Question: {question}I found this answer “{answer}”. Is that correct? Yes or No?Yes/No
COPAmch.{premise} so/because {candidate}
+ +# B Experimental Setting + +# B.1 Hyperparameters + +We will introduce the hyperparameters searching grids in Table 7 and 8, which can help people reproduce our results. + +Table 6: The prompts of the datasets we used in our LLaMA experiments. + +
ExperimentHyperparametersValues
MeZOBatch size16
Learning rate{5e-7, 1e-6, 2e-6}
ε1e-3
MeZO-RandomBatch size16
Learning rate{1e-6, 2e-6, 3e-6, 4e-6, 5e-6}
ε1e-3
S-MeZOBatch size16
Learning rate{1e-6, 2e-6, 3e-6, 4e-6, 5e-6}
ε1e-3
FT with AdamBatch size8
Learning Rates{1e-5, 5e-5, 8e-5}
+ +Table 7: The hyperparameter searching grids for LLaMA-7b experiments. + +
ExperimentHyperparametersValues
MeZOBatch size16
Learning rate{1e-8, 2e-8, 3e-8, 5e-8, 1e-7, 5e-7, 1e-6, 2e-6}
ε1e-3
MeZO-RandomBatch size16
Learning rate{1e-6, 2e-6, 3e-6, 4e-6, 5e-6}
ε1e-3
S-MeZOBatch size16
Learning rate{1e-6, 2e-6, 3e-6, 4e-6, 5e-6}
ε1e-3
FT with AdamBatch size8
Learning Rates{1e-5, 5e-5, 8e-5}
+ +# B.2 The Setting of Threshold + +We determine thresholds using a principled sparsity-based approach. Specifically, we use a percentile-based method where the threshold is set based on a target sparsity level. For example, with $80\%$ sparsity, we sort the weight values of each layer and set the threshold at the 80th percentile. Importantly, this threshold is determined once before training begins and remains fixed throughout the optimization process. We then introduce the sparsity of each task in SuperGLUE when we fine-tune LLaMA-7b. The setting is shown in the Table 9. + +Table 8: The hyperparameter searching grids for Mistral-7b experiments. + +
MethodSST-2RTEBoolQWICMultiRC
LLaMA + Sparse MeZO0.700.750.800.800.80
Mistral + Sparse MeZO0.700.600.600.700.60
+ +# B.3 Sparse Rate + +In this section, we provide the sensitivity analysis of sparsity. Table 10 summarizes the experimental results with different sparsity values. We observe that significant performance gains are achieved when using sparsity values from 0.5 to 0.8. Moreover, for most tasks, a sparsity value of 0.8 typically yields the best performance. For example, on RTE, S-MeZO improves accuracy from $71.7\%$ to $82.3\%$ when $r = 0.8$ . Similarly, it achieves a performance gain of $6.6\%$ on BoolQ (from $75.9\%$ to $82.5\%$ ). + +Table 9: Sparsity in SuperGLUE when we fine-tune LLaMA-7b and Mistral. + +
Sparsity0.50.60.70.8
RTE77.1 ↑ 5.479.0 ↑ 7.379.5 ↑ 7.882.3 ↑ 10.6
BoolQ79.1 ↑ 3.280.3 ↑ 4.481.1 ↑ 5.282.5 ↑ 6.6
WIC63.8 ↑ 2.463.6 ↑ 2.263.4 ↑ 2.064.3 ↑ 2.9
+ +Table 10: The effects of Sparsity for Fine-tuning LLaMA-7b with S-MeZO. The performance of vanilla MeZO is 71.7 on RTE, 75.9 on BoolQ and 61.4 on WIC. + +# C The Performance of Fine-Tuning Mistral-7b on SuperGLUE + +In this section, we provide the experimental results of MeZO and Sparse MeZO on Mistral-7b. From the results in Table 11, we observe that S-MeZO can achieve consistent improvement on various sub-tasks. + +
ModelMethodBoolQRTEWICMultiRCSST-2COPAAverage
Mistral-7bZero-Shot69.355.250.057.155.584.061.9
Mistral-7bICL76.778.061.471.394.690.078.7
Mistral-7bLoRA84.887.468.283.995.691.085.2
Mistral-7bFT86.787.171.286.195.691.286.3
Mistral-7bMeZO81.680.963.282.793.886.781.5
Mistral-7bMeZO - LoRA83.580.160.782.693.886.981.3
Mistral-7bR-MeZO84.078.763.283.192.484.180.9
Mistral-7bS-MeZO85.384.564.384.994.286.183.2
+ +To further illustrate the performance gain of our proposed Sparse MeZO, we provide error bars for the results on LLaMA2-7b. + +Table 11: Accuracy of Fine-Tuning Mistral-7b on SuperGLUE (1,000 examples). ICL: In-Context Learning, FT: full-parameter fine-tuning with Adam, R-MeZO: MeZO with Random Mask. + +
ModelMethodBoolQRTEWICMultiRCSST-2COPAAverage
LLaMA-7bZero-Shot65.149.550.655.879.759.760.1
LLaMA-7bICL67.454.552.758.781.284.466.5
LLaMA-7bFT84.5 ± 0.083.6 ± 0.968.4 ± 1.380.2 ± 1.495.7 ± 0.385.0 ± 0.882.9 ± 0.8
LLaMA-7bMeZO75.9 ± 1.171.7 ± 1.561.4 ± 1.869.8 ± 0.794.6 ± 0.386.3 ± 0.976.6 ± 1.1
LLaMA-7bR-MeZO76.9 ± 0.775.2 ± 1.762.1 ± 0.468.1 ± 2.094.6 ± 0.284.3 ± 1.776.9 ± 1.1
LLaMA-7bS-MeZO80.9 ± 1.680.7 ± 1.464.9 ± 1.573.3 ± 1.295.0 ± 0.386.7 ± 0.780.3 ± 1.2
+ +Table 12: Accuracy of Fine-Tuning LLaMA-7b on SuperGLUE (1,000 examples). ICL: In-Context Learning, FT: full-parameter fine-tuning with Adam, R-MeZO: MeZO with Random Mask. + +# D Comparison between MeZO and SGD + +In this section, we provide the Probability of Loss Increase with MeZO on Different Batch in Figure 4(a) and Probability of Loss Increase with SGD on Different Batch in Figure 4(b). We calculate the probability of loss increase for each epoch. + +![](images/c27343b90194cccf583a9fd67d694c0e85a09a1a751099af18849ccf605d4533.jpg) +(a) RTE-MeZO + +![](images/cbdad11a510f784b31bf92bcc902f1f68fa71ded6ff551b3cdd080c9c0757a4b.jpg) +(b) RTE-SGD +Figure 4: (a) Probability of Loss Increase with MeZO on Different Batch. (b) Probability of Loss Increase with SGD on Different Batch. We calculate the probability of loss increment for each epoch. + +# E The experimental Results on OPT + +We also provide the experimental results on OPT. As shown in the Table 13, Sparse MeZO can consistently improve the performance of vanilla MeZO on the three tasks of SuperGLUE. + +
ModelMethodBoolQRTEWIC
OPT-13bZero Shot59.059.655.0
OPT-13bICL66.962.150.5
OPT-13bMeZO72.175.562.2
OPT-13bR-MeZO72.375.261.7
OPT-13bS-MeZO73.877.663.7
+ +Table 13: Accuracy of Fine-Tuning OPT on SuperGLUE (1,000 examples). ICL: In-Context Learning, R-MeZO: MeZO with Random Mask. + +# F Pesudo-Code + +In this section, we provide the pseudo-code which is mentioned in Algorithm 1: GetMask (Algorithm 3) to compare each parameter against its threshold $h_i$ and create the mask $m$ , PerturbParameters (Algorithm 2) to generate a Gaussian noise sample $z \sim \mathcal{N}(\mathbf{0}, I_d)$ and apply the mask $m$ to produce a sparse perturbation $\hat{z} = m \odot z$ . + +# Algorithm 2 PerturbParameters + +Input: $\theta$ represents pre-trained LLM weight, perturbation scale $\epsilon$ , random seed $s$ , mask $m$ . + +Reset random seed $s$ + +for $\theta_{i}\in \theta$ do + +$$ +z _ {i} \sim \mathcal {N} (0, 1) +$$ + +$$ +\theta_ {i} \leftarrow \theta_ {i} + m _ {i} * \epsilon z _ {i} +$$ + +end for + +Algorithm 3 GetMask +Input: $\theta$ represents pre-trained LLM weight, threshold $h$ ( $h_i$ represents threshold of each layer). Output: Mask $m$ +for $i \leftarrow$ Layer 1 to Layer $N$ do + for $\theta_{i,j} \in \theta_i$ do + if $\theta_{i,j} \leq h_i$ then + $\theta_{i,j} = 1$ + else + $\theta_{i,j} = 0$ + end if + end for +end for + +# G Convergence Analysis of Sparse-MeZO + +In this section, we will explain why Sparse-MeZO can accelerate the convergence, which is based on the theory from (Ohta et al., 2020). We can define a sub-network in pre-trained large language models, which is determined by the sparse mask $m$ . The main idea of our proof is that if we follow the updated role in Sparse-MeZO, the gradient norm on the sub-network can be smaller than $\sigma^2$ after $\mathcal{O}\left(\frac{\hat{d}L}{\sigma^2}\right)$ steps, where $\hat{d}$ is the number of parameters in the sub-network. Therefore, ZO can use fewer steps to converge when we only focus on a sub-network. Some related work has illustrated that only tuning the sub-network can achieve comparable performance, which will be empirically verified in our experiments. + +Firstly, we assume the loss function $\mathcal{L}(\pmb {\theta};\pmb {x})$ is Lipschitz Continuous: + +Assumption 1 (Lipschitz Continuous). + +$$ +\left\| \nabla \mathcal {L} (\boldsymbol {\theta}; \boldsymbol {x}) - \nabla \mathcal {L} \left(\boldsymbol {\theta} ^ {\prime}, \boldsymbol {x}\right) \right\| \leq \frac {L (l)}{2} \| \boldsymbol {\theta} - \boldsymbol {\theta} ^ {\prime} \| ^ {2}, \tag {3} +$$ + +where $\nabla \mathcal{L}(\pmb{\theta};\pmb{x})$ denotes the true first-order gradient of $\pmb{\theta}$ on $\pmb{x}$ and $L(l)$ represents the Lipschitz constant of $\mathcal{L}(\cdot)$ . Given $\mathcal{L}_{\hat{z}}(\pmb{\theta}) = \mathbb{E}_{\hat{z}}[\mathcal{L}(\pmb{\theta} + \epsilon \hat{\pmb{z}})]$ and the above Assumption 1, we can obtain the relationship between sparse gradient $\widehat{\nabla}_{\pmb{\theta}}\mathcal{L}_{\hat{z}}(\pmb{\theta})$ and the expectation of estimated sparse ZO gradient $g_{\hat{z}}(\pmb{\theta})$ : + +Lemma 1. $ZO$ gradient $\pmb{g}_{\hat{z}}(\pmb {\theta})$ is unbiased estimation of $\widehat{\nabla}_{\pmb{\theta}}\mathcal{L}_{\hat{z}}(\pmb {\theta})$ + +$$ +\begin{array}{l} \widehat {\nabla} _ {\boldsymbol {\theta}} \mathcal {L} _ {\hat {z}} (\boldsymbol {\theta}) = \boldsymbol {m} \odot \nabla_ {\boldsymbol {\theta}} \mathcal {L} _ {\hat {z}} (\boldsymbol {\theta}) \\ = \boldsymbol {m} \odot \nabla_ {\boldsymbol {\theta}} \mathbb {E} _ {\hat {z}} [ \mathcal {L} (\boldsymbol {\theta} + \epsilon \hat {z}) ] \\ = \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\boldsymbol {\theta} + \epsilon \hat {z}) - \mathcal {L} (\boldsymbol {\theta} - \epsilon \hat {z})}{2 \epsilon} \hat {z} \right] \tag {4} \\ = \mathbb {E} _ {\hat {z}} [ g _ {\hat {z}} (\boldsymbol {\theta}) ], \\ \end{array} +$$ + +where $g_{\hat{z}}(\theta) = \frac{\mathcal{L}(\theta + \epsilon\hat{z}) - \mathcal{L}(\theta - \epsilon\hat{z})}{2\epsilon}\hat{z}$ . We can find that $g_{\hat{z}}(\theta)$ is unbiased estimation of $\widehat{\nabla}_{\pmb{\theta}}\mathcal{L}_{\hat{z}}(\pmb{\theta})$ . Then, based on the equation $\widehat{\nabla}_{\pmb{\theta}}\mathcal{L}_{\hat{z}}(\pmb{\theta}) = \mathbb{E}_{\hat{z}}[g_{\pmb{z}}(\pmb{\theta})]$ in Lemma 1, we can use the distance $\| \widehat{\nabla}_{\pmb{\theta}}\mathcal{L}_{\hat{z}}(\pmb{\theta}) - \nabla_{\pmb{\theta}}\mathcal{L}_{m}(\pmb{\theta})\|$ to analyze the relationship between the true sparse gradient $\nabla_{\pmb{\theta}}\mathcal{L}_{m}(\pmb{\theta}) = m \odot \nabla_{\pmb{\theta}}\mathcal{L}(\pmb{\theta})$ and sparse gradient $\widehat{\nabla}_{\pmb{\theta}}\mathcal{L}_{\hat{z}}(\pmb{\theta})$ : + +Lemma 2. Let $\mathcal{L}$ be Lipschitz Continuous, we have: + +$$ +\left\| \nabla_ {\boldsymbol {\theta}} \mathcal {L} _ {m} (\boldsymbol {\theta}) \right\| ^ {2} \leq 2 \left\| \widehat {\nabla} _ {\boldsymbol {\theta}} \mathcal {L} _ {\hat {z}} (\boldsymbol {\theta}) \right\| ^ {2} + \frac {\epsilon^ {2} L ^ {2} (l)}{2} (\hat {d} + 4) ^ {3}. \tag {5} +$$ + +where $\nabla_{\pmb{\theta}}\mathcal{L}_m(\pmb {\theta}) = \pmb {m}\odot \nabla_{\pmb{\theta}}\mathcal{L}(\pmb {\theta}),\hat{d} = \sum_{i = 1}^{i = d}m_i$ is the number of selected parameters in mask $m$ , $L(l)$ represents the Lipschitz constant. Finally, we can obtain the convergence rate of Sparse-MeZO. + +Theorem 1. Assuming a sequence of generated parameters $\{\pmb {\theta}_t\}_{t\geq 0}$ in Sparse-MeZO. We can have: + +$$ +\mathbb {E} _ {\hat {z}, x} \left[ \left\| \nabla_ {\boldsymbol {\theta}} \mathcal {L} _ {m} \left(\boldsymbol {\theta} _ {T}\right) \right\| ^ {2} \right] \leq \sigma^ {2} \tag {6} +$$ + +for any $T = \mathcal{O}\left(\frac{\hat{d}L}{\sigma^2}\right)$ + +where $L(l) \leq L$ for all $\mathcal{L}(\theta_t)$ . This theorem illustrates that the presence of pronounced sparsity patterns, along with the smoothness of the objective function, can significantly enhance the rate of convergence, potentially achieving a linear acceleration. + +# H The Proof of Lemma 1 + +Let $\mathcal{L}_z(\theta)$ be the expectation of $\mathcal{L}(\theta +\epsilon m\odot z)$ + +$$ +\begin{array}{r l} \mathcal {L} _ {\hat {z}} (\theta): = & \mathbb {E} _ {z} [ \mathcal {L} (\theta + \epsilon m \odot z) ] \\ - & \mathbb {E} _ {\hat {z}} [ \mathcal {L} (\theta + \epsilon \hat {z}) ]. \end{array} \tag {7} +$$ + +We can obtain the Lemma: + +$$ +\begin{array}{l} \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) = m \odot \nabla_ {\theta} \mathcal {L} _ {\hat {z}} (\theta) \\ = m \odot \mathbb {E} _ {z} \left[ \nabla_ {\theta} \mathcal {L} (\theta + \epsilon m \odot z) \right] \\ = \mathbb {E} _ {z} \left[ \frac {\mathcal {L} (\theta + \epsilon m \odot z) - \mathcal {L} (\theta - \epsilon m \odot z)}{2 \epsilon} m \odot z \right] \tag {8} \\ = \mathbb {E} _ {\hat {z}} [ \frac {\mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta - \epsilon \hat {z})}{2 \epsilon} \hat {z} ] \\ \end{array} +$$ + +Proof: + +$$ +\begin{array}{l} \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) = \widehat {\nabla} _ {\theta} \mathbb {E} _ {\hat {z}} [ \mathcal {L} (\theta + \epsilon \hat {z}) ] \\ = \widehat {\nabla} _ {\theta} \int_ {\widehat {z}} \operatorname {p d f} _ {\widehat {z}} (z) \mathcal {L} (\theta + \epsilon z) d z \\ = m \odot \nabla_ {\theta} \int_ {\hat {z}} \operatorname {p d f} _ {\hat {z}} (z) \mathcal {L} (\theta + \epsilon z) d z \\ = m \odot \int_ {\hat {z}} \nabla_ {\theta} \operatorname {p d f} _ {\hat {z}} (z) \mathcal {L} (\theta + \epsilon z) d z \\ = \frac {1}{k} m \odot \int_ {\hat {z}} \nabla_ {\theta} e ^ {- \frac {1}{2} \| z \| ^ {2}} \mathcal {L} (\theta + \epsilon z) d z \\ = \frac {1}{k} m \odot \int_ {\hat {y}} \nabla_ {\theta} e ^ {- \frac {1}{2} \| \frac {y - \theta}{\epsilon} \| ^ {2}} \mathcal {L} (y) \frac {1}{\epsilon^ {n}} d y \tag {9} \\ = \frac {1}{k} m \odot \int_ {\hat {y}} \frac {y - \theta}{\epsilon^ {2}} e ^ {- \frac {1}{2 \epsilon^ {2}} \| y - \theta \| ^ {2}} \mathcal {L} (y) \frac {1}{\epsilon^ {n}} d y \\ = \frac {1}{k} m \odot \int_ {\hat {z}} \frac {z}{\epsilon} e ^ {- \frac {1}{2} \| z \| ^ {2}} \mathcal {L} (\theta + \epsilon z) d z \\ = m \odot \int_ {\hat {z}} \operatorname {p d f} _ {\hat {z}} (z) \mathcal {L} (\theta + \epsilon z) \frac {z}{\epsilon} d z \\ = \mathbb {E} _ {\hat {z}} [ m \odot \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} ] \\ = \mathbb {E} _ {\hat {z}} [ \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} ] \\ \end{array} +$$ + +where we can define $y = \theta + \epsilon z$ , $\hat{y} = \theta + \epsilon m \odot z$ , $k = \sqrt{(2\pi)^{\hat{d}}}$ and $\hat{d}$ is the number of 1 in $m$ . + +Therefore, we can obtain the gradient $\nabla_{\theta}\mathcal{L}_m(\theta)$ is equal to $\mathbb{E}_{\hat{z}}[\frac{\mathcal{L}(\theta + \epsilon\hat{z})}{\epsilon}\hat{z} ]$ + +In addition, we will prove $\mathbb{E}_{\hat{z}}[\frac{\mathcal{L}(\theta + \epsilon\hat{z})}{\epsilon}\hat{z}]$ is also equal to $\mathbb{E}_{\hat{z}}[\frac{\mathcal{L}(\theta + \epsilon\hat{z}) - L(\theta)}{\epsilon}\hat{z}]$ : + +$$ +\begin{array}{l} \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta)}{\epsilon} \hat {z} \right] \\ = \frac {1}{k} \int_ {\hat {z}} \frac {\mathcal {L} (\theta + \epsilon z) - \mathcal {L} (\theta)}{\epsilon} z e ^ {- \frac {1}{2} \| z \| ^ {2}} d z \tag {10} \\ = \frac {1}{k} \int_ {\hat {\epsilon}} \frac {\mathcal {L} (\theta + \epsilon z)}{\epsilon} z e ^ {- \frac {1}{2} \| z \| ^ {2}} d z - \frac {\mathcal {L} (\theta)}{\epsilon k} \int_ {\hat {z}} z e ^ {- \frac {1}{2} \| z \| ^ {2}} d z \\ = \mathbb {E} _ {\hat {z}} [ \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} ] \\ \end{array} +$$ + +After that, we can get the relationship between $\mathbb{E}_{\hat{z}}[\frac{\mathcal{L}(\theta) - \mathcal{L}(\theta - \epsilon\hat{z})}{\epsilon}\hat{z} ]$ and $\mathbb{E}_{\hat{z}}[\frac{\mathcal{L}(\theta + \epsilon\hat{z})}{\epsilon}\hat{z} ]$ + +$$ +\begin{array}{l} \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta) - \mathcal {L} (\theta - \epsilon \hat {z})}{\epsilon} \hat {z} \right] = \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon (- \hat {z})) - \mathcal {L} (\theta)}{\epsilon} (- \hat {z}) \right] \\ = \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon \hat {z} - \mathcal {L} (\theta))}{\epsilon} \hat {z} \right] \tag {11} \\ = \mathbb {E} _ {\hat {z}} [ \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} ]. \\ \end{array} +$$ + +Based on the Equation 10 and Equation 11, we can obtain: + +$$ +\begin{array}{l} \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta - \epsilon \hat {z})}{2 \epsilon} \hat {z} \right] \\ = \frac {1}{2} \left(\mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} - \frac {\mathcal {L} (\theta)}{\epsilon} \hat {z} + \frac {\mathcal {L} (\theta)}{\epsilon} \hat {z} - \frac {\mathcal {L} (\theta - \epsilon \hat {z})}{\epsilon} \hat {z} \right]\right) \\ = \frac {1}{2} \left(\mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta)}{\epsilon} \hat {z} \right] + \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta) - \mathcal {L} (\theta - \epsilon \hat {z})}{\epsilon} \hat {z} \right]\right) \tag {12} \\ = \frac {1}{2} \left(\mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} \right] + \mathbb {E} _ {\hat {z}} \left[ \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} \right]\right) \\ = \mathbb {E} _ {\hat {z}} [ \frac {\mathcal {L} (\theta + \epsilon \hat {z})}{\epsilon} \hat {z} ] \\ = \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) \\ \end{array} +$$ + +Finally, we can obtain the relationship between $\mathbb{E}_{\hat{z}}\left[\frac{\mathcal{L}(\theta + \epsilon\hat{z}) - \mathcal{L}(\theta - \epsilon\hat{z})}{2\epsilon}\hat{z}\right]$ and $\widehat{\nabla}_{\theta}\mathcal{L}_{\hat{z}}(\theta)$ and finish the proof. + +# I The Proof of Lemma 2 + +$$ +\left\| \nabla_ {\boldsymbol {\theta}} \mathcal {L} _ {m} (\boldsymbol {\theta}) \right\| ^ {2} \leq 2 \left\| \widehat {\nabla} _ {\boldsymbol {\theta}} \mathcal {L} _ {\hat {z}} (\boldsymbol {\theta}) \right\| ^ {2} + \frac {\epsilon^ {2} L ^ {2} (l)}{2} (\hat {d} + 4) ^ {3}. \tag {13} +$$ + +Proof: + +We can first define the distance between $\widehat{\nabla}_{\pmb{\theta}}\mathcal{L}_{\hat{z}}(\pmb {\theta}) = \mathbb{E}_{\hat{z}}[\pmb {g}_{\hat{z}}(\pmb {\theta})]$ and sparse FO gradient $\nabla \mathcal{L}_m(\theta)$ as: + +$$ +\begin{array}{l} \left\| \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) - \nabla_ {\theta} \mathcal {L} _ {m} (\theta) \right\| \\ = \| \frac {1}{k} \int_ {z} (\frac {\mathcal {L} (\theta + \epsilon z) - \mathcal {L} (\theta - \epsilon z)}{2 \epsilon} - \langle \nabla_ {\theta} \mathcal {L} _ {m} (\theta), z \rangle) z e ^ {- \frac {1}{2} \| z \| ^ {2}} d \hat {z} \| \\ = \| \frac {1}{k} \int_ {z} (\frac {\mathcal {L} (\theta + \epsilon z) - \mathcal {L} (\theta)}{\epsilon} - \langle m \odot \nabla_ {\theta} \mathcal {L} (\theta), z \rangle) z e ^ {- \frac {1}{2} \| z \| ^ {2}} d \hat {z} \| \\ \leq \frac {1}{k \epsilon} \int_ {z} | \mathcal {L} (\theta + \epsilon z) - \mathcal {L} (\theta) - \epsilon \langle \nabla_ {\theta} \mathcal {L} (\theta), \epsilon \rangle | \| m \odot z \| e ^ {- \frac {1}{2} \| z \| ^ {2}} d \hat {z} \tag {14} \\ \leq \frac {\epsilon L (l)}{2 k} \int_ {\epsilon} \| z \| ^ {2} \| m \odot z \| e ^ {- \frac {1}{2} \| z \| ^ {2}} d \hat {z} \\ = \frac {\epsilon L (l)}{2} \mathbb {E} _ {\hat {z}} [ \| \hat {z} \| ^ {3} ] \\ \leq \frac {\epsilon L (l)}{2} (\hat {d} + 3) ^ {\frac {3}{2}} \\ \end{array} +$$ + +where $\hat{d}$ is the number of selected parameters with mask $m$ . The last inequality holds because the Equation (25) in (Ohta et al., 2020). In addition, $\| \pmb{a} + \pmb{b} \|^2 \leq 2 \| \pmb{a} \|^2 + 2 \| \pmb{b} \|^2$ , we can define $\pmb{a} = \pmb{a} - \pmb{b}$ and obtain that $\| \pmb{a} \|^2 \leq 2 \| \pmb{a} - \pmb{b} \|^2 + 2 \| \pmb{b} \|^2$ . Let $\pmb{a} = \nabla_{\theta} \mathcal{L}_m(\theta)$ and $\pmb{b} = \widehat{\nabla}_{\theta} \mathcal{L}_{\hat{z}}(\theta)$ , we can obtain: + +$$ +\begin{array}{l} \left\| \nabla_ {\theta} \mathcal {L} _ {m} (\theta) \right\| ^ {2} \leq 2 \left\| \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) - \nabla_ {\theta} \mathcal {L} _ {m} (\theta) \right\| ^ {2} + 2 \left\| \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) \right\| ^ {2} \\ \leq \frac {\epsilon^ {2} L ^ {2} (l)}{2} (\hat {d} + 3) ^ {3} + 2 \| \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) \| ^ {2} \tag {15} \\ \leq \frac {\epsilon^ {2} L ^ {2} (l)}{2} (\hat {d} + 4) ^ {3} + 2 \| \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) \| ^ {2} \\ \end{array} +$$ + +# J The Proof of Theorem 1 + +Proof: + +$$ +\begin{array}{l} \mathcal {L} _ {\hat {z}} (\theta) - \mathcal {L} (\theta) = \mathbb {E} _ {\hat {z}} [ \mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta) ] \\ = \mathbb {E} _ {\hat {z}} [ \mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta) - \epsilon \langle \nabla \mathcal {L} (\theta), \hat {z} \rangle ] \\ = \frac {1}{k} \int_ {\hat {z}} [ \mathcal {L} (\theta + \epsilon z) - \mathcal {L} (\theta) - \epsilon \langle \nabla \mathcal {L} (\theta), z \rangle ] e ^ {- \frac {1}{2} \| z \| ^ {2}} d z \\ \leq \frac {1}{k} \int_ {\hat {z}} \frac {\epsilon^ {2} L (l)}{2} \| z \| ^ {2} e ^ {- \frac {1}{2} \| z \| ^ {2}} d z \tag {16} \\ = \frac {\epsilon^ {2} L (l)}{2} \mathbb {E} _ {\hat {z}} [ \| \hat {z} \| ^ {2} ] \\ \leq \frac {\epsilon^ {2} L (l)}{2} \hat {d} \\ \end{array} +$$ + +The first inequality holds because Lipschitz Continuous: $|\mathcal{L}(\theta') - \mathcal{L}(\theta) - \langle \nabla \mathcal{L}(\theta), \theta' - \theta \rangle| \leq \frac{L(l)}{2} \| \theta' - \theta \|^2$ , where $\theta' = \theta + \epsilon z$ . The second inequality holds because $\mathbb{E}_{\hat{z}}[\| \hat{z} \|^2] = \hat{d}$ , where $\hat{d}$ is the number of 1 in mask $m$ . + +$$ +\begin{array}{l} \left[ \left(\mathcal {L} _ {\hat {z}} (\theta) - \mathcal {L} (\theta)\right) - \left(\mathcal {L} _ {\hat {z}} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta + \epsilon \hat {z})\right) \right] ^ {2} \\ \leq 2 \left[ \mathcal {L} _ {\hat {z}} (\theta) - \mathcal {L} (\theta) \right] ^ {2} + 2 \left[ \mathcal {L} _ {\hat {z}} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta + \epsilon \hat {z}) \right] ^ {2} \\ \leq \frac {\epsilon^ {4} L ^ {2} (l)}{2} \tilde {d} ^ {2} + \frac {\epsilon^ {4} L ^ {2} (l)}{2} \tilde {d} ^ {2} \tag {17} \\ = \epsilon^ {4} L ^ {2} (l) \hat {d} ^ {2} \\ \end{array} +$$ + +The first inequality is due to $\| \pmb{a} + \pmb{b}\|^2 \leq 2\|\pmb{a}\|^2 + 2\|\pmb{b}\|^2$ , where $\pmb{a} = \mathcal{L}_{\hat{z}}(\theta) - \mathcal{L}(\theta)$ , $\pmb{b} = \mathcal{L}_{\hat{z}}(\theta + \epsilon \hat{z}) - \mathcal{L}(\theta + \epsilon \hat{z})$ . The second inequality is due to the Equation 16. + +$$ +\begin{array}{l} \left[ \mathcal {L} _ {\hat {z}} (\theta + \epsilon \hat {z}) - \mathcal {L} _ {\hat {z}} (\theta) \right] ^ {2} \leq 2 \left[ \mathcal {L} _ {\hat {z}} (\theta + \epsilon \hat {z}) - \mathcal {L} _ {\hat {z}} (\theta) - \epsilon \langle \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta), \hat {z} \rangle \right] ^ {2} + 2 \left[ \epsilon \langle \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta), \hat {z} \rangle \right] ^ {2} \\ \leq \frac {\epsilon^ {4} L ^ {2} (l)}{2} \| \hat {z} \| ^ {4} + 2 \epsilon^ {2} \left\langle \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta), \hat {z} \right\rangle^ {2} \tag {18} \\ \leq \frac {\epsilon^ {4} L ^ {2} (l)}{2} \| \hat {z} \| ^ {4} + 2 \epsilon^ {2} \| \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) \| ^ {2} \| \hat {z} \| ^ {2} \\ \end{array} +$$ + +The first inequality is due to $\| \pmb{a} + \pmb{b} \|^2 \leq 2 \| \pmb{a} \|^2 + 2 \| \pmb{b} \|^2$ . The second inequality holds because Lipschitz Continuous: $|\mathcal{L}(\theta') - \mathcal{L}(\theta) - \langle \nabla \mathcal{L}(\theta), \theta' - \theta \rangle| \leq \frac{L(l)}{2} \| \theta' - \theta \|^2$ , where $\theta' = \theta + \epsilon \hat{z}$ . + +$$ +\begin{array}{l} [ \mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta) ] ^ {2} \\ \leq 2 \left[ \left(\mathcal {L} _ {\hat {z}} (\theta) - \mathcal {L} (\theta)\right) - \left(\mathcal {L} _ {\hat {z}} \left(\theta + \epsilon \hat {z}\right) - \mathcal {L} \left(\theta + \epsilon \hat {z}\right)\right) \right] ^ {2} + 2 \left[ \mathcal {L} _ {\hat {z}} \left(\theta + \epsilon \hat {z}\right) - \mathcal {L} _ {\hat {z}} (\theta) \right] ^ {2} \tag {19} \\ \leq 2 \epsilon^ {4} L ^ {2} (l) \hat {d} ^ {2} + \epsilon^ {4} L ^ {2} (l) \| \hat {z} \| ^ {4} + 4 \epsilon^ {2} \| \widehat {\nabla} _ {\theta} \mathcal {L} _ {\hat {z}} (\theta) \| ^ {2} \| \hat {z} \| ^ {2} \\ \end{array} +$$ + +The first inequality is due to $\| \pmb{a} + \pmb{b} \|^2 \leq 2 \| \pmb{a} \|^2 + 2 \| \pmb{b} \|^2$ . The last inequality holds because Equation 17 and Equation 18. + +$$ +\begin{array}{l} \mathbb {E} _ {z, x} [ \| g _ {\hat {z}} (\theta) \| ^ {2} ] = \mathbb {E} _ {\hat {z}} [ \| \frac {\mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta - \epsilon \hat {z})}{2 \epsilon} \hat {z} \| ^ {2} ] \\ = \mathbb {E} _ {\hat {z}} [ \| \frac {\mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta)}{2 \epsilon} \hat {z} + \frac {\mathcal {L} (\theta) - \mathcal {L} (\theta - \epsilon \hat {z})}{2 \epsilon} \hat {z} \| ^ {2} ] \\ \leq \mathbb {E} _ {\hat {z}} \left[ 2 \| \frac {\mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta)}{2 \epsilon} \hat {z} \| ^ {2} + 2 \| \frac {\mathcal {L} (\theta) - \mathcal {L} (\theta - \epsilon \hat {z})}{2 \epsilon} \hat {z} \| ^ {2} \right] \tag {20} \\ = \mathbb {E} _ {\hat {z}} \left[ \frac {1}{2 \epsilon^ {2}} \left[ \mathcal {L} (\theta + \epsilon \hat {z}) - \mathcal {L} (\theta) \right] ^ {2} \cdot \| \hat {z} \| ^ {2} + \frac {1}{2 \epsilon^ {2}} \left[ \mathcal {L} (\theta) - \mathcal {L} (\theta - \epsilon \hat {z}) \right] ^ {2} \cdot \| \hat {z} \| ^ {2} \right] \\ \leq \mathbb {E} _ {\hat {z}} [ 2 \epsilon^ {2} L ^ {2} (l) \hat {d} ^ {2} \| \hat {z} \| ^ {2} + \epsilon^ {2} L ^ {2} (l) \| \hat {z} \| ^ {6} + 4 \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} (\theta) \| ^ {2} \| \hat {z} \| ^ {4} ] \\ \leq 2 \epsilon^ {2} L ^ {2} (l) \hat {d} ^ {3} + \epsilon^ {2} L ^ {2} (l) (\hat {d} + 6) ^ {3} + 4 (\hat {d} + 4) ^ {2} \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} (\theta) \| ^ {2} \\ \leq 3 \epsilon^ {2} L ^ {2} (l) (\hat {d} + 4) ^ {3} + 4 (\hat {d} + 4) ^ {2} \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} (\theta) \| ^ {2} \\ \end{array} +$$ + +The first inequality holds because $\| \pmb{a} + \pmb{b} \|^2 \leq 2 \| \pmb{a} \|^2 + 2 \| \pmb{b} \|^2$ , where $\pmb{a} = \frac{\mathcal{L}(\theta + \epsilon \hat{z}) - \mathcal{L}(\theta)}{2\epsilon} \hat{z}$ , $\pmb{b} = \frac{\mathcal{L}(\theta) - \mathcal{L}(\theta - \epsilon \hat{z})}{2\epsilon} \hat{z}$ . The second inequality is due to the Equation 19. The third inequality holds because $\mathbb{E}_{\hat{z}}[\| \hat{z} \|^2] = \hat{d}, \mathbb{E}_{\hat{z}}[\| \hat{z} \|^p] \leq (\hat{d} + p)^{\frac{p}{2}}$ for $p \geq 2$ . The last inequality holds because $2\hat{d}^3 + (\hat{d} + 6)^3 \leq 3(\hat{d} + 4)^3$ . + +Based on the assumption about Lipschitz Continuous, we can obtain: $|\mathcal{L}(\theta_{t + 1}) - \mathcal{L}(\theta_t) - \langle \nabla \mathcal{L}(\theta_t),\theta_{t + 1} - \theta_t\rangle |\leq \frac{L(l)}{2}\| \theta_{t + 1} - \theta_t\| ^2.$ + +Then, we can obtain: + +$$ +\begin{array}{l} \mathcal {L} _ {\hat {z}} \left(\theta_ {t + 1}\right) - \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) - \left\langle \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right), \theta_ {t + 1} - \theta_ {t} \right\rangle \leq \left| \mathcal {L} _ {\hat {z}} \left(\theta_ {t + 1}\right) - \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) - \left\langle \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right), \theta_ {t + 1} - \theta_ {t} \middle | \right. \right. \\ \leq \frac {L (l)}{2} \| \theta_ {t + 1} - \theta_ {t} \| ^ {2} \tag {21} \\ \end{array} +$$ + +Based on the equation, we can follow the update rule: $\theta_{t + 1} = \theta_t - \eta_t g_{\hat{z}}(\theta_t)$ and we can find: + +$$ +\begin{array}{l} \mathcal {L} _ {\hat {z}} \left(\theta_ {t + 1}\right) \leq \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) + \left\langle \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right), \theta_ {t + 1} - \theta_ {t} \right\rangle + \frac {L (l)}{2} \left\| \theta_ {t} - \theta_ {t + 1} \right\| ^ {2} \tag {22} \\ = \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) - \eta_ {t} \langle \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right), g _ {\hat {z}} \left(\theta_ {t}\right) \rangle + \frac {\left(\eta_ {t}\right) ^ {2} L (l)}{2} \| g _ {\hat {z}} \left(\theta_ {t}\right) \| ^ {2} \\ \end{array} +$$ + +where $\eta_t$ represents the learning rate at step $t$ . Then, we can take the expectation of Equation 22 for $\hat{z}$ and input $x$ : + +$$ +\begin{array}{l} \mathbb {E} _ {\hat {z}, x} \left[ \mathcal {L} _ {\hat {z}} \left(\theta_ {t + 1}\right) \right] \\ \leq \mathbb {E} _ {\hat {z}, x} \left[ \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) \right] - \eta_ {t} \mathbb {E} _ {\hat {z}, x} \left[ \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) \| ^ {2} \right] + \frac {\left(\eta_ {t}\right) ^ {2} L (l)}{2} \mathbb {E} _ {\hat {z}, x} \left[ \| g _ {z} \left(\theta_ {t}\right) \| ^ {2} \right] \\ \leq \mathbb {E} _ {\hat {z}, x} \left[ \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) \right] - \eta_ {t} \mathbb {E} _ {\hat {z}, x} \left[ \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) \| ^ {2} \right] + \frac {\left(\eta_ {t}\right) ^ {2} L (l)}{2} \left(4 \left(\hat {d} _ {t} + 4\right) ^ {2} \mathbb {E} _ {\hat {z}, x} \left[ \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) \| ^ {2} \right] + 3 \epsilon^ {2} L ^ {2} (l) \left(\hat {d} _ {t} + 4\right) ^ {3}\right) \tag {23} \\ \end{array} +$$ + +The first inequality is due to the Equation 8 and Equation 22. The second inequality holds because Equation 20 provides the result about $\mathbb{E}_{\hat{z},x}[\| g_z(\theta_t)\|^2$ + +Then, we can select learning rate $\eta_t = \frac{1}{4(\hat{d}_t + 4)L(l)}$ and obtain: + +$$ +\mathbb {E} _ {\hat {z}, x} \left[ \mathcal {L} _ {\hat {z}} \left(\theta_ {t + 1}\right) \right] \leq \mathbb {E} _ {\hat {z}, x} \left[ \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) \right] - \frac {1}{8 \left(\hat {d} _ {t} + 4\right) L (l)} \mathbb {E} _ {\hat {z}, x} \left[ \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {t}\right) \| ^ {2} \right] + \frac {3 \epsilon^ {2}}{3 2} L (l) \left(\hat {d} _ {t} + 4\right) \tag {24} +$$ + +Then, taking the sum of Equation 24 over the index from $T + 1$ to 0, we can have that: + +$$ +\mathbb {E} _ {\hat {z}, x} \left[ \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {T}\right) \| ^ {2} \right] \leq 8 (\hat {d} + 4) L \left[ \frac {\mathcal {L} _ {\hat {z}} \left(\theta_ {0}\right) - \mathcal {L} _ {\hat {z}} ^ {*}}{T + 1} + \frac {3 \epsilon^ {2}}{3 2} L (\hat {d} + 4) \right] \tag {25} +$$ + +where $L(l) \leq L$ for all $\mathcal{L}(\pmb{\theta}_t)$ . Thus, based on Lemma 2, we can have: + +$$ +\begin{array}{l} \mathbb {E} _ {\hat {z}, x} \left[ \| \nabla \mathcal {L} _ {m} \left(\theta_ {T}\right) \| ^ {2} \right] \leq \frac {\epsilon^ {2} L ^ {2}}{2} (\hat {d} + 4) ^ {3} + 2 \mathbb {E} _ {\hat {z}, x} \left[ \| \widehat {\nabla} \mathcal {L} _ {\hat {z}} \left(\theta_ {T}\right) \| ^ {2} \right] \tag {26} \\ \leq 1 6 (\hat {d} + 4) L \frac {\mathcal {L} _ {\hat {z}} \left(\theta_ {0}\right) - \mathcal {L} _ {\hat {z}} ^ {*}}{T + 1} + \frac {\epsilon^ {2} L ^ {2}}{2} (\hat {d} + 4) ^ {2} (\hat {d} + \frac {1 1}{2}) \\ \end{array} +$$ + +The second inequality is due to the Equation 25. To obtain $\sigma$ -accurate solution: $\mathbb{E}_{\hat{z},x}[\| \nabla \mathcal{L}_m(\theta_T)\|^2 ]\leq \sigma^2$ , we can define $\epsilon = \Omega (\frac{\sigma}{\hat{d}^{\frac{3}{2}}L})$ + +$$ +\begin{array}{l} 1 6 (\hat {d} + 4) L \frac {\mathcal {L} _ {\hat {z}} \left(\theta_ {0}\right) - \mathcal {L} _ {\hat {z}} ^ {*}}{T + 1} + \mathcal {O} \left(\epsilon^ {2} L ^ {2} \hat {d} ^ {3}\right) = 1 6 (\hat {d} + 4) L \frac {\mathcal {L} _ {\hat {z}} \left(\theta_ {0} - \mathcal {L} _ {\hat {z}} ^ {*}\right)}{T + 1} + \mathcal {O} \left(\sigma^ {2}\right) \tag {27} \\ T = \mathcal {O} (\frac {\hat {d} L}{\sigma^ {2}}) \\ \end{array} +$$ + +Finally, we can finish the proof of the theorem. 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However, existing methods suffer from significant information loss due to the separation of stages such as quantization and sequence modeling, hindering their ability to achieve the modeling precision and accuracy of sequential dense retrieval techniques. Integrating generative and dense retrieval methods remains a critical challenge. To address this, we introduce the Cascaded Organized Bi-Represented generAtive retrieval (COBRA) framework, which innovatively integrates sparse semantic IDs and dense vectors through a cascading process. Our method alternates between generating these representations by first generating sparse IDs, which serve as conditions to aid in the generation of dense vectors. End-to-end training enables dynamic refinement of dense representations, capturing both semantic insights and collaborative signals from user-item interactions. During inference, COBRA employs a coarse-to-fine strategy, starting with sparse ID generation and refining them into dense vectors via the generative model. We further propose BeamFusion, an innovative approach combining beam search with nearest neighbor scores to enhance inference flexibility and recommendation diversity. Extensive experiments on public datasets and offline tests validate our method's robustness. Online A/B tests on a real-world advertising platform with over 200 million daily users demonstrate substantial improvements in key metrics, highlighting COBRA's practical advantages. + +# 1 Introduction + +Recommendation systems are vital components of modern digital ecosystems, providing personalized item suggestions that align with user preferences across e-commerce platforms, streaming services, and social networks [1-3]. Recent advancements have focused on sequential recommendation methods, which leverage the sequential nature of user interactions to enhance recommendation performance [4-7]. Notable models like SASRec [8] and BERT4Rec [9] have demonstrated the effectiveness of sequence models in capturing user behavior patterns. + +The emergence of generative models has further expanded the capabilities of recommendation systems [10-12]. Unlike traditional sequential recommendation methods, generative models can directly predict target items based on user behavior sequences [13-15]. These models handle complex user-item interactions and offer emerging abilities such as reasoning and few-shot learning, which significantly improve recommendation accuracy and diversity [16-18]. Among these, TIGER [19] + +![](images/7a2d20197b5c73d7a00ed952d938b42db9c2720ce239e8e8ff69e63df935f749.jpg) +Figure 1: Comparison of generative recommendation paradigms. The left side illustrates traditional generative retrieval approaches, exemplified by TIGER, which uses a sequence of sparse IDs as input within a Transformer encoder-decoder architecture to directly predict the sparse ID of the next item. The right side depicts the proposed COBRA(Cascaded Organized Bi-Represented generAtive retrieval), which integrates sparse IDs for coarse semantics and dense vectors for fine details. The cascaded representation is processed by a Transformer decoder that sequentially predicts the sparse ID followed by the dense vector. + +![](images/b8e4e6ce72f9aae8d475af729ee3d24bb909d9392b3542f260ad7f3a85536cf6.jpg) + +is a pioneering approach in generative retrieval for recommendation systems. As depicted in Figure 1(Lower Left), TIGER leverages a Residual Quantized Variational AutoEncoder (RQ-VAE) [20] to encode item content features into hierarchical semantic IDs, allowing the model to share knowledge across semantically similar items without the need for individual item embeddings. Beyond TIGER, several other methods have been proposed to further explore the integration of generative models with recommendation systems. LC-Rec [21] aligns semantic and collaborative information using RQ-VAE with a series of alignment tasks. ColaRec [22] combines collaborative filtering signals with content information by deriving generative identifiers from a pretrained recommendation model. IDGenRec [23] leverages large language models to generate unique, concise, and semantically rich textual identifiers for recommended items, showing strong potential in zero-shot settings. + +Despite these innovations, existing generative recommendation methods still face several challenges compared to sequential dense retrieval methods [24, 25]. Sequential dense retrieval methods, which rely on dense embeddings for each item, offer high accuracy and robustness but require substantial storage and computational resources. In contrast, generative methods, while efficient, often struggle with fine-grained similarity modeling [26]. To effectively leverage the strengths of both retrieval paradigms, we propose Cascaded Organized Bi-Represented generAtive retrieval(COBRA), a framework that synergizes generative and dense retrieval. Figure 1(Right) illustrates the cascaded sparse-dense representations in COBRA. The proposed method introduces a cascaded generative retrieval framework alternating between generating sparse IDs and dense vectors. This approach mitigates the information loss inherent in ID-based methods. Specifically, COBRA's input is a sequence of cascaded representations composed of sparse IDs and dense vectors corresponding to items in the user's interaction history. During training, dense representations are learned through contrastive learning objectives in an end-to-end manner. By first generating the sparse ID and then the dense representation, COBRA reduces the learning difficulty of dense representations and promotes mutual learning between the two representations. During inference, COBRA employs a coarse-to-fine generation process, starting with sparse ID that provides a high-level categorical sketch capturing the categorical essence of the item. The generated ID is then appended to the input sequence and fed back into the model to predict the dense vector that captures the fine-grained details, enabling more precise and personalized recommendations. To ensure flexible inference, we introduce BeamFusion, a sampling technique combining beam search with nearest neighbor retrieval scores, ensuring controllable diversity in the retrieved items. Unlike TIGER, which relies solely on sparse IDs, COBRA harnesses the strengths of both sparse and dense representations. + +Our main contributions are as follows: + +- Cascaded Bi-Represented Retrieval Framework: We introduce COBRA, a framework that alternates between generating sparse semantic IDs and dense vectors. It addresses information loss in ID-based methods and reduces the difficulty of representation learning by using sparse IDs as conditions for generating dense vectors. +- Learnable Dense Representations via End-to-End Training: COBRA uses the original item data as input to generate dense representations through end-to-end training. Unlike static embeddings, COBRA's dense vectors are dynamically learned, capturing semantic information and fine-grained details. +- Coarse-to-Fine Generation Process: During inference, COBRA first generates sparse IDs which are then fed back into the model to produce refined dense representations, enhancing the granularity of the dense vectors. We also introduce BeamFusion for more diverse recommendations. +- Comprehensive Empirical Validation: Extensive experiments on benchmark datasets show COBRA surpasses current methods in recommendation accuracy, proving its effectiveness in balancing precision and diversity. + +# 2 Related Work + +Sequential Dense Recommendation. Early sequential recommendation systems leveraged RNNs and CNNs to model user behavior sequences [27, 28]. The introduction of Transformer-based methods, such as SASRec [8] and BERT4Rec [9], greatly improved the capability to capture complex user dynamics. Recent models focus on cross-domain transferability and the integration of textual features through contrastive learning [29-31]. + +Generative Recommendation. Generative approaches have shifted the field from discriminative ranking to directly generating item identifiers [32, 19, 33]. Some models treat recommendation as a language modeling task [33], while others generate semantically meaningful or structured identifiers [19, 21, 34]. Hybrid methods, such as LIGER [26], combine generative and dense retrieval to overcome the limitations of each approach. However, how to more flexibly integrate these paradigms remains an open problem. A more comprehensive review is provided in Appendix A. + +# 3 Methodology + +This section introduces the Cascaded Organized Bi-Represented generAtive Retrieval (COBRA) framework, which integrates cascaded sparse-dense representations and coarse-to-fine generation to enhance recommendation performance. Figure 2 illustrates the overall framework of COBRA. + +# 3.1 Sparse-Dense Representation + +Sparse Representation. COBRA generates sparse IDs using a Residual Quantized Variational Autoencoder (RQ-VAE), inspired by the approach in TIGER [19]. For each item, we extract its attributes to generate a textual description, which is embedded into a dense vector space and quantized to produce sparse IDs. These IDs capture the categorical essence of items, forming the basis for subsequent processing. For the sake of brevity, the subsequent methodology descriptions will assume that the sparse ID consists of a single level. However, it should be noted that this approach can be easily extended to accommodate scenarios involving multiple levels. + +Dense Representation. To capture nuanced attribute information, we develop an end-to-end trainable dense encoder, encoding item textual contents. Each item's attributes are flattened into a text sentence, prefixed with a [CLS] token, and fed into a Transformer-based text encoder Encoder. The dense representation $\mathbf{v}_t$ is extracted from the output corresponding to the [CLS] token, capturing fine-grained details of the item's textual content. As illustrated in the lower part of Figure 2, we incorporate position embeddings and type embeddings to model the positional and context of tokens within the sequence. These embeddings are added to the token embeddings, enhancing the model's ability to distinguish between different tokens and their positions in the sequence. + +Cascaded Representation. The cascaded representation integrates sparse IDs and dense vectors within a unified generative model. Specifically, for each item, we combine its sparse ID $ID_{t}$ and dense vector $\mathbf{v}_t$ to form a cascaded representation $(ID_{t},\mathbf{v}_{t})$ . This approach leverages the strengths of both representations, providing a more comprehensive characterization of items: sparse IDs provide + +![](images/1969ad56b387cac98cdfc916baf245dda17477c80515c2efb72a482e6201be32.jpg) +Figure 2: The architecture of COBRA. The model employs a cascaded sparse-dense representation approach, where sparse IDs are generated via Residual Quantization and dense vectors are produced by a trainable Transformer Encoder. These representations serve as inputs to a Transformer Decoder, which alternates between predicting sparse IDs and dense vectors. The predicted outputs are used to compute the loss functions $\mathcal{L}_{\mathrm{sparse}}$ and $\mathcal{L}_{\mathrm{dense}}$ . For the sake of simplicity, the figure illustrates an example with a single level of sparse ID. + +a stable categorical foundation through discrete constraints, while dense vectors maintain continuous feature resolution, ensuring that the model captures both high-level semantics and fine-grained details. + +# 3.2 Sequential Modeling + +Probabilistic Decomposition. The probability distribution modeling of the target item is factorized into two stages, leveraging the complementary strengths of sparse and dense representations. Specifically, instead of directly predicting the next item $s_{t + 1}$ based on the historical interaction sequence $S_{1:t}$ , COBRA predicts the sparse ID $ID_{t + 1}$ and the dense vector $\mathbf{v}_{T + 1}$ separately: + +$$ +P \left(I D _ {t + 1}, \mathbf {v} _ {t + 1} \mid S _ {1: t}\right) = P \left(I D _ {t + 1} \mid S _ {1: t}\right) P \left(\mathbf {v} _ {t + 1} \mid I D _ {t + 1}, S _ {1: t}\right) \tag {1} +$$ + +where $P(ID_{t+1}|S_{1:t})$ represents the probability of generating the sparse ID $ID_{t+1}$ based on the historical sequence $S_{1:t}$ , capturing the categorical essence of the next item. $P(\mathbf{v}_{t+1}|ID_{t+1}, S_{1:t})$ represents the probability of generating the dense vector $\mathbf{v}_{t+1}$ given the sparse ID $ID_{t+1}$ and the historical sequence $S_{1:t}$ , capturing the fine-grained details of the next item. This decomposition allows COBRA to leverage both the categorical information provided by sparse IDs and the fine-grained details captured by dense vectors. + +Sequential Modeling with a Unified Generative Model. For sequential modeling, we utilize a unified generative model based on the Transformer architecture to effectively capture sequential dependencies in user-item interactions. The Transformer receives an input sequence of cascaded representations, with each item represented by its sparse ID and dense vector. + +The sparse ID, denoted as $ID_{t}$ , is transformed into a dense vector space through an embedding layer: $\mathbf{e}_t = \operatorname{Embed}(ID_t)$ . This embedding $\mathbf{e}_t$ is concatenated with the dense vector $\mathbf{v}_t$ to form the model's input at each time step: $\mathbf{h}_t = [\mathbf{e}_t; \mathbf{v}_t]$ . + +Our Transformer Decoder model comprises multiple layers, each featuring self-attention mechanisms and feedforward networks. As depicted in the upper part of Figure 2, the input sequence to the Decoder consists of cascaded representations. To enhance modeling of sequential and contextual information, these representations are augmented with item position and type embeddings. For brevity, mathematical formulations in the following sections focus on the cascaded sequence representation, omitting explicit notation for position and type embeddings. The Decoder processes this enriched input to generate contextualized representations for predicting the subsequent sparse ID and dense vector. + +Sparse ID Prediction. Given history interaction sequence $S_{1:t}$ , to predict the sparse ID $ID_{t+1}$ , the Transformer input sequence is: $\mathbf{S}_{1:t} = [\mathbf{h}_1, \mathbf{h}_2, \dots, \mathbf{h}_t] = [\mathbf{e}_1, \mathbf{v}_1, \mathbf{e}_2, \mathbf{v}_2, \dots, \mathbf{e}_t, \mathbf{v}_t]$ . where each $\mathbf{h}_i$ is a concatenation of the sparse ID embedding and the dense vector for the $i$ -th item. The Transformer processes this sequence to generate contextualized representations, subsequently used to predict the next sparse ID and dense vector. Specifically, the Transformer decoder processes the sequence $\mathbf{S}_{1:t}$ , producing a sequence of vectors $\mathbf{y}_t = \text{TransformerDecoder}(\mathbf{S}_{1:t})$ . The logits for sparse ID prediction are derived as: $\mathbf{z}_{t+1} = \text{SparseHead}(\mathbf{y}_t)$ . where $\mathbf{z}_{t+1}$ represents the logits for the predicted sparse ID $ID_{t+1}$ . + +Dense Vector Prediction. For predicting the dense vector $\mathbf{v}_{t + 1}$ , the Transformer input sequence can be represented as: $\bar{\mathbf{S}}_{1:t} = [\mathbf{S}_{1:t},\mathbf{e}_{t + 1}] = [\mathbf{e}_1,\mathbf{v}_1,\mathbf{e}_2,\mathbf{v}_2,\dots ,\mathbf{e}_t,\mathbf{v}_t,\mathbf{e}_{t + 1}]$ . The Transformer decoder processes $\bar{\mathbf{S}}_{1:t}$ to output the predicted dense vector: $\hat{\mathbf{v}}_{t + 1} = \mathrm{TransformerDecoder}(\bar{\mathbf{S}}_{1:t})$ + +# 3.3 End-to-End Training + +In COBRA, the end-to-end training process is designed to optimize both sparse and dense representation prediction jointly. The training process is governed by a composite loss function that combines losses for sparse ID prediction and dense vector prediction. + +Sparse ID Loss. The sparse ID prediction loss, denoted as $\mathcal{L}_{\mathrm{sparse}}$ , ensures the model's proficiency in predicting the next sparse ID based on the historical sequence $S_{1:t}$ : + +$$ +\mathcal {L} _ {\text {s p a r s e}} = - \sum_ {t = 1} ^ {T - 1} \log \left(\frac {\exp \left(z _ {t + 1} ^ {I D _ {t + 1}}\right)}{\sum_ {j = 1} ^ {C} \exp \left(z _ {t + 1} ^ {j}\right)}\right) \tag {2} +$$ + +where $T$ is the length of the historical sequence, $ID_{t+1}$ is the sparse ID corresponding to interacted item at time step $t+1$ , $z_{t+1}^{ID_{t+1}}$ represents the predicted logit of ground truth sparse ID $ID_{t+1}$ at time step $t+1$ , generated by the Transformer Decoder, and $C$ denotes the set of all sparse IDs. + +Dense vector Loss. The dense vector prediction loss $\mathcal{L}_{\mathrm{dense}}$ focuses on refining the dense vectors, enabling them to discern between similar and dissimilar items. The loss is defined as: + +$$ +\mathcal {L} _ {\text {d e n s e}} = - \sum_ {t = 1} ^ {T - 1} \log \frac {\exp \left(\cos \left(\hat {\mathbf {v}} _ {t + 1} , \mathbf {v} _ {t + 1}\right)\right)}{\sum_ {\text {i t e m} _ {j} \in \text {B a t c h}} \exp \left(\cos \left(\hat {\mathbf {v}} _ {t + 1}, \mathbf {v} _ {\text {i t e m} _ {j}}\right)\right)} \tag {3} +$$ + +where $\hat{\mathbf{v}}_t$ is the predicted dense vector, $\mathbf{v}_t$ is the ground truth dense vector for the positive item, and $\mathbf{v}_{item_j}$ represents the dense vectors of items within the batch. The term $\cos (\hat{\mathbf{v}}_{t + 1} \cdot \mathbf{v}_{t + 1})$ represents the cosine similarity between the predicted and ground truth dense vectors. The dense vectors are generated by an end-to-end trainable encoder denoted by Encoder, which is optimized during the training process. This ensures that the dense vectors are dynamically refined and adapted to the specific requirements of the recommendation task. + +Overall Loss. The overall loss function is formulated as: $\mathcal{L} = \mathcal{L}_{\mathrm{sparse}} + \mathcal{L}_{\mathrm{dense}}$ . The dual-objective loss function enables a balanced optimization process, where the model dynamically refines dense vectors guided by sparse IDs. This end-to-end training approach captures both high-level semantics and feature-level information, optimizing sparse and dense representations jointly for improved performance. + +# 3.4 Coarse-to-Fine Generation + +During the inference phase, COBRA implements the coarse-to-fine generation procedure, involving the sequential generation of sparse IDs followed by the refinement of dense vectors in a cascaded manner, as illustrated in Figure 3. The coarse-to-fine generation process in COBRA is designed to capture both the categorical essence and fine-grained details of user-item interactions. This process involves three main stages: + +![](images/b8d869363719b68e46ab78ddf14595e0033f2cf93c902c4a30f88c6754d199b6.jpg) +Figure 3: Illustration of the Coarse-to-Fine Generation process. During inference, $M$ sparse IDs are generated via Beam Search, and appended to the sequence. Dense vectors are then generated and used in ANN to obtain candidate items. BeamFusion combines beam scores and similarity scores to rank candidates, from which the top $K$ items are selected. + +Sparse ID Generation. Given a user sequence $S_{1:T}$ , we utilize the ID probability distribution modeled by the Transformer Decoder, $\hat{ID}_{T+1} \sim P(i_{T+1}|S_{1:T})$ , and employ the BeamSearch algorithm to derive the top $M$ IDs. The formulation is as follows: + +$$ +\left\{\mathbf {I} \hat {\mathbf {D}} _ {T + 1} ^ {k} \right\} _ {k = 1} ^ {M} = \operatorname {B e a m S e a r c h} (\operatorname {T r a n s f o r m e r D e c o d e r} (\mathbf {S} _ {1: T}), M) \tag {4} +$$ + +where $k \in \{1, 2, \dots, M\}$ . Each generated ID is associated with a beam score $\phi_{\hat{\mathbf{ID}}_{T + 1}^k}$ . + +**Dense Vector Refinement.** Each generated sparse ID $\hat{\mathbf{ID}}_{T+1}^{\kappa}$ is subsequently converted into an embedding and appended to the previous cascaded sequence embedding $\mathbf{S}_{1:T}$ . Then the corresponding dense vector $\hat{\mathbf{v}}_{T+1}^{k}$ is generated: + +$$ +\hat {\mathbf {v}} _ {T + 1} ^ {k} = \operatorname {T r a n s f o r m e r D e c o d e r} \left(\left[ \mathbf {S} _ {1: T}, \operatorname {E m b e d} \left(\mathbf {I} \hat {\mathbf {D}} _ {T + 1} ^ {k}\right) \right]\right) \tag {5} +$$ + +After that, we employ Approximate Nearest Neighbor (ANN) search to retrieve the top $N$ candidate items: + +$$ +\mathcal {A} _ {k} = \operatorname {A N N} \left(\hat {\mathbf {I D}} _ {T + 1} ^ {k}, \mathcal {C} \left(\hat {\mathbf {I D}} _ {T + 1} ^ {k}\right), N\right) \tag {6} +$$ + +where $\mathcal{C}(\hat{\mathbf{ID}}_{T + 1}^k)$ is the set of candidate items associated with sparse $\mathrm{ID}\hat{\mathbf{ID}}_{T + 1}^k$ , and $N$ represents the number of top items to be retrieved. + +BeamFusion Mechanism. In order to achieve a balance between precision and diversity, we devise a globally comparable score for items corresponding to each sparse ID. This score is capable of reflecting both the differences among different sparse IDs and the fine-grained difference among items under the same sparse ID. To accomplish this, we propose the BeamFusion mechanism: + +$$ +\Phi \left(\hat {\mathbf {v}} _ {T + 1} ^ {k}, \hat {\mathbf {I D}} _ {T + 1} ^ {k}, \mathbf {a}\right) = \operatorname {S o f t m a x} \left(\tau \phi_ {\hat {\mathbf {I D}} _ {T + 1} ^ {k}}\right) \times \operatorname {S o f t m a x} \left(\psi \cos \left(\hat {\mathbf {v}} _ {T + 1} ^ {k}, \mathbf {a}\right)\right) \tag {7} +$$ + +where $\mathbf{a}$ represents the candidate item, $\tau$ and $\psi$ are coefficients, and $\phi_{\hat{\mathbf{ID}}_{T + 1}^k}$ denotes the beam score obtained during the beam search process. Finally, we rank all candidate items based on their BeamFusion Scores and select the top $K$ items as the final recommendations: + +$$ +\mathcal {R} = \operatorname {T o p K} \left(\bigcup_ {k = 1} ^ {M} \mathcal {A} _ {k}, \Phi , K\right) \tag {8} +$$ + +where $\mathcal{R}$ denotes the set of final recommendations, and $\mathrm{TopK}(\cdot)$ represents the operation of selecting the top $K$ items with the highest BeamFusion Scores. For a detailed algorithmic description, please refer to the pseudocode provided in Appendix E. + +# 3.5 Theoretical Justification + +In our framework, each item is characterized by a hybrid sparse-dense representation $(ID,\mathbf{v})$ , where $ID$ denotes the sparse ID and $\mathbf{v}$ represents the dense vector. A critical consideration is how to model the joint conditional distribution $P(ID,\mathbf{v}|S)$ given a user sequence $S$ . This distribution can be modeled in two primary ways: + +Independent Modeling This approach assumes that $ID$ and $\mathbf{v}$ are predicted independently given $S$ : + +$$ +P (I D, \mathbf {v} | S) = P (I D | S) \cdot P (\mathbf {v} | S). \tag {9} +$$ + +Cascaded Modeling (Ours) This approach, which we adopt, factorizes the joint distribution by first predicting $ID$ , and subsequently predicting $\mathbf{v}$ conditioned on both $ID$ and $S$ : + +$$ +P (I D, \mathbf {v} | S) = P (I D | S) \cdot P (\mathbf {v} | I D, S). \tag {10} +$$ + +We posit that the cascaded formulation is theoretically superior as it explicitly captures the dependency between $ID$ and $\mathbf{v}$ given $S$ . We formalize this advantage in terms of information entropy. + +Theorem 1 (Superiority of Cascaded Modeling). Let $H_{\text{indep}}$ and $H_{\text{ Cascaded}}$ denote the total entropies of the probability distributions defined by the independent (Eq. 9) and cascaded (Eq. 10) formulations, respectively. Then, + +$$ +H _ {c a s c a d e d} (I D, \mathbf {v} | S) \leq H _ {i n d e p} (I D, \mathbf {v} | S) \tag {11} +$$ + +with equality holding if and only if $\mathbf{v}$ and $ID$ are conditionally independent given $S$ . + +Theorem 1 demonstrates that the cascaded modeling consistently produces a joint distribution with lower information entropy. This implies a more compact and informative representation that facilitates model learning. The detailed proof is provided in Appendix F. + +# 4 Experiment + +This section presents a comprehensive evaluation of the COBRA framework using both public and industrial datasets. Our experiments focus on assessing COBRA's ability to improve recommendation accuracy and diversity, while also validating its practical effectiveness through offline and online evaluations. + +# 4.1 Public Dataset Experiments + +Datasets. In our experiments, we evaluate the performance of COBRA using the Amazon Product Reviews dataset [35, 36]. Our analysis focuses on three specific subsets: "Beauty," "Sports and Outdoors," and "Toys and Games." For each subset, we construct item embeddings leveraging attributes such as title, price, category, and description. To ensure data quality, we apply a 5-core filtering process, eliminating items with fewer than five user interactions and users with fewer than five item interactions. Detailed statistics of the datasets are presented in Appendix C.1. + +Evaluation Metrics. For the evaluation of recommendation accuracy and ranking quality, we employ Recall@K and NDCG@K, specifically at $K = 5$ and $K = 10$ . These metrics provide insights into the system's ability to accurately recommend relevant items and maintain a high-quality ranking order. + +Implementation Details. In our approach, we adopt a method for generating semantic IDs similar to the one used in [19]. However, unlike [19], which uses a different configuration, we employ a 3-level semantic ID structure, where each level corresponds to a codebook size of 32. These semantic IDs are generated using the T5 model. COBRA is implemented with a lightweight architecture, featuring a 1-layer encoder and a 2-layer decoder. + +Baselines. To comprehensively evaluate the performance of our proposed COBRA method, we compare it with the following recommendation methods (which are described briefly in Appendix B): P5 [33], Caser [28], HGN [37], GRU4Rec [27], SASRec [8], FDSA [38], BERT4Rec [9], $\mathbf{S}^3$ -Rec [39], and TIGER [19]. + +Results. COBRA consistently surpasses all baseline models across various metrics, as presented in Table 1. On the "Beauty" dataset, COBRA achieves a Recall@5 of 0.0537 and a Recall@10 of 0.0725, exceeding the previous strongest baseline model (TIGER) by $18.3\%$ and $11.9\%$ , respectively. For the "Sports and Outdoors" dataset, COBRA records a Recall@5 of 0.0305 and an NDCG@5 of 0.0215, outperforming TIGER by $15.5\%$ and $18.8\%$ , respectively. On the "Toys and Games" dataset, COBRA attains a Recall@10 of 0.0462 and an NDCG@10 of 0.0515, surpassing TIGER by $24.5\%$ and $19.2\%$ , respectively. + +Table 1: Performance comparison on public datasets. The best metric for each dataset is highlighted in bold, while the second-best is underlined. + +
MethodR@5N@5R@10N@10
BeautyP50.01630.01070.02540.0136
Caser0.02050.01310.03470.0176
HGN0.03250.02060.05120.0266
GRU4Rec0.01640.00990.02830.0137
BERT4Rec0.02030.01240.03470.0170
FDSA0.02670.01630.04070.0208
SASRec0.03870.02490.06050.0318
S3-Rec0.03870.02440.06470.0327
TIGER0.04540.03210.06480.0384
COBRA0.05370.03950.07250.0456
SportsP50.00610.00410.00950.0052
Caser0.01160.00720.01940.0097
HGN0.01890.01200.03130.0159
GRU4Rec0.01290.00860.02040.0110
BERT4Rec0.01150.00750.01910.0099
FDSA0.01820.01220.02880.0156
SASRec0.02330.01540.03500.0192
S3-Rec0.02510.01610.03850.0204
TIGER0.02640.01810.04000.0225
COBRA0.03050.02150.04340.0257
ToysP50.00700.00500.01210.0066
Caser0.01660.01070.02700.0141
HGN0.03210.02210.04970.0277
GRU4Rec0.00970.00590.01760.0084
BERT4Rec0.01160.00710.02030.0099
FDSA0.02280.01400.03810.0189
SASRec0.04630.03060.06750.0374
S3-Rec0.04430.02940.07000.0376
TIGER0.05210.03710.07120.0432
COBRA0.06190.04620.07810.0515
+ +Ablation Study. To validate the necessity of COBRA's key components and understand their individual contributions, we compare the full model against three variants. COBRA w/o ID removes sparse IDs, relying solely on dense vectors. COBRA w/o Dense removes dense vectors, using only sparse IDs. COBRA w/o BeamFusion removes the BeamFusion module during + +Table 2: Ablation study on public datasets (Recall@10). + +
MethodBeautySportsToys
COBRA0.07250.04340.0781
COBRA w/o Dense0.06560.03310.0713
COBRA w/o ID0.06810.03650.0653
COBRA w/o BeamFusion0.07140.04130.0769
+ +inference, using top-1 sparse ID and nearest-neighbor retrieval for top- $k$ results. As shown in Table 2, the removal of any key component leads to a consistent performance drop. COBRA w/o Dense shows a significant decline, highlighting the limitations of using only discrete sparse IDs, which fail to capture fine-grained semantic nuances. COBRA w/o ID also underperforms, demonstrating the importance of sparse IDs in offering a structural framework that supports coarse-to-fine generation. COBRA w/o BeamFusion also exhibits a performance drop. + +# 4.2 Industrial-scale Experiments + +Dataset. To comprehensively evaluate the proposed COBRA method, we utilize a large-scale industrial dataset from a major information feed platform, which contains 5 million users and 2 million advertisements across diverse recommendation scenarios. Advertisements are represented via attributes such as title, industry labels, brand, and campaign text, encoded into two-level sparse IDs + +Table 3: Performance comparison on industrial dataset + +
MethodR@50R@100R@200R@500R@800
COBRA0.11800.17370.24700.37160.4466
COBRA w/o ID0.06110.09640.14740.24660.3111
COBRA w/o Dense0.06900.10320.17380.27090.3273
COBRA w/o BeamFusion0.08560.12540.17320.24550.2855
+ +and dense vectors to capture multi-granularity semantic information. A more detailed description of the dataset can be found in Appendix C.2. + +Evaluation Metrics. For offline evaluation, we employ Recall@K as the evaluation metric, testing with $K \in \{50, 100, 200, 500, 800\}$ . This metric provides a measure of the model's ability to accurately retrieve relevant recommendations at various thresholds. + +Implementation Details. COBRA is built upon a Transformer-based architecture. In this framework, the text encoder processes advertisement text into sequences, which are then handled by the sparse ID head to predict 2-level semantic IDs configured as $32 \times 32$ . + +Results. For further analysis on the industrial dataset, we also compare COBRA against the variants defined in the previous section, i.e., COBRA w/o ID, w/o Dense, and w/o Beam-Fusion. Notably, the COBRA w/o Dense variant employs finer-grained 3-level semantic IDs $(256\times 256\times 256)$ to ensure its sufficient fine-grained modeling capacity, compensating for its lack of dense vectors. As shown in Table 3, COBRA consistently outperforms all its variants across all evaluated metrics. At $K = 500$ , COBRA achieves a Recall@500 of 0.3716, representing a $42.2\%$ improvement over the COBRA w/o Dense variant. When $K = 800$ , COBRA attains a Recall@800 of 0.4466, reflecting a $43.6\%$ improvement over the COBRA w/o ID + +![](images/b59dde63a9bb2f8010848a37135caaba18c72ca1c80a9dc20d002b1f42dd8337.jpg) +Figure 4: Cosine Similarity Matrix (full comparison in Appendix 6) + +variant and a $36.1\%$ enhancement compared to COBRA w/o BeamFusion. At relatively smaller values of $K$ , the absence of dense or ID representations results in more pronounced performance declines, underscoring the importance of cascaded representations for achieving granularity and precision. Conversely, as the recall size $K$ increases, the performance advantages associated with BeamFusion become increasingly evident, demonstrating its effectiveness in practical industrial recall systems. The results further underscore the contributions of specific components: Excluding sparse IDs leads to a recall reduction ranging from $26.7\%$ to $41.5\%$ , highlighting the critical role of semantic categorization. The removal of dense vector results in a performance drop between $30.3\%$ and $48.3\%$ , underscoring the importance of fine-grained modeling. Eliminating BeamFusion results in a recall decrease of $27.5\%$ to $36.1\%$ , emphasizing the significance of fusion strategy. + +# 4.3 Further Analysis + +Analysis of Representation Learning. The heatmap in Figure 4 demonstrates COBRA's strong intra-ID cohesion and inter-ID separation, indicating effective capture of both item-specific features and categorical semantics. Quantitative verification through difference analysis is provided in Appendix D. Further validation of COBRA's embeddings is achieved through visualizing the distribution of advertisement embeddings in a two-dimensional space using t-SNE. By randomly sampling 10,000 advertisements, distinct clustering centers for various categories are observed. Figure 5a reveals that advertisements are effectively clustered by category, indicating strong cohesion within categories. The clusters in purple, teal, light green, and dark green correspond primarily to advertisements for novels, games, legal services, and clothing, respectively. This demonstrates that the advertisement representations effectively capture semantic information. + +![](images/e77f4f0b17554d9fd79e0e480c9515d1052b082c45c7aca5fe10653d9c3521f6.jpg) +(a) Embedding Visualization using t-SNE + +![](images/4aff4dc1e1e2ef1142d18a2159bca377dcac7bbd4d1e2e6f96150f3a8cce06de.jpg) +(b) Recall-Diversity Trade-off +Figure 5: (a) t-SNE visualization of 10,000 randomly sampled advertisement embeddings. (b) Recall@2000 and diversity metrics under different $\tau$ values. + +Recall-Diversity Equilibrium. To analyze the trade-off between accuracy and diversity in COBRA, we examine recall-diversity curves, which depict how Recall@2000 and diversity metrics evolve with the coefficient $\tau$ in the BeamFusion mechanism, while keeping $\phi$ fixed. As depicted in Figure 5b, increasing $\tau$ generally leads to a decrease in diversity. COBRA achieves an optimal balance between recall and diversity at $\tau = 0.9$ . At this point, the model maintains high accuracy while ensuring a sufficiently diverse set of retrieved items. Specifically, when $M = 50$ , compared to $\tau = 1.0$ , setting $\tau = 0.5$ results in a $4.99\%$ decrease in recall, but brings more than double the diversity. Meanwhile, $\tau = 0.9$ leads to a $0.12\%$ increase in recall and an $18.80\%$ relative improvement in diversity. This fine-grained control over $\tau$ and $\phi$ allows for adjusting the emphasis on accuracy or diversity according to specific business objectives. Platforms prioritizing exploration can reduce $\tau$ to enhance diversity. This flexibility distinguishes COBRA from models with fixed retrieval strategies, making it adaptable to various recommendation scenarios. + +Online Results. To validate COBRA's real-world effectiveness, we conducted online A/B tests on a major information feed platform. The experiment covered $10\%$ of user traffic, ensuring statistical significance. The primary evaluation metrics were conversion and Average Revenue Per User (ARPU), which directly reflect user engagement and economic value. Within the traffic segment exposed to our proposed strategy, COBRA achieved a $3.60\%$ increase in conversion and a $4.15\%$ increase in ARPU. These results demonstrate that COBRA's hybrid architecture not only improves recommendation quality in offline evaluations but also drives measurable business outcomes in production environments. + +# 5 Conclusion + +In this work, we introduced COBRA, a generative recommendation framework that combines sparse and dense representations for enhanced accuracy and diversity. COBRA employs a coarse-to-fine generation process, starting with a sparse ID to capture the categorical essence of an item and refining it with a dense vector. 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Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The paper clearly demonstrates the performance improvement of the new method on recommendation tasks, and the experimental section provides ample support for these claims. Therefore, the main assertions in the abstract and introduction accurately reflect the contributions and scope of the paper. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigourousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: LLM is not the central component of our method; our core contribution is a generative framework. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. + +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Related Work + +Sequential Dense Recommendation. Sequential dense recommendation utilize user interaction histories to learn representations for personalization [8, 39, 29], capturing both long-term and short-term preferences [40-43]. GRU4Rec [27] first applied RNNs [44] to model sequential behavior, effectively handling temporal dependencies. Caser [28] adopted CNNs [45] to extract local sequential patterns from behavior matrices. Transformer-based models, such as SASRec [8] and BERT4Rec [9], leverage self-attention to model complex user behavior, with SASRec focusing on autoregressive predictions and BERT4Rec adopting bidirectional context encoding. Advanced architectures like PinnerFormer [46] and FDSA [38] further enhance user modeling by integrating multi-source features and capturing long-range dependencies. Recent efforts (e.g., ZESRec [30], UniSRec [31], RecFormer [29]) have moved towards cross-domain recommendation and richer feature integration, often via contrastive learning and unified language-sequence modeling. + +Generative Recommendation. With the emergence of large generative models, recommendation is increasingly framed as a generation task [32, 47-51]. P5 [33] reformulates various recommendation tasks as language generation, enabling unified modeling and prompting strategies. TIGER [19] applies residual quantized autoencoders to produce semantic item identifiers, which transformers then generate from user sequences. LC-Rec [21] aligns these semantic IDs with collaborative signals, while IDGenRec [23] leverages large language models for unique, dense textual identifiers. SEATER [34] and ColaRec [22] focus on aligning semantic and collaborative spaces or maintaining semantic consistency via structured indexes. Despite their advantages, generative models relying on discrete IDs may lose fine-grained preference information [52], and natural language generation may be less aligned with recommendation-specific signals [53]. Hybrid approaches like LIGER [26] jointly generate sparse IDs and dense vectors, narrowing the gap between retrieval paradigms. Nevertheless, challenges remain in achieving optimal flexibility and representation granularity. + +# B Baselines + +To rigorously evaluate the effectiveness of our proposed COBRA method, we benchmark it against a range of recommendation methods. + +- P5 [33]: This innovative approach transforms recommendation tasks into natural language sequences, utilizing the capabilities of language models to unify various recommendation scenarios. +- Caser [28]: Caser employs convolutional layers to effectively capture sequential user behavior patterns, modeling high-order Markov Chains. +- HGN [37]: The Hierarchical Gating Network(HGN) is designed to capture both long-term and short-term user interests through a sophisticated gating architecture, facilitating personalized recommendations. +- GRU4Rec [27]: As a pioneering RNN-based approach, GRU4Rec leverages gated recurrent units to model user behaviors in sequential recommendation tasks, paving the way for subsequent developments in the field. +- SASRec [8]: As a Transformer-based model, SASRec focuses on long-term dependencies in user interactions, employing self-attention mechanisms for precise sequential recommendations. +- FDSA [38]: By integrating item features with embeddings, the Feature-level Deeper Self-Attention(FDSA) Network enriches the input sequence, leveraging self-attentive mechanisms to enhance recommendation quality. +- BERT4Rec [9]: Utilizing a bidirectional self-attention framework with a cloze task objective, BERT4Rec overcomes the limitations of traditional uni-directional models, offering robust recommendation capabilities. +- $\mathbf{S}^3$ -Rec [39]: $\mathbf{S}^3$ -Rec leverages contrastive learning to bolster the recommendation process, employing self-supervised techniques to enhance sequential recommendation performance. +- TIGER [19]: TIGER employs RQ-VAE for encoding item content features and leverages a Transformer for generative retrieval, showcasing a novel approach to incorporating content features in recommendation tasks. + +These methods collectively represent the forefront of recommendation technology, embodying diverse methodologies from sequential and dense recommendation to generative approaches. + +# C Dataset Statistics + +# C.1 Public Datasets. + +Table 4: Summary of dataset statistics for three real-world benchmarks. + +
DatasetUsersItemsAvg. LengthMed. Length
Beauty22,36312,1018.876
Sports and Outdoors35,59818,3578.326
Toys and Games19,41211,9248.636
+ +Our study utilizes the Amazon Product Reviews dataset [35, 36], which spans user reviews and product information from May 1996 to July 2014. To comprehensively explore the effectiveness of recommendation methods, we selected three categories: "Beauty," "Sports and Outdoors," and "Toys and Games." Table 4 provides a concise summary of these datasets. During data preprocessing, we constructed users' historical item interaction sequences based on review timestamps, excluding users with fewer than five reviews. For evaluation, we adopted the widely-used leave-one-out strategy: the last item in each user's sequence served as the test sample, the second-to-last as the validation sample, and the remaining items as training data. The "Beauty" dataset contains 22,363 users and 12,101 items, featuring an average sequence length of approximately 8.87, with a median of 6. The "Sports and Outdoors" dataset comprises 35,598 users and 18,357 items, with an average sequence length of 8.32 and a median of 6. Similarly, the "Toys and Games" dataset includes 19,412 users and 11,924 items, with an average sequence length of about 8.63 and a median of 6. + +# C.2 Industrial Dataset Details. + +To thoroughly evaluate the proposed COBRA method, we employ a large-scale industrial dataset derived from user interaction logs on a major information feed platform. This dataset covers multiple recommendation scenarios, including list-page, dual-column, and short-video recommendations, and contains approximately 5 million users and 2 million advertisements, providing a comprehensive reflection of real-world user behaviors and advertising content. + +Advertisers and advertisements are described by attributes such as title, industry labels, brand, and campaign text. To effectively capture both coarse-grained and fine-grained semantic information, these attributes are encoded into two-level sparse IDs alongside dense vector representations. This dual encoding enables COBRA to model user preferences and item characteristics more accurately. + +The dataset is divided into two parts: the training set $D_{\mathrm{train}}$ and the test set $D_{\mathrm{test}}$ . The training set consists of user interaction logs collected over the first 60 days, covering recommendation content and user behaviors during this period. The test set is constructed from logs recorded on the day immediately following the training period and serves as a benchmark for model performance evaluation. This chronological split ensures the temporal consistency of training and testing processes, improving the reliability of the evaluation. + +# D Supplementary Similarity Analysis + +The COBRA model exhibits significant intra-ID cohesion and inter-ID separation, as demonstrated in the top heatmap of Figure 4. This suggests that COBRA's dense embeddings proficiently capture detailed item characteristics while preserving semantic consistency within categories. Conversely, the model variant without sparse IDs (Figure 6a) shows weaker category separation, underscoring the importance of sparse IDs in maintaining semantic structure. The difference matrix in Figure 6b quantitatively confirms that incorporating sparse IDs enhances both cohesion and separation. + +![](images/fcd93d57565605cce1bccea4fd7a4523b13d5d8c52ca39abeab7fb0c7c68e3b0.jpg) +(a) Without ID (COBRA w/o ID) + +![](images/60f85711da61ea64b322c7d49f29f644ac0b7fd28e5b5c24557bcc55ad27d0d3.jpg) +(b) Difference Matrix +Figure 6: Complete similarity matrix comparison: (a) Weaker separation without ID, (b) Quantitative improvement from sparse IDs + +# E Pseudocode for Coarse-to-Fine Generation + +Algorithm 1 Inference with BeamFusion +1: Input: input_seq $\mathbf{S}_{1:T}$ , beam_size $M$ , nn_num $N$ , recall_num $K$ , candidate items a +2: Output: Top $K$ recommendations $\mathcal{R}$ +3: +4: procedure FORWARD(..) +5: Sparse ID Generation: +6: for each ID hierarchy do +7: Run transformer forward pass +8: Compute ID logits and scores $\{\hat{\mathbf{ID}}_{T + 1}^{k}\}_{k = 1}^{M}$ +9: Update beam scores and decoder input +10: end for +11: Dense Vector Refinement: +12: Obtain final decoder output $\hat{\mathbf{v}}_{T + 1}^{k}$ +13: Compute similarity scores $\cos (\hat{\mathbf{v}}_{T + 1}^{k},\mathbf{a})$ +14: Filter logits using generated ID +15: BeamFusion Mechanism: +16: Combine beam scores $\phi_{\hat{\mathbf{ID}}_{T + 1}^k}$ with similarity scores $\cos (\hat{\mathbf{v}}_{T + 1}^{k},\mathbf{a})$ +17: Select top $K$ candidates based on fused scores +18: Prepare Results: +19: Collect final ID sequences, retrieved items, and scores +20: end procedure + +# F Proof of Theorem 1 + +We calculate the information entropy (for discrete variable $ID$ ) and differential entropy (for continuous variable $\mathbf{v}$ ) under both probability distributions given by Eq. 9 and Eq. 10. We denote the discrete entropy by $H(\cdot)$ and the differential entropy by $h(\cdot)$ . + +# Independent Modeling (Eq. 9): + +$$ +\begin{array}{l} H _ {i n d e p} = - \mathbb {E} _ {P _ {i n d e p}} [ \log P (I D, \mathbf {v} | S) ] \\ = - \mathbb {E} _ {P _ {\text {i n d e p}}} [ \log (P (I D | S) \cdot P (\mathbf {v} | S)) ] \\ = - \mathbb {E} _ {P _ {\text {i n d e p}}} [ \log P (I D | S) ] - \mathbb {E} _ {P _ {\text {i n d e p}}} [ \log P (\mathbf {v} | S) ] \\ = H (I D | S) + h (\mathbf {v} | S) \tag {12} \\ \end{array} +$$ + +Hence, under independent modeling, the total entropy decomposes additively over $ID$ and $\mathbf{v}$ . + +# Cascaded Modeling (Eq. 10): + +$$ +\begin{array}{l} H _ {c a s c a d e d} = - \mathbb {E} _ {P _ {c a s c a d e d}} [ \log P (I D, \mathbf {v} | S) ] \\ = - \mathbb {E} _ {P _ {\text {c a s c a d e d}}} [ \log (P (I D | S) \cdot P (\mathbf {v} | I D, S)) ] \\ = - \mathbb {E} _ {P _ {c a s c a d e d}} [ \log P (I D | S) ] - \mathbb {E} _ {P _ {c a s c a d e d}} [ \log P (\mathbf {v} | I D, S) ] \\ = H (I D | S) + h (\mathbf {v} | I D, S) \tag {13} \\ \end{array} +$$ + +Hence, cascaded modeling captures the full joint uncertainty between $ID$ and $\mathbf{v}$ . + +Comparison: By the property of conditional entropy [54]: + +$$ +h (\mathbf {v} | S) \geq h (\mathbf {v} | I D, S) \tag {14} +$$ + +Equality holds if and only if $\mathbf{v}$ and $ID$ are conditionally independent given $S$ . + +Substituting this inequality into Eq. 12 and 13 yields: + +$$ +H _ {i n d e p} \geq H _ {c a s c a d e d} \tag {15} +$$ + +which confirms Theorem 1. + +# G Computational Cost Analysis + +We define $L$ as the sequence length, $T$ as the number of tokens per item and $D$ as the embedding dimension. + +Table 5: Training Complexity. (Please add citations for TIGER/SASRec) + +
ComponentCOBRATIGER [19]SASRec [8]
Item Feature EncoderO(L·T2·D+L·T·D2)N/AN/A
Sequential ModelO(L2·D+L·D2)O(L2·D+L·D2)O(L2·D+L·D2)
+ +Table 6: Inference Complexity. (Please add citations for TIGER/SASRec) + +
ComponentCOBRATIGER [19]SASRec [8]
Item Feature EncoderO(1) (cached)N/AN/A
Sequential ModelO(L2·D + L·D2)O(L2·D + L·D2)O(L2·D + L·D2)
+ +To ensure practical efficiency, we employ several key techniques during implementation. These include sequence Packing and FlashAttention to minimize computational waste. Furthermore, encoder caching is utilized to decouple the encoder computation, significantly speeding up the inference process. 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Most existing algorithms are designed to achieve optimal worst-case regret in either the data-rich regime, where polynomial dependence on the ambient dimension is unavoidable, or the data-poor regime, where dimension-independence is possible at the cost of worse dependence on the number of rounds. In contrast, the sparse Information Directed Sampling (IDS) algorithm satisfies a Bayesian regret bound that has the optimal rate in both regimes simultaneously. In this work, we explore the use of Sparse Optimistic Information Directed Sampling (SOIDS) to achieve the same adaptivity in the worst-case setting, without Bayesian assumptions. Through a novel analysis that enables the use of a time-dependent learning rate, we show that SOIDS can optimally balance information and regret. Our results extend the theoretical guarantees of IDS, providing the first algorithm that simultaneously achieves optimal worst-case regret in both the data-rich and data-poor regimes. We empirically demonstrate the good performance of SOIDS. + +# 1 Introduction + +In stochastic linear bandits, one assumes that the mean reward associated with each action is linear in an unknown $d$ -dimensional parameter vector [Abe and Long, 1999, Auer, 2002, Dani et al., 2008, Abbasi-Yadkori et al., 2011]. Under standard conditions, it is known that the minimax regret in this setting is of the order $\mathcal{O}(d\sqrt{T})$ [Dani et al., 2008, Rusmevichientong and Tsitsiklis, 2010]. Numerous follow-up works have investigated the possibility of reduced regret under various structural assumptions on the unknown parameter vector, the noise, or the shape of the decision set [Valko et al., 2014, Chu et al., 2011, Kirschner and Krause, 2018], [Lattimore and Szepesvári, 2020, Chapter 22]. One such assumption is that the unknown parameter vector is sparse, which means that it has only $s \ll d$ non-zero components. This setting is called sparse linear bandits and $s$ is referred to as the sparsity level. In this setting, previous work has established the existence of algorithms with regret scaling as $\mathcal{O}(\sqrt{sdT})$ [Abbasi-Yadkori et al., 2012]. This result is complemented by a lower bound, which says that this rate cannot be improved as long as $T \geq d^{\alpha}$ for some $\alpha > 0$ [Lattimore and Szepesvári, 2020]. We refer to this scenario as the data-rich regime. Since this bound scales polynomially with the dimension $d$ , many researchers have considered this to be a negative result, interpreting it as a sign that sparsity cannot be effectively exploited in linear bandit problems. This interpretation has been challenged by a more recent observation that, when the action set admits an exploratory distribution, simple "explore-then-commit" algorithms enjoy regret bounds of order $\mathcal{O}((sT)^{\frac{2}{3}})$ [Hao et al., 2020, Jang et al., 2022]. These bounds scale only logarithmically with the dimension, and constitute a major improvement over the previously mentioned rate in the data-poor regime, where $T \ll \left(\frac{d}{s}\right)^3$ . Most known algorithms are specialized to either the data-poor or data-rich regime, and perform poorly in the other one. A notable exception is the sparse Information Directed Sampling algorithm introduced in Hao et al. [2021], which performs almost optimally in both regimes. However, Hao et al. [2021] only provide Bayesian performance guarantees for sparse + +IDS. These results hold on average, assuming that the problem instance is drawn at random from a known prior distribution. + +In this work, we lift this assumption and develop an algorithm that can adapt to both regimes in a "frequentist" sense: we assume that the true parameter is fixed and unknown to the learner, and provide guarantees that hold for any given instance. The algorithm is an adaptation of the recently proposed Optimistic Information Directed Sampling (OIDS) algorithm of Neu, Papini, and Schwartz [2024], which itself is an adaptation of the classic Bayesian IDS algorithm originally proposed by Russo and Van Roy [2017]. Within the Bayesian setting, it has been shown that IDS can exploit various types of problem structure, and adapt to the hardness of the given instance [Hao and Lattimore, 2022, Hao et al., 2022]. These results have been complemented by the recent work of Neu, Papini, and Schwartz [2024], which showed that similar improvements can be achieved without Bayesian assumptions, via a simple adjustment of the standard IDS method. In this paper, we continue this line of work. We show that a sparse version of OIDS enjoys a worst-case regret bound that matches the optimal rate in both regimes simultaneously. + +Our contribution is as follows: + +- We extend the analysis of the optimistic posterior to allow the use of time-dependent learning rates and history-dependent learning rates. Time-dependent learning rates allow us to drop the requirement that the horizon is known in advance, and are essential for worst-case regret bounds that can adapt to both regimes. History-dependent learning rates allow us to update the learning rate based on data observed by the agent instead of some loose theoretical constant, a necessity for efficient algorithms. +- We demonstrate that the Sparse Optimistic Information Directed Sampling (SOIDS) algorithm recovers almost optimal rates in both the data-poor and data-rich regimes. This is the first algorithm to do so in a frequentist setting. + +# 2 Preliminaries + +Sparse linear bandits. We consider the following decision-making game, in which a learning agent interacts with an environment over a sequence of $T$ rounds. At the start of each round $t$ , the learner selects an action $A_{t} \in \mathcal{A} \subset \mathbb{R}^{d}$ according to a randomized policy $\pi_t \in \Delta(\mathcal{A})$ . In response, the environment generates a stochastic reward $Y_{t} = r(A_{t}) + \epsilon_{t}$ , where $r: \mathcal{A} \to \mathbb{R}$ is a fixed reward function and $\epsilon_{t}$ is zero-mean, conditionally 1-sub-Gaussian noise. We assume that the action set $\mathcal{A}$ is finite, and that the reward function can be written as + +$$ +r (a) = \left\langle \theta_ {0}, a \right\rangle , +$$ + +where $\theta_0\in \mathbb{R}^d$ is an unknown parameter vector. We make the mild boundedness assumptions that $\max_{a\in \mathcal{A}}\| a\|_{\infty}\leq 1$ and $\| \theta_0\| _1\leq 1$ . We study the special case of this problem in which the parameter vector $\theta_0$ is $s$ -sparse in the sense that at most $s\ll d$ of its components are non-zero. In other words, we assume that $\theta_0$ belongs to the following sparse parameter space: + +$$ +\Theta = \left\{\theta \in \mathbb {R} ^ {d}: \sum_ {j = 1} ^ {d} \mathbb {I} _ {\{\theta_ {j} \neq 0 \}} \leq s, \| \theta \| _ {1} \leq 1 \right\}. +$$ + +We assume that the sparsity level $s$ is known to the agent. The performance of the agent is evaluated in terms of the regret, which is defined as + +$$ +R _ {T} = T \max _ {a \in \mathcal {A}} \left\langle \theta_ {0}, a \right\rangle - \mathbb {E} \left[ \sum_ {t = 1} ^ {T} r \left(A _ {t}, \theta_ {0}\right) \right], \tag {1} +$$ + +where the expectation is taken with respect to both the random choices of the agent and the random noise in the observed rewards. We note that the regret is implicitly a function of the true parameter $\theta_0$ . Our focus is on proving regret bounds that hold for arbitrary choices of $\theta_0 \in \Theta$ . + +The data-rich and data-poor regimes. As mentioned in the introduction, it is known there exist algorithms for sparse linear bandits with worst-case regret of the order $\mathcal{O}(\sqrt{sdT})$ [Abbasi-Yadkori et al., 2012]. This regret bound is only meaningful when the dimension $d$ is smaller than the number of rounds $T$ , a situation referred to as the data-rich regime. Under the assumption that there exists an exploratory policy, Hao et al. [2020] showed that there is a simple algorithm that satisfies a + +problem-dependent regret bound, which can be meaningful in the so-called data-poor regime, where $d$ is much larger than $T$ . Formally, we say that there exists an exploratory policy if the action set $\mathcal{A}$ is such that + +$$ +C _ {\min } := \max _ {\mu \in \Delta (\mathcal {A})} \sigma_ {\min } \left(\int_ {\mathcal {A}} a a ^ {\top} d \mu (a)\right) > 0, +$$ + +which is equivalent to the condition that $\mathcal{A}$ spans $\mathbb{R}^d$ . The exploratory policy is the distribution on $\mathcal{A}$ that achieves the maximum (which is guaranteed to exist when $\mathcal{A}$ is finite). The Explore the Sparsity Then Commit (ESTC) algorithm was shown to satisfy a regret bound of the order $\mathcal{O}(s^{2/3}T^{2/3}C_{\min}^{-2/3})$ [Hao et al., 2020]. The transition between the $T^{2/3}$ rate in the data-poor regime and the $\sqrt{T}$ rate in the data-rich regime also appears in an existing lower bound of the order $\Omega(\min(s^{1/3}T^{2/3}C_{\min}^{-1/3}), \sqrt{dT})$ [Hao et al., 2020]. + +Adapting to both regimes. Recently, Hao et al. [2021] showed that the sparse Information Directed Sampling (IDS) algorithm performs well in both regimes. Under the sparse optimal action condition (Definition 1), IDS satisfies a regret bound of the order $\mathcal{O}(\min (\sqrt{dT\Delta},(sT)^{2 / 3}\Delta^{1 / 3}C_{\min}^{-1 / 3}))$ , where $\Delta \propto \min (\log (|\mathcal{A}|),s\log (dT / s))$ . This is simultaneously optimal in both the data-rich and data-poor regimes. However, this result is limited to the Bayesian setting. This is because IDS uses the Bayesian posterior to quantify uncertainty, which is only meaningful if $\theta_0$ really is a random draw from the prior. + +The sparse optimal action condition. Part of our analysis requires that a certain technical condition is satisfied. This condition comes from prior work [Hao et al., 2021], and is used to bound the regret in the data-poor regime (cf. Lemma 7). + +Definition 1. For a given prior $Q_1^+$ , an action set $\mathcal{A}$ has sparse optimal actions if with probability 1 over the random draw of $\theta$ from $Q_1^+$ , there exists $a' \in \arg \max_{a \in \mathcal{A}} r(a, \theta)$ such that $\| a' \|_0 \leq s$ . + +We use a prior that only assigns positive probability to $s$ -sparse vectors, which means the sparse optimal action property is satisfied whenever the action set is an $\ell_p$ -ball. Note that the hard instances in both the $\sqrt{sdT}$ lower bound in Theorem 24.3 of Lattimore and Szepesvári [2020] and the $s^{2/3}T^{2/3}$ lower bound in Theorem 5 of Jang et al. [2022] satisfy the sparse optimal action property1. Therefore, imposing this additional condition does not trivialize the problem. + +Notation. We conclude this section by introducing some additional notation that will be used in the subsequent sections. For any candidate parameter vector (or model) $\theta \in \mathbb{R}^d$ , we let $r(a,\theta) = \langle \theta ,a\rangle$ denote the corresponding linear reward function. In addition, we define $a^{*}(\theta) = \arg \max_{a\in \mathcal{A}}r(a,\theta)$ (with ties broken arbitrarily) and $r^{*}(\theta) = r(a^{*}(\theta),\theta)$ to be the optimal action and maximum reward for the model $\theta$ . The gap of an action $a$ for a model $\theta$ is $\Delta (a,\theta) = r^{*}(\theta) - r(a,\theta)$ . Similarly, the gap for a policy $\pi \in \Delta (\mathcal{A})$ and a model distribution $Q\in \Delta (\Theta)$ is $\Delta (\pi ,Q) = \int_{\mathcal{A}\times \Theta}\Delta (a,\theta)\mathrm{d}\pi \otimes Q(a,\theta)$ , and we let $\Delta_t = \Delta (\pi_t,\theta_0)$ denote the gap of the policy played by the agent in round $t$ under the true model $\theta_0$ . Using this notation, the regret can be written as $R_{T} = \mathbb{E}[\sum_{t = 1}^{T}\Delta_{t}]$ . We define the unnormalized Gaussian likelihood function $p(y|\theta ,a) = \exp (-\frac{(y - \langle\theta,a\rangle)^2}{2})$ . Finally, we let $\mathcal{F}_t = \sigma (A_1,Y_1,\ldots ,A_t,Y_t)$ denote the $\sigma$ -algebra generated by the interaction between the agent and the environment up to the end of round $t$ . + +# 3 Sparse Optimistic Information Directed Sampling + +We develop an extension of the Optimistic Information Directed Sampling (OIDS) algorithm proposed by Neu, Papini, and Schwartz [2024]. The main difference between OIDS and IDS is that the Bayesian posterior is replaced by an appropriately adjusted optimistic posterior. For an arbitrary prior $Q_{1}^{+} \in \Delta(\Theta)$ , the optimistic posterior is defined by the following update rule: + +$$ +\frac {d Q _ {t + 1} ^ {+}}{d Q _ {1} ^ {+}} (\theta) \propto \prod_ {s = 1} ^ {t} \left(p \left(Y _ {s} \mid \theta , A _ {s}\right)\right) ^ {\eta} \cdot \exp \left(\lambda_ {t} \sum_ {s = 1} ^ {t} \Delta \left(A _ {s}, \theta\right)\right). \tag {2} +$$ + +Here, $\eta$ is a positive constant that should be thought of as "large", and $(\lambda_{t})_{t}$ is a decreasing sequence of positive real numbers that decays to 0, and should be thought of as "small". We allow $\lambda_{t}$ to be computed by the algorithm at the end of the round $t$ . In other words, any $\mathcal{F}_t$ -measurable $\lambda_t$ is admissible. Note that when $\eta = 1$ and $\lambda_{t} = 0$ , the optimistic posterior coincides with the Bayesian posterior. While this construction is closely related to the optimistic posterior update described in Zhang [2022] and Neu, Papini, and Schwartz [2024], there are a few important differences. First, the $\Delta (A_s,\theta)$ term appearing in the adjustment serves as an alternative to their proposal of using $r^* (\theta)$ for the same purpose. Intuitively this serves to "overestimate" the true gaps with the optimistic posterior, driving exploration towards parameters that promise rewards much higher than whatever would have been accrued by the agent. In contrast, the adjustment of Zhang [2022] drives exploration towards parameters $\theta$ with high optimal reward regardless of how well the agent would have performed under the same $\theta$ meaning that it unduly assigns mass to uninteresting parameter choices, where any policy is guaranteed to work well anyway. Intuition aside, this adjustment greatly simplifies our analysis of the optimistic posterior as compared to the analysis of Zhang [2022] and Neu, Papini, and Schwartz [2024]. An important additional novelty is that our update features a time-dependent exploration parameter $\lambda_{t}$ , which is crucial for the adaptive regret bounds that we seek in this work. To describe the OIDS algorithm, we must first define the surrogate information gain and the surrogate regret. For any round $t$ and any policy $\pi \in \Delta (\mathcal{A})$ , the surrogate information gain is defined as + +$$ +\overline {{\mathbf {I G}}} _ {t} (\pi) = \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi (a) \int_ {\Theta} \left(\langle \theta - \bar {\theta} (Q _ {t} ^ {+}), a \rangle\right) ^ {2} \mathrm {d} Q _ {t} ^ {+} (\theta), +$$ + +where for any $Q \in \Delta(\Theta)$ , $\bar{\theta}(Q) = \mathbb{E}_{\theta \sim Q}[\theta]$ is the mean parameter under distribution $Q$ . The surrogate regret is defined as + +$$ +\widehat {\Delta} _ {t} (\pi) = \sum_ {a \in \mathcal {A}} \pi (a) \int_ {\Theta} \Delta (a, \theta) \mathrm {d} Q _ {t} ^ {+} (\theta). +$$ + +For any policy $\pi$ and any $\gamma \geq 2$ , we define the surrogate generalized information ratio as + +$$ +\overline {{\mathrm {I R}}} _ {t} ^ {(\gamma)} (\pi) = \frac {(\widehat {\Delta} _ {t} (\pi)) ^ {\gamma}}{\overline {{\mathrm {I G}}} _ {t} (\pi)} = 2 \cdot \frac {\left(\sum_ {a \in \mathcal {A}} \pi (a) \int_ {\Theta} \langle \theta , a ^ {*} (\theta) - a \rangle d Q _ {t} ^ {+} (\theta)\right) ^ {\gamma}}{\sum_ {a \in \mathcal {A}} \pi (a) \int_ {\Theta} (\langle \theta - \bar {\theta} (Q _ {t} ^ {+}), a \rangle) ^ {2} d Q _ {t} ^ {+} (\theta)}. \tag {3} +$$ + +We can at last define the Sparse Optimistic Information Directed Sampling (SOIDS) algorithm. In each round $t$ , the policy played by SOIDS is defined to be the distribution on $\mathcal{A}$ that minimizes the 2-information ratio: + +$$ +\pi_ {t} ^ {\left(\mathbf {S O I D S}\right)} = \underset {\pi \in \Delta (\mathcal {A})} {\arg \min } \overline {{\mathbb {R}}} _ {t} ^ {(2)} (\pi). \tag {4} +$$ + +The choice of $\gamma = 2$ is motivated by the fact that the minimizer of the 2-information ratio is an approximate minimizer of surrogate generalized information ratio for all $\gamma \geq 2$ . + +Lemma 1. For all $\gamma \geq 2$ + +$$ +\overline {{I R}} _ {t} ^ {(\gamma)} (\pi_ {t} ^ {(\mathbf {S O I D S})}) \leq 2 ^ {\gamma - 2} \min _ {\pi \in \Delta (\mathcal {A})} \overline {{I R}} _ {t} ^ {(\gamma)} (\pi). +$$ + +This fact was discovered for the Bayesian IDS policy by Lattimore and György [2021] and remains true for the SOIDS policy. We provide a proof in Appendix F.2 for completeness. Finally, we remark that the "sparse" part of the name SOIDS refers to the choice of the prior $Q_{1}^{+}$ . We use the subset selection prior from Section 3 of Alquier and Lounici [2011], which is described in Appendix B.2. + +# 4 Main results + +In this section, we state our main results. First, we relate the true regret of any policy sequence to the surrogate regret of the same policy sequence. We then use the fact that the surrogate regret is controlled by both the 2- and 3-information ratios. This, combined with Lemma 1, allows us to show that with properly tuned parameters, SOIDS has optimal worst-case regret in both the data-poor and data-rich regimes. Finally, we show that SOIDS can be tuned in a data-dependent manner, such that its regret bound scales with the cumulative observed information ratio instead of the time horizon. + +# 4.1 General bound for the optimistic posterior + +We start with a generic worst-case regret bound relating the true regret of any algorithm to its surrogate regret. Since the surrogate regret is defined with respect to the optimistic posterior, which is known to the learner, it can be controlled with standard Bayesian techniques. This result is an extension of the bounds stated in Neu et al. [2024], Zhang [2022]. To our knowledge, it is the first result of its kind which is compatible with time-dependent or data-dependent learning rates. The stated result is specialized to the setting of sparse linear bandits, but the techniques used to deal with time-dependent and data-dependent learning rates are applicable beyond this setting. + +Theorem 1. Assume that the optimistic posterior is computed with $\eta = \frac{1}{4}$ and a sequence of decreasing learning rates $\lambda_{t}$ satisfying $\forall t\geq 1,\lambda_t\leq \frac{1}{2}$ . Set $\lambda_0 = \frac{1}{2}$ . If the learning rates do not depend on the history, then the regret of any sequence of policies $\pi_t$ satisfies + +$$ +R _ {T} \leq \mathbb {E} \left[ \frac {5 + 2 s \log \frac {e d T}{s}}{\lambda_ {T - 1}} - \sum_ {t = 1} ^ {T} \frac {3}{3 2} \cdot \frac {\overline {{I G}} _ {t} (\pi_ {t})}{\lambda_ {t - 1}} + 2 \sum_ {t = 1} ^ {T} \widehat {\Delta} _ {t} (\pi_ {t}) \right]. \tag {5} +$$ + +Otherwise, if the learning rates depend on the history, let $C_{1,T}$ be a deterministic upper bound on $\frac{1}{\lambda_t} - \frac{1}{\lambda_{t-1}}$ valid for all $t \leq T$ , and $C_{2,T}$ be a deterministic upper bound on $\frac{1}{\lambda_{T-1}}$ . The regret of any sequence of policies $\pi_t$ satisfies + +$$ +R _ {T} \leq \mathbb {E} \left[ \frac {2 + s \log \frac {4 e ^ {3} d ^ {2} T ^ {3} C _ {1 , T} ^ {2} C _ {2 , T}}{s ^ {2}}}{\lambda_ {T - 1}} - \sum_ {t = 1} ^ {T} \frac {3}{3 2} \cdot \frac {\overline {{I G}} _ {t} (\pi_ {t})}{\lambda_ {t - 1}} + 2 \sum_ {t = 1} ^ {T} \widehat {\Delta} _ {t} (\pi_ {t}) \right] + 2. (6) +$$ + +# 4.2 Adapting to both regimes + +We show that the SOIDS algorithm with properly tuned parameters achieves the optimal regret rate in both the data-rich and data-poor regimes. + +Theorem 2. Assume that the sparse optimal action condition in Definition 1 is satisfied. Let $\lambda_t^{(2)} = \sqrt{\frac{3C_{t+1}}{128d(t+1)}}$ and $\lambda_t^{(3)} = \frac{1}{4\cdot6^{\frac{1}{3}}}\left(\frac{C_{t+1}\sqrt{C_{\min}}}{(t+1)\sqrt{s}}\right)^{\frac{2}{3}}$ , with $C_t = 5 + 2s\log \frac{edt}{s}$ . If $\lambda_t = \min \left(\frac{1}{2}, \max (\lambda_t^{(2)}, \lambda_t^{(3)})\right)$ , then the regret of SOIDS satisfies + +$$ +\begin{array}{l} R _ {T} \leq \min \left(2 7 \sqrt {\left(5 + 2 s \log \frac {e d T}{s}\right) d T}, 3 0 \left(5 + 2 s \log \frac {e d T}{s}\right) ^ {\frac {1}{3}} \left(\frac {T \sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}}\right) + \mathcal {O} (\sqrt {s} \log \frac {d}{\sqrt {s}}) \tag {7} \\ = \min \left(\mathcal {O} \left(\sqrt {s d T \log {\frac {e d T}{s}}}\right), \mathcal {O} \left((s T) ^ {\frac {2}{3}} \left(\log {\frac {e d T}{s}}\right) ^ {\frac {1}{3}}\right)\right), \\ \end{array} +$$ + +where $\mathcal{O}(\sqrt{s}\log \frac{d}{\sqrt{s}})$ represents an absolute constant independent of $T$ . + +We observe that our algorithm enjoys both the $\widetilde{\mathcal{O}} (\sqrt{sdT})$ and the $\widetilde{\mathcal{O}} ((sT)^{\frac{2}{3}})$ regret rates. Unlike the Bayesian regret bound for the sparse IDS algorithm of Hao et al. [2021], our regret bound holds in a "worst-case" sense for any value of $\theta_0\in \Theta$ . To our knowledge, this makes our method the first algorithm to achieve optimal worst-case regret in both the data-poor and data-rich regimes + +# 4.3 Instance-dependent guarantees + +The bounds presented in the previous sections are minimax in nature, meaning they hold uniformly over all problem instances. We present a bound in which the scaling with respect to the horizon $T$ is replaced with the cumulative surrogate-information ratio, which could be much smaller than $T$ in "easier" instances, leading to better guarantees. + +Theorem 3. Assume that the sparse optimal action condition in Definition 1 is satisfied, and + +$$ +\text {t h a t} s \leq \frac {d}{2}. \text {L e t} \lambda_ {t} ^ {(2)} = \sqrt {\frac {s}{2 d + \sum_ {s = 1} ^ {t} \overline {{I R}} _ {s} ^ {(2)} (\pi_ {s})}} a n d \lambda_ {t} ^ {(3)} = \left(\frac {s}{\frac {3 \sqrt {6} s}{\sqrt {C _ {\min}}} + \sum_ {s = 1} ^ {t} \sqrt {\overline {{I R}} _ {s} ^ {(3)} (\pi_ {s})}}\right) ^ {\frac {2}{3}}. I f +$$ + +$\lambda_{t} = \max (\lambda_{t}^{(3)},\lambda_{t}^{(2)})$ , then the regret of SOIDS satisfies + +$$ +\begin{array}{l} R _ {T} \leq \left(\frac {2}{s} + \frac {8 0}{3} + 5 \log \frac {e d T}{s}\right) \min \left(\sqrt {s \left(2 d + \sum_ {t = 1} ^ {T - 1} \overline {{I R}} _ {t} ^ {(2)} (\pi_ {t})\right)}, s ^ {\frac {1}{3}} \left(\frac {3 \sqrt {6} s}{\sqrt {C _ {\operatorname* {m i n}}}} + \sum_ {t = 1} ^ {T} \sqrt {\overline {{I R}} _ {t} ^ {(3)} (\pi_ {t})}\right) ^ {\frac {2}{3}}\right) \tag {8} \\ = \mathcal {O} \left(\log \frac {e d T}{s} \min \left(\sqrt {s \left(2 d + \sum_ {t = 1} ^ {T - 1} \overline {{I R}} _ {t} ^ {(2)} (\pi_ {t})\right)}, s ^ {\frac {1}{3}} \left(\frac {3 \sqrt {6} s}{\sqrt {C _ {\mathrm {m i n}}}} + \sum_ {t = 1} ^ {T} \sqrt {\overline {{I R}} _ {t} ^ {(3)} (\pi_ {t})}\right) ^ {\frac {2}{3}}\right)\right). \\ \end{array} +$$ + +This type of result is only possible because our novel analysis of the optimistic posterior (cf. Theorem 1) can handle history-dependent learning rates. A full proof is provided in Appendix D. This result shows that (with appropriate choices of the learning rates) SOIDS is fully adaptive to which of the two regimes is best. Because our analysis requires decreasing learning rates, we are forced to leave the $\log(T)$ terms out of the learning rates, and our logarithmic term has a worse power than in the bound of Theorem 2. An interesting open question is whether it is possible to improve the dependency on this logarithmic term while still using data-dependent learning rates. + +# 5 Analysis + +We now provide an outline of the proofs of the main results. + +# 5.1 Proof of Theorem 1 + +A key observation is that the optimistic posterior can be interpreted as a learner playing an auxiliary online learning game over distributions $\Delta(\Theta)$ . The loss of that game is a weighted sum of negative log-likelihood and estimation error losses. We define + +$$ +L _ {t} ^ {(1)} (\theta) = \sum_ {s = 1} ^ {t} \log \left(\frac {1}{p (Y _ {s} | \theta , A _ {s})}\right) = \sum_ {s = 1} ^ {t} \frac {1}{2} \left(\langle \theta , A _ {s} \rangle - Y _ {s}\right) ^ {2} +$$ + +to be the cumulative negative log-likelihood loss of $\theta$ and + +$$ +L _ {t} ^ {(2)} (\theta) = \sum_ {s = 1} ^ {t} - \Delta (A _ {s}, \theta) +$$ + +to be the cumulative estimation error loss of $\theta$ . In addition, we define the regularizer $\Phi : \Delta(\Theta) \to \mathbb{R}$ by the mapping $P \mapsto \mathcal{D}_{\mathrm{KL}}(P \| Q_1^+)$ , which is the KL-divergence with respect to the prior $Q_1^+$ . With those notations, the optimistic posterior corresponds to an instance of the Follow the Regularized Leader (FTRL) algorithm introduced by Hazan and Kale [2010] and Abernethy et al. [2008]. FTRL is a standard method in online convex optimization that balances cumulative loss minimization with a regularization term to enforce stability and guarantee controlled regret. The update can be reframed as + +$$ +Q_{t + 1}^{+} = \operatorname *{arg min}_{P\in \Delta (\Theta)}\langle P,\eta L_{t}^{(1)} + \lambda_{t}L_{t}^{(2)}\rangle +\Phi (P). +$$ + +This formulation enables the application of tools from convex analysis and online learning, such as Fenchel duality, to derive regret bounds for this auxiliary online learning game and to understand the interplay between the two losses under the learning rates $\eta$ and $\lambda_{t}$ . We now focus on the case in which the learning rates $\lambda_{t}$ don't depend on the history and relegate the analysis of history-dependent learning rates to Appendix C. The following lemma provides a bound on the average regret when the model $\theta_0$ is drawn from an arbitrary comparator distribution $P$ . + +Lemma 2. Let $P \in \Delta(\Theta)$ be any comparator, then the following bound holds + +$$ +\sum_ {t = 1} ^ {T} \Delta \left(A _ {t}, P\right) \leq \frac {\Phi (P) + \Phi^ {*} \left(\eta \left(L _ {T} ^ {(1)} \left(\theta_ {T}\right) - L _ {T} ^ {(1)} (\cdot)\right) - \lambda_ {T} L _ {T} ^ {(2)} (\cdot)\right)}{\lambda_ {T}} + \frac {\eta}{\lambda_ {T}} \left(\langle P, L _ {T} ^ {(1)} \rangle - L _ {T} ^ {(1)} \left(\theta_ {T}\right)\right). \tag {9} +$$ + +Here $\theta_{t} = \arg \min_{\theta \in \Theta}L_{t}^{(1)}(\theta)$ denotes the maximum likelihood estimator at time $t$ , and $\Phi^{*}(L) = \log \int_{\Theta}\exp (L(\theta))dQ_{1}^{+}(\theta)$ is the Fenchel dual of the regularizer $\Phi$ . A complete proof of this result is provided in appendix B.1.1. We aim to choose a comparator $P$ and the prior $Q_{1}^{+}$ such that $P$ is concentrated around $\theta_0$ and the KL divergence $\mathcal{D}_{\mathrm{KL}}\left(P\| Q_1^+\right)$ is controlled. If the parameter space were finite, the natural choice would be to take $P$ as a Dirac on $\theta_0$ and $Q_{1}^{+}$ as a uniform distribution on the whole parameter space; more care is necessary here. Choosing $Q_{1}^{+}$ as a subset-selection prior and $P$ as a uniform distribution on a sparse neighborhood of $\theta_0$ satisfies both requirements. + +Lemma 3. The subset-selection prior $Q_1^+ \in \Delta(\Theta)$ verifies that for any $\epsilon > 0$ and $\theta \in \Theta$ , there is a comparator $P_\theta \in \Delta(\Theta)$ satisfying both + +$$ +\forall \theta^ {\prime} \in \operatorname {s u p p} (P _ {\theta}), \| \theta - \theta^ {\prime} \| _ {1} \leq \epsilon a n d \mathcal {D} _ {K L} \left(P _ {\theta} \big \| Q _ {1} ^ {+}\right) \leq s \log \frac {2 e d}{\epsilon s}. +$$ + +The proof of this lemma, as well as the exact choice of the prior $Q_{1}^{+}$ and the comparator $P(\theta_0)$ , are provided in Appendix B.2. In Appendix I (cf. Lemma 21), we establish that both $L_T^{(2)}(\cdot)$ and $\mathbb{E}\left[L_T^{(1)}(\cdot)\right]$ are $2T$ -Lipschitz with respect to the $\ell_1$ -norm. Hence, + +$$ +\mathbb {E} \left[ \frac {\left| P \cdot L _ {T} ^ {(1)} - L _ {T} ^ {(1)} \left(\theta_ {0}\right) \right|}{\lambda_ {T}} \right] \leq \frac {2 T \epsilon}{\lambda_ {T}}, \quad \text {a n d} \quad \sum_ {t = 1} ^ {T} \left| \Delta \left(\theta_ {0}, A _ {t}\right) - \Delta \left(P, A _ {t}\right) \right| \leq 2 T \epsilon . +$$ + +Combining these with Lemma 2, we obtain the following bound on the cumulative regret: + +$$ +\begin{array}{l} R _ {T} \leq \mathbb {E} \left[ \frac {s \log \frac {2 e d}{\epsilon s} + 2 T (\lambda_ {T} + \eta) \epsilon}{\lambda_ {T}} + \frac {\Phi^ {*} (- \eta (L _ {T} ^ {(1)} (\cdot) - L _ {T} ^ {(1)} (\theta_ {T})) - \lambda_ {T} L _ {T} ^ {(2)} (\cdot))}{\lambda_ {T}} \right] \\ + \mathbb {E} \left[ \frac {\eta}{\lambda_ {T}} (L _ {T} ^ {(1)} (\theta_ {0}) - L _ {T} ^ {(1)} (\theta_ {T})) \right]. \\ \end{array} +$$ + +The first term balances model complexity and approximation via $\epsilon$ . In the usual FTRL analysis, $\lambda \rightarrow \frac{\phi^{*}(\lambda L)}{\lambda}$ is non-decreasing for any $L \in \mathbb{R}^{\Theta}$ , and the term involving $\Phi^{*}$ can be telescoped. Things are more complex here because only some part of the loss is weighted by the time varying learning rate $\lambda_{T}$ . Through a careful analysis involving the maximum likelihood estimator, we can decompose the $\Phi^{*}$ term into a telescoping sum and a remainder term. + +# Lemma 4. + +$$ +\begin{array}{l} \mathbb {E} \left[ \frac {\Phi^ {*} (\eta (L _ {T} ^ {(1)} (\theta_ {T}) - L _ {T} ^ {(1)} (\cdot)) - \lambda_ {T} L _ {T} ^ {(2)} (\cdot))}{\lambda_ {T}} \right] \\ \leq \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\Phi^ {*} \left(\eta \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) L _ {t} ^ {(1)} (\cdot)\right) - \lambda_ {t - 1} L _ {t} ^ {(2)} (\cdot)\right)}{\lambda_ {t - 1}} - \frac {\Phi^ {*} \left(\eta \left(L _ {t - 1} ^ {(1)} \left(\theta_ {0}\right) - L _ {t - 1} ^ {(1)} (\cdot)\right) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\cdot)\right)}{\lambda_ {t - 1}} \right] (10) \\ + \frac {\eta (6 + s \log \frac {e d T}{s})}{\lambda_ {T}}. (11) \\ \end{array} +$$ + +A detailed proof of this result is provided in Appendix B.1.2. Finally, the remaining sum can be handled by looking at the explicit formula for $\Phi^{*}$ . The terms related to the likelihood and the gap estimates can be separated using Hölder's inequality, as is done in Zhang [2022] and Neu, Papini, and Schwartz [2024]. More explicitly, by now choosing $\eta = \frac{1}{4}$ , we obtain the following lemma. + +# Lemma 5. + +$$ +\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\Phi^ {*} (\eta (L _ {t} ^ {(1)} (\theta_ {0}) - L _ {t} ^ {(1)} (\cdot)) - \lambda_ {t - 1} L _ {t} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} - \frac {\Phi^ {*} (\eta (L _ {t - 1} ^ {(1)} (\theta_ {0}) - L _ {t - 1} ^ {(1)} (\cdot)) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} \right] \\ \leq \mathbb {E} \left[ - \sum_ {t = 1} ^ {T} \frac {3 \overline {{I G}} _ {t} (\pi_ {t})}{3 2 \lambda_ {t - 1}} + 2 \sum_ {t = 1} ^ {T} \widehat {\Delta} (\pi_ {t}) \right]. \tag {12} \\ \end{array} +$$ + +A full proof of this result is provided in Appendix B.1.4. Combining Lemmas 2, 3, 4, 5 and setting $\epsilon = \frac{2}{T}$ , we obtain the desired regret bound stated in Theorem 1. + +# 5.2 Proof of Theorem 2 + +We show how Theorem 1 can be combined with bounds on the surrogate regret to control the true regret. The first important fact is that the surrogate regret of any policy can always be controlled in terms of the 2 or the 3-surrogate information ratio of that policy. + +Lemma 6. Let $\lambda >0$ , then we have that for any policy $\pi \in \Delta (\mathcal{A})$ + +$$ +\widehat {\Delta} _ {t} (\pi) \leq \frac {\overline {{I G}} _ {t} (\pi)}{\lambda} + \min \left(\frac {1}{4} \lambda \overline {{I R}} _ {t} ^ {(2)} (\pi), c _ {3} ^ {*} \sqrt {\lambda \overline {{I R}} _ {t} ^ {(3)} (\pi)}\right), +$$ + +where $c_3^* < 2$ is an absolute constant defined in Lemma 27. + +This is a consequence of a simple generalization of the AM-GM inequality and is proved in Appendix F.1. Combining the previous lemma with $\lambda = \frac{64}{3}\lambda_{t - 1}$ and Theorem 1, we can further upper bound the regret of a sequence of policies $(\pi_t)_t$ as + +$$ +\begin{array}{l} R _ {T} \leq \mathbb {E} \left[ \frac {5 + 2 s \log \frac {e d T}{s}}{\lambda_ {T - 1}} - \sum_ {t = 1} ^ {T} \frac {3 \overline {{\mathrm {I G}}} _ {t} (\pi_ {t})}{3 2 \lambda_ {t - 1}} + 2 \sum_ {t = 1} ^ {T} \widehat {\Delta} _ {t} (\pi_ {t}) \right] \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{\lambda_ {T - 1}} + \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \lambda_ {t - 1} \overline {{\mathrm {I R}}} _ {t} ^ {(2)} \left(\pi_ {t}\right), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} \overline {{\mathrm {I R}}} _ {t} ^ {(3)} \left(\pi_ {t}\right)}\right) \right], \tag {13} \\ \end{array} +$$ + +where $C_T = 5 + 2s\log \frac{edT}{s}$ . Usually, bounds on the 2-information ratio can be converted to $\mathcal{O}(\sqrt{T})$ bounds and bounds on the 3-information ratio can be converted to $\mathcal{O}(T^{\frac{2}{3}})$ bounds. Hence we will use the 2-information ratio to control the regret in the data-rich regime and the 3-information ratio to control the regret in the data-poor regime. Due to Lemma 1, the SOIDS policy minimizes the 2-information ratio and approximately minimizes the 3-information ratio. As a result, if there exists a "forerunner" algorithm with bounded 2-information ratio or 3-information ratio, SOIDS inherits these bounds automatically. In particular, we can use a different forerunner for each regime and SOIDS will match the regret guarantees of the best forerunner in each regime. + +This forerunner-based technique is widely used to analyze IDS based algorithms and has been applied to a variety of Bayesian settings [Russo and Van Roy, 2017, Hao et al., 2021, Hao and Latimore, 2022] and some frequentist settings [Kirschner and Krause, 2018, Kirschner et al., 2020, 2021]. An advantage of the OIDS framework is that since the surrogate quantities are defined with respect to the optimistic posterior, the analysis of the surrogate information ratio is virtually identical to the corresponding analysis of the information ratio in the Bayesian setting. + +The forerunner we consider for the 2-information ratio is the Feel-Good Thompson Sampling (FGTS) algorithm of Zhang [2022]. For the 3-information ratio, we consider a mixture of the FGTS policy and an exploratory policy. The following lemma provides bounds on the surrogate information ratios of the SOIDS algorithm. + +Lemma 7. The 2- and 3-surrogate-information ratio of the SOIDS algorithm satisfy for any $t \geq 0$ + +$$ +\overline {{I R}} _ {t} ^ {(2)} \left(\pi_ {t} ^ {\left(\mathbf {S O I D S}\right)}\right) \leq \overline {{I R}} _ {t} ^ {(2)} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) \leq 2 d \tag {14} +$$ + +and + +$$ +\overline {{I R}} _ {t} ^ {(3)} \left(\pi_ {t} ^ {\left(\mathbf {S O I D S}\right)}\right) \leq 2 \overline {{I R}} _ {t} ^ {(3)} \left(\pi_ {t} ^ {\left(\mathbf {m i x}\right)}\right) \leq \frac {5 4 s}{C _ {\min }}. \tag {15} +$$ + +The explicit definition of both forerunner algorithms, as well as the proof of this lemma, are deferred to Appendix F.3. Finally, it remains to pick the learning rate $\lambda_{t}$ . The following lemma describes the appropriate learning rate for the data-poor and the data-rich regimes separately. + +Lemma 8. The choice of learning rate $\lambda_t^{(2)} = \sqrt{\frac{3C_{t+1}}{128d(t+1)}}$ guarantees + +$$ +\frac {C _ {T}}{\lambda_ {T - 1} ^ {(2)}} + \frac {3 2}{3} \sum_ {t = 1} ^ {T} \lambda_ {t - 1} ^ {(2)} \overline {{I R}} _ {t} ^ {(2)} (\pi_ {t} ^ {(\mathbf {S O I D S})}) \leq 1 6 \sqrt {\frac {2}{3} C _ {T} d T}. +$$ + +The choice of learning rate $\lambda_{t}^{(3)} = \frac{1}{4\cdot 6^{\frac{1}{3}}}\left(\frac{C_{t + 1}\sqrt{C_{\min}}}{(t + 1)\sqrt{s}}\right)^{\frac{2}{3}}$ guarantees + +$$ +\frac {C _ {T}}{\lambda_ {T - 1} ^ {(3)}} + \frac {1 6}{3} c _ {3} ^ {*} \sum_ {t = 1} ^ {T} \sqrt {3 \lambda_ {t - 1} ^ {(3)} \overline {{I R}} _ {t} ^ {(3)} (\pi_ {t} ^ {(\mathbf {S O I D S})})} \leq 1 2 \cdot 6 ^ {\frac {1}{3}} \left(\frac {s \cdot C _ {T}}{C _ {\mathrm {m i n}}}\right) ^ {\frac {1}{3}} T ^ {\frac {2}{3}}. +$$ + +![](images/cd1510c624eba458f324e6da75a6f292b611b250365d04f7458a40e3fa8c3e90.jpg) +Figure 1: Cumulative regret for $d = 20$ (left) 40 (middle) and 100 (right). We plot the mean ± standard deviation over 10 repetitions. + +![](images/0d0e7c5a8856b4b75f4fc18eb063d97187785092cb9f738e1490e03123c8e75b.jpg) + +![](images/3b206300b2617a45b67bd75def508dee217208944f06b77aabb949d2ece8e07c.jpg) + +The proof is deferred to Appendix G.2. The expression for the constant $c_3^* \leq 2$ can be found in Appendix I.2 (cf. Lemma 27). It remains to analyze what happens when the learning rate $\lambda_t = \min\left(\frac{1}{2}, \max\left(\lambda_t^{(2)}, \lambda_t^{(3)}\right)\right)$ is chosen. We defer this to Appendix G.4. + +# 6 Experiments + +We aim to verify that, in both the data-rich and data-poor regimes simultaneously, the regret of SOIDS is comparable with the regret of existing algorithms that achieve near optimal worst-case regret in either the data-rich or the data-poor regime. Our baseline for the data-rich regime is the online-to-confidence-set (OTCS) method proposed by Abbasi-Yadkori et al. [2012], which has worst case regret of the order $\sqrt{sdT}$ . For a tougher comparison, we run this method with the confidence sets from Theorem 4.7 of Clerico et al. [2025], which have much smaller constant factors than those used by Abbasi-Yadkori et al. [2012]. Our baseline for the data-poor regime is the Explore the Sparsity Then Commit (ESTC) algorithm proposed by Hao et al. [2020], which has worst-case regret of the order $(sT)^{2/3}$ . For reference, we also compare with LinUCB Abbasi-Yadkori et al. [2011], which does not adapt to sparsity. + +It is generally difficult to run the SOIDS algorithm exactly because the surrogate information ratio contains expectations w.r.t. the optimistic posterior. In our implementation of SOIDS, we use the empirical Bayesian sparse sampling procedure of Hao et al. [2021] to draw approximate samples from the optimistic posterior, and then approximate the surrogate information ratio via sample averages. We provide further details regarding the implementations of each method in Appendix J. + +For each $d \in \{20, 40, 100\}$ , $\theta_0$ is the $s$ -sparse vector in $\mathbb{R}^d$ , with $s = d / 10$ , in which first $s$ components are $10 / s$ and the remaining components are zero. The action set consists of 200 random draws from the uniform distribution on $[-1, 1]^d$ . The noise variance is 1 and we run each method 10 times. In Figure 1, we report the cumulative regret over $T = 1000$ steps. As $d$ is varied from 20 to 100, we appear to transition from the data-rich regime to the data-poor regime: for $d = 20$ , the OTCS method is the best performing baseline, whereas for $d = 100$ , ETCS is the best performing baseline. As our theoretical results would suggest, SOIDS performs well in both regimes. + +# 7 Conclusion + +There remain several interesting questions that our work leaves open for future research, such as the possibility of improving the logarithmic terms in the instance-dependent regret bound (as mentioned earlier in Section 4). We highlight another question below. + +In our experiments, we have made use of an approximate implementation of OIDS adapted from Hao et al. [2021]. The initial success we have seen in our experiments suggests that this approximation might be viable in more challenging settings, and worthy of an attempt at a solid theoretical analysis. More broadly, the results indicate a potential advantage of IDS-style methods over DEC-inspired methods [Foster et al., 2022b, Kirschner et al., 2023]. Indeed, we are not aware of any general methods for approximating the optimization problems that the E2D algorithm of Foster et al. 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URL http://arxiv.org/abs/1912.13213. +Paat Rusmevichientong and John N Tsitsiklis. Linearly parameterized bandits. Mathematics of Operations Research, 35(2):395-411, 2010. +Daniel Russo and Benjamin Van Roy. An information-theoretic analysis of thompson sampling. Journal of Machine Learning Research, 17:68:1-68:30, 2016. URL https://jmlr.org/papers/v17/14-087.html. +Daniel Russo and Benjamin Van Roy. Learning to Optimize via Information-Directed Sampling, 2017. URL http://arxiv.org/abs/1403.5556. + +Michal Valko, Rémi Munos, Branislav Kveton, and Tomáš Kocák. Spectral bandits for smooth graph functions. volume 32 of JMLR Workshop and Conference Proceedings, pages 46-54, 2014. URL http://proceedings.mlr.press/v32/valko14.html. +M.J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. 2019. ISBN 978-1-108-49802-9. URL https://books.google.es/books?id=8C8nuQEACAAJ. +Xue Wang, Mingcheng Wei, and Tao Yao. Minimax concave penalized multi-armed bandit model with high-dimensional covariates. In International Conference on Machine Learning, pages 5200-5208. PMLR, 2018. +Tong Zhang. Feel-good thompson sampling for contextual bandits and reinforcement learning. SIAM Journal on Mathematics of Data Science, 4(2):834-857, 2022. doi: 10.1137/21M140924X. URL https://doi.org/10.1137/21m140924x. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. + +Please read the checklist guidelines carefully for information on how to answer these questions. 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If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Related work + +The first algorithms and regret bounds for sparse linear bandits were designed for the data-rich regime. Abbasi-Yadkori et al. [2012] developed an online-to-confidence-set conversion for linear models, which converts any algorithm for online linear regression into a linear bandit algorithm whose regret depends on the regret of the online regression algorithm. When the SeqSEW algorithm [Gerchinovitz, 2013] is used in this conversion, the result is a sparse linear bandit algorithm with a regret bound of the order $\mathcal{O}(\sqrt{sdT})$ (ignoring logarithmic factors). Lattimore and Szepesváři [2020] established a matching lower bound for the data-rich regime, showing that this rate cannot be improved. + +More recently, several works have studied the data-poor regime, in which the dimension $d$ is much larger than the number of rounds $T$ . Hao et al. [2020] showed that an explore-then-commit algorithm satisfies a regret bound of the order $O((sT)^{2/3}C_{\min}^{-2/3})$ , and established a lower bound of order $\Omega(\min(s^{1/3}T^{2/3}C_{\min}^{-1/3}, \sqrt{dT})$ . Subsequently, Jang et al. [2022] proposed the PopArt estimator for sparse linear regression, and showed that an explore-then-commit algorithm that uses this estimator achieves a regret bound of the order $\mathcal{O}(s^{2/3}T^{2/3}H_{\star}^{2/3})$ , where $H_{\star}$ is another problem-dependent quantity that satisfies $H_{\star}^2 \leq C_{\min}^{-1}$ . In addition, Jang et al. [2022] established a lower bound of order $\Omega(s^{2/3}T^{2/3}C_{\min}^{-1/3})$ , showing that the optimal rate for the data-poor regime is $s^{2/3}T^{2/3}$ . Hao et al. [2021] showed that sparse IDS has a Bayesian regret bound that is optimal for both regimes. + +A number of works have considered sparse contextual linear bandits, in which the action set $\mathcal{A}$ changes in each round $t$ . In the case where the actions sets are chosen by an adaptive adversary, the upper and lower bounds of the order $\sqrt{sdT}$ by Abbasi-Yadkori et al. [2012] and Lattimore and Szepesvári [2020] respectively still hold. Under the assumption that the action sets are generated randomly, and such that either a uniform or greedy policy is (with high probability) exploratory, several methods have been shown to achieve nearly dimension-free regret bounds Bastani and Bayati [2020], Wang et al. [2018], Kim and Paik [2019], Oh et al. [2021], Chakraborty et al. [2023]. + +The concept of balancing instantaneous regret and information gain through the information ratio was first introduced by Russo and Roy [2016] in the context of analyzing Thompson Sampling. Building upon this, the Information-Directed Sampling (IDS) algorithm was proposed by Russo and Van Roy [2017] to directly minimize the information ratio, thereby optimizing the trade-off between regret and information gain. These foundational ideas have since been extended and applied to a variety of settings including bandits [Bubeck and Sellke, 2022], contextual bandits [Neu et al., 2022, Hao et al., 2022], reinforcement learning [Hao and Lattimore, 2022], and sparse linear bandits [Hao et al., 2021]. However, these works are primarily situated in the Bayesian framework and focus on Bayesian regret bounds that hold only in expectation with respect to the prior distribution. + +A key challenge in extending these methods to the frequentist setting lies in estimating the instantaneous regret and define a meaningful notion of information gain. Both of those things are naturally possible in Bayesian analysis but difficult when the true model is unknown. Moreover, Bayesian posteriors may inadequately represent model uncertainty from a frequentist perspective. We highlight three strands of research that have attempted to address this challenge: + +Confidence-set based information ratio approaches: Works such as Kirschner and Krause [2018], Kirschner et al. [2020], and Kirschner et al. [2021] extend the notion of the information ratio to frequentist settings by constructing high-probability confidence sets for the instantaneous regret and information gain. These results are mostly limited to setting with some linear structure. + +Distributionally robust and worst-case information-regret trade-offs: The Decision-to-Estimation-Coefficient(DEC) line of work of [Foster et al., 2022b, Foster and Rakhlin, 2020, Foster et al., 2022c,a, Kirschner et al., 2023] explores the frequentist setting by analyzing worst-case trade-offs between regret and information gain. One limitation is that the DEC is an inherently worst-case measure of comlexity. Moreover, algorithms based on the DEC usually require solving complex min-max optimization problems at each time step, making their practical implementation challenging and unclear. + +Optimistic posterior approaches for frequentist guarantees: The approach most closely related to our work modifies the Bayesian posterior to provide frequentist guarantees. Introduced by Zhang [2022], the optimistic posterior is a modification of the Bayesian posterior which enables frequentist regret bounds for a variant of Thompson Sampling. Subsequently, Neu et al. [2024] studied the optimistic posterior framework in greater depth, defining a frequentist analog of the information + +ratio to extend IDS to frequentist settings. A notable limitation of these works is their restriction to constant learning rates in the optimistic posterior, which limits adaptivity, an issue that we address in this paper. + +# B Analysis of the Optimistic posterior + +This section provides further details about the prior underlying the optimistic posterior and guarantees on the posterior updates. + +# B.1 Follow the regularized leader analysis + +The main step in our analysis of the optimistic posterior is to leverage the follow the regularized leader formulation of our optimistic posterior update + +$$ +Q_{t + 1}^{+} = \operatorname *{arg min}_{P\in \Delta (\Theta)}\langle P,\eta L_{t}^{(1)} + \lambda_{t}L_{t}^{(2)}\rangle +\Phi (P). +$$ + +# B.1.1 Proof of Lemma 2 + +As is usual in the analysis of the follow the regularized leader algorithm, we introduce the Fenchel conjugate $\Phi^{*}:\mathbb{R}^{\Theta}\to \mathbb{R}$ of the regularization function $\Phi = \mathcal{D}_{\mathrm{KL}}\left(\cdot \| Q_1^+\right)$ , which takes values $\Phi^{*}(L) = \sup_{P\in \Delta (\Theta)}\{\langle P,L\rangle -\Phi (P)\}$ . The Fenchel-Young inequality guarantees that for any $P\in \Delta (\Theta)$ and $L\in \mathbb{R}^{\Theta}$ , + +$$ +\langle P, L \rangle \leq \Phi (P) + \Phi^ {*} (L). +$$ + +We introduce the maximum likelihood estimator $\theta_t = \arg \min_{\theta \in \Theta} L_t^{(1)}(\theta)$ and the function + +$$ +L (\cdot) = - \eta (L _ {T} ^ {(1)} (\cdot) - L _ {T} ^ {(1)} (\theta_ {T})) - \lambda_ {T} L _ {T} ^ {(2)} (\cdot). +$$ + +Since $\lambda_{T}$ is never used by the algorithm, we can assume that $\lambda_{T} = \lambda_{T - 1}$ . The role of the maximum likelihood estimator is to ensure that the term $L_{t}^{(1)}(\theta) - L_{t}^{(1)}(\theta_{t})$ is always non-negative. The Fenchel-Young inequality tells us that + +$$ +\eta \left(L _ {T} ^ {(1)} (\theta_ {T}) - \big < P, L _ {T} ^ {(1)} \big >\right) - \lambda_ {T} \big < P, L _ {T} ^ {(2)} \big > \leq \Phi (P) + \Phi^ {*} \left(- \eta (L _ {T} ^ {(1)} (\cdot) - L _ {T} ^ {(1)} (\theta_ {T})) - \lambda_ {T} L _ {T} ^ {(2)} (\cdot)\right). +$$ + +Noticing that $\left\langle P,L_T^{(2)}\right\rangle = -\sum_{t = 1}^T\Delta (P,A_t)$ and rearranging the terms concludes the proof. + +# B.1.2 Proof of Lemma 4 + +We start by rewriting the potential function in the form of the following telescopic sum: + +$$ +\begin{array}{l} \frac {\Phi^ {*} (- \eta (L _ {T} ^ {(1)} (\cdot) - L _ {T} ^ {(1)} (\theta_ {T})) - \lambda_ {T} L _ {T} ^ {(2)} (\cdot))}{\lambda_ {T}} \\ = \sum_ {t = 1} ^ {T} \frac {\Phi^ {*} (- \eta (L _ {t} ^ {(1)} (\cdot) - L _ {t} ^ {(1)} (\theta_ {t})) - \lambda_ {t} L _ {t} ^ {(2)} (\cdot))}{\lambda_ {t}} - \frac {\Phi^ {*} (- \eta (L _ {t - 1} ^ {(1)} (\cdot) - L _ {t - 1} ^ {(1)} (\theta_ {t - 1})) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\cdot))}{\lambda_ {t - 1}}. \\ \end{array} +$$ + +In the usual follow-the-regularized-leader analysis, we use the fact that $\lambda \mapsto \frac{\phi^{*}(\lambda L)}{\lambda}$ is nondecreasing for any $L\in \mathbb{R}^{\Theta}$ . Here however, only one component of the linear loss is scaled by $\lambda_{t}$ , and so the standard FTRL analysis fails. Crucially, because we introduced the maximum likelihood estimator $\theta_t$ , we have that $L_{t}^{(1)}(\cdot) - L_{t}^{(1)}(\theta_{t})\geq 0$ . We can therefore use the following lemma that guarantees that a scaled and shifted dual is monotonic. + +Lemma 9. Let $\Phi \geq 0$ and $\Phi^{*}$ be a convex function and its dual as defined previously. For $L_{1}, L_{2} \in \mathbb{R}^{\Theta}$ such that $L_{1} \geq 0$ , the mapping $\lambda \mapsto \frac{\Phi^{*}(-L_{1} + \lambda L_{2})}{\lambda}$ is non-decreasing on $\mathbb{R}^{+\ast}$ . + +Proof. By definition, we have + +$$ +\begin{array}{l} \frac {\Phi^ {*} (- L _ {1} + \lambda L _ {2})}{\lambda} = \frac {\sup _ {P \in \Delta (\Theta)} \langle P , - L _ {1} + \lambda L _ {2} \rangle - \Phi (P)}{\lambda} \\ = \sup _ {P \in \Delta (\Theta)} \langle P, L _ {2} \rangle - \frac {\langle P , L _ {1} \rangle + \Phi (P)}{\lambda}. \\ \end{array} +$$ + +For any $P \in \Delta(\Theta)$ , we have that $\Phi(P) + \langle P, L_1 \rangle \geq 0$ and the term inside the supremum is non-decreasing with respect to lambda. Since the supremum of non-decreasing functions is also non-decreasing, this concludes the proof. + +Applying the previous lemma, we upper bound the previous sum by replacing each $\lambda_{t}$ factor by $\lambda_{t - 1}$ (using the convention $\lambda_0 = 1 / 2$ ), and then we replace the maximum likelihood estimator $\theta_t$ by $\theta_0$ inside $\Phi^*$ to obtain + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \frac {\Phi^ {*} (- \eta (L _ {t} ^ {(1)} (\cdot) - L _ {t} ^ {(1)} (\theta_ {t})) - \lambda_ {t} L _ {t} ^ {(2)} (\cdot))}{\lambda_ {t}} - \frac {\Phi^ {*} (- \eta (L _ {t - 1} ^ {(1)} (\cdot) - L _ {t - 1} ^ {(1)} (\theta_ {t - 1})) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} \\ \leq \sum_ {t = 1} ^ {T} \frac {\Phi^ {*} (- \eta (L _ {t} ^ {(1)} (\cdot) - L _ {t} ^ {(1)} (\theta_ {t})) - \lambda_ {t - 1} L _ {t} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} - \frac {\Phi^ {*} (- \eta (L _ {t - 1} ^ {(1)} (\cdot) - L _ {t - 1} ^ {(1)} (\theta_ {t - 1})) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} \\ = \sum_ {t = 1} ^ {T} \frac {\Phi^ {*} (- \eta (L _ {t} ^ {(1)} (\cdot) - L _ {t} ^ {(1)} (\theta_ {0})) - \lambda_ {t - 1} L _ {t} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} - \frac {\Phi^ {*} (- \eta (L _ {t - 1} ^ {(1)} (\cdot) - L _ {t - 1} ^ {(1)} (\theta_ {0})) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} \\ + \frac {\eta}{\lambda_ {t - 1}} (L _ {t} ^ {(1)} (\theta_ {t}) - L _ {t} ^ {(1)} (\theta_ {0}) + L _ {t - 1} ^ {(1)} (\theta_ {0}) - L _ {t - 1} ^ {(1)} (\theta_ {t - 1})). \\ \end{array} +$$ + +It remains to bound the difference of the negative log likelihood of the true parameter and the maximum likelihood estimator. This is done via the following result (whose proof we relegate to appendix E.1.1). + +Lemma 10. For any $t \geq 1$ , we have + +$$ +0 \leq \mathbb {E} \left[ L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right) \right] \leq \inf _ {\rho} \left\{2 \rho t + s \log \frac {e d (1 + 2 / \rho)}{s} \right\} \leq 6 + s \log \frac {e d t}{s} \tag {16} +$$ + +Using this lemma, we can further bound the previously considered expression as the following telescopic sum: + +$$ +\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\eta}{\lambda_ {t - 1}} \left(L _ {t} ^ {(1)} \left(\theta_ {t}\right) - L _ {t} ^ {(1)} \left(\theta_ {0}\right) + L _ {t - 1} ^ {(1)} \left(\theta_ {0}\right) - L _ {t - 1} ^ {(1)} \left(\theta_ {t - 1}\right)\right) + \frac {\eta}{\lambda_ {T}} \left(L _ {T} ^ {(1)} \left(\theta_ {0}\right) - L _ {T} ^ {(1)} \left(\theta_ {T}\right)\right) \right] \\ = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\eta}{\lambda_ {t - 1}} \left(L _ {t} ^ {(1)} \left(\theta_ {t}\right) - L _ {t} ^ {(1)} \left(\theta_ {0}\right)\right) - \sum_ {t = 1} ^ {T} \frac {\eta}{\lambda_ {t}} \left(L _ {t} ^ {(1)} \left(\theta_ {t}\right) - L _ {t} ^ {(1)} \left(\theta_ {0}\right)\right) \right] \\ \leq \eta \cdot \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right)\right) \left(\frac {1}{\lambda_ {t}} - \frac {1}{\lambda_ {t - 1}}\right) \right. \\ \leq \frac {\eta (6 + s \log \frac {e d T}{s})}{\lambda_ {T}}. \\ \end{array} +$$ + +Here, the first inequality comes from the non-negativity of $L_{t}^{(1)}(\theta_0) - L_{t}^{(1)}(\theta_t)$ by definition of $\theta_t$ and the second one is from Lemma 10 just above and a telescoping argument. Finally we obtain the claim of Lemma 4. + +# B.1.3 Controlling the losses separately + +The focus of this section is to understand how to control $\Phi^{*}(-L)$ where $L$ is either the negative-likelihood loss or the estimation-error loss. We start by analyzing the negative-likelihood loss. As was done in Neu, Papini, and Schwartz [2024], we will relate the negative-likelihood loss to the surrogate information gain. + +For this analysis, we define the true information gain as + +$$ +\mathrm {I G} _ {t} (\pi) = \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi (a) \int_ {\Theta} \left(\left\langle \theta - \theta_ {0}, a \right\rangle\right) ^ {2} \mathrm {d} Q _ {t} ^ {+} (\theta), \tag {17} +$$ + +and note that, by linearity reward function, the surrogate information gain is always smaller than the true information gain. This is stated formally below. + +Proposition 1. For any policy $\pi \in \Delta (\mathcal{A})$ and any $t\geq 1$ we have that + +$$ +\overline {{I G}} _ {t} (\pi) \leq I G _ {t} (\pi) \tag {18} +$$ + +The proof is provided in Appendix I.1. This result can then be used to relate the surrogate and the true information gain to the negative-likelihood loss. This result and its proof are identical to the proof of Lemma 17 in Neu, Papini, and Schwartz [2024]. + +Lemma 11. Assume that the noise $\epsilon_{t}$ is conditionally $I$ -sub-Gaussian, then for any $t \geq 1$ , $\eta, \alpha \geq 0$ such that $\gamma = \frac{\eta\alpha}{2} (1 - \eta\alpha) > 0$ , the following inequality holds + +$$ +\begin{array}{l} \mathbb {E} \left[ \log \int_ {\Theta} \left(\frac {p \left(Y _ {t} \mid \theta , A _ {t}\right)}{p \left(Y _ {t} \mid \theta_ {0} , A _ {t}\right)}\right) ^ {\eta \alpha} \mathrm {d} Q _ {t} ^ {+} (\theta) \right] \leq - 2 \gamma (1 - 2 \gamma) \mathbb {E} \left[ I G _ {t} \left(\pi_ {t}\right) \right] (19) \\ \leq - 2 \gamma (1 - 2 \gamma) \mathbb {E} \left[ \overline {{I G}} _ {t} \left(\pi_ {t}\right) \right]. (20) \\ \end{array} +$$ + +In particular, the constant $2\gamma (1 - 2\gamma)$ can be maximized to the value $\frac{3}{16}$ by the choice $\eta \alpha = \frac{1}{2}$ . + +Proof. By the tower rule of expectation and Jensen's inequality applied to the logarithm, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \log \int_ {\Theta} \left(\frac {p \left(Y _ {t} | \theta , A _ {t}\right)}{p \left(Y _ {t} | \theta_ {0} , A _ {t}\right)}\right) ^ {\eta \alpha} \mathrm {d} Q _ {t} ^ {+} (\theta) \right] = \mathbb {E} \left[ \mathbb {E} \left[ \log \int_ {\Theta} \left(\frac {p \left(Y _ {t} | \theta , A _ {t}\right)}{p \left(Y _ {t} | \theta_ {0} , A _ {t}\right)}\right) ^ {\eta \alpha} \mathrm {d} Q _ {t} ^ {+} (\theta) \middle | \mathcal {F} _ {t - 1}, A _ {t} \right] \right] \\ \leq \mathbb {E} \left[ \log \mathbb {E} \left[ \int_ {\Theta} \left(\frac {p \left(Y _ {t} \mid \theta , A _ {t}\right)}{p \left(Y _ {t} \mid \theta_ {0} , A _ {t}\right)}\right) ^ {\eta \alpha} \mathrm {d} Q _ {t} ^ {+} (\theta) \middle | \mathcal {F} _ {t - 1}, A _ {t} \right] \right] \\ = \mathbb {E} \left[ \log \int_ {\Theta} \mathbb {E} \left[ \exp \left(- \eta \alpha \left(\frac {\left(Y _ {t} - \langle \theta , A _ {t} \rangle\right) ^ {2}}{2} - \frac {\left(Y _ {t} - \langle \theta_ {0} , A _ {t} \rangle\right) ^ {2}}{2}\right)\right) \mid \mathcal {F} _ {t - 1}, A _ {t} \right] d Q _ {t} ^ {+} (\theta) \right]. \\ \end{array} +$$ + +Now, we fix some $\theta \in \Theta$ and to simplify the notation, we let $r_0 = \langle \theta_0,A_t\rangle$ and $r = \langle \theta ,A_t\rangle$ . Using some elementary manipulations and the conditional sub-Gaussian tail behaviour of $\epsilon_{t}$ and $Y_{t} = r_{0} + \epsilon_{t}$ which implies that for any $(\mathcal{F}_{t - 1},A_t)$ -measurable $\zeta_t$ , $\mathbb{E}\left[\exp (Y_t\zeta_t)|\mathcal{F}_{t - 1},A_t\right] = \exp (r_0\zeta_t)\mathbb{E}\left[\exp (\epsilon_t\zeta_t)|\mathcal{F}_{t - 1},A_t\right]\leq \exp (r_0\zeta_t)\exp \left(\frac{\zeta_t^2}{2}\right)$ , we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \exp \left(- \eta \alpha \left(\frac {\left(Y _ {t} - r\right) ^ {2}}{2} - \frac {\left(Y _ {t} - r _ {0}\right) ^ {2}}{2}\right)\right) \Bigg | \mathcal {F} _ {t - 1}, A _ {t} \right] \\ = \mathbb {E} \left[ \exp \left(- \frac {\eta \alpha}{2} \left(2 Y _ {t} - r - r _ {0}\right) \left(r _ {0} - r\right)\right) \mid \mathcal {F} _ {t - 1}, A _ {t} \right] \\ = \exp \left(\eta \alpha \frac {r _ {0} ^ {2} - r ^ {2}}{2}\right) \mathbb {E} \left[ \exp \left(\eta \alpha Y _ {t} (r - r _ {0})\right) \mid \mathcal {F} _ {t - 1}, A _ {t} \right] \\ \leq \exp \left(\eta \alpha \frac {r _ {0} ^ {2} - r ^ {2}}{2}\right) \cdot \exp \left(\eta \alpha r _ {0} (r - r _ {0})\right) \exp \left(\frac {\eta^ {2} \alpha^ {2}}{2} (r - r _ {0}) ^ {2}\right) \\ = \exp \left(- (r - r _ {0}) ^ {2} \cdot \frac {\eta \alpha}{2} (1 - \eta \alpha)\right). \\ \end{array} +$$ + +Further, defining $\gamma = \frac{\eta\alpha}{2} (1 - \eta \alpha)$ , we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \exp \left(- \eta \alpha \left(\frac {\left(Y _ {t} - r\right) ^ {2}}{2} - \frac {\left(Y _ {t} - r _ {0}\right) ^ {2}}{2}\right)\right) \mid \mathcal {F} _ {t - 1}, A _ {t} \right] \\ \leq \exp (- \gamma (r - r _ {0}) ^ {2}) \\ \leq 1 - \gamma (r - r _ {0}) ^ {2} + \frac {\gamma^ {2}}{2} (r - r _ {0}) ^ {4} \\ \leq 1 - \gamma (r - r _ {0}) ^ {2} + 2 \gamma^ {2} (r - r _ {0}) ^ {2} \\ \leq 1 - \gamma (1 - 2 \gamma) (r - r _ {0}) ^ {2}. \\ \end{array} +$$ + +Here, we used the elementary inequality $\exp(x) \leq 1 + x + \frac{x^2}{2}$ for $x \leq 0$ and then used $|r - r_0| \leq 2$ . Finally, using that $\log x \leq x - 1$ for any $x > 0$ , and taking the integral over $\Theta$ , we get that + +$$ +\begin{array}{l} \mathbb {E} \left[ \log \int_ {\Theta} \left(\frac {p \left(Y _ {t} | \theta , A _ {t}\right)}{p \left(Y _ {t} | \theta_ {0} , A _ {t}\right)}\right) ^ {\eta \alpha} \mathrm {d} Q _ {t} ^ {+} (\theta) \right] \leq - \gamma (1 - 2 \gamma) \mathbb {E} \left[ \sum_ {a \in \mathcal {A}} \pi_ {t} (A) \int_ {\Theta} \left(\langle \theta - \theta_ {0}, a \rangle\right) ^ {2} \mathrm {d} Q _ {t} ^ {+} (\theta) \right] \\ = - 2 \gamma (1 - 2 \gamma) \mathbb {E} \left[ \mathbf {I G} _ {t} \left(\pi_ {t}\right) \right]. \\ \end{array} +$$ + +Rearranging and combining the result with Proposition 1 yields the claim of the lemma. + +We now turn our focus to the estimation error loss and relate it to the surrogate regret through the following lemma, whose proof is a straightforward application of Lemma 23. + +Lemma 12. For any $t \geq 1, \beta > 1$ , if $\beta \lambda_{t-1} \leq 1$ , we have + +$$ +\mathbb {E} \left[ \frac {1}{\beta \lambda_ {t - 1}} \log \int_ {\Theta} \exp \left(\beta \lambda_ {t - 1} \Delta \left(A _ {t}, \theta\right)\right) \mathrm {d} Q _ {t} ^ {+} (\theta) \right] \leq \mathbb {E} \left[ 2 \widehat {\Delta} _ {t} \left(\pi_ {t}\right) \right]. \tag {21} +$$ + +# B.1.4 Separation of the two losses: proof of Lemma 5 + +We make use of the fact that the Fenchel dual of $\Phi$ can be explicitly written as $\Phi^{*}(L) = \log \int_{\Theta} \exp(L(\theta)) \, \mathrm{d}Q_{1}^{+}(\theta)$ . As a result, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\Phi^ {*} (- \eta (L _ {t} ^ {(1)} (\cdot) - L _ {t} ^ {(1)} (\theta_ {0})) - \lambda_ {t - 1} L _ {t} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} - \frac {\Phi^ {*} (- \eta (L _ {t - 1} ^ {(1)} (\cdot) - L _ {t - 1} ^ {(1)} (\theta_ {0})) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\cdot))}{\lambda_ {t - 1}} \right] \\ = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {1}{\lambda_ {t - 1}} \log \frac {\int_ {\Theta} \left(\frac {p \left(Y _ {t} \mid \theta , a _ {t}\right)}{p \left(Y _ {t} \mid \theta_ {0} , A _ {t}\right)}\right) ^ {\eta} \exp \left(\lambda_ {t - 1} \Delta (A _ {t} , \theta)\right) \exp \left(- \eta L _ {t - 1} ^ {(1)} (\theta) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\theta)\right) \mathrm {d} Q _ {1} ^ {+} (\theta)}{\int_ {\Theta} \exp \left(- \eta L _ {t - 1} ^ {(1)} (\theta) - \lambda_ {t - 1} L _ {t - 1} ^ {(2)} (\theta)\right) \mathrm {d} Q _ {1} ^ {+} (\theta)} \right] \\ = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {1}{\lambda_ {t - 1}} \log \int_ {\Theta} \left(\frac {p \left(Y _ {t} \mid \theta , A _ {t}\right)}{p \left(Y _ {t} \mid \theta_ {0} , A _ {t}\right)}\right) ^ {\eta} \exp \left(\lambda_ {t - 1} \Delta (A _ {t}, \theta)\right) \mathrm {d} Q _ {t} ^ {+} (\theta) \right] \\ \leq \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {1}{\alpha \lambda_ {t - 1}} \log \int_ {\Theta} \left(\frac {p (Y _ {t} | \theta , A _ {t})}{p (Y _ {t} | \theta_ {0} , A _ {t})}\right) ^ {\eta \alpha} + \frac {1}{\beta \lambda_ {t - 1}} \log \int_ {\Theta} \exp (\beta \lambda_ {t - 1} \Delta (A _ {t}, \theta)) d Q _ {t} ^ {+} (\theta) \right], \\ \end{array} +$$ + +where the last equality is by definition of the optimistic posterior and the last inequality follows from using Hölder's inequality with the two real numbers $\alpha, \beta > 1$ that satisfy $\frac{1}{\alpha} + \frac{1}{\beta} = 1$ . Combining Lemma 11 and Lemma 12 with the choice $\alpha = \beta = 2$ , the fact that $\eta = \frac{1}{4}$ and the last inequality yields the claim of the lemma. + +# B.2 Choice of the prior and comparator distribution: proof of Lemma 3 + +In order to construct the prior $Q_{1}^{+}$ and the comparator $P$ for the regret analysis, we need to take into account two criteria: that $\mathcal{D}_{\mathrm{KL}}\left(P\big||Q_1^+\right)$ be controlled and that $|\langle P,L\rangle -L(\theta_0)|$ be small. The comparator will be a function of the unknown parameter $\theta_0$ , and thus we denote it by $P_{\theta_0}$ . + +As for the prior, it should take into account the sparsity level of the unknown $\theta_0$ , but should have no access to its support. We first design a distribution $\Pi$ over the set of all subsets of $[d] = \{1,\dots ,d\}$ which have cardinality at most $s$ . We choose the distribution such that: a) the probability assigned to each subset depends only on its cardinality; b) the probability assigned to the set of all subsets of size $k$ is proportional to $2^{-k}$ , where $1\leq k\leq s$ . In other words, we prefer smaller subsets and have no preference over which indices in $[d]$ are included. The distribution that satisfies these requirements is + +$$ +\Pi (S) = \frac {2 ^ {- | S |}}{\binom {d} {| S |} \sum_ {k = 1} ^ {s} 2 ^ {- k}}. \tag {22} +$$ + +For $S = \emptyset$ , we set $\Pi(S) = 0$ . Doing so only complicates matters if the support of $\theta_0$ is empty (i.e., $\theta_0 = 0$ ). However, in this case, the reward function is 0 everywhere, which means any algorithm would have 0 regret. We therefore continue under the assumption that $\theta_0 \neq 0$ . The most important property of this distribution, which we will use later, is that for any subset $S$ of cardinality $s$ , $\log(1 / \Pi(S)) \leq s \log(2ed / s)$ . For each subset $S$ , we define $Q_S$ to be the uniform distribution on $\Theta_S$ . The prior is defined to be + +$$ +Q _ {1} ^ {+} = \sum_ {S \subset [ d ]: | S | \leq s} \Pi (S) Q _ {S}. +$$ + +As for the comparator distribution $P_{\theta_0}$ , we would ideally like to take a Dirac measure on $\theta_0$ , but this would make the KL divergence appearing in the bound blow up. Thus, we pick a comparator $P_{\theta_0}$ which dilutes its mass around $\theta_0$ . For any $\bar{\theta} \in \Theta$ , with support $\bar{S}$ , and any $\epsilon \in (0,1)$ , we define the set $(1 - \epsilon)\bar{\theta} + \epsilon \Theta_{\bar{S}} = \{(1 - \epsilon)\bar{\theta} + \epsilon \theta' : \theta' \in \Theta_{\bar{S}}\} \subset \Theta_{\bar{S}}$ . We choose $P_{\theta_0}$ to be the uniform distribution on $(1 - \epsilon)\theta_0 + \epsilon \Theta_{S_0}$ , where $S_0$ is the support of $\theta_0$ . We now bound $\Phi(P_{\theta_0}) = \mathcal{D}_{\mathrm{KL}}(P_{\theta_0} \| Q_1^+)$ for this choice of $P_{\theta_0}$ in the following lemma, from which the claim of Lemma 3 then directly follows. + +Lemma 13. For any $\bar{\theta} \in \Theta$ , let $\bar{S}$ denote its support, and let $|\bar{S}| = s$ . If, for $\epsilon \in (0,1)$ , $P_{\bar{\theta}} = \mathcal{U}((1 - \epsilon)\bar{\theta} + \epsilon \Theta_{\bar{S}})$ and $Q_1^+ = \sum_{S \subset [d]:|S| = s} \Pi(S) Q_S$ , then $\mathcal{D}_{KL}(P_{\bar{\theta}} \| Q_1^+) \leq s \log \frac{2ed}{\epsilon s}$ . + +Proof. We notice that $(1 - \epsilon)\bar{\theta} +\epsilon \Theta_{\bar{S}}$ is an $s$ -dimensional $\ell_1$ -ball of radius $\epsilon$ , which is contained in $\Theta_{\bar{S}}$ . Therefore, on the support of $P_{\bar{\theta}}$ $\frac{\mathrm{d}P_{\bar{\theta}}}{\mathrm{d}Q_{\bar{S}}}$ is equal to the ratio of the volumes of a unit $\ell_1$ -ball and an $\ell_1$ -ball of radius $\epsilon$ , which is $(1 / \epsilon)^{s}$ . Thus, + +$$ +\mathcal {D} _ {\mathrm {K L}} \left(P _ {\bar {\theta}} \| Q _ {1} ^ {+}\right) = \int \log \frac {\mathrm {d} P _ {\bar {\theta}}}{\sum_ {S} \Pi (S) \mathrm {d} Q _ {S}} \mathrm {d} P _ {\bar {\theta}} \leq \int \log \frac {\mathrm {d} P _ {\bar {\theta}}}{\Pi (\bar {S}) \mathrm {d} Q _ {\bar {S}}} \mathrm {d} P _ {\bar {\theta}} \leq s \log \frac {1}{\epsilon} + \log \frac {1}{\Pi (\bar {S})}. +$$ + +Using the definition of $\Pi$ and the bound $\binom{d}{s} \leq \left(\frac{ed}{s}\right)^s$ on the binomial coefficient, we have + +$$ +\log \frac {1}{\Pi (\bar {S})} = \log \binom {d} {s} + s \log (2) + \log \sum_ {k = 1} ^ {s} 2 ^ {- k} \leq s \log \frac {2 e d}{s}. +$$ + +Combining everything, we obtain + +$$ +\mathcal {D} _ {\mathrm {K L}} \left(P _ {\theta} \| Q _ {1} ^ {+}\right) \leq s \log \frac {1}{\epsilon} + s \log \frac {2 e d}{s} = s \log \frac {2 e d}{\epsilon s}, \tag {23} +$$ + +as advertised. + +![](images/96f4717d80eda17d267c444d3bc4e647f8a66f37cb28b9b86306ead78588bef8.jpg) + +# C Proof of the history-dependent part of Theorem 1 + +Following the original analysis, we arrive again at (9). + +$$ +\sum_ {t = 1} ^ {T} \Delta (A _ {t}, P) \leq \frac {\Phi (P)}{\lambda_ {T}} + \frac {\Phi^ {*} (- \eta L _ {T} ^ {(1)} (\cdot) + \eta L _ {T} ^ {(1)} (\theta_ {T}) - \lambda_ {T} L _ {T} ^ {(2)} (\cdot))}{\lambda_ {T}} + \frac {\eta}{\lambda_ {T}} (\langle P, L _ {T} ^ {(1)} \rangle - L _ {T} ^ {(1)} (\theta_ {T})), +$$ + +where $P \in \Delta(\Theta)$ can be any comparator distribution. Lemma 3 is still valid and we can choose the same prior as before. We can still choose a comparator distribution supported on an $\epsilon$ -ball around $\theta_0$ . However, because $\lambda_t$ depends on the history, we can no longer upper bound $\mathbb{E}\left[\frac{|P \cdot L_T^{(1)} - L_T^{(1)}(\theta_0)|}{\lambda_{T-1}}\right]$ by $\mathbb{E}\left[\frac{2T\epsilon}{\lambda_T}\right]$ . Using Lemma 21, we still have that $L_T^{(2)}(\cdot)$ is $2T$ -Lipschitz and $\mathbb{E}\left[L_T^{(1)}(\cdot)\right]$ is $2T$ -Lipschitz. Hence, + +$$ +\mathbb {E} \left[ \frac {\left| P \cdot L _ {T} ^ {(1)} - L _ {T} ^ {(1)} \left(\theta_ {0}\right) \right|}{\lambda_ {T - 1}} \right] \leq 2 T \epsilon C _ {2, T}, \quad \text {a n d} \quad \sum_ {t = 1} ^ {T} \left| \Delta \left(\theta_ {0}, a _ {t}\right) - \Delta \left(P, a _ {t}\right) \right| \leq 2 T \epsilon , +$$ + +where we used $C_{2,T}$ , a deterministic upper bound on $\frac{1}{\lambda_{T - 1}}$ . Exactly the same telescoping of $\Phi^*$ can be done, however because the learning rate is history-dependent, the difference between the negative log likelihood of $\theta_0$ and $\theta_t$ must be treated with more care. We have the following lemma + +Lemma 14. Let $C_{1,T}$ be a deterministic upper bound on $\left(\frac{1}{\lambda_{t+1}} - \frac{1}{\lambda_t}\right)$ that holds for all $t < T$ , then + +$$ +\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\eta}{\lambda_ {t - 1}} \left(L _ {t} ^ {(1)} \left(\theta_ {t}\right) - L _ {t} ^ {(1)} \left(\theta_ {0}\right) + L _ {t - 1} ^ {(1)} \left(\theta_ {0}\right) - L _ {t - 1} ^ {(1)} \left(\theta_ {t - 1}\right)\right) + \frac {\eta}{\lambda_ {T}} \left(L _ {T} ^ {(1)} \left(\theta_ {0}\right) - L _ {T} ^ {(1)} \left(\theta_ {T}\right)\right) \right] \\ \leq \mathbb {E} \left[ \frac {\eta (1 5 + 3 s \log \frac {2 e ^ {2} d T ^ {2} C _ {1 , T} ^ {2}}{s})}{2 \lambda_ {T - 1}} \right]. \tag {24} \\ \end{array} +$$ + +A complete proof of that result can be found in appendix E.2.1. + +Finally, as was the case in the history independent version the telescoping sum can be handled by looking at the explicit formula for $\Phi^{*}$ and Lemma 5 still holds. Applying Lemma 5 and setting $\epsilon = \frac{1}{TC_{2,T}}$ yields the claim of the theorem. + +# D Proof of Theorem 3 + +We turn our attention to data-dependent bounds (that will scale with the cumulative information ratio rather than the time horizon). Combining the second part of Theorem 1 with Lemma 6 and the choice $\lambda = \frac{64}{3}\lambda_{t - 1}$ , we have that for any non-increasing sequence of learning rates $\lambda_t$ satisfying $\lambda_0\leq \frac{1}{2}$ , the following holds + +$$ +R _ {T} \leq \mathbb {E} \left[ \frac {C _ {T}}{\lambda_ {T - 1}} + \min \left(\sum_ {t = 1} ^ {T} \frac {3 2}{3} \lambda_ {t - 1} \overline {{\mathbf {I}}} \mathbf {R} _ {t} ^ {(2)} (\pi_ {t}), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} \overline {{\mathbf {I}}} \overline {{\mathbf {R}}} _ {t} ^ {(3)} (\pi_ {t})}\right) \right], \tag {25} +$$ + +where $C_T = 2 + s \log \frac{4e^3 d^2 T^3 C_{1,T}^2 C_{2,T}^2}{s^2}$ and $C_{1,T}$ , respectively $C_{2,T}$ are deterministic upper bounds on $\frac{1}{\lambda_t} - \frac{1}{\lambda_{t-1}}$ , respectively $\frac{1}{\lambda_{T-1}}$ . + +We let $\lambda_t^{(2)} = \sqrt{\frac{s}{2d + \sum_{s=1}^t \overline{\mathbb{R}}_s^{(2)}(\pi_s)}}$ and $\lambda_t^{(3)} = \left(\frac{s}{\sqrt{\frac{3\sqrt{6}s}{\sqrt{C_{\min}}} + \sum_{s=1}^t \sqrt{\overline{\mathbb{R}}_s^{(3)}(\pi_s)}}}\right)^{\frac{2}{3}}$ , and verify that $\lambda_t = \max(\lambda_t^{(2)}, \lambda_t^{(3)})$ is decreasing and always smaller than $\frac{1}{2}$ . We also verify that $C_{1,T} = C_{2,T} = \sqrt{\frac{dT}{s}}$ are valid upper bounds. As a result, we have the following upper bound + +$$ +C _ {T} = 2 + s \log \frac {4 e ^ {3} d ^ {2} T ^ {3} C _ {1 , T} ^ {2} C _ {2 , T}}{s ^ {2}} \leq 2 + s \log 4 e ^ {3} T ^ {4. 5} \left(\frac {d}{s}\right) ^ {3. 5} \leq 2 + 5 s \log \left(\frac {e d T}{s}\right). \tag {26} +$$ + +We now focus on bounding the sum containing the information ratios. Applying Lemma 7, we obtain that for all $t \geq 1$ , $\overline{\mathbb{R}}_t^{(2)}(\pi_t) \leq 2d$ and for any $T \geq 1$ + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \lambda_ {t - 1} ^ {(2)} \overline {{\mathbb {R}}} _ {t} ^ {(2)} (\pi) = \sqrt {s} \sum_ {t = 1} ^ {T} \frac {\overline {{\mathbb {R}}} _ {t} ^ {(2)} (\pi_ {t})}{\sqrt {2 d + \sum_ {s = 1} ^ {t - 1} \overline {{\mathbb {R}}} _ {s} ^ {(2)} (\pi_ {s})}} \\ \leq \sqrt {s} \sum_ {t = 1} ^ {T} \frac {\overline {{\mathrm {I R}}} _ {t} ^ {(2)} (\pi_ {t})}{\sqrt {\sum_ {s = 1} ^ {t} \overline {{\mathrm {I R}}} _ {s} ^ {(2)} (\pi_ {s})}} \\ \leq 2 \sqrt {s \sum_ {t = 1} ^ {T} \overline {{\mathbb {R}}} _ {t} ^ {(2)} (\pi_ {t})} \\ \leq 2 \sqrt {s \left(2 d + \sum_ {t = 1} ^ {T - 1} \overline {{\mathbf {I R}}} _ {t} ^ {(2)} (\pi_ {t})\right)}, \\ \end{array} +$$ + +where we applied Lemma 19 with the function $f(x) = \frac{1}{\sqrt{x}}$ and $a_{i} = \overline{\mathrm{IR}}_{i}^{(2)}(\pi_{i})$ to get the second inequality. This can be seen as a generalization of the usual $\sum_{t=1}^{T} \frac{1}{\sqrt{t}} \leq 2\sqrt{T}$ inequality. We now define $R_{T}^{(2)} = \sqrt{s\left(2d + \sum_{t=1}^{T-1} \overline{\mathrm{IR}}_{t}^{(2)}(\pi_{t})\right)}$ , the data-dependent regret rate associated to the 2-surrogate-information ratio. + +We now turn our attention to the 3-information ratio. Applying Lemma 7 we obtain that for all $t \geq 1$ , $\overline{\mathrm{IR}}_t^{(3)}(\pi_t) \leq 54\frac{s}{C_{\min}} \leq 54\frac{s^2}{C_{\min}}$ and for any $T \geq 1$ + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \sqrt {\lambda_ {t - 1} ^ {(3)} \overline {{\mathrm {I R}}} _ {t} ^ {(3)} (\pi_ {t})} = s ^ {\frac {1}{3}} \sum_ {t = 1} ^ {T} \frac {\sqrt {\overline {{\mathrm {I R}}} _ {t} ^ {(3)} (\pi_ {t})}}{\left(\frac {3 \sqrt {6} s}{\sqrt {C _ {m i n}}} + \sum_ {s = 1} ^ {t - 1} \sqrt {\overline {{\mathrm {I R}}} _ {s} ^ {(3)} (\pi_ {s})}\right) ^ {\frac {1}{3}}} \\ \leq s ^ {\frac {1}{3}} \sum_ {t = 1} ^ {T} \frac {\sqrt {\overline {{\mathbb {R}}} _ {t} ^ {(3)} (\pi_ {t})}}{\left(\sum_ {s = 1} ^ {t} \sqrt {\overline {{\mathbb {R}}} _ {s} ^ {(3)} (\pi_ {s})}\right) ^ {\frac {1}{3}}} \\ \leq \frac {3}{2} s ^ {\frac {1}{3}} \left(\sum_ {t = 1} ^ {T} \sqrt {\overline {{\mathbb {R}}} _ {t} ^ {(3)} (\pi_ {t})}\right) ^ {\frac {2}{3}} \\ \leq \frac {3}{2} s ^ {\frac {1}{3}} \left(\frac {3 \sqrt {6} s}{\sqrt {C _ {m i n}}} + \sum_ {t = 1} ^ {T - 1} \sqrt {\overline {{\mathbb {R}}} _ {t} ^ {(3)} (\pi_ {t})}\right), \\ \end{array} +$$ + +where we applied Lemma 19 with the function $f(x) = \frac{1}{x^{\frac{1}{3}}}$ and $a_{i} = \sqrt{\overline{\mathrm{IR}}_{i}^{(3)}(\pi_{i})}$ to get the second inequality. This can be seen as a generalization of the usual $\sum_{t=1}^{T} \frac{1}{t^{\frac{1}{3}}} \leq \frac{3}{2} T^{\frac{2}{3}}$ . We now define $R_{T}^{(3)} = s^{\frac{1}{3}}\left(\frac{3\sqrt{6}s}{\sqrt{C_{min}}} + \sum_{t=1}^{T-1} \sqrt{\overline{\mathrm{IR}}_{t}^{(3)}(\pi_{t})}\right)^{\frac{2}{3}}$ , the data-dependent regret rate associated to the 3-surrogate-information ratio. We now consider the last time that the learning rates $\lambda_{t}^{(3)}$ and $\lambda_{t}^{(2)}$ have been used. More specifically, we denote $T_{2} = \max \{t \leq T, \lambda_{t-1}^{(2)} \geq \lambda_{t-1}^{(3)}\}$ , and $T_{3} = \max \{t \leq T, \lambda_{t-1}^{(3)} \geq \lambda_{t-1}^{(2)}\}$ . Coming back to the bound of Equation 25 and using the definition $\lambda_{t} = \max (\lambda_{t}^{(2)}, \lambda_{t}^{(3)})$ , the following bound holds + +$R_{T}$ + +$$ +\begin{array}{l} \leq \mathbb {E} \left[ \frac {C _ {T}}{\lambda_ {T - 1}} + \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \lambda_ {t - 1} \overline {{\mathrm {I R}}} _ {t} ^ {(2)} (\pi_ {t}), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} \overline {{\mathrm {I R}}} _ {t} ^ {(3)} (\pi_ {t})}\right) \right] \\ \leq \mathbb {E} \left[ C _ {T} \min \left(\frac {1}{\lambda_ {T - 1} ^ {(2)}}, \frac {1}{\lambda_ {T - 1} ^ {(3)}}\right) + \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \max (\lambda_ {t - 1} ^ {(2)}, \lambda_ {t - 1} ^ {(3)}) \overline {{\mathbf {I R}}} _ {t} ^ {(2)} (\pi_ {t}), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \max (\lambda_ {t - 1} ^ {(2)} , \lambda_ {t - 1} ^ {(3)}) \overline {{\mathbf {I R}}} _ {t} ^ {(3)} (\pi_ {t})}\right) \right]. \\ \end{array} +$$ + +We can now separate the sum obtained at the last line based on which learning rate was used at time $t$ . + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \max \left(\lambda_ {t - 1} ^ {(2)}, \lambda_ {t - 1} ^ {(3)}\right) \overline {{\mathbb {R}}} _ {t} ^ {(2)} \left(\pi_ {t}\right), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \max \left(\lambda_ {t - 1} ^ {(2)} , \lambda_ {t - 1} ^ {(3)}\right) \overline {{\mathbb {R}}} _ {t} ^ {(3)} \left(\pi_ {t}\right)}\right) \\ \leq \sum_ {\lambda_ {t - 1} ^ {(2)} \geq \lambda_ {t - 1} ^ {(3)}} \frac {3 2}{3} \lambda_ {t - 1} ^ {(2)} \overline {{\mathrm {I R}}} _ {t} ^ {(2)} (\pi_ {t}) + \sum_ {\lambda_ {t - 1} ^ {(3)} \geq \lambda_ {t - 1} ^ {(2)}} \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} ^ {(3)} \overline {{\mathrm {I R}}} _ {t} ^ {(3)} (\pi_ {t})} \\ \leq \sum_ {t = 1} ^ {T _ {2}} \frac {3 2}{3} \lambda_ {t - 1} ^ {(2)} \overline {{\mathrm {I R}}} _ {t} ^ {(2)} (\pi_ {t}) + \sum_ {t = 1} ^ {T _ {3}} \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} ^ {(3)} \overline {{\mathrm {I R}}} _ {t} ^ {(3)} (\pi_ {t})}. \\ \end{array} +$$ + +We further bound $\sum_{t=1}^{T_2} \frac{32}{3} \lambda_{t-1}^{(2)} \overline{\mathbb{R}}_t^{(2)}(\pi_t) \leq \frac{64}{3} R_{T_2}^{(2)}$ and $\sum_{t=1}^{T_3} \frac{16}{3} c_3^* \sqrt{3 \lambda_{t-1}^{(3)} \overline{\mathbb{R}}_t^{(3)}(\pi_t)} \leq \frac{16}{3} R_{T_3}^{(3)}$ . (Using the explicit value $c_3^* = \frac{2}{3^{\frac{3}{2}}}$ ). + +The crucial observation is that which of $\lambda_T^{(3)}$ or $\lambda_T^{(2)}$ is bigger will determine whether $R_T^{(2)}$ or $R_T^{(3)}$ is the term of leading order (up to some constants). More specifically, Let $T$ be such that $\lambda_{T - 1}^{(2)}\geq \lambda_{T - 1}^{(3)}$ + +which means that $\sqrt{\frac{s}{2d + \sum_{t=1}^{T-1}\overline{\mathrm{IR}}_t^{(2)}(\pi_t)}} \geq \left(\frac{s}{\frac{3\sqrt{6}s}{\sqrt{C_{\min}}} + \sum_{t=1}^{T-1}\sqrt{\overline{\mathrm{IR}}_t^{(3)}(\pi_t)}}\right)^{\frac{2}{3}}$ . Rearranging, this implies that $\sqrt{s\big(2d + \sum_{s=1}^{T-1}\overline{\mathrm{IR}}_t^{(2)}(\pi_t)\big)} \leq s^{\frac{2}{3}}\left(\frac{3\sqrt{6}s}{\sqrt{C_{\min}}} + \sum_{t=1}^{T-1}\sqrt{\overline{\mathrm{IR}}_t^{(3)}(\pi_t)}\right)^{\frac{2}{3}}$ , which means that $R_T^{(2)} \leq R_T^{(3)}$ . Following the exact same steps, we also have that $\lambda_{T-1}^{(3)} \geq \lambda_{T-1}^{(2)}$ implies that $R_T^{(3)} \leq R_T^{(2)}$ . We apply this to the time $T_2$ in which $\lambda_{T_2-1}^{(2)} \geq \lambda_{T_2-1}^{(3)}$ by definition. We have that $R_{T_2}^{(2)} \leq R_{T_2}^{(3)}$ and putting this together with the previous bound, we have + +$$ +\begin{array}{l} R _ {T} \leq \mathbb {E} \left[ \frac {C _ {T}}{\lambda_ {T - 1} ^ {(3)}} + \frac {6 4}{3} R _ {T _ {2}} ^ {(2)} + \frac {1 6}{3} R _ {T _ {3}} ^ {(3)} \right] \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{s} R _ {T} ^ {(3)} + \frac {6 4}{3} R _ {T _ {2}} ^ {(2)} + \frac {1 6}{3} R _ {T _ {3}} ^ {(3)} \right] \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{s} R _ {T} ^ {(3)} + \frac {6 4}{3} R _ {T _ {2}} ^ {(3)} + \frac {1 6}{3} R _ {T _ {3}} ^ {(3)} \right] \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{s} R _ {T} ^ {(3)} + \frac {6 4}{3} R _ {T} ^ {(3)} + \frac {1 6}{3} R _ {T} ^ {(3)} \right] \\ \leq \mathbb {E} \left[ \left(\frac {C _ {T}}{s} + \frac {8 0}{3}\right) R _ {T} ^ {(3)} \right], \\ \end{array} +$$ + +where we use the fact that $T \mapsto R_T^{(2)}$ and $T \mapsto R_T^{(3)}$ are non-decreasing and $T_2 \leq T, T_3 \leq T$ . + +Similarly by definition of $T_3$ , we have that $\lambda_{T_3 - 1}^{(3)} \geq \lambda_{T_3 - 1}^{(2)}$ and we can conclude that $R_{T_3}^{(3)} \leq R_{T_3}^{(2)}$ . Putting this together, with the previous bound, we have + +$$ +\begin{array}{l} R _ {T} \leq \mathbb {E} \left[ \frac {C _ {T}}{\lambda_ {T - 1} ^ {(2)}} + \frac {6 4}{3} R _ {T _ {2}} ^ {(2)} + \frac {1 6}{3} R _ {T _ {3}} ^ {(3)} \right] \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{s} R _ {T} ^ {(2)} + \frac {6 4}{3} R _ {T _ {2}} ^ {(2)} + \frac {1 6}{3} R _ {T _ {3}} ^ {(3)} \right] \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{s} R _ {T} ^ {(2)} + \frac {6 4}{3} R _ {T _ {2}} ^ {(2)} + \frac {1 6}{3} R _ {T _ {3}} ^ {(2)} \right] \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{s} R _ {T} ^ {(2)} + \frac {6 4}{3} R _ {T} ^ {(2)} + \frac {1 6}{3} R _ {T} ^ {(2)} \right] \\ \leq \mathbb {E} \left[ (\frac {C _ {T}}{s} + \frac {8 0}{3}) R _ {T} ^ {(2)} \right], \\ \end{array} +$$ + +where we use the fact that $T \to R_T^{(2)}$ and $T \to R_T^{(3)}$ are non-decreasing and $T_2 \leq T, T_3 \leq T$ . Putting both of those bounds together with Equation 26 yields the claim of the Theorem. + +# E Maximum likelihood estimation + +The focus of this section is to bound the difference between the log-likelihoods associated with the true parameter and the maximum likelihood estimator (MLE). We start by establishing an upper bound that holds in expectation which suffices to handle history-independent learning rates. Then, we move on to high-probability bounds that will allow us to deal with data-dependent learning rates. + +# E.1 Bound in expectation + +We start with the case in which the maximum likelihood estimator is computed on a finite subset of the parameter space $\Theta$ . + +Lemma 15. Let $t \geq 1$ , and $\Theta'$ be a finite subset of $\Theta$ , we define the MLE over $\Theta'$ as + +$$ +\theta_{MLE,t}(\Theta^{\prime}) = \operatorname *{arg min}_{\theta \in \Theta^{\prime}}L_{t}^{(1)}(\theta). +$$ + +Then, + +$$ +\mathbb {E} \left[ L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {M L E, t} \left(\Theta^ {\prime}\right)\right) \right] \leq \log | \Theta^ {\prime} | \tag {27} +$$ + +Proof. By the concavity of the logarithm and Jensen's inequality, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ L _ {t} ^ {(1)} (\theta_ {0}) - L _ {t} ^ {(1)} (\theta_ {\mathrm {M L E}, t} (\Theta^ {\prime})) \right] \leq \log \mathbb {E} \left[ \prod_ {s = 1} ^ {t} \frac {p (Y _ {s} | \theta_ {\mathrm {M L E} , t} (\Theta^ {\prime}) , A _ {s})}{p (Y _ {s} | \theta_ {0} , A _ {s})} \right] \\ = \log \mathbb {E} \left[ \max _ {\theta \in \Theta^ {\prime}} \prod_ {s = 1} ^ {t} \frac {p \left(Y _ {s} \mid \theta , A _ {s}\right)}{p \left(Y _ {s} \mid \theta_ {0} , A _ {s}\right)} \right] \leq \log \mathbb {E} \left[ \sum_ {\theta \in \Theta^ {\prime}} \prod_ {s = 1} ^ {t} \frac {p \left(Y _ {s} \mid \theta , A _ {s}\right)}{p \left(Y _ {s} \mid \theta_ {0} , A _ {s}\right)} \right] \\ = \log \sum_ {\theta \in \Theta^ {\prime}} \mathbb {E} \left[ \prod_ {s = 1} ^ {t} \frac {p \left(Y _ {s} \mid \theta , A _ {s}\right)}{p \left(Y _ {s} \mid \theta_ {0} , A _ {s}\right)} \right] \\ \end{array} +$$ + +By Lemma 25, we have that $\exp \left(L_t^{(1)}(\theta_0) - L_t^{(1)}(\theta)\right) = \prod_{s=1}^t \frac{p(Y_s|\theta, A_s)}{p(Y_s|\theta_0, A_s)}$ is a non-negative supermartingale with respect to the filtration $\mathcal{F}_t' = \sigma(\mathcal{F}_{t-1}, A_t)$ . That implies that each term in the sum is upper bounded by 1. Hence, + +$$ +\mathbb {E} \left[ L _ {t} ^ {(1)} (\theta_ {0}) - L _ {t} ^ {(1)} (\theta_ {\mathrm {M L E}, t} (\Theta^ {\prime})) \right] \leq \log \sum_ {\theta \in \Theta^ {\prime}} 1 = \log | \Theta^ {\prime} |, +$$ + +which proves the claim. + +![](images/7793a33b09b61eccc73e0e53954c0b72fc09fc235ef3429ab868a8643a0f5bc9.jpg) + +To extend the previous bound to the full parameter space, we use a covering argument. A subset $\Theta' \subset \Theta$ is said to be a valid $\rho$ -covering of $\Theta$ with respect to the $\ell_1$ norm if for every $\theta \in \Theta$ , there exists a $\theta' \in \Theta'$ such that $\| \theta - \theta' \|_1 \leq \rho$ . We denote by $\mathcal{N}(\Theta, \| \cdot \|_1, \rho)$ the smallest possible cardinality of a valid $\rho$ covering. We have the following bound on this quantity. + +Lemma 16. For every $\rho >0$ + +$$ +\log \mathcal {N} (\Theta , \| \cdot \| _ {1}, \rho) \leq \log \binom {d} {s} (1 + \frac {2}{\rho}) ^ {s} \leq s \log \frac {e d (1 + 2 / \rho)}{s}. +$$ + +Proof. For each subset $S \subset [d]$ of cardinality $|S| = s$ , there is a surjective isometric embedding from $(\Theta_S, \| \cdot \|_1)$ to $(\mathbb{B}_1^s(1), \| \cdot \|_1)$ . In particular, to embed $\theta \in \Theta_S$ into $\mathbb{B}_1^s(1)$ , one can simply remove all the components of $\theta$ corresponding to indices not in $S$ . Therefore, for every $\rho > 0$ , $\mathcal{N}(\Theta_S, \| \cdot \|_1, \rho) \leq \mathcal{N}(\mathbb{B}_1^s(1), \| \cdot \|_1, \rho)$ . Moreover, via a standard argument, we have $\mathcal{N}(\mathbb{B}_1^s(1), \| \cdot \|_1, \rho) \leq (1 + \frac{2}{\rho})^s$ (see, e.g., Lemma 5.7 in Wainwright, 2019). Now, let $\Theta_{S,\rho}$ denote any minimal $\rho$ -covering of $\Theta_S$ and notice that for an arbitrary $\theta \in \Theta$ with support $S$ , there exists a subset $\tilde{S}$ such that $S \subseteq \tilde{S}$ and $|\tilde{S}| = s$ . Therefore, there exists $\tilde{\theta} \in \Theta_{\tilde{S},\rho}$ such that $\| \theta - \tilde{\theta} \|_1 \leq \rho$ . Hence, $\cup_{S \subset [d]:|S| = s} \Theta_{S,\rho}$ forms a valid $\rho$ -covering of $\Theta$ and its cardinality is bounded by + +$$ +\mathcal{N}(\Theta ,\| \cdot \|_{1},\rho)\leq \big|\cup_{S\subset [d]:|S| = s}\Theta_{S,\rho}\big|\leq \sum_{S\subset [d]:|S| = s}\left(1 + \frac{2}{\rho}\right)^{s} = \binom {d}{s}\left(1 + \frac{2}{\rho}\right)^{s}. +$$ + +and we conclude by the elementary inequality $\binom{d}{s} \leq \left(\frac{de}{s}\right)^s$ . + +![](images/a62b7d44f8bb1966f70cff0ddd9330f1f380329d2c3e24340027b8ea6ef9d662.jpg) + +# E.1.1 Proof of Lemma 10 + +We bound the difference between the log-likelihood of the true parameter and that of the maximum likelihood estimator on the full parameter space. To this end, let $\rho > 0$ and $\Theta'$ be a minimal valid $\rho$ -cover of $\Theta$ as is defined in Lemma 16, and $\theta' \in \Theta'$ be such that $\| \theta' - \theta_t \| \leq \rho$ , which exists by + +definition of a $\rho$ -covering. Then, + +$$ +\begin{array}{l} \mathbb {E} \left[ L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right) \right] = \mathbb {E} \left[ L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {\mathrm {M L E}, t} \left(\Theta^ {\prime}\right)\right) \right] \\ + \mathbb {E} \left[ L _ {t} ^ {(1)} \left(\theta_ {\mathrm {M L E}, t} \left(\Theta^ {\prime}\right)\right) - L _ {t} ^ {(1)} \left(\theta^ {\prime}\right) \right] \\ + \mathbb {E} \left[ L _ {t} ^ {(1)} \left(\theta^ {\prime}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right) \right] \\ \leq \log (\mathcal {N} (\Theta , \| \cdot \| _ {1}, \rho)) + 0 + 2 \rho t, \\ \end{array} +$$ + +where the first term is bounded by Lemma 15, the second term is non-positive by definition of the maximum likelihood estimator because $\theta' \in \Theta'$ and the third term is bounded because the mapping $\theta \mapsto \mathbb{E}\left[L_t^{(1)}(\theta)\right]$ is $2t$ -Lipschitz with respect to the 1-norm by Lemma 21. Finally applying Lemma 16 and setting $\rho = \frac{2}{t}$ yields the desired bound. + +# E.2 High-probability bounds + +We begin with the case where the maximum likelihood estimator is computed over a finite subset of the parameter space $\Theta$ and provide a corresponding high-probability bound. + +Lemma 17. Let $\Theta'$ be a finite subset of $\Theta$ , we define $\theta_{MLE,t}(\Theta') = \arg \min_{\theta \in \Theta'} L_t^{(1)}(\theta)$ . Then + +$$ +\mathbb {P} \left[ \exists t \geq 1, L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {M L E, t} \left(\Theta^ {\prime}\right)\right) \geq \log \frac {\left| \Theta^ {\prime} \right|}{\delta} \right] \leq \delta . \tag {28} +$$ + +Proof. Fix $\theta \in \Theta'$ . By Lemma 25, we have that $\exp\left(L_t^{(1)}(\theta_0) - L_t^{(1)}(\theta)\right) = \prod_{s=1}^t \frac{p(Y_s|\theta,A_s)}{p(Y_s|\theta_0,A_s)}$ is a non-negative supermartingale with respect to the filtration $(\mathcal{F}_t')_t$ , where $\mathcal{F}_t' = \sigma(\mathcal{F}_t, A_{t+1})$ , allowing us to invoke Ville's inequality to get the following guarantee: + +$$ +\mathbb {P} \left[ \exists t \geq 1, \exp \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} (\theta)\right) \geq \frac {1}{\delta} \right] \leq \delta . +$$ + +Taking the logarithm and a union bound on $\Theta^{\prime}$ yields the desired result. + +![](images/32009fba646700be39ab6a8a780635cc33bcb8d47e66bdc23b87dbdce09737cd.jpg) + +We now provide a bound on the expected product of a bounded random variable with the difference in log-likelihood between the true parameter and the maximum likelihood estimator. + +Lemma 18. Let $B \in \mathbb{R}$ and $X$ be a random variable satisfying $0 \leq X \leq B$ almost surely. Then for any $t \geq 1$ , + +$$ +\begin{array}{l} \mathbb {E} \left[ X \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right)\right) \right] \leq \inf _ {\delta , \rho > 0} \left\{\mathbb {E} \left[ X s \log \frac {e d \left(1 + \frac {2}{\rho}\right)}{s \delta^ {\frac {1}{s}}} \right] + \frac {5}{2} B \rho t + B \delta s \log \frac {e ^ {1 + \frac {1}{s}} d \left(1 + \frac {2}{\rho}\right)}{s \delta^ {\frac {1}{s}}} \right\} \\ \leq 5 + s \log \frac {2 e ^ {2} d T ^ {2} B ^ {2}}{s} \mathbb {E} \left[ X + \frac {1}{T} \right]. \tag {29} \\ \end{array} +$$ + +Proof. Let $\delta, \rho > 0$ and $\Theta'$ be a minimal valid $\rho$ -cover of $\Theta$ as defined in Lemma 16, $N = |\Theta'|$ , let $\theta' = \theta_{\mathrm{MLE},t}(\Theta')$ and let $\bar{\theta} \in \Theta'$ be such that $\left\| \bar{\theta} - \theta_t \right\| \leq \rho$ , which exists by definition of a valid $\rho$ -cover. We have the following decomposition: + +$$ +\begin{array}{l} \mathbb {E} \left[ X \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right)\right) \right] \leq \mathbb {E} \left[ X \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta^ {\prime}\right)\right) \mathbf {1} _ {\left\{L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta^ {\prime}\right) \leq \log \frac {N}{\delta} \right\}} \right] \\ + B \mathbb {E} \left[ \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta^ {\prime}\right)\right) \mathbf {1} _ {\left\{L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta^ {\prime}\right) > \log \frac {N}{\delta} \right\}} \right] \\ + B \mathbb {E} \left[ \left(L _ {t} ^ {(1)} (\bar {\theta}) - L _ {t} ^ {(1)} \left(\theta_ {t}\right)\right) \right] + B \mathbb {E} \left[ \left(L _ {t} ^ {(1)} \left(\theta^ {\prime}\right) - L _ {t} ^ {(1)} (\bar {\theta})\right) \right]. \\ \end{array} +$$ + +The first term is upper bounded by $\mathbb{E}\left[X\log \frac{N}{\delta}\right]$ , the third term is upper bounded by $\frac{5}{2} B\rho t$ because $\| \theta -\theta^{\prime}\|_{1}$ is uniformly bounded by $\rho$ and by Lemma 21. The fourth term is non-positive because $\theta^\prime$ minimizes the negative log likelihood on $\Theta^{\prime}$ . Finally, we turn our attention to the second term. + +To simplify the computations, we let $Y = L_{t}^{(1)}(\theta_{0}) - L_{t}^{(1)}(\theta^{\prime})$ , and compute $\mathbb{E}\left[Y\mathbf{1}_{\{Y > \log \frac{N}{\delta}\}}\right]$ . Conditioning on whether $\epsilon$ is larger or smaller than $\log \frac{N}{\delta}$ yields the following identity + +$$ +\mathbb {P} \left[ Y \mathbf {1} _ {\{Y \geq \log \frac {N}{\delta} \}} \geq \epsilon \right] = \left\{ \begin{array}{l l} \mathbb {P} \left[ Y \geq \epsilon \right] & \text {i f} \epsilon \geq \log \frac {N}{\delta}, \\ \mathbb {P} \left[ Y \geq \log \frac {N}{\delta} \right] & \text {o t h e r w i s e}. \end{array} \right. +$$ + +We can now upper bound the expectation as follows + +$$ +\begin{array}{l} \mathbb {E} \left[ Y \mathbf {1} _ {\{Y \geq \log \frac {N}{\delta} \}} \right] = \int_ {0} ^ {\infty} \mathbb {P} \left[ Y \mathbf {1} _ {\{Y \geq \log \frac {N}{\delta} \}} \right\rbrace \geq \epsilon \biggr ] d \epsilon \\ = \log \frac {N}{\delta} \mathbb {P} \left[ Y \geq \log \frac {N}{\delta} \right] + \int_ {\log \frac {N}{\delta}} ^ {\infty} \mathbb {P} \left[ Y \geq \epsilon \right] d \epsilon \\ = \log {\frac {N}{\delta}} \mathbb {P} \left[ Y \geq \log {\frac {N}{\delta}} \right] + \int_ {0} ^ {\delta} {\frac {1}{\delta^ {\prime}}} \mathbb {P} \left[ Y \geq \log {\frac {N}{\delta^ {\prime}}} \right] d \delta^ {\prime} \\ \leq \delta \log \frac {N}{\delta} + \delta , \\ \end{array} +$$ + +where we used the change of variable $\epsilon = \log \frac{N}{\delta'}$ and used $\mathbb{P}\left[Y \geq \log \frac{N}{\delta}\right] \leq \delta$ by Lemma 17. Finally, putting everything together and using $N \leq \mathcal{N}(\Theta, \| \cdot \|_1, \rho) \leq \left(\frac{ed(1 + \frac{2}{\rho})}{s}\right)^s$ , by Lemma 16, we get + +$$ +\mathbb {E} \left[ X \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right)\right) \right] \leq \mathbb {E} \left[ X s \log \frac {e d \left(1 + \frac {2}{\rho}\right)}{s \delta^ {\frac {1}{s}}} \right] + \frac {5}{2} B \rho t + B \delta s \log \frac {e ^ {1 + \frac {1}{s}} d \left(1 + \frac {2}{\rho}\right)}{s \delta^ {\frac {1}{s}}}. +$$ + +To balance the trade-off between the approximation error and the covering complexity, we choose $\rho = \frac{2}{BT}$ , and $\delta = \frac{1}{BT}$ which yields the desired form of the logarithmic factors. Substituting these into the bound completes the proof. + +# E.2.1 Proof of Lemma 14 + +As was noted in the analysis, since $\lambda_T$ is not used by the algorithm, we can replace $\lambda_T$ by $\lambda_{T-1}$ in our computations. We have + +$$ +\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\eta}{\lambda_ {t - 1}} \left(L _ {t} ^ {(1)} \left(\theta_ {t}\right) - L _ {t} ^ {(1)} \left(\theta_ {0}\right) + L _ {t - 1} ^ {(1)} \left(\theta_ {0}\right) - L _ {t - 1} ^ {(1)} \left(\theta_ {t - 1}\right)\right) + \frac {\eta}{\lambda_ {T}} \left(L _ {T} ^ {(1)} \left(\theta_ {0}\right) - L _ {T} ^ {(1)} \left(\theta_ {T}\right)\right) \right] \\ = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \frac {\eta}{\lambda_ {t - 1}} \left(L _ {t} ^ {(1)} \left(\theta_ {t}\right) - L _ {t} ^ {(1)} \left(\theta_ {0}\right)\right) - \sum_ {t = 1} ^ {T} \frac {\eta}{\lambda_ {t}} \left(L _ {t} ^ {(1)} \left(\theta_ {t}\right) - L _ {t} ^ {(1)} \left(\theta_ {0}\right)\right) \right] \\ = \eta \cdot \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right)\right) \left(\frac {1}{\lambda_ {t}} - \frac {1}{\lambda_ {t - 1}}\right) \right]. \\ \end{array} +$$ + +Let $C_{1,T}$ be a deterministic upper bound on $\left(\frac{1}{\lambda_{t + 1}} - \frac{1}{\lambda_t}\right)$ . Applying Lemma 18 to $X = \left(\frac{1}{\lambda_{t + 1}} - \frac{1}{\lambda_t}\right)$ and telescoping, we get + +$$ +\begin{array}{l} \eta \cdot \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left(L _ {t} ^ {(1)} \left(\theta_ {0}\right) - L _ {t} ^ {(1)} \left(\theta_ {t}\right)\right) \left(\frac {1}{\lambda_ {t}} - \frac {1}{\lambda_ {t - 1}}\right) \right] \\ \cdot \leq \eta \left(5 + s \log \frac {2 e ^ {2} d t ^ {2} C _ {1 , T} ^ {2}}{s}\right) \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left(\frac {1}{\lambda_ {t}} - \frac {1}{\lambda_ {t - 1}}\right) + \frac {1}{T} \right] \\ \leq \eta \left(5 + s \log \frac {2 e ^ {2} d t ^ {2} C _ {1 , T} ^ {2}}{s}\right) \mathbb {E} \left[ \left(\frac {1}{\lambda_ {T}} + 1\right) \right] \\ \leq \mathbb {E} \left[ \frac {\eta (1 5 + 3 s \log \frac {2 e ^ {2} d t ^ {2} C _ {1 , T} ^ {2}}{s})}{2 \lambda_ {T - 1}} \right], \\ \end{array} +$$ + +where in the last step, we used $1 \leq \frac{1}{2\lambda_T}$ which implies $\frac{1}{\lambda_T} + 1 \leq \frac{3}{2\lambda_T}$ . This finishes the proof. + +![](images/cd63b2de672aa4022bac127253c542988d8a7442775ea4d5afbee8221743d5ab.jpg) + +# F Bounding the surrogate information ratio + +# F.1 Proof of Lemma 6 + +The surrogate regret of a policy is directly related to its 2- and 3-information ratio by definition + +$$ +\widehat {\Delta} _ {t} (\pi) = \sqrt {\overline {{\mathbf {I G}}} _ {t} (\pi) \overline {{\mathbf {I R}}} _ {t} ^ {(2)} (\pi)} = \left(\overline {{\mathbf {I G}}} _ {t} (\pi) \overline {{\mathbf {I R}}} _ {t} ^ {(3)} (\pi)\right) ^ {\frac {1}{3}}. +$$ + +By the AM-GM inequality, we have that for any $\lambda > 0$ , the surrogate regret is controlled as follows + +$$ +\widehat {\Delta} _ {t} (\pi) \leq \frac {\overline {{\mathbf {I G}}} _ {t} (\pi)}{\lambda} + \frac {\lambda}{4} \overline {{\mathbf {I R}}} _ {t} ^ {(2)} (\pi). +$$ + +Similarly, by Lemma 27 which generalizes the AM-GM inequality, we can obtain the following regret bound + +$$ +\widehat {\Delta} _ {t} (\pi) \leq \frac {\overline {{\mathbf {I G}}} _ {t} (\pi)}{\lambda} + c _ {3} ^ {*} \sqrt {\lambda \overline {{\mathbf {I R}}} _ {t} ^ {(3)} (\pi)}, +$$ + +where $c_3^* < 2$ is an absolute constant defined in Lemma 27. This concludes the proof. + +![](images/f4f32f246d94445d1b68966ae7dd6d82549e27dc157b1bad56abbc30a2860efc.jpg) + +# F.2 Proof of Lemma 1 + +The proof of Lemma 1 is essentially the same as the proof of Lemma 5.6 in Hao et al. [2021], but we state it here for completeness. Throughout this proof, we use $\langle p,f\rangle = \sum_{a\in \mathcal{A}}p(a)f(a)$ to denote the inner product between a signed measure $p$ on $\mathcal{A}$ and a function $f:\mathcal{A}\to \mathbb{R}$ . Using this notation, we can, for example, write the generalized surrogate information ratio as $\overline{\mathrm{IR}}_t^{(\gamma)}(\pi) = \langle \pi ,\overline{\mathrm{IR}}_t^{(\gamma)}\rangle$ + +We define $\pi_t^{(\gamma)} \in \arg \min_{\pi \in \Delta(\mathcal{A})} \overline{\mathrm{IR}}_t^{(\gamma)}(\pi)$ to be any minimizer of the generalized surrogate information ratio with parameter $\gamma \geq 2$ . First, we observe that + +$$ +\nabla_ {\pi} \overline {{\mathbf {I}}} \overline {{\mathbf {R}}} _ {t} ^ {(2)} (\boldsymbol {\pi}) = \frac {2 \langle \boldsymbol {\pi} , \widehat {\Delta} _ {t} \rangle \widehat {\Delta} _ {t}}{\langle \boldsymbol {\pi} , \overline {{\mathbf {I}}} \overline {{\mathbf {G}}} _ {t} \rangle} - \frac {(\langle \boldsymbol {\pi} , \widehat {\Delta} _ {t} \rangle) ^ {2} \overline {{\mathbf {I}}} \overline {{\mathbf {G}}} _ {t}}{(\langle \boldsymbol {\pi} , \overline {{\mathbf {I}}} \overline {{\mathbf {G}}} _ {t} \rangle) ^ {2}}. +$$ + +Therefore, from the first-order optimality condition for convex constrained minimization (and the fact that $\overline{\mathbb{R}}_t^{(2)}$ is convex on $\Delta (\mathcal{A})$ ), we have + +$$ +\forall \pi \in \Delta (\mathcal {A}), 0 \leq \langle \pi - \pi_ {t} ^ {(\mathbf {S O I D S})}, \nabla_ {\pi} \overline {{\mathbb {R}}} _ {t} ^ {(2)} (\pi_ {t} ^ {(\mathbf {S O I D S})}) \rangle . +$$ + +In particular, + +$$ +0 \leq \frac {2 \langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \widehat {\Delta} _ {t} \rangle \langle \pi_ {t} ^ {(\gamma)} - \pi^ {(\mathbf {S O I D S})} , \widehat {\Delta} _ {t} \rangle}{\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \overline {{\mathbf {I G}}} _ {t} \rangle} - \frac {(\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \widehat {\Delta} _ {t} \rangle) ^ {2} \langle \pi_ {t} ^ {(\gamma)} - \pi^ {(\mathbf {S O I D S})} , \overline {{\mathbf {I G}}} _ {t} \rangle}{(\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \overline {{\mathbf {I G}}} _ {t} \rangle) ^ {2}}. +$$ + +This inequality is equivalent to + +$$ +2 \langle \pi_ {t} ^ {(\gamma)}, \widehat {\Delta} _ {t} \rangle \geq \langle \pi_ {t} ^ {(\mathbf {S O I D S})}, \widehat {\Delta} _ {t} \rangle \left(1 + \frac {\langle \pi_ {t} ^ {(\gamma)} , \overline {{\mathbf {I G}}} _ {t} \rangle}{\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \overline {{\mathbf {I G}}} _ {t} \rangle}\right) \geq \langle \pi_ {t} ^ {(\mathbf {S O I D S})}, \widehat {\Delta} _ {t} \rangle . +$$ + +From this inequality, we obtain + +$$ +\begin{array}{l} \frac {(\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \widehat {\Delta} _ {t} \rangle) ^ {\gamma}}{\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \overline {{\mathbf {I G}}} _ {t} \rangle} = \frac {(\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \widehat {\Delta} _ {t} \rangle) ^ {2} (\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \widehat {\Delta} _ {t} \rangle) ^ {\gamma - 2}}{\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \overline {{\mathbf {I G}}} _ {t} \rangle} \\ \leq \frac {(\langle \pi_ {t} ^ {(\gamma)} , \widehat {\Delta} _ {t} \rangle) ^ {2} (\langle \pi_ {t} ^ {(\mathbf {S O I D S})} , \widehat {\Delta} _ {t} \rangle) ^ {\gamma - 2}}{\langle \pi_ {t} ^ {(\gamma)} , \overline {{\mathbf {I G}}} _ {t} \rangle} \\ \leq 2 ^ {\gamma - 2} \frac {(\langle \pi_ {t} ^ {(\gamma)} , \widehat {\Delta} _ {t} \rangle) ^ {\gamma}}{\langle \pi_ {t} ^ {(\gamma)} , \overline {{\mathbf {I G}}} _ {t} \rangle} = 2 ^ {\gamma - 2} \min _ {\pi \in \Delta (\mathcal {A})} \overline {{\mathbf {I R}}} _ {t} ^ {(\gamma)} (\pi), \\ \end{array} +$$ + +thus proving the claim. + +![](images/54c89d7d8c0fb20e037f363e0d56521ae034cf68d604f21fb975ab73ad50dbe2.jpg) + +# F.3 Proof of Lemma 7 + +This section is focused on bounding the information ratios of the sparse optimistic information directed sampling policy. As is widely done in the information directed sampling literature, we will introduce a "forerunner" algorithm with controlled surrogate information ratio. By Lemma 1, the SOIDS policy will then automatically inherit the bound of the forerunner. + +As one of our forerunners, we will make use of the Feel-Good Thompson Sampling (FGTS) algorithm introduced by Zhang [2022]. Letting $\widetilde{\theta}_t \sim Q_t^+$ , the FGTS policy is defined as + +$$ +\pi_ {t} ^ {\left(\mathbf {F G T S}\right)} (a) = \mathbb {P} _ {t} \left[ a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a \right]. \tag {30} +$$ + +Which can be seen as the policy obtained by sampling a parameter $\widetilde{\theta}_t \sim Q_t^+$ and then picking the optimal action under this parameter. Compared to the usual Thompson Sampling policy, this boils down to replacing the Bayesian posterior by the optimistic posterior. Whenever the optimal action for $\theta$ is non-unique, we define $a^*(\theta)$ to be any optimal action with minimal 0-norm. If there are multiple optimal actions with minimal 0-norm, ties can be broken arbitrarily. + +# F.3.1 Bounding the two information ratio + +We will now prove the first part of Lemma 7, by showing that the information ratio of the FGTS policy is bounded by the dimension. The proof is exactly the same as in the Bayesian setting as is done in Proposition 5 of Russo and Roy [2016], Lemma 7 of Neu et al. [2022] or in Lemma 5.7 of Hao et al. [2021], except the Bayesian posterior is replaced with the optimistic posterior. We provide the proof here for completeness. + +Since we defined the surrogate information gain in terms of the model $\theta$ , as opposed to the optimal action $a^{*}(\theta)$ , we follow the proof of Lemma 7 in Neu et al. [2022]. For brevity, we let $\alpha_{a} = \pi_{t}^{(\mathbf{FGTS})}(a) = \mathbb{P}_{t}\left[a^{*}(\widetilde{\theta}_{t}) = a\right]$ . We define the $|\mathcal{A}| \times |\mathcal{A}|$ matrix $M$ by + +$$ +M _ {a, a ^ {\prime}} = \sqrt {\alpha_ {a} \alpha_ {a ^ {\prime}}} (\mathbb {E} _ {t} [ r (a, \widetilde {\theta} _ {t}) | a ^ {*} (\widetilde {\theta} _ {t}) = a ^ {\prime} ] - r (a, \bar {\theta} (Q _ {t} ^ {+}))). +$$ + +Next, we relate the surrogate information gain and the surrogate regret to the Frobenius norm and the trace of $M$ . First, we can lower bound the surrogate information gain of FGTS as + +$$ +\begin{array}{l} \overline {{\mathbf {I G}}} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) = \frac {1}{2} \sum_ {a \in \mathcal {A}} \alpha_ {a} \int_ {\Theta} \left(r (a, \bar {\theta} \left(Q _ {t} ^ {+}\right)) - r (a, \theta)\right) ^ {2} \mathrm {d} Q _ {t} ^ {+} (\theta) \\ = \frac {1}{2} \sum_ {a \in \mathcal {A}} \alpha_ {a} \int_ {\Theta} \sum_ {a ^ {\prime} \in \mathcal {A}} \mathbf {1} _ {\{a ^ {*} (\theta) = a ^ {\prime} \}} (r (a, \bar {\theta} (Q _ {t} ^ {+})) - r (a, \theta)) ^ {2} d Q _ {t} ^ {+} (\theta) \\ = \frac {1}{2} \sum_ {a \in \mathcal {A}} \sum_ {a ^ {\prime} \in \mathcal {A}} \alpha_ {a} \int_ {\Theta} \mathbf {1} _ {\{a ^ {*} (\theta) = a ^ {\prime} \}} d Q _ {t} ^ {+} (\theta) \mathbb {E} _ {t} [ (r (a, \bar {\theta} (Q _ {t} ^ {+})) - r (a, \widetilde {\theta} _ {t}) | a ^ {*} (\widetilde {\theta} _ {t}) = a ^ {\prime} ] \\ \geq \frac {1}{2} \sum_ {a \in \mathcal {A}} \sum_ {a ^ {\prime} \in \mathcal {A}} \alpha_ {a} \alpha_ {a ^ {\prime}} \left(r (a, \bar {\theta} (Q _ {t} ^ {+})) - \mathbb {E} _ {t} [ r (a, \widetilde {\theta} _ {t}) | a ^ {*} (\widetilde {\theta} _ {t}) = a ^ {\prime} ]\right) ^ {2} \\ = \frac {1}{2} \sum_ {a \in \mathcal {A}} \sum_ {a ^ {\prime} \in \mathcal {A}} M _ {a, a ^ {\prime}} ^ {2} = \frac {1}{2} \| M \| _ {F} ^ {2}. \\ \end{array} +$$ + +Next, we can re-write the surrogate regret of FGTS as + +$$ +\begin{array}{l} \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) = \int_ {\Theta} r \left(a ^ {*} (\theta), \theta\right) \mathrm {d} Q _ {t} ^ {+} (\theta) - \sum_ {a \in \mathcal {A}} \alpha_ {a} \int_ {\Theta} r (a, \theta) \mathrm {d} Q _ {t} ^ {+} \tag {31} \\ = \int_ {\Theta} \sum_ {a \in \mathcal {A}} \mathbf {1} _ {\{a ^ {*} (\theta) = a \}} r (a ^ {*} (\theta), \theta) d Q _ {t} ^ {+} (\theta) - \sum_ {a \in \mathcal {A}} \alpha_ {a} r (a, \bar {\theta} (Q _ {t} ^ {+})) \\ = \sum_ {a \in \mathcal {A}} \alpha_ {a} \mathbb {E} _ {t} \left[ r (a, \widetilde {\theta} _ {t}) | a ^ {*} (\widetilde {\theta} _ {t}) = a \right] - \sum_ {a \in \mathcal {A}} \alpha_ {a} r (a, \bar {\theta} \left(Q _ {t} ^ {+}\right)) \\ = \operatorname {t r} (M). \\ \end{array} +$$ + +Using Fact 10 from Russo and Roy [2016], we bound $\overline{\mathrm{IR}}_t^{(2)}(\pi_t^{(\mathbf{FGTS})})$ as + +$$ +\overline {{\mathrm {I R}}} _ {t} ^ {(2)} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) = \frac {\left(\widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right)\right) ^ {2}}{\overline {{\mathrm {I G}}} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right)} \leq \frac {2 \left(\operatorname {t r} (M)\right) ^ {2}}{\| M \| _ {F} ^ {2}} \leq 2 \cdot \operatorname {r a n k} (M). +$$ + +All the remains is to show that $M$ has rank at most $d$ . Enumerate the actions as $\mathcal{A} = \{a_1, \ldots, a_{|\mathcal{A}|}\}$ , and let $\mu_i = \mathbb{E}_t[\widetilde{\theta}_t | a^*(\widetilde{\theta}_t) = a_i]$ . By linearity of expectation (and of the reward function), we can write + +$$ +M _ {i, j} = \sqrt {\alpha_ {i} \alpha_ {j}} \langle \mu_ {i} - \bar {\theta} (Q _ {t} ^ {+}), a _ {j} \rangle . +$$ + +Therefore, $M$ can be factorized as + +$$ +M = \left[ \begin{array}{c} \sqrt {\alpha_ {1}} (\mu_ {1} - \bar {\theta} (Q _ {t} ^ {+})) ^ {\top} \\ \vdots \\ \sqrt {\alpha_ {| \mathcal {A} |}} (\mu_ {| \mathcal {A} |} - \bar {\theta} (Q _ {t} ^ {+})) ^ {\top} \end{array} \right] \left[ \begin{array}{c c c} \sqrt {\alpha_ {1}} a _ {1} & \dots & \sqrt {\alpha_ {| \mathcal {A} |}} a _ {| \mathcal {A} |} \end{array} \right]. +$$ + +Since $M$ is the product of a $K\times d$ matrix and a $d\times K$ matrix, it must have rank at most $\min (K,d)$ + +# F.3.2 Bounding the three information ratio + +To bound the 3 information ratio we follow Hao et al. [2021] and we introduce the exploratory policy + +$$ +\mu = \underset {\pi \in \Delta (\mathcal {A})} {\arg \max } \sigma_ {\min } \left(\sum_ {a \in \mathcal {A}} \pi (a) a a ^ {\top}\right). \tag {32} +$$ + +We define the mixture policy $\pi_t^{\mathrm{(mix)}} = (1 - \gamma)\pi_t^{\mathrm{(FGTS)}} + \gamma \mu$ where $\gamma \geq 0$ will be determined later. First, we lower bound the surrogate information gain of the mixture policy in the same way that we lower bounded the surrogate information gain of the FGTS policy previously. This time, we obtain the lower bound + +$$ +\begin{array}{l} \overline {{\mathrm {I G}}} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {m i x}\right)}\right) \geq \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi_ {t} ^ {\left(\mathbf {m i x}\right)} (a) \sum_ {a ^ {\prime} \in \mathcal {A}} \mathbb {P} _ {t} \left(a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a ^ {\prime}\right) \left(r \left(a, \bar {\theta} \left(Q _ {t} ^ {+}\right)\right) - \mathbb {E} _ {t} \left[ r \left(a, \widetilde {\theta} _ {t}\right) \mid a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a ^ {\prime} \right]\right) ^ {2} \\ = \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi_ {t} ^ {(\mathrm {m i x})} (a) \sum_ {a ^ {\prime} \in \mathcal {A}} \mathbb {P} _ {t} (a ^ {*} (\widetilde {\theta} _ {t}) = a ^ {\prime}) \langle \mu_ {a ^ {\prime}} - \bar {\theta} (Q _ {t} ^ {+}), a \rangle^ {2}, \\ \end{array} +$$ + +where $\mu_{a'} = \mathbb{E}_t[\widetilde{\theta}_t|a^*(\widetilde{\theta}_t) = a']$ . From the inequality $\pi_t^{(\mathrm{mix})}(a) \geq \gamma \mu(a)$ , and the definition of $C_{\min}$ , we have + +$$ +\begin{array}{l} \overline {{\mathrm {I G}}} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {m i x}\right)}\right) \geq \frac {\gamma}{2} \sum_ {a ^ {\prime} \in \mathcal {A}} \mathbb {P} _ {t} \left(a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a ^ {\prime}\right) \sum_ {a \in \mathcal {A}} \mu (a) \left(\mu_ {a ^ {\prime}} - \bar {\theta} \left(Q _ {t} ^ {+}\right)\right) ^ {\top} a a ^ {\top} \left(\mu_ {a ^ {\prime}} - \bar {\theta} \left(Q _ {t} ^ {+}\right)\right) \\ \geq \frac {\gamma}{2} \sum_ {a ^ {\prime} \in \mathcal {A}} \mathbb {P} _ {t} (a ^ {*} (\widetilde {\theta} _ {t}) = a ^ {\prime}) C _ {\mathrm {m i n}} \| \mu_ {a ^ {\prime}} - \bar {\theta} (Q _ {t} ^ {+}) \| _ {2} ^ {2}. \\ \end{array} +$$ + +Using the expression for the surrogate regret of FGTS in (31), we obtain + +$$ +\begin{array}{l} \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) = \sum_ {a \in \mathcal {A}} \mathbb {P} _ {t} \left(a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a\right) \left(\mathbb {E} _ {t} \left[ \langle \widetilde {\theta} _ {t}, a \rangle \mid a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a \right] - \langle \bar {\theta} \left(Q _ {t} ^ {+}\right), a \rangle\right) \\ \leq \sqrt {\sum_ {a \in \mathcal {A}} \mathbb {P} _ {t} (a ^ {*} (\widetilde {\theta} _ {t}) = a) (\mathbb {E} _ {t} [ \langle \widetilde {\theta} _ {t} , a \rangle | a ^ {*} (\widetilde {\theta} _ {t}) = a ] - \langle \bar {\theta} (Q _ {t} ^ {+}) , a \rangle) ^ {2}}, \\ \end{array} +$$ + +where in the last we used the Cathy-Schwarz inequality. Due to the sparse optimal action property, all actions for which $\mathbb{P}_t(a^* (\widetilde{\theta}_t) = a) > 0$ have at most $s$ non-zero elements. Therefore, + +$$ +\sum_ {a \in \mathcal {A}} \mathbb {P} _ {t} (a ^ {*} (\widetilde {\theta} _ {t}) = a) (\mathbb {E} _ {t} [ \langle \widetilde {\theta} _ {t}, a \rangle | a ^ {*} (\widetilde {\theta} _ {t}) = a ] - \langle \bar {\theta} (Q _ {t} ^ {+}), a \rangle) ^ {2} \leq \sum_ {a \in \mathcal {A}} \mathbb {P} _ {t} (a ^ {*} (\widetilde {\theta} _ {t}) = a) s \| \mu_ {a} - \bar {\theta} (Q _ {t} ^ {+}) \| _ {2} ^ {2}. +$$ + +This, combined with the lower bound on $\overline{\mathrm{IG}}_t(\pi_t^{\mathrm{(mix)}})$ means that + +$$ +\begin{array}{l} \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) \leq \sqrt {\sum_ {a \in \mathcal {A}} \mathbb {P} _ {t} \left(a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a\right) s \left\| \mu_ {a} - \bar {\theta} \left(Q _ {t} ^ {+}\right) \right\| _ {2} ^ {2}} \\ = \sqrt {\frac {2 s}{\gamma C _ {\operatorname* {m i n}}} \frac {\gamma}{2} \sum_ {a \in \mathcal {A}} \mathbb {P} _ {t} \left(a ^ {*} \left(\widetilde {\theta} _ {t}\right) = a\right) C _ {\operatorname* {m i n}} \| \mu_ {a} - \bar {\theta} \left(Q _ {t} ^ {+}\right) \| _ {2} ^ {2}} \\ \leq \sqrt {\frac {2 s}{\gamma C _ {\mathrm {m i n}}} \overline {{\mathbf {I G}}} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {m i x}\right)}\right)}. \\ \end{array} +$$ + +Choosing $\gamma = 1$ , this tells us that + +$$ +\left(\widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right)\right) ^ {2} \leq \frac {2 s}{C _ {\min }} \overline {{\mathbf {I G}}} _ {t} (\mu). +$$ + +We bound the information ratio in three cases. First, suppose that $\widehat{\Delta}_t(\mu) \leq \widehat{\Delta}_t(\pi_t^{(\mathbf{FGTS})})$ . In this case, + +$$ +\overline {{\mathbb {R}}} _ {t} ^ {(3)} (\mu) = \frac {\widehat {\Delta} _ {t} (\mu) \left(\widehat {\Delta} _ {t} (\mu)\right) ^ {2}}{\overline {{\mathbb {G}}} _ {t} (\mu)} \leq \frac {2 \left(\widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right)\right) ^ {2}}{\overline {{\mathbb {G}}} _ {t} (\mu)} \leq \frac {4 s}{C _ {\min }}. +$$ + +Next, we consider the case where $\widehat{\Delta}_t(\mu) > \widehat{\Delta}_t(\pi_t^{(\mathbf{FGTS})})$ . For any $\gamma \in (0,1]$ , + +$$ +\overline {{\mathbb {R}}} _ {t} ^ {(3)} (\pi_ {t} ^ {(\mathbf {m i x})}) = \frac {((1 - \gamma) \widehat {\Delta} _ {t} (\pi_ {t} ^ {(\mathbf {F G T S})}) + \gamma \widehat {\Delta} _ {t} (\mu)) ^ {3}}{(1 - \gamma) \overline {{\mathbb {I G}}} _ {t} (\pi_ {t} ^ {(\mathbf {F G T S})}) + \gamma \overline {{\mathbb {I G}}} _ {t} (\mu)} \leq \frac {((1 - \gamma) \widehat {\Delta} _ {t} (\pi_ {t} ^ {(\mathbf {F G T S})}) + \gamma \widehat {\Delta} _ {t} (\mu)) ^ {3}}{\gamma \overline {{\mathbb {I G}}} _ {t} (\mu)}. +$$ + +We define $f(\gamma) = ((1 - \gamma)\widehat{\Delta}_t(\pi_t^{(\mathbf{FGTS})}) + \gamma \widehat{\Delta}_t(\mu))^3 /(\gamma \overline{\mathrm{IG}}_t(\mu))$ to be the RHS of the previous equation. One can verify that the derivative of $f(\gamma)$ is + +$$ +f ^ {\prime} (\gamma) = \frac {((1 - \gamma) \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) + \gamma \widehat {\Delta} _ {t} (\mu)) ^ {2}}{\gamma^ {2} \overline {{\mathbf {I G}}} _ {t} (\mu)} \left[ 2 \gamma \left(\widehat {\Delta} _ {t} (\mu) - \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right)\right) - \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right) \right], +$$ + +and that $f(\gamma)$ is minimised w.r.t. $\gamma > 0$ at $\widehat{\gamma}$ , where $\widehat{\gamma}$ is the positive solution of $f'(\widehat{\gamma}) = 0$ , which is + +$$ +\widehat {\gamma} = \frac {\widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right)}{2 \left(\widehat {\Delta} _ {t} (\mu) - \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {\left(\mathbf {F G T S}\right)}\right)\right)}. +$$ + +That $\widehat{\gamma}$ is always positive follows from the fact that $\widehat{\Delta}_t(\mu) > \widehat{\Delta}_t(\pi_t^{(\mathbf{FGTS})})$ . If $\widehat{\gamma} \leq 1$ , then we can take the forerunner to be the mixture policy with $\gamma = \widehat{\gamma}$ . In this case, + +$$ +\begin{array}{l} \overline {{\mathbf {I}}} \mathbf {R} _ {t} ^ {(3)} (\boldsymbol {\pi} _ {t} ^ {(\mathbf {m i x})}) = \frac {(\frac {3}{2}) ^ {3} 2 (\widehat {\Delta} _ {t} (\mu) - \widehat {\Delta} _ {t} (\boldsymbol {\pi} _ {t} ^ {(\mathbf {F G T S})})) \widehat {\Delta} _ {t} (\boldsymbol {\pi} _ {t} ^ {(\mathbf {F G T S})}) ^ {2}}{\overline {{\mathbf {I G}}} _ {t} (\mu)} \\ \leq \frac {\left(\frac {3}{2}\right) ^ {3} 8 s}{C _ {\min}} = \frac {2 7 s}{C _ {\min}}. \\ \end{array} +$$ + +Otherwise, if $\widehat{\gamma} > 1$ , then + +$$ +\widehat {\Delta} _ {t} (\mu) \leq \frac {3}{2} \widehat {\Delta} _ {t} \left(\pi_ {t} ^ {(\mathbf {F G T S})}\right). +$$ + +In this case, we can take the forerunner to be $\mu$ . The surrogate 3-information ratio can then be upper bounded as + +$$ +\overline {{\mathbf {I}}} \mathbf {R} _ {t} ^ {(3)} (\mu) = \frac {\widehat {\Delta} _ {t} (\mu) (\widehat {\Delta} _ {t} (\mu)) ^ {2}}{\overline {{\mathbf {I}}} \mathbf {G} _ {t} (\mu)} \leq \frac {2 \left(\frac {3}{2}\right) ^ {2} \left(\widehat {\Delta} _ {t} \left(\pi_ {t} ^ {(\mathbf {F G T S})}\right)\right) ^ {2}}{\overline {{\mathbf {I}}} \mathbf {G} _ {t} (\mu)} \leq \frac {\left(\frac {3}{2}\right) ^ {2} 4 s}{C _ {\min }} = \frac {9 s}{C _ {\min }}. +$$ + +Therefore, one can always find a value of $\gamma \in (0,1]$ such that + +$$ +\overline {{\mathrm {I R}}} _ {t} ^ {(3)} \left(\pi_ {t} ^ {\left(\mathbf {m i x}\right)}\right) \leq \frac {2 7 s}{C _ {\min }}. +$$ + +# G Choosing the learning rates + +This section is focused on the choice of the learning rates required to obtain the bound of Theorem 2. + +# G.1 Technical tools + +We start by a collection of technical results to help with choosing a time-dependent learning rate. + +Lemma 19. Let $a_i \geq 0$ and $f: [0, \infty) \to [0, \infty)$ be a nonincreasing function. Then + +$$ +\sum_ {t = 1} ^ {T} a _ {t} f \left(\sum_ {i = 0} ^ {t} a _ {i}\right) \leq \int_ {a _ {0}} ^ {\sum_ {t = 0} ^ {T} a _ {t}} f (x) d x. \tag {33} +$$ + +The proof follows from elementary manipulations comparing sums and integrals. The result is taken from Lemma 4.13 of Orabona [2019], where a complete proof is also supplied. The following lemma ensures that the learning rates are non-increasing. + +Lemma 20. Let $C_1 > e, C_2 > 0$ and define $\lambda_t = \frac{\log(C_1t)}{C_2t}$ , then $\lambda_t$ is a non-decreasing sequence. + +Proof. Let $t > 0$ , we have + +$$ +\frac {\log (C _ {1} (t + 1))}{\log (C _ {1} t)} = \frac {\log \left(C _ {1} t \left(\frac {t + 1}{t}\right)\right)}{\log (C _ {1} t)} = \frac {\log (C _ {1} t) + \log \left(\frac {t + 1}{t}\right)}{\log (C _ {1} t)} \leq 1 + \frac {1}{t \log (C _ {1} t)} \leq 1 + \frac {1}{t}, +$$ + +where the first inequality uses $\log (1 + x) \leq x$ for any $x > -1$ and the second inequality uses $\log (C_1 t) \geq \log (C_1) \geq 1$ because we assumed $C_1 \geq e$ . Since $\frac{C_2(t + 1)}{C_2 t} = 1 + \frac{1}{t}$ , we can conclude that the sequence $\lambda_t$ is non-increasing. + +# G.2 Data-rich regime: Proof of Lemma 8 + +We start by focusing on the data rich regime, and we bound the following part of the regret bound given in Equation (13): + +$$ +\frac {C _ {T}}{\lambda_ {T - 1}} + \frac {3 2}{3} \sum_ {t = 1} ^ {T} \lambda_ {t - 1} \overline {{\mathrm {I R}}} _ {t} ^ {(2)} (\pi_ {t}). +$$ + +Here, $C_T = 5 + 2s\log \frac{edT}{s}$ . To proceed, we let $\lambda_{t} = \alpha \sqrt{\frac{C_{t+1}}{d(t+1)}}$ , where $\alpha > 0$ is a constant that we will optimize later. Because $t \mapsto C_t$ is increasing, we get that $\lambda_{t-1} \leq \alpha \sqrt{\frac{C_T}{dt}}$ . By Lemma 7, we know that for all $t \geq 1$ , $\overline{\mathrm{IR}}_t^{(2)}(\pi_t) \leq 2d$ , hence + +$$ +\begin{array}{l} \frac {C _ {T}}{\lambda_ {T - 1}} + \frac {3 2}{3} \sum_ {t = 1} ^ {T} \lambda_ {t - 1} \overline {{\mathbf {I}}} \mathbf {R} _ {t} ^ {(2)} (\pi_ {t}) \leq \frac {1}{\alpha} \sqrt {C _ {T} d T} + \frac {6 4}{3} \alpha \sqrt {C _ {T}} \sum_ {t = 1} ^ {T} \frac {d}{\sqrt {d t}} \\ \leq \frac {1}{\alpha} \sqrt {C _ {T} d T} + \frac {1 2 8}{3} \alpha \sqrt {C _ {T} d T} \\ \leq \left(\frac {1}{\alpha} + \frac {1 2 8}{3} \alpha\right) \sqrt {C _ {T} d T} \\ \leq 1 6 \sqrt {\frac {2}{3} C _ {T} d T}, \\ \end{array} +$$ + +where the second line uses the standard inequality $\sum_{t=1}^{T} \frac{1}{\sqrt{t}} \leq 2\sqrt{T}$ , and the last line is obtained by optimizing the expression $\left(\frac{1}{\alpha} + \frac{128}{3}\alpha\right)$ with the optimal choice $\alpha = \sqrt{\frac{3}{128}}$ which yields the value $16\sqrt{\frac{2}{3}}$ . This concludes the proof of the claim. + +# G.3 Data-poor regime: proof of Lemma 8 + +We now focus on the data-poor regime and specifically on bounding the following part of the bound given in Equation (13): + +$$ +\frac {C _ {T}}{\lambda_ {T - 1}} + \frac {1 6}{3} c _ {3} ^ {*} \sum_ {t = 1} ^ {T} \sqrt {3 \lambda_ {t - 1} \overline {{\mathbb {R}}} _ {t} ^ {(3)} (\pi_ {t})}. +$$ + +Here, $C_T = 5 + 2s\log \frac{edT}{s}$ . Now, we let $\lambda_{t} = \alpha \left(\frac{C_{t + 1}\sqrt{C_{\min}}}{(t + 1)\sqrt{s}}\right)^{\frac{2}{3}}$ , where $\alpha >0$ is a constant that we will optimize later. Because $t\to C_t$ is increasing, we get that $\lambda_{t - 1}\leq \alpha \left(\frac{C_T\sqrt{C_{\min}}}{ts}\right)^{\frac{2}{3}}$ . By Lemma 7, the 3-surrogate-information ratio is bounded for all $t\geq 1$ as $\overline{\mathrm{IR}}_t^{(3)}(\pi_t)\leq \frac{54s}{C_{\min}}$ . Hence, the following holds: + +$$ +\begin{array}{l} \frac {C _ {T}}{\lambda_ {T - 1}} + \frac {1 6}{3} c _ {3} ^ {*} \sum_ {t = 1} ^ {T} \sqrt {3 \lambda_ {t - 1} \overline {{\mathbb {R}}} _ {t} ^ {(3)} (\pi_ {t})} \leq \frac {1}{\alpha} (C _ {T}) ^ {\frac {1}{3}} \left(\frac {T \sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}} + 4 8 c _ {3} ^ {*} \sqrt {2 \alpha} (C _ {T}) ^ {\frac {1}{3}} \left(\frac {\sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}} \sum_ {t = 1} ^ {T} \frac {1}{t ^ {\frac {1}{3}}} \\ \leq \frac {1}{\alpha} (C _ {T}) ^ {\frac {1}{3}} \left(\frac {T \sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}} + 7 2 c _ {3} ^ {*} \sqrt {2 \alpha} (C _ {T}) ^ {\frac {1}{3}} \left(\frac {T \sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}} \\ \leq \left(\frac {1}{\alpha} + 7 2 c _ {3} ^ {*} \sqrt {2 \alpha}\right) \left(C _ {T}\right) ^ {\frac {1}{3}} \left(\frac {T \sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}} \\ \leq 1 2 \cdot 6 ^ {\frac {1}{3}} \left(C _ {T}\right) ^ {\frac {1}{3}} \left(\frac {T \sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}}. \\ \end{array} +$$ + +Here, we have applied Lemma 19 with the function $f(x) = x^{\frac{1}{3}}$ and $a_{i} = 1$ to bound $\sum_{t=1}^{T} t^{-1/3} \leq \frac{3}{2} T^{\frac{2}{3}}$ in the second line, the last line comes from the choice $\alpha = \frac{1}{4 \cdot 6^{\frac{1}{3}}}$ which optimizes the constant $\left( \frac{1}{\alpha} + 72c_{3}^{*}\sqrt{2\alpha} \right)$ (as per Lemma 27). This proves the statement. + +# G.4 Joint learning rates, end of the proof of Theorem 2 + +In the section below, we present the technical derivation related to choosing the choice of learning rate $\lambda_{t} = \min \left(\frac{1}{2},\max (\lambda_{t}^{(2)},\lambda_{t}^{(3)})\right)$ , where $\lambda_t^{(2)} = \sqrt{\frac{3C_{t + 1}}{128d(t + 1)}}$ and $\lambda_t^{(3)} = \frac{1}{4\cdot6^{\frac{1}{3}}}\left(\frac{C_{t + 1}\sqrt{C_{\mathrm{min}}}}{(t + 1)\sqrt{s}}\right)^{\frac{2}{3}}$ , with $C_t = 5 + 2s\log \frac{edt}{s}$ . This choice interpolates between the data-rich and data-poor regimes. As a first step, we start by confirming via Lemma 20 that both $\lambda_t^{(2)}$ and $\lambda_t^{(3)}$ are non-increasing and the bound of Theorem 1 holds with our choice of $\lambda_t$ . + +First, note that our choice of learning rates ensures that $\lambda_t \leq \frac{1}{2}$ holds as long as $T$ is larger than an absolute constant, and thus we focus on this case here (and relegate the complete details of establishing this absolute constant to Appendix G.5). To proceed, we define the (constant-free) regret rates $R_t^{(2)} = \sqrt{C_t dt}$ and $R_t^{(3)} = \left(t\sqrt{s\frac{C_t}{C_{\min}}}\right)^{\frac{2}{3}}$ and note that they correspond to the regret bounds obtained when using the respective learning rates $\lambda_t^{(2)}$ and $\lambda_t^{(3)}$ , as per Lemma 8. + +We now consider the last time that the learning rates $\lambda_{t}^{(3)}$ and $\lambda_{t}^{(2)}$ have been used. More specifically, we denote $T_{2} = \max \{t\leq T,\lambda_{t - 1}^{(2)}\geq \lambda_{t - 1}^{(3)}\}$ , and $T_{3} = \max \{t\leq T,\lambda_{t - 1}^{(3)}\geq \lambda_{t - 1}^{(2)}\}$ . Combining the bound of Equation 13 and using the definition $\lambda_{t} = \min (\frac{1}{2},\max (\lambda_{t}^{(2)},\lambda_{t}^{(3)}))$ , the following bound + +holds + +$$ +\begin{array}{l} R _ {T} \\ \leq \mathbb {E} \left[ \frac {C _ {T}}{\lambda_ {T - 1}} + \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \lambda_ {t - 1} \overline {{\mathbf {I R}}} _ {t} ^ {(2)} (\pi_ {t}), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} \overline {{\mathbf {I R}}} _ {t} ^ {(3)} (\pi_ {t})}\right) \right] \\ = \mathbb {E} \left[ \frac {C _ {T}}{\min (\frac {1}{2} , \max (\lambda_ {T - 1} ^ {(2)} , \lambda_ {T - 1} ^ {(3)}))} \right. \\ + \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \min (\frac {1}{2}, \max (\lambda_ {t - 1} ^ {(2)}, \lambda_ {t - 1} ^ {(3)})) \overline {{\mathbb {R}}} _ {t} ^ {(2)} (\pi_ {t}), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \min (\frac {1}{2} , \max (\lambda_ {t - 1} ^ {(2)} , \lambda_ {t - 1} ^ {(3)})) \overline {{\mathbb {R}}} _ {t} ^ {(3)} (\pi_ {t})}\right) \Biggr ] \\ \leq \mathbb {E} \left[ C _ {T} \min \left(\frac {1}{\lambda_ {T - 1} ^ {(2)}}, \frac {1}{\lambda_ {T - 1} ^ {(3)}}\right) + \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \max (\lambda_ {t - 1} ^ {(2)}, \lambda_ {t - 1} ^ {(3)}) \overline {{\mathbf {I R}}} _ {t} ^ {(2)} (\pi_ {t}), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \max (\lambda_ {t - 1} ^ {(2)} , \lambda_ {t - 1} ^ {(3)}) \overline {{\mathbf {I R}}} _ {t} ^ {(3)} (\pi_ {t})}\right) \right]. \\ \end{array} +$$ + +We can now separate the sum obtained at the last line based on which learning rate was used at time $t$ . + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \min \left(\frac {3 2}{3} \max \left(\lambda_ {t - 1} ^ {(2)}, \lambda_ {t - 1} ^ {(3)}\right) \overline {{\mathbb {R}}} _ {t} ^ {(2)} \left(\pi_ {t}\right), \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \max \left(\lambda_ {t - 1} ^ {(2)} , \lambda_ {t - 1} ^ {(3)}\right) \overline {{\mathbb {R}}} _ {t} ^ {(3)} \left(\pi_ {t}\right)}\right) \\ \leq \sum_ {\lambda_ {t - 1} ^ {(2)} \geq \lambda_ {t - 1} ^ {(3)}} \frac {3 2}{3} \lambda_ {t - 1} ^ {(2)} \overline {{\mathbf {I}}} \overline {{\mathbf {R}}} _ {t} ^ {(2)} (\pi_ {t}) + \sum_ {\lambda_ {t} ^ {(3)} \geq \lambda_ {t} ^ {(2)}} \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} ^ {(3)} \overline {{\mathbf {I}}} \overline {{\mathbf {R}}} _ {t} ^ {(3)} (\pi_ {t})} \\ \leq \sum_ {t = 1} ^ {T _ {2}} \frac {3 2}{3} \lambda_ {t - 1} ^ {(2)} \overline {{\mathbf {I R}}} _ {t} ^ {(2)} (\pi_ {t}) + \sum_ {t = 1} ^ {T _ {3}} \frac {1 6}{3} c _ {3} ^ {*} \sqrt {3 \lambda_ {t - 1} ^ {(3)} \overline {{\mathbf {I R}}} _ {t} ^ {(3)} (\pi_ {t})}. \\ \end{array} +$$ + +Following exactly the same step as in the proof of Lemma 8, we further bound $\sum_{t=1}^{T_2} \frac{32}{3} \lambda_{t-1}^{(2)} \overline{\mathbb{R}}_t^{(2)}(\pi_t) \leq 8 \sqrt{\frac{2}{3}} R_{T_2}^{(2)}$ and $\sum_{t=1}^{T_3} \frac{16}{3} c_3^* \sqrt{3 \lambda_{t-1}^{(3)} \overline{\mathbb{R}}_t^{(3)}(\pi_t)} \leq 8 \cdot 6^{\frac{1}{3}} R_{T_3}^{(3)}$ . + +The crucial observation is that which of $\lambda_T^{(3)}$ or $\lambda_T^{(2)}$ is bigger will determine whether $R_T^{(2)}$ or $R_T^{(3)}$ is the term of leading order (up to some constants). More specifically, Let $T$ be such that $\lambda_{T - 1}^{(2)}\geq \lambda_{T - 1}^{(3)}$ which means that $\sqrt{\frac{3C_T}{128dT}}\geq \frac{1}{4\cdot6^{\frac{1}{3}}}\left(\frac{C_T\sqrt{C_{\min}}}{T\sqrt{s}}\right)^{\frac{2}{3}}$ . Rearranging, this implies that $\sqrt{C_TdT}\leq \frac{6^{\frac{5}{6}}}{4}\left(T\sqrt{s\frac{C_T}{C_{\min}}}\right)^{\frac{2}{3}}$ , which means that $R_T^{(2)}\leq \frac{6^{\frac{5}{6}}}{4} R_T^{(3)}$ . Following the exact same steps, we also have that $\lambda_{T - 1}^{(3)}\geq \lambda_{T - 1}^{(2)}$ implies that $R_T^{(3)}\leq \frac{4}{6^{\frac{5}{6}}} R_T^{(2)}$ . We apply this to the time $T_2$ in which $\lambda_{T_2 - 1}^{(2)}\geq \lambda_{T_2 - 1}^{(3)}$ by definition. We have that $R_{T_2}^{(2)}\leq \frac{6^{\frac{5}{6}}}{4} R_{T_2}^{(3)}$ and putting this together with the previous bound, we have + +$$ +\begin{array}{l} R _ {T} \leq \frac {C _ {T}}{\lambda_ {T - 1} ^ {(3)}} + 8 \sqrt {\frac {2}{3}} R _ {T _ {2}} ^ {(2)} + 8 \cdot 6 ^ {\frac {1}{3}} R _ {T _ {3}} ^ {(3)} \\ \leq 4 \cdot 6 ^ {\frac {1}{3}} R _ {T} ^ {(3)} + 8 \sqrt {\frac {2}{3}} \cdot \frac {6 ^ {\frac {5}{6}}}{4} R _ {T _ {2}} ^ {(2)} + 8 \cdot 6 ^ {\frac {1}{3}} R _ {T _ {3}} ^ {(3)} \\ \leq 4 \cdot 6 ^ {\frac {1}{3}} R _ {T} ^ {(3)} + 4 \cdot 6 ^ {\frac {1}{3}} R _ {T _ {2}} ^ {(3)} + 8 \cdot 6 ^ {\frac {1}{3}} R _ {T _ {3}} ^ {(3)} \\ \leq 4 \cdot 6 ^ {\frac {1}{3}} R _ {T} ^ {(3)} + 4 \cdot 6 ^ {\frac {1}{3}} R _ {T} ^ {(3)} + 8 \cdot 6 ^ {\frac {1}{3}} R _ {T} ^ {(3)} \\ \leq 1 6 \cdot 6 ^ {\frac {1}{3}} R _ {T} ^ {(3)}, \\ \end{array} +$$ + +where we use the fact that $T \to R_T^{(3)}$ is increasing and $T_2 \leq T, T_3 \leq T$ . + +Using the same argument as before, we have that $\lambda_{T_3 - 1}^{(3)}\geq \lambda_{T_3 - 1}^{(2)}$ , and we can conclude that $R_{T_3}^{(3)}\leq \frac{4}{6^{\frac{5}{6}}} R_{T_3}^{(2)}$ . + +Putting this together, with the previous bound, we have + +$$ +\begin{array}{l} R _ {T} \leq \frac {C _ {T}}{\lambda_ {T - 1} ^ {(2)}} + 8 \sqrt {\frac {2}{3}} R _ {T _ {2}} ^ {(2)} + 8 \cdot 6 ^ {\frac {1}{3}} R _ {T _ {3}} ^ {(3)} \\ \leq 8 \sqrt {\frac {2}{3}} R _ {T} ^ {(2)} + 8 \sqrt {\frac {2}{3}} R _ {T _ {2}} ^ {(2)} + 8 \cdot 6 ^ {\frac {1}{3}} \cdot \frac {4}{6 ^ {\frac {5}{6}}} R _ {T _ {3}} ^ {(3)} \\ \leq 8 \sqrt {\frac {2}{3}} R _ {T} ^ {(2)} + 8 \sqrt {\frac {2}{3}} R _ {T _ {2}} ^ {(2)} + 1 6 \sqrt {\frac {2}{3}} R _ {T _ {3}} ^ {(2)} \\ \leq 8 \sqrt {\frac {2}{3}} R _ {T} ^ {(2)} + 8 \sqrt {\frac {2}{3}} R _ {T} ^ {(2)} + 1 6 \sqrt {\frac {2}{3}} R _ {T} ^ {(2)} \\ \leq 3 2 \sqrt {\frac {2}{3}} R _ {T} ^ {(2)}, \\ \end{array} +$$ + +where we use the fact that $T \to R_T^{(3)}$ is increasing and $T_2 \leq T, T_3 \leq T$ . Evaluating the constants numerically yields $16 \cdot 6^{\frac{1}{3}} \approx 29.07 \leq 30$ and $32\sqrt{\frac{2}{3}} \approx 26.13 \leq 27$ . + +# G.5 Upper bound on the learning rates + +We now consider the case where the learning rates exceed $\frac{1}{2}$ , and show that this only holds for small values of $T$ . First, we have that $\lambda_{T - 1}^{(2)} \leq \frac{1}{2}$ if + +$$ +\sqrt {\frac {3 C _ {T}}{1 2 8 d T}} \leq \frac {1}{2}. +$$ + +Rearranging the inequality and recalling $C_T = 5 + 2s\log \frac{edT}{s}$ , this is equivalent to + +$$ +T \geq \frac {1 5}{3 2 d} + \frac {3 s}{1 6 d} \log \frac {e d T}{s}. +$$ + +Using the loose inequality $\log \frac{edT}{s} \leq \frac{dT}{s}$ , we get that this condition is satisfied for any $T \geq 1$ . Similarly, we have that $\lambda_{T-1}^{(3)} \leq \frac{1}{2}$ if + +$$ +\frac {1}{4 \cdot 6 ^ {\frac {1}{3}}} \left(\frac {C _ {T} \sqrt {C _ {\mathrm {m i n}}}}{T \sqrt {s}}\right) ^ {\frac {2}{3}} \leq \frac {1}{2}. +$$ + +We note that + +$$ +C _ {\min } = \max _ {\mu \in \Delta (A)} \sigma_ {\min } \left(\mathbb {E} _ {A \sim \mu} \left[ A A ^ {\top} \right]\right) \leq \max _ {\mu \in \Delta (A)} \frac {\operatorname {T r} \left(\mathbb {E} _ {A \sim \mu} \left[ A A ^ {\top} \right]\right)}{d} \leq 1, +$$ + +where the first inequality uses that the trace of a matrix is always bigger than $d$ -times its smallest eigenvalue and the second inequality uses the fact that for any vector $a$ , we have $\mathrm{Tr}(aa^{\top}) = \sum_{i=1}^{d} a_i^2 \leq d \max_i |a_i| \leq d$ because we assumed that all the actions are bounded in infinity norm. Hence the previous inequality will be satisfied if + +$$ +\frac {1}{4 \cdot 6 ^ {\frac {1}{3}}} \left(\frac {C _ {T}}{T \sqrt {s}}\right) ^ {\frac {2}{3}} \leq \frac {1}{2}. +$$ + +Rearranging the inequality, this is equivalent to + +$$ +T \geq 4 \sqrt {\frac {3}{s}} C _ {t} = 8 \sqrt {3 s} \log (e T) + \sqrt {3 s} \left(\frac {2 0}{s} + 8 \log \frac {d}{s}\right). +$$ + +Applying Lemma 24 with $a = 8\sqrt{3s}$ and $b = \sqrt{3s}\left(\frac{20}{s} + 8\log \left(\frac{d}{s}\right)\right)$ , we find that the previous inequality is satisfied for all + +$$ +T \geq 2 a \log e a + 2 b = 4 0 \sqrt {\frac {3}{s}} + 1 6 \sqrt {3 s} \log \frac {8 e \sqrt {3} d}{\sqrt {s}}. +$$ + +Thus, letting $T_{\mathrm{min}} = 40\sqrt{\frac{3}{s}} + 16\sqrt{3s}\log \frac{8e\sqrt{3}d}{\sqrt{s}}$ be the constant given above, both learning rates stay upper bounded by $\frac{1}{2}$ for all $T \geq T_{\mathrm{min}}$ and the upper bound on the regret given the previous subsection holds. Otherwise, we upper bound the instantaneous regret by 2 and this leads to an additional $2T_{\mathrm{min}} = \mathcal{O}(\sqrt{s}\log \frac{d}{\sqrt{s}})$ in the regret. Putting this together with the bound proved in the previous section, we thus have that the following regret bound is valid for any $T \geq 1$ : + +$$ +R _ {T} \leq \min \left(2 7 \sqrt {\left(5 + 2 s \log \frac {e d T}{s}\right) d T}, 3 0 \left(5 + 2 s \log \frac {e d T}{s}\right) ^ {\frac {1}{3}} \left(\frac {T \sqrt {s}}{\sqrt {C _ {\operatorname* {m i n}}}}\right) ^ {\frac {2}{3}}\right) + \mathcal {O} \left(\sqrt {s} \log \frac {d}{\sqrt {s}}\right). +$$ + +This concludes the proof of Theorem 2. + +![](images/dfa66005e5181f2f82d5208bc3c5982bbdef6472e34dae298b746184ac93deee.jpg) + +# I Technical Results + +We state and prove the remaining technical results. + +Lemma 21. Let $\pi \in \Delta (\mathcal{A})$ , the function $\theta \mapsto \Delta (\pi ,\theta)$ is 2-Lipschitz with respect to the 1 norm. Let $t\geq 1$ , the function $\theta \to \mathbb{E}\left[\log \left(\frac{1}{p_t(Y_t|\theta,A_t)}\right)\right]$ is 2-Lipschitz with respect to the 1 norm. Moreover if $\theta, \theta^{\prime}$ are random variables, then + +$$ +\left| \mathbb {E} \left[ \log \left(\frac {1}{p _ {t} (Y _ {t} | \theta , A _ {t})}\right) - \log \left(\frac {1}{p _ {t} (Y _ {t} | \theta^ {\prime} , A _ {t})}\right) \right] \right| \leq \frac {5}{2} \sup \| \theta - \theta^ {\prime} \| _ {1}. +$$ + +Proof. Let $\theta, \theta' \in \Theta$ , we have + +$$ +\begin{array}{l} \left| r (\pi , \theta) - r (\pi , \theta^ {\prime}) \right| = \left| \sum_ {a \in \mathcal {A}} \pi (a) \langle \theta - \theta^ {\prime}, a \rangle \right| \\ \leq \sum_ {a \in \mathcal {A}} \pi (a) | \langle \theta - \theta^ {\prime}, a \rangle | \\ \leq \sum_ {a \in \mathcal {A}} \pi (a) \| \theta - \theta^ {\prime} \| _ {1} \| a \| _ {\infty} \\ \leq \left\| \theta - \theta^ {\prime} \right\| _ {1}. \\ \end{array} +$$ + +Similarly, + +$$ +\left| r ^ {*} (\theta) - r ^ {*} (\theta^ {\prime}) \right| = \left| \max _ {a \in \mathcal {A}} r (a, \theta) - \max _ {a \in \mathcal {A}} r (a, \theta^ {\prime}) \right| \leq \max _ {a \in \mathcal {A}} \left| r (\theta , a) - r (a, \theta^ {\prime}) \right| \leq \| \theta - \theta^ {\prime} \| _ {1}. +$$ + +Finally + +$$ +\left| \Delta (\pi , \theta) - \Delta (\pi , \theta^ {\prime}) \right| = \left| r ^ {*} (\theta) - r ^ {*} \left(\theta^ {\prime}\right) + r (\pi , \theta^ {\prime}) - r (\pi , \theta) \right| \leq 2 \left\| \theta - \theta^ {\prime} \right\| _ {1}. +$$ + +Let us write $r = \langle \theta, A_t \rangle$ , $r' = \langle \theta', A_t \rangle$ and $r_0 = \langle \theta_0, A_t \rangle$ . We have that $Y_t = r_0 + \epsilon_t$ where $\epsilon_t$ is 1-sub-Gaussian. For the negative log-likelihood, we then have + +$$ +\begin{array}{l} \mathbb {E} \left[ \log \left(\frac {1}{p (Y _ {t} | \theta , A _ {t})}\right) - \log \left(\frac {1}{p (Y _ {t} | \theta^ {\prime} , A _ {t})}\right) \right] = \frac {1}{2} \mathbb {E} \left[ (\langle \theta , A _ {t} \rangle - Y _ {t}) ^ {2} - (\langle \theta^ {\prime}, A _ {t} \rangle - Y _ {t}) ^ {2} \right] \\ = \frac {1}{2} \mathbb {E} \left[ (r - Y _ {t}) ^ {2} - (r ^ {\prime} - Y _ {t}) ^ {2} \right] \\ = \frac {1}{2} \mathbb {E} \left[ \left(r - r ^ {\prime}\right) \left(r + r ^ {\prime} - 2 Y _ {t}\right) \right] \\ = \frac {1}{2} \mathbb {E} \left[ \left(r - r ^ {\prime}\right) \left(r + r ^ {\prime} - 2 r _ {0}\right) \right] + \frac {1}{2} \mathbb {E} \left[ \left(r - r ^ {\prime}\right) \epsilon_ {t} \right] \\ \leq 2 \mathbb {E} \left[ \| \theta - \theta^ {\prime} \| _ {1} \right] + \frac {1}{2} \mathbb {E} \left[ \| \theta - \theta^ {\prime} \| _ {1} | \epsilon_ {t} | \right], \\ \end{array} +$$ + +where we use the fact that $|r + r' - 2r_0| \leq 4$ . + +Now if $\theta, \theta'$ are fixed, since $\mathbb{E}[\epsilon_t] = 0$ , we get + +$$ +\mathbb {E} \left[ \log \left(\frac {1}{p (Y _ {t} | \theta , A _ {t})}\right) - \log \left(\frac {1}{p (Y _ {t} | \theta^ {\prime} , A _ {t})}\right) \right] \leq 2 \| \theta - \theta^ {\prime} \| _ {1} +$$ + +If $\theta, \theta'$ are random variables, the subgaussianity of $\epsilon_t$ implies that $\mathbb{E}\left[\epsilon_t^2\right] \leq 1$ and by Cauchy-Schwarz we have + +$$ +\mathbb {E} \left[ \| \theta - \theta^ {\prime} \| _ {1} | \epsilon_ {t} | \right] \leq \sqrt {\mathbb {E} \left[ \| \theta - \theta^ {\prime} \| _ {1} ^ {2} \right] \mathbb {E} \left[ \epsilon_ {t} ^ {2} \right]} \leq \sup \| \theta - \theta^ {\prime} \| _ {1}. +$$ + +This yields the second claim of the Lemma. + +![](images/92a361efd83adf62f0b0f020d176c1cbadb9f9ee99728eff0a337237008bc947.jpg) + +Lemma 22. (Hoeffding's Lemma) Let $X$ be a bounded real random variable such that $X \in [a, b]$ almost surely. Let $\eta \neq 0$ , then we have + +$$ +\frac {1}{\eta} \log \mathbb {E} [ \exp (\eta X) ] \leq \mathbb {E} [ X ] + \frac {\eta (b - a) ^ {2}}{8}. \tag {34} +$$ + +Proof. See for instance Chapter 2 in Boucheron et al. [2013]. + +![](images/34a09a9faf42984f82e384717b61f1cb191c0515c87cda62810b6c879dc91810.jpg) + +We now provide a data dependent version of Hoeffding's lemma that is used in the analysis of the gaps in the optimistic posterior. + +Lemma 23. (A data dependent version of Hoeffding's Lemma) Let $X$ be a real random variable and $\eta \neq 0$ be such that $\eta X \leq 1$ almost surely, then we have + +$$ +\frac {1}{\eta} \log \mathbb {E} [ \exp (\eta X) ] \leq \mathbb {E} [ X ] + \eta \mathbb {E} [ X ^ {2} ] \leq 2 \mathbb {E} [ X ]. \tag {35} +$$ + +Proof. Using the elementary inequalities $\log (x) \leq x - 1$ for $x > 0$ and $e^x \leq 1 + x + x^2$ for $x \leq 1$ , we get that + +$$ +\begin{array}{l} \frac {1}{\eta} \log \mathbb {E} [ \exp (\eta X) ] \leq \frac {1}{\eta} \mathbb {E} [ \exp (\eta X) - 1 ] \\ \leq \frac {1}{\eta} \mathbb {E} \left[ \eta X + \eta^ {2} X ^ {2} \right] \\ \leq \mathbb {E} [ X ] + \eta \mathbb {E} \left[ X ^ {2} \right]. \\ \end{array} +$$ + +![](images/8e2c4e8dce5eacea6779d1efd402e331b0314b7229c86afbf63103bceabad3ee.jpg) + +The following lemmas help us to analyze when the learning rates are smaller or bigger than $\frac{1}{2}$ . + +Lemma 24. Let $a \geq 1, b \geq 0$ , then, the equation $t \geq a \log et + b$ is verified for any $t \geq 2a \log ea + 2b$ . + +Proof. We let $f(t) = t - a\log et - b$ , we have that $f^{\prime}(t)\geq 0$ on $[a, + \infty)$ and $f(a)\leq 0$ . Hence $f(t) = 0$ has a unique solution $\alpha$ on $[a,\infty)$ such that $f(t)\geq 0$ if $t\geq \alpha$ . We now focus on upper bounding $\alpha$ . The equation $f(\alpha) = 0$ is equivalent to + +$$ +\log \alpha = \frac {\alpha - b}{a} - 1. +$$ + +Now taking the exponential and reordering this is also equivalent to + +$$ +\frac {- \alpha}{a} \exp \left(\frac {- \alpha}{a}\right) = \frac {- \exp \left(- \frac {a + b}{a}\right)}{a}. +$$ + +Let + +$$ +\begin{array}{l} g: (- \infty , - 1 ] \longrightarrow [ - \frac {1}{e}, 0) \\ x \longmapsto x e ^ {x}. \\ \end{array} +$$ + +The previous equation can be rewritten $g\left(\frac{-\alpha}{a}\right) = -\frac{\exp\left(-\frac{a + b}{a}\right)}{a}$ . We define $W_{-1}:[-\frac{1}{e},0)\longrightarrow (-\infty ,1]$ as the (functional) inverse of $g$ . The function $g$ is the -1 branch of the Lambert W function. We have that for any $x\leq -1$ , $W_{-1}(xe^{x}) = x$ and that for any $y\geq e$ , $-W_{-1}(-\frac{1}{y})\leq 2\log (y)$ . Since $g$ is decreasing on its domain, $W_{-1}$ is well-defined and decreasing. Moreover, for any $x\leq -1$ + +, $W_{-1}(g(x)) = x$ . In particular, we have that $\alpha = -aW_{-1}\left(-\frac{\exp\left(-\frac{a + b}{a}\right)}{a}\right)$ . We will use that formulation to find an upper bound on $\alpha$ . + +We fix some $y \geq e$ . We have $-2\log(y) \leq -1$ hence $W_{-1}\left(-2\log(y)e^{(-2\log(y))}\right) = -2\log(y)$ , which means that $2\log(y) = -W_{-1}\left(-\frac{1}{y^*}\right)$ where $y^* = \frac{e^{(2\log(y))}}{2\log(y)} = \frac{y^2}{2\log(y)}$ . + +Because of the elementary inequality $2\log (x) \leq x$ for $x > 0$ , we conclude that $y \leq y^{*}$ . Since $y \mapsto -W_{-1}\left(-\frac{1}{y}\right)$ is an increasing function we finally have that for any $y \geq e$ + +$$ +- W _ {- 1} \left(- \frac {1}{y}\right) \leq - W _ {- 1} \left(- \frac {1}{y ^ {*}}\right) = 2 \log (y). +$$ + +Applying this to $y = a\exp \left(\frac{a + b}{a}\right)\geq e$ , we get + +$$ +\alpha = - a W _ {- 1} \left(\frac {- 1}{y}\right) \leq 2 a \log (y) = 2 a \log e a + 2 b. +$$ + +Since any $t \geq \alpha$ will satisfy $f(t) \geq 0$ , this concludes our proof. + +![](images/cdd25de84cfdcae32c8655625bf3043075624f1fdb0fc43ffa3649eba06c3d90.jpg) + +Lemma 25. Let $\theta \in \Theta$ , then $M_t = \exp(L_t^{(1)}(\theta_0) - L_t^{(1)}(\theta)) = \prod_{s=1}^t \frac{p(Y_t | \theta, A_t)}{p(Y_t | \theta_0, A_t)}$ is a supermartingale with respect to the filtration $(\mathcal{F}_t)_t$ . + +Proof. We have + +$$ +\begin{array}{l} \mathbb {E} \left[ \frac {p \left(Y _ {t} \mid \theta , A _ {t}\right)}{p \left(Y _ {t} \mid \theta_ {0} , A _ {t}\right)} \Bigg | \mathcal {F} _ {t - 1}, A _ {t} \right] = \mathbb {E} \left[ \exp \left(\frac {\left(\left\langle \theta_ {0} , A _ {t} \right\rangle - Y _ {t}\right) ^ {2} - \left(\left\langle \theta , A _ {t} \right\rangle - Y _ {t} ^ {2}\right)}{2}\right) \Bigg | \mathcal {F} _ {t - 1}, A _ {t} \right] \\ = \mathbb {E} \left[ \exp \left(\frac {\epsilon_ {t} ^ {2} - \left(\left\langle \theta - \theta_ {0} , A _ {t} \right\rangle - \epsilon_ {t}\right) ^ {2}}{2}\right) \Bigg | \mathcal {F} _ {t - 1}, A _ {t} \right] \\ = \exp \left(- \frac {\left(\left\langle \theta - \theta_ {0} , A _ {t} \right\rangle\right) ^ {2}}{2}\right) \mathbb {E} \left[ \exp \left(\epsilon_ {t} \left\langle \theta - \theta_ {0}, A _ {t} \right\rangle\right) \mid \mathcal {F} _ {t - 1}, A _ {t} \right] \\ \leq \exp \left(- \frac {(\langle \theta - \theta_ {0} , A _ {t} \rangle) ^ {2}}{2}\right) \cdot \exp \left(\frac {(\langle \theta - \theta_ {0} , A _ {t} \rangle) ^ {2}}{2}\right) \\ = 1, \\ \end{array} +$$ + +where the inequality comes from the conditional subgaussianity of $\epsilon_t$ . Finally, by the tower rule of conditional expectations + +$$ +\mathbb {E} \left[ \frac {p \left(Y _ {t} \mid \theta , A _ {t}\right)}{p \left(Y _ {t} \mid \theta_ {0} , A _ {t}\right)} \Big | \mathcal {F} _ {t - 1} \right] = \mathbb {E} \left[ \mathbb {E} \left[ \frac {p \left(Y _ {t} \mid \theta , A _ {t}\right)}{p \left(Y _ {t} \mid \theta_ {0} , A _ {t}\right)} \Big | \mathcal {F} _ {t - 1}, A _ {t} \right] \Big | \mathcal {F} _ {t - 1} \right] \leq 1. +$$ + +![](images/b8b787655f9d8fbdb58670830dacd705c8ebe198ca88e73008ad1b61724afe99.jpg) + +# I.1 Proof of Proposition 1 + +This is coming from the fact that the mean is the constant minimizing the mean squared error. We remind the reader of the definition of the surrogate information gain and the true information gain for a policy $\pi \in \Delta (\mathcal{A})$ + +$$ +\overline {{\mathbf {I G}}} _ {t} (\pi) = \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi (a) \int_ {\Theta} \left(\left\langle \theta - \bar {\theta} \left(Q _ {t} ^ {+}\right), a \right\rangle\right) ^ {2} \mathrm {d} Q _ {t} ^ {+} (\theta), \tag {36} +$$ + +where $\bar{\theta} (Q_t^+) = \mathbb{E}_{\theta \sim Q_t^+}\left[\theta \right]$ is the mean parameter under the optimistic posterior $Q_{t}^{+}$ . + +$$ +\mathrm {I G} _ {t} (\pi) = \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi (a) \int_ {\Theta} \left(\left\langle \theta , a \right\rangle - \left\langle \theta_ {0}, a \right\rangle\right) ^ {2} \mathrm {d} Q _ {t} ^ {+} (\theta), \tag {37} +$$ + +Let's fix $a \in \mathcal{A}$ , we have that + +$$ +\begin{array}{l} \left(\langle \theta - \theta_ {0}, a \rangle\right) ^ {2} = \left(\langle \theta - \bar {\theta} \left(Q _ {t} ^ {+}\right) + \bar {\theta} \left(Q _ {t} ^ {+}\right) - \theta_ {0}, a \rangle\right) ^ {2} \\ = \left(\left\langle \theta - \bar {\theta} \left(Q _ {t} ^ {+}\right), a \right\rangle\right) ^ {2} + 2 \left\langle \theta - \bar {\theta} \left(Q _ {t} ^ {+}\right), a \right\rangle \left\langle \bar {\theta} \left(Q _ {t} ^ {+}\right) - \theta_ {0}, a \right\rangle + \left(\left\langle \bar {\theta} \left(Q _ {t} ^ {+}\right) - \theta_ {0}, a \right\rangle\right) ^ {2} \\ \geq \left(\left\langle \theta - \bar {\theta} \left(Q _ {t} ^ {+}\right), a \right\rangle\right) ^ {2} + 2 \left\langle \theta - \bar {\theta} \left(Q _ {t} ^ {+}\right), a \right\rangle \left\langle \bar {\theta} \left(Q _ {t} ^ {+}\right) - \theta_ {0}, a \right\rangle \\ \end{array} +$$ + +Now using that $\bar{\theta} (Q_t^+) = \int_{\Theta}\theta dQ_t^+ (\theta)$ and integrating, we get + +$$ +\int_ {\Theta} \left(\left\langle \theta - \theta_ {0}, a \right\rangle\right) ^ {2} d Q _ {t} ^ {+} (\theta) \geq \int_ {\Theta} \left(\left\langle \theta - \bar {\theta} \left(Q _ {t} ^ {+}\right), a \right\rangle\right) ^ {2} d Q _ {t} ^ {+} (\theta). +$$ + +Multiplying by $\pi(a)$ and summing over actions, we get the claim of the lemma. + +# I.2 Generalization of the AM-GM inequality + +Dealing with the generalized information ratio requires bounding the cubic root of products. While one could use Hölder's inequality to deal directly with products, we find it more flexible to use a variational form of this inequality. In all that follows, we let $p > 1$ be a real number and $q$ be such that $\frac{1}{p} + \frac{1}{q} = 1$ . It is not hard to check that $q = \frac{p}{p - 1}$ . We start by stating a direct consequence of the Fenchel-Young Inequality which can be seen as an extension of the AM-GM inequality. + +Lemma 26. Let $x, y \geq 0$ , then + +$$ +x y \leq \frac {x ^ {p}}{p} + \frac {y ^ {q}}{q}. \tag {38} +$$ + +With equality if and only if $px^{p - 1} = y$ + +Proof. One can check that the Fenchel dual of the function + +$$ +f: \mathbb {R} ^ {+} \longrightarrow \mathbb {R} ^ {+} +$$ + +$$ +x \longmapsto \frac {x ^ {p}}{p} +$$ + +is exactly $f^{*}(y) = \frac{1}{q} y^{q}$ (for non-negative $y$ ). Then the Lemma is a direct consequence of the Fenchel Young inequality and of its equality case. + +Refining a bit this Lemma, we get the following variational form of the previous inequality : + +Lemma 27. Let $x, y \geq 0, \lambda > 0$ , then + +$$ +\sqrt [ p ]{x y} \leq \frac {x}{\lambda} + c _ {p} ^ {*} (\lambda y) ^ {\frac {1}{p - 1}} \tag {39} +$$ + +where $c_{p}^{*} = (p - 1)\frac{1}{p}^{\frac{p}{p - 1}}$ with equality if and only if $x = y = 0$ or $\lambda = p\frac{x^{\frac{p - 1}{p}}}{y^{\frac{1}{p}}}$ . + +Proof. We apply the previous lemma to $\sqrt[p]{\frac{px}{\lambda}}$ and $\sqrt[p]{\frac{\lambda y}{p}}$ . + +In order to go from the variational form to the product form, we may use the following result. + +Lemma 28. Let $\alpha, \beta > 0$ , then + +$$ +\inf _ {\lambda > 0} \frac {\alpha}{\lambda} + \beta \lambda^ {\frac {1}{p - 1}} = c _ {p} \alpha^ {\frac {1}{p}} \beta^ {\frac {p - 1}{p}}, \tag {40} +$$ + +where $c_{p} = p\frac{1}{p - 1}\frac{\frac{p - 1}{p}}{p}$ satisfies $c_{p}\cdot c_{p}^{*}\frac{p - 1}{p} = 1$ , and the minimum is reached at $\lambda^{*} = (p - 1)^{\frac{p - 1}{p}}\frac{\alpha^{\frac{p - 1}{p}}}{\beta^{\frac{p - 1}{p}}}$ . + +Proof. Applying the previous Lemma to $x = \alpha$ and $y = c_{p}^{\frac{p}{p - 1}}\beta^{p - 1}$ yields the result. + +Remark An alternative is to pick $\lambda$ to make both terms equal resulting in the same result but with 2 as a leading constant. Now + +$$ +\begin{array}{l} c _ {p} = p ^ {\frac {1}{p}} \frac {p}{p - 1} ^ {\frac {p - 1}{p}} \\ = \exp \left(\frac {1}{p} \log p + \frac {p - 1}{p} \log \frac {p}{p - 1}\right) \\ \leq \frac {1}{p} \cdot p + \frac {p - 1}{p} \cdot \frac {p}{p - 1} \\ = 2. \\ \end{array} +$$ + +With equality if and only if $p = 2$ . So, the choice of $c_{p}$ always yields a better leading constant. However, $c_{3} \simeq 1.88$ so one could argue that the gain is small. Since we will usually use Lemma 27, $c_{p}^{*}$ will naturally appear and $c_{p}$ will cancel it, ultimately making the leading constant as simple as possible. + +# J Experimental details + +Here, we describe our implementation of the SOIDS algorithm in more detail, as well as the hyperparameters of all the methods used in our experiments. To run the SOIDS algorithm, one must minimise $\overline{\mathrm{IR}}_t^{(2)}(\pi)$ w.r.t. $\pi$ in each round $t$ . This is not straightforward, because $\overline{\mathrm{IR}}_t^{(2)}(\pi)$ contains expectations w.r.t. the optimistic posterior $Q_{t}^{+}$ . When we use the Spike-and-Slab prior in Appendix B.2, we are not aware of any efficient method that can be used to maximise $\overline{\mathrm{IR}}_t^{(2)}(\pi)$ . Instead, we draw (approximate) samples $\theta^{(1)},\ldots ,\theta^{(M)}$ from $Q_{t}^{+}$ to produce the estimates $\widetilde{\Delta}_t(\pi)$ and $\widetilde{\mathrm{IG}}_t(\pi)$ for the surrogate regret and the surrogate information respectively, where + +$$ +\widetilde {\Delta} _ {t} (\pi) = \sum_ {a \in \mathcal {A}} \pi (a) \frac {1}{M} \sum_ {i = 1} ^ {M} \Delta (a, \theta^ {(i)}), \qquad \widetilde {\mathrm {I G}} _ {t} (\pi) = \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi (a) \frac {1}{M} \sum_ {i = 1} ^ {M} \left(\langle \theta^ {(i)} - \bar {\theta} _ {M}, a \rangle\right) ^ {2}. +$$ + +Here, $\bar{\theta}_M$ is the sample mean $\frac{1}{M}\sum_{i=1}^{M}\theta^{(i)}$ . We then maximise the approximate surrogate information ratio $\widetilde{\mathrm{IR}}_t^{(2)}(\pi)$ , where + +$$ +\widetilde {\mathrm {I R}} _ {t} ^ {(2)} (\pi) = \frac {(\widetilde {\Delta} _ {t} (\pi)) ^ {2}}{\widetilde {\mathrm {I G}} _ {t} (\pi)}. +$$ + +To draw the samples $\theta^{(1)},\ldots ,\theta^{(M)}$ , we use the empirical Bayesian sparse sampling procedure proposed by Hao et al. [2021], which is designed to draw samples from the Bayesian posterior. To sample from the optimistic posterior, we incorporate the optimistic adjustment into the likelihood. This method replaces the theoretically sound spike-and-slab prior with a relaxation in which the "spikes" are Laplace distributions with small variance, and the "slabs" are Gaussian distributions with large variance. In particular, the density of this prior is + +$$ +q _ {1} (\theta) = \sum_ {\gamma \in \{0, 1 \} ^ {d}} p (\gamma) \prod_ {j = 1} ^ {d} [ \gamma_ {j} \psi_ {1} (\theta_ {j}) + (1 - \gamma_ {j}) \psi_ {0} (\theta_ {j}) ]. +$$ + +Here, $\psi_{1}(\theta)$ is the density function of a univariate Gaussian distribution, with mean 0 and variance $\rho_{1}$ , and $\psi_{0}$ is the density function of a univariate Laplace distribution, with mean 0 and scale parameter $\rho_{0}$ . $p(\gamma)$ is a product of Bernoulli distributions with mean $\beta$ . In our experiments, we always use $\rho_{1} = 10$ , $\rho_{0} = 0.1$ and $\beta = 0.1$ . Also, we set the learning rates to $\eta = 1/2$ and $\lambda_{t} = \min \left(\frac{1}{2}, \frac{1}{10} \max \left(\sqrt{\frac{s \log(edt/s)}{dt}}, \left(\frac{\log(edt/s)}{t}\right)^{2/3}\right)\right)$ . + +Implementing the OTCS baseline exactly would require us to compute the means of the distributions played by an exponentially weighted average forecaster with a sparsity prior. These distributions are the same as the optimistic posterior, except $\lambda_t = 0$ (i.e. there is no optimistic adjustment). In our implementation of the OTCS baseline, we draw samples using the same empirical Bayesian sparse sampling procedure, and then replace the exact means with the sample means. We use the same + +choices for the parameters $\eta$ , $\rho_{1}$ , $\rho_{0}$ and $\beta$ . We set the radii of the confidence sets to the values given in Theorem 4.7 of Clerico et al. [2025] + +For the LinUCB baseline, we set the radii of the confidence sets to the values given in Theorem 2 of Abbasi-Yadkori et al. [2011]. For the ESTC baseline, we set the exploration length $T_{1}$ to 50 when $d = 20$ , 100 when $d = 40$ and $d = 100$ . These values were chosen based on a small amount of trial and error. The theoretically motivated values in Theorem 4.2 of Hao et al. [2020] are much larger than these values. Also for ESTC, we set the LASSO regularisation parameter to $\lambda = 4\sqrt{\log(d) / T_1}$ , which is the value given in Theorem 4.2 of Hao et al. 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In such settings, the standard Polyak step size performs poorly, requiring an increasing number of iterations to achieve optimal statistical precision—even when, the problem remains well conditioned and/or the achievable precision itself does not degrade with problem size. We trace this limitation to a mismatch in how smoothness is measured: in high dimensions, it is no longer effective to estimate the Lipschitz smoothness constant. Instead, it is more appropriate to estimate the smoothness restricted to specific directions relevant to the problem (restricted Lipschitz smoothness constant). Sparse Polyak overcomes this issue by modifying the step size to estimate the restricted Lipschitz smoothness constant. We support our approach with both theoretical analysis and numerical experiments, demonstrating its improved performance. + +# 1 Introduction + +Consider the high-dimensional statistical estimation problem + +$$ +\min _ {\mathbb {R} ^ {d} \ni \theta : \| \theta \| _ {0} \leq s} f (\theta) = \frac {1}{n} \sum_ {i = 1} ^ {n} \ell \left(z _ {i}, \theta\right), \tag {1} +$$ + +with data points $z_{i} \in \mathbb{R}^{d}, i = 1 \dots n$ . We focus on the regime in which the dimensionality grows much faster than the sample size, i.e. $\frac{d}{n} \to \infty$ . To obtain consistent estimates in this setting, it is necessary to assume that the true solution exhibits some low-dimensional structure-such as sparsity. In (1) sparsity is enforced through the $\ell_0$ constraint, which guarantees that $\theta$ will have at most $s$ non-zero elements. This constraint renders the problem in (1) non-convex and, in general, NP-hard, regardless of the objective function $f$ Natarajan (1995). Nevertheless, under certain assumptions on the data, various algorithms have been developed to efficiently find approximate solutions to (1). Notably, under suitable assumptions that hold for a variety of statistical models, the Iterative Hard Thresholding (IHT) algorithm has been shown to efficiently find sufficiently accurate solutions to (1). The IHT algorithm results from applying projected gradient descent to (1) and reads + +$$ +\theta_ {t + 1} = \mathrm {H T} _ {s} (\theta_ {t} - \gamma \nabla f (\theta_ {t})) +$$ + +where $\mathrm{HT}_s$ denotes the hard thresholding operator. $\mathrm{HT}_s$ retains the $s$ largest-magnitude components of its input and sets the remaining to zero. Here, $\gamma > 0$ denotes the step-size which ought to be chosen as $\gamma = 2 / (3\bar{L})$ Jain et al. (2014), where $\bar{L}$ denotes the restricted Lipschitz smoothness (RSS) constant, and can be interpreted as the Lipschitz smoothness constant of $f$ when restricted to sparse directions. + +For a variety of statistical models, such as Generalized Linear Models (GLMs), $\bar{L}$ remains constant as long as $\frac{s\log(d)}{n}$ remains constant, even if $\frac{d}{n} \to \infty$ . This insight underpins the rate invariance of IHT: under suitable conditions, the number of iterations required to achieve (near) optimal statistical precision remains constant even as both $d$ and $n$ grow. + +Analogous to the Lipschitz smoothness constant, the RSS constant must be estimated in practice. Thus, the natural question in this context, and the starting point to the work in this paper is: (i) Do already existing approaches to adaptively tune $\gamma$ via the estimation of the Lipschitz smoothness constant work in the high-dimensional setting? Our criteria to determine whether a step-size rule works in the high dimension, additional to convergence will be determined by the answers to the following questions: (ii) Can they achieve the same or better guarantees than by choosing the optimal fixed step-size? (iii) Can they guarantee rate invariance as $\frac{d}{n} \to \infty$ ? + +# 1.1 Related works + +Over the past three decades, numerous methods have been proposed to solve the problem in (1), including Matching Pursuit Mallat and Zhang (1993), Orthogonal Matching Pursuit Pati et al. (1993), and CoSaMP Needell and Tropp (2009). Iterative Hard Thresholding (IHT) was first introduced in Blumensath and Davies (2009) with many variants proposed since. While initial convergence guarantees for IHT required the Restricted Isometry Property (RIP)-a condition often too stringent in practice-Jain et al. (2014) extended IHT's convergence guarantees to problems fulfilling the restricted strong convexity (RSC) and RSS; establishing linear convergence to near optimal statistical precision. Observe that in the M-estimation context, convergence to arbitrary precision is unnecessary and convergence to the best achievable statistical precision is preferred. However, for the results in Jain et al. (2014) + +to hold, if the optimal parameter to recover is $s^*$ -sparse, $s \geq \mathcal{O}(\bar{\kappa}^2 s^*)$ is required, where $\bar{\kappa}$ is the restricted condition number. Khanna and Kyrillidis (2018) proposed an accelerated version of IHT which was extended to the stochastic setting in Zhou et al. (2018). Zhou et al. (2018) establishes that with a sufficiently large mini-batch size, acceleration can be achieved and the faster rate requires only $s \geq \mathcal{O}(\bar{\kappa} s^*)$ . Axiotis and Sviridenko (2022) proposed a variant of IHT with an adaptively chosen weighted $\ell_2$ penalty for which only $s \geq \mathcal{O}(\bar{\kappa} s^*)$ is required. They further establish that for IHT $s \geq \mathcal{O}(\bar{\kappa} s^*)$ is in fact a necessary condition to achieve near optimal statistical precision. Li et al. (2016) and Shen and Li (2018) introduced variants of IHT that incorporate variance reduction. Shen and Li (2017b) and Yuan et al. (2018) propose Partial Hard Thresholding and Gradient Hard Thresholding pursuit respectively, with a focus on support recovery under high SNR assumptions. Yuan and Li (2021) establishes generalization bounds for solutions found via IHT. Further, Zhang et al. (2025) establishes that IHT, for a range of $\bar{\kappa}$ , finds solutions that can be shown to achieve the oracle estimation rate under a high SNR condition. + +With no exception, the discussed works establishing convergence under the RSS require knowledge of the RSS constant $\bar{L}$ . Knowledge of $\bar{L}$ is crucial for the requirement $s \geq \mathcal{O}(\bar{\kappa}^2 s^*)$ being sufficient for convergence to optimal statistical precision. More generally, convergence can be established whenever $s \geq \mathcal{O}\left(\frac{s^*}{\bar{\mu}^2\gamma^2}\right)$ , with $\gamma \leq \frac{1}{L}$ . Consequently, overestimating $\bar{L}$ forces a smaller step size $\gamma$ , which in turn slows down convergence and leads to denser solutions. + +![](images/13055c74e280d62bf50d724ecb3137a22a3e283bf08d1fe8ecafd3f2870884cd.jpg) +Figure 1: Performance of Polyak's step size (dashed) and Sparse Polyak (solid) on logistic regression problems with increasing $d$ and $n$ . The quantities $s$ , $s^*$ , $\bar{\kappa}$ and $\frac{\log(d)}{n}$ remain constant. With Polyak's step-size the performance degrades as $d$ increases whereas Sparse Polyak exhibits rate invariance, i.e. the number of iterations to achieve (near) optimal statistical precision does not change. + +Several recent works have proposed adaptive step-size schemes based on estimating the local Lipschitz constant Malitsky and Mishchenko (2020, 2024); Latafat et al. (2024). With the goal to jointly exploit the function's local regularity and the algorithm's trajectory, Mishkin et al. (2024) derive new convergence results for gradient descent as a function of the objective function's local Lipschitz smoothness and strong convexity parameters. They additionally establish that Polyak's step-size obtains fast and path-dependent rates. However, their results do not naturally extend to constrained problems, particularly non-convex ones like (1). + +Polyak's original step-size rule, first proposed in Polyak (1969), has gathered renewed attention in the machine learning community Ren et al. (2022); Hazan and Kakade (2019); Loizou et al. (2021); Wang et al. (2023); Zamani and Glineur (2024), but it remains ill-suited for non-convex constrained settings such as (1) unless additional assumptions are imposed. Some efforts have been made to adapt Polyak's step size to constrained problems. For instance, Cheng and Li (2012) addresses box constraints, while Devanathan and Boyd (2024) consider convex constraints. + +To this day, except for Li et al. (2024), and works that employ inexact line-search strategies Xiao and Zhang (2013); Wang et al. (2014), there has been no study of adaptive step-size schemes in the high-dimensional context. The work in Li et al. (2024) handles (1) in the stochastic setting and proposes the use of Polyak's step-size with no modification. The results in Li et al. (2024) are limited even with no stochasticity, as they imply bounds on the restricted condition number, i.e. $\bar{\kappa} \leq \frac{\eta}{\eta - 1}$ where $\eta = (\sqrt{5} + 1)^{2}/4$ . Further, it is worth mentioning that the notion of RSS assumed in Li et al. (2024) is more restrictive than that assumed in the present paper and Jain et al. (2014). Further, as shown in Fig. 1 Polyak's step-size with no modification presents a performance that degrades as the size of the problem increases even if $\bar{\kappa}$ and the optimal statistical precision (of the order of $\mathcal{O}\left(\frac{s^{*} \log d}{n}\right)$ for this particular statistical model) remain constant. This effect is highly undesirable in the high-dimensional setting and as shown in the present work, can be avoided with a suitable modification of Polyak's step-size. + +# 1.2 Major contributions + +Our main contribution is the first adaptive step-size rule that performs well in high-dimensions and preserves the rate invariance property. To develop this scheme, we address question (i) posed in the introduction. We observe that adaptive step-size rules that estimate the Lipschitz smoothness constant do not necessarily work well in high-dimensions. This is because, in many common statistical models, the Lipschitz smoothness constant scales as $\mathcal{O}(d)$ with high probability. Consequently, we answer questions (i) through (iii) in the negative, by demonstrating empirically (c.f. Fig. 1) that estimating the Lipschitz smoothness constant via Polyak's step-size (dashed line) does not yield rate invariance as $d$ grows even if $\frac{\log(d)}{n}$ is held constant. + +To overcome this limitation, we design an adaptive step-size rule that estimates the restricted Lipschitz smoothness constant instead. With this modification, we answer questions (ii) and (iii) in the affirmative. This is captured in Theorem 1 and its Corollaries, which particularize the results of Theorem 1 to relevant statistical models. In Sections 3, 6, and in Appendix D we provide theoretical and empirical evidence, both on synthetic and real data, that our proposed method outperforms Polyak's step-size for high-dimensional M-estimation tasks and achieves rate invariance. Moreover, we theoretically and empirically show that Sparse Polyak converges to optimal statistical precision at least as fast as IHT with the optimal fixed step-size $\gamma = \mathcal{O}(1 / \bar{L})$ . We also establish sufficient conditions under which we can guarantee support recovery, and particularize our results for specific statistical learning models in Section 4. + +Our guarantees are derived under standard assumptions, identical to those in Jain et al. (2014); Shen and Li (2017b); Yuan et al. (2018). To guarantee convergence to optimal statistical precision we require the knowledge of $f(\theta^{*})$ where $\theta^{*}$ denotes the true parameter. While $f(\theta^{*})$ is known to be of the order $\mathcal{O}\left(\frac{\log d}{n}\right)$ , its exact value is typically unknown. Consequently, we provide in Appendix B a double loop method that estimates a surrogate to $f(\theta^{*})$ and allows convergence to optimal statistical precision as long as lower bound to $f(\theta^{*})$ is known. Observe that in our context $f(\theta^{*}) \geq 0$ making 0 a valid lower bound. + +In addition, we prove linear convergence for statistical models that do not satisfy the regularity conditions in Jain et al. (2014); Axiotis and Sviridenko (2022); Yuan and Li (2021) but instead fulfill + +the weaker condition in Loh and Wainwright (2015). This extends the results in Jain et al. (2014); Yuan and Li (2021) to cover additional GLMs with an adaptive step-size rule. We provide these additional results in Appendix C.2. + +Our proof technique may be of independent interest as it provides a clear pathway to establishing convergence of the IHT algorithm. We believe this is key in extending theoretical guarantees to adaptive step-size schemes to solve (1). We provide a sketch of the proof of our main results in Section 5 and the formal proof in Appendix A. + +Finally, although our analysis applies only to Polyak's step-size, we conjecture that other adaptive step-size rules such as Barzilai and Borwein (1988), Zhou et al. (2025), Malitsky and Mishchenko (2020) may also suffer from similar performance degradation in high dimensions, and would therefore require analogous re-engineering to be effective. + +Notation. Throughout this paper, we adopt the following notations. For vectors $x,\in \mathbb{R}^d$ , we denote by $x_{i}$ the $i^{\mathrm{th}}$ element. We denote the $\ell_{\infty}$ -norm of $x$ as $\| x\|_{\infty}$ , the Euclidean norm as $\| x\|$ , the $\ell_1$ -norm as $\| x\|_{1}$ , and the $\ell_0$ -norm as $\| x\|_{0}$ . Recall $\| x\|_{0} = |\{i:x_{i}\neq 0\} |$ . Also, we let $|x|_{\min}$ denote the minimal entry of $x$ in the sense of absolute value. The inner product between two vectors is denoted as $\langle x,y\rangle$ . Matrices such as $X\in \mathbb{R}^{d\times d}$ , are capitalized. For any matrix $\Sigma$ , we denote its largest singular value by $\sigma_{\max}(\Sigma)$ and its smallest singular value by $\sigma_{\min}(\Sigma)$ . Similarly, if $\Sigma \in \mathbb{R}^{d\times d}$ diagonalizes, we use $\lambda_{\max}(\Sigma)$ and $\lambda_{\min}(\Sigma)$ to denote the largest and smallest eigenvalues of $\Sigma$ respectively. The Frobenius norm of $X$ is given by $\| X\| _F$ , and the nuclear norm by $\| X\| _*$ . + +# 2 Setup and background + +We make the following assumptions regarding the objective function $f$ . + +Assumption 1 (RSC Agarwal et al. (2012)). The objective function $f$ is $(\mu, \tau)$ -restricted strongly convex in $\mathbb{R}^d$ , i.e. + +$$ +\frac {\mu}{2} \| \theta_ {1} - \theta_ {2} \| ^ {2} - \frac {\tau}{2} \| \theta_ {1} - \theta_ {2} \| _ {1} ^ {2} \leq f (\theta_ {1}) - f (\theta_ {2}) - \langle \nabla f (\theta_ {2}), \theta_ {1} - \theta_ {2} \rangle , \forall \theta_ {1}, \theta_ {2} \in \mathbb {R} ^ {d}. \tag {2} +$$ + +Assumption 2 (RSSAgarwal et al. (2012)). The objective function $f$ is $(L, \tau)$ -restricted smooth in $\mathbb{R}^d$ , i.e. + +$$ +f (\theta_ {1}) - f (\theta_ {2}) - \langle \nabla f (\theta_ {2}), \theta_ {1} - \theta_ {2} \rangle \leq \frac {L}{2} \| \theta_ {1} - \theta_ {2} \| ^ {2} + \frac {\tau}{2} \| \theta_ {1} - \theta_ {2} \| _ {1} ^ {2}, \forall \theta_ {1}, \theta_ {2} \in \mathbb {R} ^ {d}. +$$ + +Assumptions 1 and 2 extend the classical notions of strong convexity and $L$ -Lipschitz smoothness. These assumptions reduce to their classical counterparts when $\tau$ is sufficiently small and the direction $\theta_{1} - \theta_{2}$ is appropriately sparse. Observe that when $\theta_{1} - \theta_{2}$ is dense and $f$ is convex, Assumption 1 becomes vacuous and the upper bound in Assumption 2 scales linearly with the problem dimension $d$ . This highlights that the RSC and RSS depend not only on the magnitude of the direction $\theta_{1} - \theta_{2}$ but also its structure. + +In high-dimensional statistical learning settings where $\frac{d}{n} \to \infty$ , standard strong convexity and smoothness assumptions fail to hold. However, many important problems still satisfy variants of the RSC and RSS, with both $\mu$ and $L$ remaining dimension-independent and with $\tau$ exhibiting only moderate dependence on $d$ , e.g., Jain et al. (2014); Agarwal et al. (2012). We leverage this in Section 4 to establish fast computational and near optimal statistical guarantees for a variety of high-dimensional statistical learning problems. + +We further highlight that our results can be generalized by adopting a weaker RSC condition, where we assume that (2) holds only for pairs $\theta$ , $\theta^{*}$ satisfying $\| \theta - \theta^{*} \| \leq 1$ , where $\theta$ denotes the ground truth, rather than requiring it to hold globally. This relaxation broadens the applicability of our approach, allowing it to accommodate a wider class of functions. Notably, for some generalized linear models (GLMs), the loss function does not necessarily satisfy (2) without imposing additional constraints. We provide a detailed discussion of these results in Appendix C.2. + +# 3 Main Result + +In this section we present our main theoretical result, which establishes convergence guarantees for Sparse Polyak (c.f. Algorithm 1). In this section, we focus on the deterministic setting of $f$ , while + +the statistical case will be addressed in Section 4. Note that IHT corresponds to projected gradient descent when applied to (1). The projection onto the $\ell_0$ is given by the hard thresholding operator, already discussed in the introduction and formally defined as follows. + +Definition 1 (Hard Thresholding Operator). For any $s > 0$ , and $z \in \mathbb{R}^d$ , we define $\mathrm{HT}_s(z)$ as the projection of $z$ on $B_0(s) := \{\theta \in \mathbb{R}^d \mid \| \theta \|_0 \leq s\}$ . i.e., + +$$ +\mathrm{HT}_{s}(z) = \operatorname *{argmin}_{\theta \in B_{0}(s)}\| \theta -z\|_{2}, +$$ + +where ties are broken lexicographically. + +# Algorithm 1 Iterative Hard Thresholding (IHT) with Polyak Step-Size + +1: Input: Function $f$ , target function value $\widehat{f}$ , sparsity parameter $s$ , number of iterations $T$ +2: Initialize: $\theta_0\in \mathbb{R}^d$ , with $\| \theta_0\| _0\leq s$ +3: for $t = 0$ to $T - 1$ do +4: Compute step-size $\gamma_{t} = \frac{\max\{f(\theta_{t}) - \widehat{f},0\}}{5\|\mathrm{HT}_{s}(\nabla f(\theta_{t}))\|^{2}}$ +5: Update: $\theta_{t + 1} = \mathrm{HT}_s(\theta_t - \gamma_t\nabla f(\theta_t))$ +6: end for +7: Output: $\theta_T$ + +The adaptive step-size rule in Algorithm 1 differs from the classical Polyak rule by replacing $\| \nabla f(\theta_t) \|^2$ with $\| \mathrm{HT}_s(\nabla f(\theta_t)) \|^2$ . This approach contrasts with the low-dimensional case and with the work in Li et al. (2024). Observe that even if the current iterate $\theta_t$ is sparse, sparsity of $\nabla f(\theta_t)$ can not be guaranteed. In fact, the worst-case relationship $\| \nabla f(\theta_t) \| \leq \sqrt{\frac{d}{s}} \| \mathrm{HT}_s(\nabla f(\theta_t)) \|$ , which holds with equality for a vector in which all coordinates are identical, may hold. As a result, using the full gradient norm can lead to using overly conservative step-sizes, slowing down convergence dramatically as $d$ increases. Unless strong additional conditions are imposed on the problem, convergence may be even be jeopardized (see discussion in Section 5 for more details). + +Before providing our main result we introduce some notation. Consider fixed values $s \geq s^* > 0$ and let $f^* \triangleq \min_{\theta: \| \theta \|_0 \leq s^*} f(\theta)$ . Assume the chosen target function value satisfies $\widehat{f} \geq f^*$ . Let $\bar{L} = L + 3\tau s$ , $\bar{\mu} = \mu - 3\tau s$ , and $\bar{\kappa} = \bar{L} / \bar{\mu}$ denote the restricted condition number. + +Theorem 1. Let $\{\theta_t\}_{t\geq 1}$ denote the sequence of iterates generated by Algorithm 1. Suppose the objective function $f$ satisfies the RSC and RSS in Assumptions 1 and 2, respectively. Let $\widehat{\theta}$ be any $s^*$ -sparse vector such that $f(\widehat{\theta}) = \widehat{f}$ , and assume $\bar{\mu} >0$ and $s\geq (240\bar{\kappa})^{2}s^{*}$ . Then, for any iterate $\theta_{t}$ such that $\| \theta_t - \widehat{\theta}\| ^2\geq \frac{36\|HT_s(\nabla f(\widehat{\theta}))\|^2}{\bar{\mu}^2}$ we can guarantee + +$$ +\left\| \theta_ {t + 1} - \widehat {\theta} \right\| ^ {2} \leq \left(1 - \frac {1}{8 0 \bar {\kappa}}\right) \left\| \theta_ {t} - \widehat {\theta} \right\| ^ {2}. +$$ + +Moreover, let $t_0 \geq 0$ be the first iteration for which $\| \theta_{t_0} - \widehat{\theta} \|^2 < \frac{36\|\mathrm{HT}_s(\nabla f(\widehat{\theta}))\|^2}{\bar{\mu}^2}$ . Then, for all $t \geq t_0$ , $\| \theta_t - \widehat{\theta}\| ^2 \leq \left(1 + \frac{1}{80\bar{\kappa}}\right)\frac{36\|\mathrm{HT}_s(\nabla f(\widehat{\theta}))\|^2}{\bar{\mu}^2}$ . + +Theorem 1 implies linear convergence at a rate scaling with $\bar{\kappa}^{-1}$ up to precision $\mathcal{O}\left(\frac{\|\mathrm{HT}_s(\nabla f(\hat{\theta}))\|^2}{\bar{\mu}^2}\right)$ . This result is near equivalent to that in Theorem 3 in Jain et al. (2014), where the RSS constant is assumed to be known, up to a constant factor. Thus, we successfully answer in the affirmative question (ii) posed in Section 1. Further, observe that, if $\bar{L}$ , $\bar{\mu}$ and $\| \mathrm{HT}_s(\nabla f(\widehat{\theta}))\|^2$ remain constant as the size of the problem grows, the rate remains unchanged and the achievable precision does so too. This implies that for a variety of statistical models the rate and final precision will remain invariant as the problem size increases, as long as the aforementioned quantities do not change. Thus, we also answer in the affirmative question (iii) posed in Section 1. We particularize the result to specific statistical models and provide further discussion in Section 4 and Appendix C. + +Here, $\widehat{f}$ is a user-defined target value that reflects the desired level of optimization, which can be set above or equal to the statistical accuracy of the problem. Such a relaxation is natural in learning + +problems, as it is often counterproductive to optimize to full precision. The continuity and restricted strong convexity of $f$ imply that an $s^*$ -sparse vector $\widehat{\theta}$ such that $f(\widehat{\theta}) = \widehat{f}$ always exists. + +The additional factor $\frac{1}{80\kappa}$ in the final precision (c.f. Theorem 1 when $t \geq t_0$ ) stems from the expansiveness of the hard thresholding operator. The removal of this factor and support recovery guarantees can be achieved under the signal to noise ratio (SNR) condition (c.f.(3)) and are provided in the following Corollary. Such a condition is widely used in hard thresholding and support recovery studies, as seen in (Shen and Li, 2017a, Prop 2) Bouchot et al. (2016); Shen and Li (2017c); Yuan et al. (2016). + +Corollary 1. Under the assumptions stated in Theorem 1, if further, the SNR condition: + +$$ +\left| \widehat {\theta} \right| _ {\min } \geq \frac {7 \| \mathrm {H T} _ {s} (\nabla f (\widehat {\theta})) \|}{\bar {\mu}} \tag {3} +$$ + +holds, for any $t \geq t_0$ , where $t_0$ is defined in Theorem 1, the support of $\theta_t$ contains that of $\widehat{\theta}$ , and the sequence $\| \theta_t - \widehat{\theta} \|^2$ is non-increasing and upper bounded by $\frac{36 \| \mathrm{HT}_s(\nabla f(\widehat{\theta})) \|^2}{\widehat{\mu}^2}$ . + +Algorithm 1 offers an approach to obtain a target accuracy that we assume known in advance to (1), without requiring precise knowledge of $L$ , $\mu$ , $\tau$ . For scenarios in which we only have access to a lower bound on the problem, Algorithm 2 serves as an alternative; in most learning problems, the bound can be simply set to 0, making the method broadly applicable. This adaptive variant of Algorithm 1 builds on the framework of Hazan and Kakade (2019), which reviews gradient descent with Polyak's step size and its double-loop counterpart. By updating the lower bound adaptively in an outer loop, the method ensures that either the accuracy $\mathcal{O}\left(\frac{\|\mathrm{HT}_s(\nabla f(\hat{\theta}))\|^2}{\hat{\mu}^2}\right)$ is attained or the updated lower bound remains valid. + +Algorithm 2 IHT with Adaptive Polyak +1: Input: Function $f$ , a lower bound $\tilde{f}_1$ , and sparsity parameter $s$ . +2: Initialize: $\theta_0 = 0 \in \mathbb{R}^d$ +3: for $k = 1$ to $K$ do +4: for $t = 0$ to $T - 1$ do +5: Compute step-size $\gamma_t = \frac{f(\theta_t) - \widetilde{f}_k}{10\|\mathrm{HT}_s(\nabla f(\theta_t))\|^2}$ +6: Update: $\theta_{t + 1} = \mathrm{HT}_s(\theta_t - \gamma_t g_t)$ +7: end for +8: $\bar{\theta}_k = \mathrm{argmin}_{t \leq T} f(\theta_t)$ +9: $\widetilde{f}_{k + 1} = \frac{f(\bar{\theta}_k) + \widetilde{f}_k}{2}$ +10: $\theta_0 = \bar{\theta}_k$ +11: end for +12: Output: $\bar{\theta} = \mathrm{argmin}_{k \leq K} f(\bar{\theta}_k)$ + +Let $s \geq s^* > 0$ , and define $f^* := \min_{\| \theta \|_0 \leq s^*} f(\theta)$ , attained by some $s^*$ -sparse vector $\theta^*$ . + +Theorem 2. Consider the iterates $\{\bar{\theta}_k\}$ generated by Algorithm 2. Assume that the function $f$ fulfills Assumptions 1 and 2. Then for $\varepsilon = (1 + \frac{1}{160\bar{\kappa}})\frac{36(\bar{L} + \bar{\mu})\|\mathrm{HT}_s(\nabla f(\theta^*))\|^2}{\bar{\mu}^2}$ , when $\bar{\mu} > 0, s \geq (480\bar{\kappa})^2 s^*$ , Algorithm 2 requires at most $\widetilde{T} := \left(1 + \log_2\frac{2(f(\theta_0) - f(\theta^*))}{\varepsilon}\right)T$ gradient evaluations to achieve $f(\bar{\theta}) - f(\theta^*) \leq \varepsilon$ and $\|\bar{\theta} - \theta^*\|^2 \leq (1 + \frac{1}{160\bar{\kappa}})\frac{36\|\mathrm{HT}_s(\nabla f(\theta^*))\|^2}{\bar{\mu}^2}$ . Here $T = \left[\frac{1}{\log(1/(1-1/160\bar{\kappa}))}\log\left(\frac{\bar{\mu}^2\|\theta_0 - \theta^*\|^2}{36(1+1/160\bar{\kappa})\|\mathrm{HT}_s(\nabla f(\theta^*))\|^2}\right)\right]$ . + +This theorem focuses on the distance of the iterates to $\theta^{*}$ . The quantity $T$ can be interpreted as the number of iterations required to reach the desired accuracy when applying Algorithm 1 with $\widehat{f} = f^{*}$ . The additional term $\mathcal{O}\left(\frac{f(\theta_0) - f(\theta^*)}{\varepsilon}\right)$ in the definition of $\widetilde{T}$ corresponds to the number of outer iterations needed to obtain a sufficiently tight lower bound for the targeted accuracy. Similar order guarantees are established in Theorems 1 and 2. The proof of Theorem 2 is provided in Appendix B. + +# 4 Statistical Guarantees + +The results in Section 3 are deterministic in nature and consequently, do not depend on a data generation model. In contrast, in this section we use Theorem 1 to provide guarantees for specific statistical models. Corollaries 2 and 3 establish the computational-statistical performance guarantees for sparse logistic regression and low-rank matrix regression respectively. We provide guarantees for additional statistical models, including sparse linear regression, in Appendix C. + +# 4.1 Logistic Regression + +We consider a dataset consisting of observations $z_{i} = (x_{i},y_{i})$ for $i = 1,\dots ,n$ , where $x_{i}\in \mathbb{R}^{d}$ denote the feature vectors, and $y_{i}\in \{0,1\}$ denote the corresponding responses. The feature vectors are organized into the design matrix + +$$ +X \triangleq (x _ {1}, \ldots , x _ {n}) ^ {\top} \in \mathbb {R} ^ {n \times d}. +$$ + +We assume that the relationship between $y_{i}$ and $x_{i}$ follows the model + +$$ +\Pr \left(y _ {i} = 1 \mid x _ {i}\right) = \frac {1}{1 + \exp \left(- x _ {i} ^ {\top} \theta^ {*}\right)}, \tag {4} +$$ + +where $\theta^{*}$ is an $s^*$ -sparse vector representing the underlying ground truth parameter. The objective function is defined as + +$$ +f (\theta) = \frac {1}{n} \sum_ {i = 1} ^ {n} \log \left(1 + \exp (x _ {i} ^ {T} \theta)\right) - y _ {i} x _ {i} ^ {T} \theta . +$$ + +We assume that each covariate vector $x_{i}$ is drawn independently from a multivariate normal distribution $\mathcal{N}(0,\Sigma)$ , where $\Sigma$ is a non-singular covariance matrix. By invoking Corollary 1 in Yuan and Li (2021), we can establish that the objective function $f(\theta)$ satisfies the RSS and RSC conditions, as formalized in the following lemma. + +Lemma 1. Consider the sparse linear logistic regression problem described above. Suppose the covariates $x_{i}$ are uniformly bounded such that $\| x_{i}\| \leq 1$ for all $i\in [n]$ . Then $f(\theta)$ is $L$ -smooth with $\bar{L} = 1$ . Moreover, with probability at least $1 - e^{-c_0n}$ , the RSC condition holds with curvature parameter $\mu \coloneqq \frac{1}{2}\exp (-4R)\sigma_{\min}(\Sigma)$ and tolerance $\tau \coloneqq c_{1}\exp (-4R)\zeta (\Sigma)\frac{\log d}{n}$ , where $R\coloneqq \| \theta^{*}\|$ , $\zeta (\Sigma) = \max_{i = 1,\dots,d}\Sigma_{ii}$ , and $c_{0},c_{1} > 0$ are universal constants. + +Corollary 2. Consider the sparse linear logistic regression problem described above. Under the assumptions of Lemma 1, further suppose that the sample size is sufficiently large so that $\bar{\mu} > 0$ . Further, assume the design matrix $X \in \mathbb{R}^{n \times d}$ is normalized such that $\| X_j / \sqrt{n} \| \leq C$ for all $j = 1, \ldots, d$ . Let $\{\theta_t\}_{t \geq 0}$ be the sequence of iterates produced by Algorithm 1 when applied to the sparse logistic regression problem. Assume the sparsity parameter satisfies $s \geq (240\bar{\kappa})^2 s^*$ , and $\widehat{f} = f(\theta^*)$ . Then, with probability at least $1 - e^{-c_0n} - \frac{2}{d}$ , the following hold: + +(i) If $\| \theta_t - \theta^*\|^2 \geq 72C^2\frac{s\log d}{n\bar{\mu}^2}$ , the iterates exhibit contraction toward $\theta^*$ , i.e., $\| \theta_{t + 1} - \theta^{*}\|^{2} \leq \left(1 - \frac{1}{80\bar{\kappa}}\right)\|\theta_t-\theta^*\|^2$ . (ii) Let $t_0$ denote the first iteration at which $\| \theta_{t_0} - \theta^*\|^2 < 72C^2\frac{s\log d}{n\bar{\mu}^2}$ . Then for all $t \geq t_0$ , the iterates remain confined in a neighborhood of $\theta^*$ : $\| \theta_t - \theta^*\|^2 \leq \left(1 + \frac{1}{80\bar{\kappa}}\right)72C^2\frac{s\log d}{n\bar{\mu}^2}$ . + +(iii) If $\theta^{*}$ satisfies the SNR condition (3), then the iterates remain confined in a neighborhood of $\theta^{*}$ $\| \theta_t - \theta^*\| ^2\leq 72C^2\frac{s\log d}{n\bar{\mu}^2}$ for all $t\geq t_0$ , and the support of $\theta^{*}$ is exactly recovered and preserved for all subsequent iterations. + +The proof of Corollary 2 is provided in Appendix A.3. + +Remark 1. The assumption $\| x_{i}\| \leq 1, \forall i\in [n]$ is required in Yuan and Li (2021) to provide performance guarantees of HT with a fixed step-size on Logistic Regression. However, we note that this assumption is extremely restrictive in the high-dimensional setting. We provide additional results that do not require $\| x_{i}\| \leq 1, \forall i\in [n]$ in Appendix C.2. Our results in Appendix C.2 further apply to additional GLMs that do not satisfy the RSC condition (2) globally, and are aligned with the results in Loh and Wainwright (2015) both in terms of sample and asymptotic convergence rates. In these cases, convergence at a rate of $(1 - c_0 / \bar{\kappa})$ can be guaranteed if the algorithm is suitably initialized. However, a stricter requirement on the sparsity level, specifically $s\geq \mathcal{O}(\bar{\kappa}^4)s^*$ , + +is necessary otherwise, and a rate of $\left(1 - 1 / c_1\bar{\kappa}^2\right)$ can be guaranteed. Here $c_{0}$ , and $c_{1}$ are universal constants. + +The result above and the result in Appendix C.2 match the observed behavior of Sparse Polyak in Fig 1. Namely, we observe that Sparse Polyak indeed achieves rate invariance, as $\frac{d}{n} \to \infty$ , the rate and final precision remain constant as long as $\frac{s \log d}{n}$ is left unchanged. + +# 4.2 Matrix Regression + +Consider the data generation model + +$$ +y _ {i} = \left\langle X _ {i}, \Theta^ {*} \right\rangle + \varepsilon_ {i}, \quad \text {f o r} i = 1, 2 \dots , n, +$$ + +where $\Theta^{*} \in \mathbb{R}^{d \times d}$ is a matrix of rank at most $s^{*}$ . We assume, $X_{i} \in \mathbb{R}^{d \times d}$ , $\operatorname{vec}(X_{i}) \sim \mathcal{N}(0, \Sigma)$ are i.i.d, and $\Sigma \succ 0$ . Further, $\varepsilon_{i} \sim N(0, \sigma^{2})$ are i.i.d. and independent of $X_{i}$ . Define $f(\Theta) = \frac{1}{2n} \sum_{i=1}^{n} (y_{i} - \langle X_{i}, \Theta \rangle)^{2}$ . + +By invoking (Agarwal et al., 2012, Lemma 7), we establish the RSS and RSC properties of $f(\Theta)$ , as formalized in the following lemma. + +Lemma 2. Consider the low-rank matrix regression problem described above. Then, with probability at least $1 - e^{-c_0n}$ , $f(\Theta)$ satisfies the RSS and RSC conditions with respect to the Frobenius norm and the nuclear norm. The corresponding parameters are given by: + +$$ +L = 2 \sigma_ {\mathrm {m a x}} (\Sigma), \quad \mu = \frac {1}{2} \sigma_ {\mathrm {m i n}} (\Sigma), \quad a n d \quad \tau = c _ {1} \zeta (\Sigma) \frac {d}{n}, +$$ + +where $\zeta(\Sigma) := \sup_{\|u\|=1, \|v\|=1} \operatorname{Var}(u^\top X_1 v)$ , and $c_0, c_1 > 0$ are universal constants. + +To enforce the suitable low-rank structure on the iterates we define + +$$ +\mathrm {P M} _ {s} (W) = \sum_ {i = 1} ^ {s} \sigma_ {i} u _ {i} v _ {i} ^ {T}, +$$ + +where $\sigma_{i}, i = 1, \ldots, s$ are the $s$ largest singular values of $W$ , and $u_{i}, v_{i}$ the corresponding singular vectors. We substitute all instances of the $\mathrm{HT}_s$ operator by $\mathrm{PM}_s$ in Algorithm 1 and 2 (c.f. Appendix B). + +Corollary 3. Consider the low rank matrix regression problem described above. Let $\{\Theta_t\}_{t\geq 0}$ be the sequence of iterates generated by Algorithm 1 when applied to a low rank matrix regression problem. Suppose that $\widehat{f} = f(\Theta^{*})$ $n$ is sufficiently large such that $\bar{\mu} >0$ , and $s\geq (240\bar{\kappa})^2 s^*$ . Then, with probability at least $1 - e^{-c_0n} - 2e^{-4d}$ , the following holds: (i) If $\| \Theta_t - \Theta^*\| _F^2\geq \frac{7200\sigma^2\zeta(\Sigma)sd}{n\bar{\mu}^2}$ then the iterates contract relative to $\Theta^{*}$ as $\| \Theta_{t + 1} - \Theta^{*}\|_{F}^{2}\leq \left(1 - \frac{1}{80\bar{\kappa}}\right)\| \Theta_{t} - \Theta^{*}\|_{F}^{2}$ (ii) Let $t_0$ be the first iteration for which $\| \Theta_{t_0} - \Theta^*\| _F^2 < \frac{7200\sigma^2\zeta(\Sigma)sd}{n\bar{\mu}^2}$ . Then, for all $t\geq t_0$ , the iterates remain in a stable neighborhood around $\Theta^{*}$ , with $\| \Theta_t - \Theta^*\| _F^2\leq \left(1 + \frac{1}{80\bar{\kappa}}\right)\frac{7200\sigma^2\zeta(\Sigma)sd}{n\bar{\mu}^2}$ + +The proof of Corollary 3 can be found in Appendix A.4. + +Remark 2. The result in Corollary 1 also extends to Algorithm 2. If $\theta^{*}$ satisfies the SNR condition (3), the iterates of Algorithm 2 recover the support after $\widetilde{T}$ gradient descent steps. + +# 5 Sketch of the Proof for Theorem 1 and Corollary 1 + +We provide a sketch of the proof of the main results of the paper. We refer the reader to the appendix for the complete proof. The proof of Theorem 1 follows the outline: (i) study the behavior of $\| \theta_{t + 1} - \widehat{\theta}\| ^2$ given $\gamma_{t}$ and $\| \theta_t - \widehat{\theta}\|$ , (ii) establish that under the assumption that $\gamma_{t}$ is sufficiently large the expansive effect of the Hard Thresholding operator can be offset by the contractive effect of the gradient update, (iii) show that $\gamma_{t}$ is sufficiently large until we reach optimal statistical precision. Both to finalize the proof of Theorem 1 and to establish Corollary 1: (iv) the iterates remain confined within a neighborhood of $\theta^{*}$ , and, given that the SNR condition holds, the support can be identified, providing further benefits to the algorithm's performance. We elaborate on points (i) through (iv). + +(i) To understand the dynamics of $\| \theta_t - \widehat{\theta}\|^2$ we analyze the combined effect of gradient descent under the RSC and RSS and the Hard Thresholding operator. + +For this we suitably apply Lemma 1 from Jain et al. (2014), exploit the RSC, and the properties of $\theta_{t + 1}$ in relation to $\mathrm{HT}_s$ to yield: + +$$ +\begin{array}{l} \left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left(\left(1 - \bar {\mu} \gamma_ {t}\right) \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} \right. (5a) \\ \left. - 2 \gamma_ {t} \left(f \left(\theta_ {t}\right) - f \left(\theta^ {*}\right)\right) + 1 0 \gamma_ {t} ^ {2} \left\| \mathrm {H T} _ {s} \left(\nabla f \left(\theta_ {t}\right)\right) \right\| ^ {2}\right). (5b) \\ \end{array} +$$ + +Setting $\gamma_{t} = \frac{f(\theta_{t}) - f(\theta^{*})}{5\|\mathrm{HT}_{s}(\nabla f(\theta_{t}))\|^{2}}$ makes (5b) nonpositive and yields a choice that is invariant with $d$ . The use of $\| \mathrm{HT}_s(\nabla f(\theta_t))\|^2$ in $\gamma_{t}$ and the RSS are critical to avoiding a step-size that scales with the Lipschitz smoothness constant (which scales as $\mathcal{O}(d)$ in the high-dimensional setting). Setting $\gamma_{t} = 1 / (40\bar{L})$ in (5) recovers the result in Jain et al. (2014), which establishes that IHT requires at most $\mathcal{O}(\bar{\kappa}^{-1}\log (1 / \varepsilon))$ to be within $\varepsilon$ -accuracy of near optimal statistical precision. + +(ii) To achieve the optimal linear rate under our choice of step size, we require that + +$$ +\left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} (1 - \bar {\mu} \gamma_ {t}) \leq 1 - c _ {0} \frac {\bar {\mu}}{\bar {L}}. \tag {6} +$$ + +Sufficient conditions for (6) are $s \geq c_1 \bar{\kappa}^2 s^*$ and $\gamma_t \geq \frac{c_2}{L}$ for some universal constants $c_0, c_1$ and $c_2$ . However, if $\gamma_t$ were to scale with the Lipschitz smoothness constant, i.e. with $d^{-1}$ we would require $s \geq d^2 s^*$ to establish linear convergence. Observe that this requirement can not be fulfilled as $s \leq d$ . + +(iii) We exploit the RSS and RSC to show that $\gamma_t \geq \frac{1}{40\bar{L}}$ when $\| \theta_t - \widehat{\theta} \|^2 \geq \frac{36 \| \nabla f(\widehat{\theta}) \|_s^2}{\bar{\mu}^2}$ . If, further, $s \geq (240\bar{\kappa})^2 s^*$ , (6) holds with $c_0 = 1/160$ , yielding $\| \theta_{t+1} - \widehat{\theta} \|^2 \leq (1 - c_0\bar{\kappa}^{-1}) \| \theta_t - \widehat{\theta} \|^2$ . + +The condition $\| \theta_t - \widehat{\theta}\|^2 \geq \frac{36\|\mathrm{HT}_s(\nabla f(\widehat{\theta}))\|^2}{\bar{\mu}^2}$ stems from (1) being a constrained problem. If the Polyak step-size had been left unaltered, additional regularity conditions are required to establish convergence in the constrained case. As established in Theorem 3 in Polyak (1969), one such condition is $\frac{f(\theta_t) - f(\widehat{\theta})}{\|\theta_t - \widehat{\theta}\|} \geq c$ for some $c > 0$ and any $\theta_t \in \mathbb{R}^d$ , which does not uniformly in high-dimensional M-estimation. For some GLMs, we can show the condition holds locally and exploit this fact to provide more general results in Appendix C.2. + +(iv) From (i)-(iii) it follows that there exists $t_0$ at which $\| \theta_{t_0} - \widehat{\theta}\| ^2\leq \frac{36\|\mathrm{HT}_s(\nabla f(\widehat{\theta}))\|^2}{\widehat{\mu}^2}$ . Since the potential expansion is at most $\left(1 + \frac{1}{80\bar{\kappa}}\right)$ Theorem 1 follows. If $\widehat{\theta}$ satisfies (3), we show that the inequality $\| \theta_{t_0} - \widehat{\theta}\| ^2\leq \frac{36\|\mathrm{HT}_s(\nabla f(\widehat{\theta}))\|^2}{\widehat{\mu}^2}$ guarantees that $\widehat{S}\subset S_{t_0}$ . From here, we establish that, this results in two possible scenarios: (a) $\gamma_t < \frac{1}{40L}$ , implying $\widehat{S}\subset S_{t_0 + 1}$ , ensuring that $\| [\widetilde{\theta}_{t_0 + 1} - \widehat{\theta} ]_{\widehat{S}_{t + 1}}\| ^2 = \| \widetilde{\theta}_{t_0 + 1} - \widehat{\theta}\| ^2$ , and eliminating the expansion term in (5); or, (b) $\gamma_t\geq \frac{1}{40L}$ , and (6) holds. Based on (a) and (b), Corollary 1 is established by induction. + +# 6 Numerical experiments + +We first consider sparse linear regression and sparse logistic regression on synthetic data. This is done to illustrate the algorithm's performance as the size of the problem grows while the problem conditioning is kept the same. In all scenarios that rely on synthetic data, we set $d \in \{5000, 10000, 20000\}$ , and $s^* = 300$ . The design matrix $X \in \mathbb{R}^{d \times n}$ is generated to reflect a time-series structure with a correlation parameter $\omega = 0.5$ . We set the sample size $n$ according to $n = \lceil \alpha s \log d \rceil$ , where $\alpha > 0$ is a constant. In this section, we use $\alpha = 5$ . For each column index $j \in \{1, \dots, n\}$ , we generate a sequence of i.i.d. standard normal variables $\varepsilon_1, \ldots, \varepsilon_{d-1}$ , and construct $x_{1,j} = \varepsilon_1 / \sqrt{1 - \omega^2}$ . The subsequent entries are generated recursively as $x_{t+1,j} = \omega x_{t,j} + \varepsilon_t$ for $t \in \{1, \dots, d-1\}$ , where $\varepsilon_t \sim \mathcal{N}(0,1)$ . The true parameter $\theta^*$ is created by sampling each entry from $\mathcal{N}(0,1)$ , and assigning nonzero values to $s^*$ randomly chosen entries. + +![](images/9de89bf5c22666749b91e70229bf8b7e379ad52b33846b9e99241f182963a51b.jpg) +Figure 2: Left and center: IHT with $\frac{2}{3L}$ (blue) vs. Algorithm 1 (red) on linear and logistic regression respectively. Right: Choice of $\widehat{f}$ on Algorithm 1. In all scenarios $\alpha = 5$ , $d = 5000$ and $s = 700$ . + +![](images/3bb2c4751827afd6d8399e8faf82e85317a9cc29b6f986000ad8056679602d0b.jpg) + +![](images/7500bfa1c5abdef4ef7777eca8ef89485d8acbd011eec77cac32c88eecf94bf0.jpg) + +In the case of linear regression, each sample $i$ (where $i \in \{1, \dots, n\}$ ) is generated according to the model: + +$$ +y _ {i} = x _ {i} ^ {T} \theta^ {*} + w _ {i}, \quad w _ {i} \sim \mathcal {N} (0, 0. 2 5). +$$ + +For logistic regression, the relationship between $y_{i}$ and $x_{i}$ follows the model (4). + +(i) Comparison to fixed step-size: In Figure 6 we present a comparison between IHT with a fixed step size (blue) and the adaptive step size used in Algorithm 1 (red) when solving linear regression (left panel) and logistic regression (center panel) respectively. When working with a constant step-size, we set the step size to $\frac{2}{3L}$ following Jain et al. (2014) for both linear and logistic regression. $\bar{L}$ for linear regression can be upper bounded as $\lambda_{\max}(\Sigma)\left(3 + \frac{2(2s + s^{*})}{s\alpha}\right)$ (Loh and Wainwright, 2015, Appendix D.1), whereas $\bar{L}$ for logistic regression is one fourth of that of linear regression. In both settings, $\lambda_{\max}(\Sigma) \leq \frac{2}{(1 - \omega)^2(1 + \omega)}$ Agarwal et al. (2012). Although both step size strategies share the same theoretical guarantees, we observe that the adaptive step size speeds-up convergence. This advantage arises because an adaptive step size can adapt to the local curvature and therefore be significantly more aggressive, allowing for faster progress towards the solution. As expected, this effect is more pronounced when solving logistic regression where the functions' curvature will depend on the point. (ii) Rate invariance: In Figure 1 (c.f. Section 1), we compare the classical Polyak step size with the step size employed in Algorithm 1 across different problem dimensions $d$ , while maintaining $\alpha = 5$ . The solid line represents the performance of Algorithm 1, whereas the dashed line corresponds to the classical Polyak step size. This demonstrates that when the condition number of $\Sigma$ remains unchanged, the complexity of the method remains almost identical under our chosen step size. In contrast, Polyak's step size leads to an increased number of iterations, even if the achievable statistical precision and $(\lambda_{\max}(\Sigma) / \lambda_{\min}(\Sigma))$ remain the same. +(iii) Choice of $\widehat{f}$ : Finally, Figure 6 (right) highlights the impact of the choice of $\widehat{f}$ . The results confirm that $\widehat{f}$ determines the best achievable accuracy. Additionally, from the formulation of $\gamma_{t}$ , we observe that $\widehat{f}$ directly influences the step size magnitude, thus is impacting the convergence rate. + +Experiments on real world data are provided in Appendix D. All experiments were conducted on a laptop equipped with 16 GB of RAM and a 12th Gen Intel Core i5-12500H 3.10 GHz CPU. + +# References + +Alekh Agarwal, Sahand Negahban, and Martin J. Wainwright. Fast global convergence of gradient methods for high-dimensional statistical recovery. The Annals of Statistics, 40(5):2452-2482, 2012. +Kyriakos Axiotis and Maxim Sviridenko. Iterative hard thresholding with adaptive regularization: Sparser solutions without sacrificing runtime. In International Conference on Machine Learning, pages 1175-1197. PMLR, 2022. +Jonathan Barzilai and Jonathan M Borwein. Two-point step size gradient methods. IMA journal of numerical analysis, 8(1):141-148, 1988. +Thomas Blumensath and Mike E Davies. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +A Main Theorems 21 + +A.1 Proof of Theorem 1 24 +A.2 Proof of Corollary 1. 24 +A.3 Proof of Corollary 2 25 +A.4 Proof of Corollary 3 26 +B Adaptive Lower Bound 26 +B.1 Proof of Theorem 2 26 +C Other Statistical Guarantees 27 +C.1 Sparse Linear Regression 27 +C.2 Generalized Linear Models 29 +D Experiments on real data 32 +D.1 Linear Regression 32 +D.2 Logistic Regression 32 + +# A Main Theorems + +In this section we provide the formal proof to the statements included in the main body of the paper. To formally establish Theorem 1 we build on (Jain et al., 2014, Lemma 1) and Lemmas 4-6. (Jain et al., 2014, Lemma 1) allows us to control the expansive properties of the Hard Thresholding operator, whereas Lemma 4 allows us to establish (5) (c.f. Section 5). Further, in Lemma 5 we establish consequences of the RSS (Assumption 2) that are instrumental in establishing a lower bound on the step-size $\gamma_{t}$ . We lower bound $\gamma_{t}$ in Lemma 6 and establish conditions under which this lower bound holds. We combine these results in the proof of Theorem 1 in Appendix A.1, followed by a formal proof of the corollaries included in the main body of the paper in the remaining sections of Appendix A. + +For simplicity, we let $\widehat{S}_t = S_t \cup \widehat{S}$ . Also, we let $g_t = \nabla f(\theta_t)$ , and $\widehat{g} = \nabla f(\widehat{\theta})$ . When discussing specific statistical models we denote by $\theta^*$ the ground truth, $g^* = \nabla f(\theta^*)$ and $S^* = \mathrm{supp}(\theta^*)$ . For any index set $S$ and vector $\theta \in \mathbb{R}^d$ , we define $[\theta]_S$ as the vector that retains the entries indexed by $S$ , while setting all other entries to zero. + +We include the following fundamental lemma, which plays a key role in our analysis. It corresponds to Lemma 1 in Jain et al. (2014), and is presented here for completeness. + +Lemma 3. For any index set $I$ , any $z \in \mathbb{R}^I$ , let $\theta = \mathrm{HT}_s(z)$ . Then for any $\widehat{\theta} \in \mathbb{R}^I$ such that $\|\widehat{\theta}\|_0 \leq s^*$ , we have + +$$ +\left\| \theta - z \right\| ^ {2} \leq \frac {| I | - s}{| I | - s ^ {*}} \left\| \widehat {\theta} - z \right\| ^ {2}. +$$ + +Lemma 4. Let $\widehat{\theta}$ be any $s^*$ -sparse vector, and $\theta_t$ be any $s$ -sparse vector. Assume that the function $f$ fulfills Assumption 1 with $\bar{\mu} = \mu - 3\tau s > 0$ . Let $\theta_{t+1} \coloneqq \mathrm{HT}_s(\theta_t - \gamma_t g_t)$ , and $\widetilde{\theta}_{t+1} \coloneqq \theta_t - \gamma_t g_t$ . For any $\gamma_t \leq \frac{f(\theta_t) - f(\widehat{\theta})}{5\|\mathrm{HT}_s(\widehat{g})\|^2}$ we have + +$$ +\begin{array}{l} \left\| \theta_ {t + 1} - \widehat {\theta} \right\| ^ {2} \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left\| \left[ \widetilde {\theta} _ {t + 1} \right] _ {\widehat {\mathcal {S}} _ {t + 1}} - \widehat {\theta} \right\| ^ {2} \\ \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left(1 - \bar {\mu} \gamma_ {t}\right) \| \theta_ {t} - \widehat {\theta} \| ^ {2}. \\ \end{array} +$$ + +Proof. For the first inequality, + +$$ +\| \theta_ {t + 1} - \widehat {\theta} \| \stackrel {(i)} {\leq} \| \theta_ {t + 1} - [ \widetilde {\theta} _ {t + 1} ] _ {\widehat {\mathcal {S}} _ {t + 1}} \| + \| [ \widetilde {\theta} _ {t + 1} ] _ {\widehat {\mathcal {S}} _ {t + 1}} - \widehat {\theta} \| \stackrel {(i i)} {\leq} \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) \| [ \widetilde {\theta} _ {t + 1} ] _ {\widehat {\mathcal {S}} _ {t + 1}} - \widehat {\theta} \|, +$$ + +where in $(i)$ we use the triangle inequality and in $(ii)$ we used Lemma 3. + +For the second inequality, we consider the expansion + +$$ +\begin{array}{l} \left\| \left[ \tilde {\theta} _ {t + 1} \right] _ {\hat {\mathcal {S}} _ {t + 1}} - \widehat {\theta} \right\| ^ {2} = \left\| \left[ \tilde {\theta} _ {t + 1} - \widehat {\theta} \right] _ {\hat {\mathcal {S}} _ {t + 1}} \right\| ^ {2} \\ = \| [ \theta_ {t} - \widehat {\theta} ] _ {\widehat {\mathcal {S}} _ {t + 1}} \| ^ {2} - 2 \gamma_ {t} \langle \theta_ {t} - \widehat {\theta}, [ g _ {t} ] _ {\widehat {\mathcal {S}} _ {t + 1}} \rangle + \gamma_ {t} ^ {2} \| [ g _ {t} ] _ {\widehat {\mathcal {S}} _ {t + 1}} \| ^ {2}, \\ \end{array} +$$ + +where + +$$ +- 2 \gamma_ {t} \langle \theta_ {t} - \widehat {\theta}, [ g _ {t} ] _ {\widehat {S} _ {t + 1}} \rangle = - 2 \gamma_ {t} \langle \theta_ {t} - \widehat {\theta}, g _ {t} \rangle + 2 \gamma_ {t} \langle \theta_ {t} - \widehat {\theta}, [ g _ {t} ] _ {S _ {t} \backslash \widehat {S} _ {t + 1}} \rangle , +$$ + +and consequently + +$$ +\left\| [ \widehat {\theta} _ {t + 1} - \widehat {\theta} ] _ {\widehat {\mathcal {S}} _ {t + 1}} \right\| ^ {2} \leq \| \theta_ {t} - \widehat {\theta} \| ^ {2} - 2 \gamma_ {t} \langle \theta_ {t} - \widehat {\theta}, g _ {t} - [ g _ {t} ] _ {\mathcal {S} _ {t} \setminus \widehat {\mathcal {S}} _ {t + 1}} \rangle + \gamma_ {t} ^ {2} \left\| [ g _ {t} ] _ {\widehat {\mathcal {S}} _ {t + 1}} \right\| ^ {2} +$$ + +Using the RSC yields + +$$ +\begin{array}{l} \left\| \left[ \widetilde {\theta} _ {t + 1} - \widehat {\theta} \right] _ {\widehat {S} _ {t + 1}} \right\| ^ {2} \leq (1 - \bar {\mu} \gamma_ {t}) \| \theta_ {t} - \widehat {\theta} \| ^ {2} - 2 \gamma_ {t} (f (\theta_ {t}) - f (\widehat {\theta})) + \gamma_ {t} ^ {2} \| \mathrm {H T} _ {s + s ^ {*}} (g _ {t}) \| ^ {2} \\ + 2 \gamma_ {t} \left\langle \theta_ {t} - \widehat {\theta}, \left[ g _ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \right\rangle . \tag {7} \\ \end{array} +$$ + +To obtain (5) we must upper bound the inner product in (7), for which we have: + +$$ +\begin{array}{l} \langle [ \theta_ {t} ] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}}, \gamma_ {t} [ g _ {t} ] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \rangle = \langle [ \theta_ {t} ] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}}, \gamma_ {t} [ g _ {t} ] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \rangle \\ = \left\langle \left[ \theta_ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} - \gamma_ {t} \left[ g _ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} + \gamma_ {t} \left[ g _ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}}, \gamma_ {t} \left[ g _ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \right\rangle \\ \leq \left\| \left[ \theta_ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} - \gamma_ {t} \left[ g _ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \right\| \left\| \gamma_ {t} \left[ g _ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \right\| + \left\| \gamma_ {t} \left[ g _ {t} \right] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \right\| ^ {2}. \\ \end{array} +$$ + +Then, + +$$ +\| [ \theta_ {t} - \gamma_ {t} g _ {t} ] _ {\mathcal {S} _ {t} \setminus \widehat {\mathcal {S}} _ {t + 1}} \| \overset {(i)} {\leq} \| [ \theta_ {t} - \gamma_ {t} g _ {t} ] _ {\mathcal {S} _ {t} \setminus \mathcal {S} _ {t + 1}} \| \overset {(i i)} {\leq} \| [ \theta_ {t} - \gamma_ {t} g _ {t} ] _ {\mathcal {S} _ {t + 1} \setminus \mathcal {S} _ {t}} \| = \| [ \gamma_ {t} g _ {t} ] _ {\mathcal {S} _ {t + 1} \setminus \mathcal {S} _ {t}} \|, +$$ + +where in $(i)$ we use that $\mathcal{S}_t\setminus \widehat{\mathcal{S}}_{t + 1}\subseteq \mathcal{S}_t\setminus \mathcal{S}_{t + 1}$ , and in $(ii)$ we exploit that $|\mathcal{S}_t\setminus \mathcal{S}_{t + 1}| = |\mathcal{S}_{t + 1}\setminus \mathcal{S}_t|$ and that $\mathcal{S}_{t + 1}$ contains the indexes of the $s$ largest elements of $\theta_t - \gamma_tg_t$ . Thus, we obtain the overall upper bound + +$$ +2 \gamma_ {t} \langle \theta_ {t} - \widehat {\theta}, [ g _ {t} ] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \rangle \leq 2 \gamma_ {t} ^ {2} \| [ g _ {t} ] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \| \| [ g _ {t} ] _ {\mathcal {S} _ {t + 1} \backslash \mathcal {S} _ {t}} \| + 2 \gamma_ {t} ^ {2} \| [ g _ {t} ] _ {\mathcal {S} _ {t} \backslash \widehat {\mathcal {S}} _ {t + 1}} \| ^ {2}, +$$ + +which together with (7) yields + +$$ +\| \left[ \widetilde {\theta} _ {t + 1} - \widehat {\theta} \right] _ {\widehat {\mathcal {S}} _ {t}} \| ^ {2} \leq (1 - \bar {\mu} \gamma_ {t}) \| \theta_ {t} - \widehat {\theta} \| ^ {2} - 2 \gamma_ {t} (f (\theta_ {t}) - f (\widehat {\theta})) + 5 \gamma_ {t} ^ {2} \| \mathrm {H T} _ {2 s} (g _ {t}) \| ^ {2}. \tag {8} +$$ + +Given the upperbound on $\gamma_{t}$ , the two right most terms together are negative, and we thus the proof is complete. + +Observe that in the proof of Lemma 4 we use the iterate $\tilde{\theta}_{t + 1}$ to treat the effect of the hard thresholding operator and gradient descent separately. We then restrict to the support $\widehat{S}_{t + 1}$ to avoid the scaling of any bound with the ambient dimension $d$ . + +Lemma 5. (RSS-gradient bound) Assume that $f$ fulfills Assumptions 1 where $\bar{\mu} = \mu - 3\tau s > 0$ , and Assumption 2. Then, for any pair $x, y$ of $s$ -sparse vectors there holds + +$$ +\frac {1}{2 (L + 3 \tau s)} \| \mathrm {H T} _ {s} (\nabla f (x) - \nabla f (y)) \| ^ {2} \leq f (y) - f (x) - \langle \nabla f (x), y - x \rangle . +$$ + +Proof. We define + +$$ +\phi (t) = f (t) - \langle \nabla f (x), t - x \rangle . +$$ + +From its formulation, we know $\phi(t)$ inherits the RSS and RSC property of $f$ . + +As a result, for any $2s$ -sparse vector $z$ , we have + +$$ +\begin{array}{l} \phi (x) \leq \phi (z) - \langle 0, z - x \rangle - \frac {\bar {\alpha}}{2} \| x - z \| ^ {2} \\ = \phi (z) - \frac {\bar {\alpha}}{2} \| x - z \| ^ {2} \\ \leq \phi (z) \\ \leq \phi (y) + \langle \nabla \phi (y), z - y \rangle + \frac {L + 3 \tau s}{2} \| z - y \| ^ {2}. \\ \end{array} +$$ + +Set $z = y - \frac{1}{L + 3\tau s}\mathrm{HT}_s(\nabla \phi (y))$ , the inequality above indicates + +$$ +\phi (x) \leq \phi (y) - \frac {1}{2 (L + 3 \tau s)} \| \mathrm {H T} _ {s} (\nabla \phi (y)) \| ^ {2}, +$$ + +which is equivalent to + +$$ +f (x) + \langle \nabla f (x), y - x \rangle + \frac {1}{2 (L + 3 \tau s)} \| \mathrm {H T} _ {s} (\nabla f (y) - \nabla f (x)) \| ^ {2} \leq f (y). +$$ + +![](images/bb7fa406ac43c6d1444c9230d8b7d87f5e1e117dc10963a9840b6a25530cf7ae.jpg) + +Lemma 6. Consider the iterates $\{\theta_t\}_{t\geq 1}$ generated by Algorithm 1 to solve (1). Assume that $f$ fulfills Assumptions 1 where $\bar{\mu} = \mu -3\tau s > 0$ , and 2. Further, denote by $\widehat{\theta}$ an arbitrary $s^*$ -sparse vector for which $f(\widehat{\theta})$ is known and desirable. Then, the step-size + +$$ +\gamma_ {t} = \frac {f (\theta_ {t}) - f (\widehat {\theta})}{5 \| \mathrm {H T} _ {s} (g _ {t}) \| ^ {2}} \geq \frac {1}{4 0 \bar {L}} +$$ + +for each $t \geq 0$ for which + +$$ +\left\| \theta_ {t} - \widehat {\theta} \right\| ^ {2} \geq \frac {1 8 \left\| \mathrm {H T} _ {s + s ^ {*}} (\widehat {g}) \right\| ^ {2}}{\bar {\mu} ^ {2}} +$$ + +$$ +f (\theta_ {t}) > f (\widehat {\theta}). +$$ + +Proof. Assume that $f(\theta_t) - f(\widehat{\theta}) > 0$ , then + +$$ +\gamma_ {t} \geq \frac {f (\theta_ {t}) - f (\widehat {\theta})}{1 0 \| \mathrm {H T} _ {s} (g _ {t} - \widehat {g}) \| ^ {2} + 1 0 \| \mathrm {H T} _ {s} (\widehat {g}) \| ^ {2}}. +$$ + +Given that $f$ fulfills Assumption 2 we may invoke Lemma 5 yielding the bound + +$$ +\gamma_ {t} \geq \frac {f (\theta_ {t}) - f (\widehat {\theta})}{2 0 \bar {L} (f (\theta_ {t}) - f (\widehat {\theta}) - \langle \widehat {g} , \theta_ {t} - \widehat {\theta} \rangle) + 1 0 \| \mathrm {H T} _ {s} (\widehat {g}) \| ^ {2}}. +$$ + +If + +$$ +1 0 \left(2 \bar {L} \langle \hat {g}, \hat {\theta} - \theta_ {t} \rangle + \| \mathrm {H T} _ {s} (\hat {g}) \| ^ {2}\right) \leq 2 0 \bar {L} (f (\theta_ {t}) - f (\hat {\theta})) \tag {9} +$$ + +we can guarantee that + +$$ +\gamma_ {t} \geq \frac {1}{4 0 \bar {L}}. +$$ + +Rearranging (9) we have that the condition can be equivalently written as + +$$ +\left\| \mathrm {H T} _ {s} (\widehat {g}) \right\| ^ {2} \leq 2 \bar {L} \left(f \left(\theta_ {t}\right) - f (\widehat {\theta}) + \langle \widehat {g}, \theta_ {t} - \widehat {\theta} \rangle\right). +$$ + +Invoking the RSC, a sufficient condition for the above is + +$$ +\left\| \mathrm {H T} _ {s} (\widehat {g}) \right\| ^ {2} \leq 2 \bar {L} \left(\frac {\bar {\mu}}{2} \left\| \theta_ {t} - \widehat {\theta} \right\| ^ {2} + 2 \langle \widehat {g}, \theta_ {t} - \widehat {\theta} \rangle\right), +$$ + +which can be guaranteed as long as + +$$ +\begin{array}{l} \left\| \mathrm {H T} _ {s} (\widehat {g}) \right\| ^ {2} \leq 2 \bar {L} \left(\frac {\bar {\mu}}{2} \left\| \theta_ {t} - \widehat {\theta} \right\| ^ {2} - 2 \left\| \mathrm {H T} _ {s + s ^ {*}} (\widehat {g}) \right\| \left\| \theta_ {t} - \widehat {\theta} \right\|\right) \\ = 2 \bar {L} \left(\frac {\bar {\mu}}{2} \| \theta_ {t} - \widehat {\theta} \| - 2 \| \mathrm {H T} _ {s + s ^ {*}} (\widehat {g}) \|\right) \| \theta_ {t} - \widehat {\theta} \|. \\ \end{array} +$$ + +To guarantee that the above holds it is sufficient to request that $\| \theta_t - \widehat{\theta}\| \geq \frac{106}{25\mu}\| \mathrm{HT}_{s + s^*}(\widehat{g})\|$ , and thus the result follows. + +![](images/85bdfce3d33f712c5d51130fe8fa93989ecc328d4b07fab6e48048f3552db625.jpg) + +# A.1 Proof of Theorem 1 + +As a consequence of Lemma 6 we distinguish three cases for any $t: (\mathbf{i}) \|\theta_t - \widehat{\theta}\|^2 \geq \frac{36\|\mathrm{HT}_s(\widehat{g})\|^2}{\bar{\mu}^2}$ and $f(\theta_t) - \widehat{f} > 0$ , (ii) $\|\theta_t - \widehat{\theta}\|^2 < \frac{36\|\mathrm{HT}_s(\widehat{g})\|^2}{\mu^2}$ and $f(\theta_t) - \widehat{f} > 0$ , or (iii) $f(\theta_t) - \widehat{f} \leq 0$ . + +In case (iii) no progress is made, i.e. $\theta_{t + 1} = \theta_t$ and by the RSC there holds + +$$ +\frac {\bar {\mu}}{2} \| \theta_ {t} - \widehat {\theta} \| ^ {2} \leq \| \theta_ {t} - \widehat {\theta} \| \| \mathrm {H T} _ {s + s ^ {*}} (\widehat {g}) \| +$$ + +and thus + +$$ +\left\| \theta_ {t + 1} - \widehat {\theta} \right\| ^ {2} = \left\| \theta_ {t} - \widehat {\theta} \right\| ^ {2} < \frac {3 6 \left\| \mathrm {H T} _ {s} (\widehat {g}) \right\| ^ {2}}{\bar {\mu} ^ {2}}. +$$ + +Further, from the above it follows that if $\| \theta_t - \widehat{\theta} \|^2 \geq \frac{36 \| \mathrm{HT}_s(\widehat{g}) \|^2}{\bar{\mu}^2}$ we can guarantee that $\gamma_t > 0$ . + +For case (i) we begin by invoking Lemma 4, which guarantees that + +$$ +\left\| \theta_ {t + 1} - \widehat {\theta} \right\| ^ {2} \leq \left(1 + 3 \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left(1 - \bar {\mu} \gamma_ {t}\right) \left\| \theta_ {t} - \widehat {\theta} \right\| ^ {2}. +$$ + +Using Lemma 6 we can guarantee a lower bound on the step-size $\gamma_t \geq \frac{1}{40L}$ . Further, under our assumption on $s$ , namely, $s \geq (240\bar{\kappa})^2 s^*$ , we can bound the contraction factor: + +$$ +\left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} (1 - \bar {\mu} \gamma_ {t}) \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) (1 - \bar {\mu} \gamma_ {t}) \leq \left(1 + \frac {1}{8 0 \bar {\kappa}}\right) \left(1 - \frac {1}{4 0 \bar {\kappa}}\right) \leq 1 - \frac {1}{8 0 \bar {\kappa}}, +$$ + +and therefore, + +$$ +\left\| \theta_ {t + 1} - \widehat {\theta} \right\| ^ {2} \leq \left(1 - \frac {1}{8 0 \bar {\kappa}}\right) \left\| \theta_ {t} - \widehat {\theta} \right\| ^ {2}. +$$ + +Thus, the first part of the theorem's statement follows, i.e. when $\| \theta_t - \widehat{\theta} \|$ is sufficiently large, we can guarantee that $\theta_{t + 1}$ will approach $\widehat{\theta}$ . We are now left with establishing the veracity of the second statement. For this, let $t_0$ be the time defined in the theorem's statement. Then, we are under case (ii) or case (iii). If we are under case (iii) there is nothing left to proof. If we are in case (ii), there are two further cases: (a) the iterates remain confined within a ball of radius $6\| \mathrm{HT}_s(\widehat{g})\| ^2 /\bar{\mu}^2$ , i.e. + +$$ +\forall t \geq t _ {0}, \| \theta_ {t} - \widehat {\theta} \| ^ {2} < \frac {3 6 \| \mathrm {H T} _ {s} (\widehat {g}) \| ^ {2}}{\bar {\mu} ^ {2}}, +$$ + +and the theorem's second statement is therefore true, or (b) there exists a time $t_1 > t_0$ at which for the first time + +$$ +\frac {3 6 \left\| \mathrm {H T} _ {s} (\widehat {g}) \right\| ^ {2}}{\bar {\mu} ^ {2}} \leq \left\| \theta_ {t _ {1}} - \widehat {\theta} \right\| ^ {2}. +$$ + +By Lemma 4 and by definition of $t_1$ we have + +$$ +\left\| \theta_ {t _ {1}} - \widehat {\theta} \right\| ^ {2} \leq \left(1 + \frac {1}{8 0 \bar {\kappa}}\right) \left(1 - \bar {\mu} \gamma_ {t}\right) \frac {3 6 \left\| \mathrm {H T} _ {s} (\widehat {g}) \right\| ^ {2}}{\bar {\mu} ^ {2}} < \left(1 + \frac {1}{8 0 \bar {\kappa}}\right) \frac {3 6 \left\| \mathrm {H T} _ {s} (\widehat {g}) \right\| ^ {2}}{\bar {\mu} ^ {2}}. +$$ + +This implies, we find ourselves again in case (i). Observe that going forward, no iterate can escape the above ball and therefore the second statement of the theorem holds. + +# A.2 Proof of Corollary 1 + +If (3) holds for $\widehat{\theta}$ , for any $\theta_t$ fulfilling $\| \theta_t - \widehat{\theta} \|^2 \leq \frac{36 \| \mathrm{HT}_s(\widehat{g}) \|^2}{\bar{\mu}^2}$ there holds + +$$ +\min _ {i \in \widehat {\mathcal {S}}} | [ \theta_ {t} ] _ {i} | \geq \frac {7 \| \mathrm {H T} _ {s} (\widehat {g}) \|}{\bar {\mu}} - \frac {6 \| \mathrm {H T} _ {s} (\widehat {g}) \|}{\bar {\mu}} > 0. \tag {10} +$$ + +This can be established by contradiction, i.e. if the condition above is violated, either $\| \theta_t - \widehat{\theta} \|^2 > \frac{36 \| \mathrm{HT}_s(\widehat{g}) \|^2}{\bar{\mu}^2}$ or (3) can not hold. Consequently, $\widehat{S} \subset S^t$ . Now we consider the iterate $\theta_{t+1}$ . Note that $\widehat{S} \subset S^{t+1}$ , if + +$$ +\gamma_ {t} \| g _ {t} \| _ {\infty} < \frac {\| \mathrm {H T} _ {s} (\widehat {g}) \|}{2 \bar {\mu}}. \tag {11} +$$ + +To see this note that for any $i \notin S_t$ we have + +$$ +\left| \left[ \theta_ {t} \right] _ {i} - \gamma_ {t} \left[ g _ {t} \right] _ {i} \right| = \gamma_ {t} \left| \left[ g _ {t} \right] _ {i} \right| < \frac {\left\| \mathrm {H T} _ {s} (\widehat {g}) \right\|}{2 \bar {\mu}}, +$$ + +and for any $i\in \widehat{\mathcal{S}}$ , given that $\| \theta_t - \widehat{\theta}\| \leq \frac{6\|\mathrm{HT}_s(\widehat{g})\|}{\bar{\mu}}$ , we have + +$$ +| [ \theta_ {t} ] _ {i} - \gamma_ {t} [ g _ {t} ] _ {i} | > \frac {7 \| \mathrm {H T} _ {s} (\widehat {g}) \|}{\bar {\mu}} - \frac {6 \| \mathrm {H T} _ {s} (\widehat {g}) \|}{2 \bar {\mu}} - \frac {\| \mathrm {H T} _ {s} (\widehat {g}) \|}{2 \bar {\mu}} = \frac {\| \mathrm {H T} _ {s} (\widehat {g}) \|}{2 \bar {\mu}}. +$$ + +Because the Hard Thresholding operator selects the $s$ largest components of $\theta_{t + 1}$ , in the selection of the elements that should go into $\mathcal{S}^{t + 1}$ the operator will not deselect elements from $\widehat{\mathcal{S}}$ in benefit of any outside of $\mathcal{S}^t$ . Thus, $\widehat{\mathcal{S}}\subset \mathcal{S}^{t + 1}$ . + +We now find conditions on $\gamma_{t}$ under which (11) can be guaranteed. Observe that + +$$ +\begin{array}{l} \gamma_ {t} \| g _ {t} \| _ {\infty} \leq \gamma_ {t} \left(\| \widehat {g} \| _ {\infty} + \| g _ {t} - \widehat {g} \| _ {\infty}\right) \\ \stackrel {(i)} {\leq} \gamma_ {t} \left(\| \widehat {g} \| _ {\infty} + \bar {L} \| \theta_ {t} - \widehat {\theta} \|\right) \stackrel {(i i)} {\leq} \gamma_ {t} \left(\| \widehat {g} \| _ {\infty} + \frac {6 \bar {L}}{\bar {\mu}} \| \mathrm {H T} _ {s} (\widehat {g}) \|\right), \\ \end{array} +$$ + +where $(i)$ follows from Lemma 5, and $(ii)$ follows from the assumption that $\| \theta_t - \widehat{\theta}\|^2\leq \frac{36\|\mathrm{HT}_s(\widehat{g})\|^2}{\bar{\mu}}$ A sufficient condition for (11) to hold is thus given by + +$$ +\gamma_ {t} < \frac {1}{2 \bar {\mu} + 1 2 \bar {L}}. +$$ + +As a result, when $\gamma_t < \frac{1}{2\bar{\mu} + 12L}$ , we have $\widehat{\mathcal{S}} \subset \mathcal{S}^{t+1}$ , and thus $\| \theta_{t+1} - \widehat{\theta} \|^2 \leq \| \theta_t - \widehat{\theta} \|^2$ by $\| \theta_{t+1} - \widehat{\theta} \|^2 = \| [\theta_{t+1} - \widehat{\theta}]_{\widehat{\mathcal{S}}_{t+1}} \|^2 \leq (1 - \bar{\mu} \gamma_t) \| \theta_t - \widehat{\theta} \|^2$ . Otherwise, $\gamma_t \geq \frac{1}{2\bar{\mu} + 12L} > \frac{1}{40L}$ , we still have $\| \theta_{t+1} - \widehat{\theta} \|^2 \leq \| \theta_t - \widehat{\theta} \|^2$ by Lemma 4. Thus, the proof is completed by induction. + +# A.3 Proof of Corollary 2 + +By invoking Lemma 1, we can guarantee that $f$ fulfills the RSC and RSM with probability at least $1 - e^{-c_0n}$ . The function $f$ is convex by construction, and assuming that Algorithm 1 is provided suitable parameters $\widehat{f}$ and $s$ as stipulated by the Corollary, we may invoke Theorem 1. Thus, with probability at least $1 - e^{-c_0n}$ , the iterates satisfy a contractive relation for $\| \theta_t - \theta^* \|^2$ until the point where $\| \theta_t - \theta^* \|^2 < \frac{36 \| \mathrm{HT}_s(g^*) \|^2}{\bar{\mu}^2}$ . + +To complete the proof, it remains to establish that, with probability at least $1 - \frac{2}{d}$ , it holds that + +$$ +\frac {3 6 \left\| \mathrm {H T} _ {s} \left(g ^ {*}\right) \right\| ^ {2}}{\bar {\mu} ^ {2}} \leq 7 2 C ^ {2} \frac {s \log d}{n \bar {\mu} ^ {2}}. \tag {12} +$$ + +The proof of this claim essentially follows the arguments in (Wainwright, 2019, Example 7.14); for completeness, we include the full details here. + +Define $\sigma(x) = \frac{1}{1 + \exp(-x)}$ . The gradient of $f$ evaluated at $\theta^{*}$ can be expressed as + +$$ +g ^ {*} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\sigma \left(x _ {i} ^ {\top} \theta^ {*}\right) - y _ {i}\right) x _ {i}. +$$ + +Recall the relation between $x_{i}$ and $y_{i}$ is governed by the model + +$$ +\operatorname * {P r} (y _ {i} = 1 \mid x _ {i}) = \frac {1}{1 + \exp (- x _ {i} ^ {\top} \theta^ {*})}. +$$ + +Under this model, it follows that each term $\sigma (x_i^\top \theta^*) - y_i$ is a zero-mean sub-Gaussian random variable with sub-Gaussian parameter $\sigma^2 = \frac{1}{4}$ . + +Thus, $\| g^{*}\|_{\infty}$ is the maximum of $d$ independent zero-mean sub-Gaussian random variables, each with variance proxy at most $\sigma^2 = \frac{C^2}{4n}$ . By standard sub-Gaussian maximal inequalities, we have + +$$ +\operatorname * {P r} \left(\| g ^ {*} \| _ {\infty} \geq C \sqrt {\frac {\log d}{2 n}} + \frac {C \delta}{2}\right) \leq 2 e ^ {- \frac {n \delta^ {2}}{2}} +$$ + +for any $\delta > 0$ . Setting $\delta = \sqrt{\frac{2\log d}{n}}$ completes the proof of the bound in (12). + +Applying the union bound allows us to claim that the guarantees provided by Theorem 1 hold with probability at least $1 - c_0 e^{-n} - \frac{2}{d}$ . Guarantees on support recovery follow then by direct application of Corollary 1. + +# A.4 Proof to Corollary 3 + +By (Jain et al., 2014, Lemma 2), we can verify the result of Theorem 1 translates to the matrix case, where the vector $\ell_2$ -norm is replaced by the Frobenius norm, the $\ell_1$ -norm is replaced by the nuclear norm, and the $\mathrm{HT}_s$ operator is substituted by $\mathrm{PM}_s$ . + +By applying Theorem 1 and Lemma 2, we conclude that, with probability at least $1 - e^{-c_0n}$ , the iterates satisfy a contractive relation for $\| \Theta_t - \Theta^*\| _F^2$ until the point where $\| \Theta_t - \Theta^*\| _F^2 < \frac{36\|\mathrm{PM}_s(g^*)\|_F^2}{\bar{\mu}^2}$ . + +To complete the proof, it remains to establish that, with probability at least $1 - \frac{2}{d}$ , + +$$ +\frac {3 6 \left\| \mathrm {P M} _ {s} \left(g ^ {*}\right) \right\| _ {F} ^ {2}}{\bar {\mu} ^ {2}} \leq \frac {7 2 0 0 \sigma^ {2} \zeta (\Sigma) s d}{n \bar {\mu} ^ {2}}. \tag {13} +$$ + +Note that $g^{*} = \frac{1}{n}\varepsilon_{i}X_{i}$ , by (Wainwright, 2019, Corollary 10.10), we have + +$$ +\Pr \left(\| g ^ {*} \| _ {2} \geq \frac {\lambda_ {n}}{2}\right) \leq 2 e ^ {- 2 n \delta^ {2}}, +$$ + +where $\lambda_{n} = 10\sigma \sqrt{\zeta(\Sigma)}\left(\sqrt{\frac{2d}{n}} +\delta\right)$ + +By setting $\delta = \sqrt{\frac{2d}{n}}$ , we have + +$$ +\| g ^ {*} \| _ {2} \leq 1 0 \sigma \sqrt {\zeta (\Sigma)} \sqrt {\frac {2 d}{n}}, +$$ + +with probability at least $1 - 2e^{-4d}$ . We thus apply the union bound to complete the proof of claim (13). + +# B Adaptive Lower Bound + +# B.1 Proof of Theorem 2 + +Let $a_{t} \coloneqq \frac{f(\theta_{t}) - f(\theta^{*})}{5\|\mathrm{HT}_{s}(g_{t})\|}$ . Suppose the step size $\gamma_{t}$ used in Algorithm 2 satisfies $\gamma_{t} = ba_{t}$ for some scalar $b \in \left[\frac{1}{2}, 1\right]$ . By invoking Lemmas 4 and 6, as long as $\| \theta_{t} - \theta^{*}\|^{2} \geq \frac{36\|\mathrm{HT}_{s}(g^{*})\|^{2}}{\bar{\mu}^{2}}$ , we have: + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \frac {1}{1 6 0 \bar {\kappa}}\right) \left(1 - \frac {b}{4 0 \bar {\kappa}}\right) \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} \leq \left(1 - \frac {1}{1 6 0 \bar {\kappa}}\right) \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2}. +$$ + +This establishes that the iterates exhibit contractive behavior until they enter a small neighborhood of the optimum. + +We now consider two possible cases depending on whether the lower bound surrogate $\widetilde{f}_k$ is valid and how the step size compares to $a_{t}$ during epoch $k$ . + +Case (i): Suppose $\widetilde{f}_k$ is a valid lower bound for $f(\theta^{*})$ , and that $\gamma_t \leq a_t$ holds for all iterations $t = 0, \dots, T(\alpha)$ within epoch $k$ . Then, the contractive relation applies repeatedly, and we obtain + +$$ +\| \theta_ {T (\alpha)} - \theta^ {*} \| ^ {2} \leq \left(1 - \frac {\bar {\mu}}{1 6 0 \bar {L}}\right) ^ {T (\alpha)} \| \theta_ {0} - \theta^ {*} \| ^ {2} \leq (1 + \alpha) \left(1 + \frac {1}{1 6 0 \bar {\kappa}}\right) \frac {3 6 \| \mathrm {H T} _ {s} (g ^ {*}) \| ^ {2}}{\bar {\mu} ^ {2}}. +$$ + +By the restricted smoothness and strong convexity assumptions, this implies that the function suboptimality satisfies $f(\theta_{T(\alpha)}) - f^{*} \leq \varepsilon (\alpha)$ , thus completing the proof for this case. + +Case (ii): Alternatively, suppose that $\widetilde{f}_k$ is a valid lower bound, but $\gamma_t > a_t$ for some $t$ in epoch $k$ . This condition implies that + +$$ +f \left(\theta_ {t}\right) - \widetilde {f} _ {k} > 2 \left(f \left(\theta_ {t}\right) - f \left(\theta^ {*}\right)\right), +$$ + +which in turn yields + +$$ +\widetilde {f} _ {k + 1} = \frac {f (\bar {\theta} _ {k}) + \widetilde {f} _ {k}}{2} \leq \frac {f (\theta_ {t}) + \widetilde {f} _ {k}}{2} < f (\theta^ {*}). +$$ + +Hence, $\widetilde{f}_{k + 1}$ is also a valid lower bound. By induction, we conclude that if Case I never occurs, then all $\widetilde{f}_k$ , for $k = 1,\dots ,K$ , remain valid lower bounds. + +Moreover, under this scenario, the sequence $f^{*} - \widetilde{f}_{k}$ decreases geometrically. In particular, + +$$ +f ^ {*} - \widetilde {f} _ {k + 1} \leq f ^ {*} - \frac {f ^ {*} + \widetilde {f} _ {k}}{2} = \frac {f ^ {*} - \widetilde {f} _ {k}}{2}. +$$ + +The geometric decrease of $f^{*} - \widetilde{f}_{k}$ in case (ii) ensures that if case (i) never occurs, then there exists some $k_{0}$ such that $f(\theta^{*}) - \widetilde{f}_{k_{0}} < \varepsilon (\alpha)$ . In that case, $\bar{\theta}_{k_0}$ is either an output corresponding to case (i) (which completes the proof) or an output under case (ii), i.e., it satisfies + +$$ +f \left(\bar {\theta} _ {k _ {0}}\right) - f \left(\theta^ {*}\right) < f \left(\theta^ {*}\right) - \widetilde {f} _ {k _ {0}} < \varepsilon (\alpha). +$$ + +# C Other Statistical Guarantees + +In this section we provide guarantees for additional statistical models not provided in the body of the paper. The results in Appendix C.1 hold for sparse linear regression. The results in Appendix C.2 apply to some GLMs and require the analysis of the behavior of Sparse Polyak under different regularity conditions. These are not entirely captured in Theorem 1. + +# C.1 Sparse Linear Regression + +In this section, we assume the dataset consists of data points $z_{i} = (x_{i},y_{i})$ for $i = 1,\dots ,n$ , where $x_{i}\in \mathbb{R}^{d}$ denote the feature vectors, and $y_{i}\in \mathbb{R}$ denote the responses. The feature vectors are aggregated into the design matrix + +$$ +\mathbb {R} ^ {n \times d} \ni X \triangleq \left( \begin{array}{c} x _ {1} ^ {\top} \\ \vdots \\ x _ {n} ^ {\top} \end{array} \right) +$$ + +and the responses are aggregated in $\mathbb{R}^n\ni y\triangleq (y_1,\ldots ,y_n)^\top$ . Let $\theta^{*}$ denote the ground truth of the statistical model, with $\| \theta^{*}\|_{0}\leq s^{*}$ , and $f^{*}$ denote the corresponding objective value. Specifically, we assume that the responses $y_{i}$ and feature vectors $x_{i}$ are related by $y_{i} = x_{i}^{T}\theta^{*} + \varepsilon_{i}$ , where $x_{i}$ are drawn from a $N(0,\Sigma)$ distribution, $\Sigma$ is non singular, $\varepsilon_{i}\sim N(0,\sigma^{2})$ , and $x_{i}$ and $\varepsilon_{i}$ are i.i.d and independent of one another. Additionally, the objective function $f(\theta) = \frac{1}{2n}\| X\theta -y\| ^2$ . + +From Agarwal et al. (2012)[Lemma 6], it follows that the RSS and RSC conditions hold with probability at least $1 - e^{-c_0n}$ with coefficients $L = 2\sigma_{\max}(\Sigma)$ , $\mu = \frac{1}{2}\sigma_{\min}(\Sigma)$ , and $\tau = c_1\zeta(\Sigma)\frac{\log d}{n}$ where $\zeta(\Sigma) = \max_{i=1,\dots,d}\Sigma_{ii}$ . Here $c_0$ and $c_1$ are universal constants. + +Corollary 4. Consider the sparse linear regression problem described above. Let $\{\theta_t\}_{t\geq 0}$ be the sequence of iterates generated by Algorithm 1 or Algorithm 2 when employed to solve a sparse linear + +regression problem. Suppose that $\widehat{f} = f^{*}$ for Algorithm 1. Assume we have sufficient samples for $\bar{\mu} >0$ , with $s\geq (240\bar{\kappa})^2 s^*$ for Algorithm 1, and $s\geq (480\bar{\kappa})^2 s^*$ for Algorithm 2. Further, assume that each column of $X$ is $C$ -normalized, i.e., $\| \frac{X_j}{\sqrt{n}}\| \leq C$ for $j = 1,\ldots ,d$ . Here $X_{j}$ denotes the $j$ -th column of $X$ . Then, with probability at least $1 - e^{-c_0n} - \frac{2}{d}$ , for any $\alpha \geq \frac{1}{80}$ after $T(\alpha)$ iterations, $\min_{t\leq T(\alpha)}\| \theta_t - \theta^*\|^2$ is upper bounded by + +$$ +\varepsilon (\alpha) = (1 + \alpha) \frac {2 8 8 C ^ {2} \sigma^ {2} s \log d}{n \bar {\mu} ^ {2}}. +$$ + +The required number of iterations $T(\alpha)$ fulfills: + +$$ +T (\alpha) \leq \left\lceil \frac {1}{\log \left(1 / \left(1 - 1 / 8 0 \bar {\kappa}\right)\right)} \log \left(\frac {\| \theta_ {0} - \theta^ {*} \| ^ {2}}{\varepsilon (\alpha)}\right)\right\rceil , a n d +$$ + +$$ +T (\alpha) \leq \left(1 + \log_ {2} \frac {4 (f (\theta_ {0}) - f (\theta^ {*}))}{\bar {\mu} \varepsilon (\alpha)}\right) \left\lceil \frac {1}{\log \left(1 / (1 - 1 / 1 6 0 \bar {\kappa})\right)} \log \left(\frac {\| \theta_ {0} - \theta^ {*} \| ^ {2}}{\varepsilon (\alpha)}\right)\right\rceil , +$$ + +for Algorithm 1 and Algorithm 2 respectively. + +Moreover, if $\theta^{*}$ satisfies the SNR condition (3), we can ensure that after $T(0)$ iterations, the error is upper bounded by $\varepsilon(0)$ , and the support of $\theta^{*}$ has been identified, i.e. $\mathcal{S}^{*} \subset \mathcal{S}^{t} \forall t \geq T(0)$ . + +Corollary 4 establishes the convergence properties of Algorithm 1 and Algorithm 2. The error term is of order $\mathcal{O}\left(\frac{\bar{\kappa}^2s^*\log d}{n\bar{\mu}^2}\right)$ , which is of the same order as that in (Jain et al., 2014, Theorem 3), where a fixed step size is considered under the assumption that $\bar{L}$ is known. + +Proof. For Algorithm 1, the proof follows the same steps as the proof of Corollary 2. The only difference is the upper bound for $\frac{36\|\mathrm{HT}_s(g^*)\|^2}{\bar{\mu}^2}$ , which we provide next. + +When the columns of $X$ are $C$ -normalized, by (Wainwright, 2019, Example 7.14), with probability $1 - \frac{2}{d}$ , + +$$ +\| g ^ {*} \| _ {\infty} ^ {2} = \| X ^ {T} \varepsilon \| _ {\infty} ^ {2} \leq 8 C ^ {2} \sigma^ {2} \frac {\log d}{n}. +$$ + +Using the union bound together with (Agarwal et al., 2012, Lemma 6), and the assumptions stated in the Corollary, yield the required assumptions for Theorem 1 to hold. Thus, this completes the proof for Algorithm 1. + +As we can see from the proof of Theorem 1 and Theorem 2, the accuracy level $\varepsilon(0)$ is determined by the point at which a lower bound on the step size can be established. According to the formulation of $\gamma_{t}$ in Algorithm 2, it is guaranteed to be at least half the step size used in Algorithm 1. This observation implies that the accuracy level $\varepsilon(0)$ can also be achieved by Algorithm 2 when the conditions of Case (i) in the proof of Theorem 2 are satisfied. + +To complete the proof for Algorithm 2, we only need to show that $f(\theta_t) - f^* \leq \frac{18\|\mathrm{HT}_s(g^*)\|^2}{\bar{\mu}}$ implies $\| \theta_t - \theta^* \|^2 \leq \varepsilon(0)$ . + +By RSC, + +$$ +f \left(\theta_ {t}\right) - f ^ {*} \geq \bar {\mu} \| \theta_ {t} - \theta^ {*} \| ^ {2} - \left\| \mathrm {H T} _ {s} \left(g ^ {*}\right) \right\| \| \theta_ {t} - \theta^ {*} \|. +$$ + +When $f(\theta_t) - f^* \leq \frac{18\|\mathrm{HT}_s(g^*)\|^2}{\bar{\mu}}$ , it implies + +$$ +\frac {\left\| \mathrm {H T} _ {s} \left(g ^ {*}\right) \right\| _ {s}}{\bar {\mu}} \left(\frac {3 \left\| \mathrm {H T} _ {s} \left(g ^ {*}\right) \right\|}{\bar {\mu}} + \left\| \theta_ {t} - \theta^ {*} \right\|\right) \geq \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2}. +$$ + +A necessary condition for the inequality above is + +$$ +\| \theta_ {t} - \theta^ {*} \| ^ {2} \leq \frac {3 6 \| \mathrm {H T} _ {s} (g ^ {*}) \| _ {s} ^ {2}}{\bar {\mu} ^ {2}} = \varepsilon (0). +$$ + +![](images/45694aa200abb32c588f8ea6add80cd63efe3c2ca01f02e1d4e6d7ddb4595425.jpg) + +# C.2 Generalized Linear Models + +In this section, we consider a dataset consisting of observations $z_{i} = (x_{i},y_{i})$ for $i = 1,\dots ,n$ , where $x_{i}\in \mathbb{R}^{d}$ denote the feature vectors, and $y_{i}\in \mathbb{R}$ denotes the corresponding responses. The feature vectors are organized into the design matrix + +$$ +X \triangleq \left( \begin{array}{c} x _ {1} ^ {\top} \\ \vdots \\ x _ {n} ^ {\top} \end{array} \right) \in \mathbb {R} ^ {n \times d}, +$$ + +and the responses are collected in the vector $y \triangleq (y_1, \ldots, y_n)^\top \in \mathbb{R}^n$ . + +For notational convenience, we let $\theta^{*}$ denote the true underlying parameter of the statistical model and $f^{*}$ the corresponding objective function value. We assume the relationship between $x_{i}$ and $y_{i}$ is characterized by the conditional distribution + +$$ +\operatorname * {P r} (y _ {i} \mid x _ {i}, \theta^ {*}, \sigma) = \exp \left\{\frac {y _ {i} x _ {i} ^ {T} \theta^ {*} - \psi (x _ {i} ^ {T} \theta^ {*})}{c (\sigma)} \right\}, +$$ + +where $\sigma > 0$ is a scale parameter, and $\psi$ is the cumulant function. Given this data generation model, we define the objective function + +$$ +f (\theta) = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\psi (x _ {i} ^ {T} \theta) - y _ {i} x _ {i} ^ {T} \theta\right). +$$ + +We assume that $\psi$ is infinitely differentiable with $\psi''(t) > 0$ and uniformly bounded for all $t \in \mathbb{R}$ . These assumptions are satisfied in a variety of settings, including logistic regression and multinomial regression Loh and Wainwright (2015). We assume the feature vectors $x_i$ are i.i.d. and drawn from a multivariate normal distribution $N(0, \Sigma)$ , where $\Sigma$ is non-singular. Under the setting described above, the RSC condition does not hold everywhere. However, in the described setting it can be shown that the following milder RSC condition holds Loh and Wainwright (2015) + +$$ +f (y) - f (x) - \langle \nabla f (x), y - x \rangle \geq \left\{ \begin{array}{l l} \frac {\mu}{2} \| y - x \| _ {2} ^ {2} - \frac {\tau}{2} \| y - x \| _ {1} ^ {2}, & \text {i f} \| y - x \| _ {2} \leq 1, \\ \| y - x \| _ {2} \left(\frac {\mu}{2} - \frac {\tau}{2} \frac {\| y - x \| _ {1} ^ {2}}{\| y - x \| _ {2} ^ {2}}\right), & \text {i f} \| y - x \| _ {2} > 1. \end{array} \right. \tag {14a} +$$ + +We now present a lemma grouping the results that we need to proceed: the RSS condition, the condition (14), and the order of achievable statistical accuracy for GLMs in our setting. This lemma aggregates results from (Loh and Wainwright, 2015, Proof of Corollary 2, Appendix D.1) and (Negahban et al., 2012, Proposition 2). Notably, while (Negahban et al., 2012, Proposition 2) is originally stated centered only around the ground truth $x = \theta^{*}$ (c.f. (14)) its proof extends to any given $x$ . This implies that while our results are currently stated for $x = \theta^{*}$ , to achieve optimal statistical precision, we can instead state equivalent results to those in Theorem 1 where $\widehat{\theta}$ is such that $f(\widehat{\theta}) = \widehat{f}$ . + +Lemma 7. For the statistical models described above, with probability at least $1 - c_{1}d^{-1} - c_{2}e^{-n}$ Assumption 2 and (14) hold, and + +$$ +\| \nabla f (\theta^ {*}) \| _ {\infty} \leq c _ {0} \sqrt {\frac {\log d}{n}}, +$$ + +where $c_{0}, c_{1}, c_{2} > 0$ are universal constants. The constants $\mu$ , and $L$ in (14) and Assumption 2, respectively, depend on $\psi$ , and $\Sigma$ . Further $\tau = c_{3} \frac{\log d}{n}$ where $c_{3} > 0$ is a universal constant. + +In the following, we keep the definition $\bar{L} = L + 3\tau s$ , and $\bar{\mu} = \mu - 3\tau s$ , where $\mu$ is that of (14). Observe that Corollary 5 is stated for Algorithm 1 for simplicity but an analogous statement for Algorithm 2 follows. + +Corollary 5. Let $\{\theta_t\}_{t\geq 0}$ denote the iterates generated by Algorithm 1 when applied to the generalized linear models described above. Set the step size rule according to + +$$ +\gamma_ {t} = \frac {\max \{f (\theta_ {t}) - \widehat {f} , 0 \}}{5 \| \mathrm {H T} _ {2 s} (g _ {t}) \| ^ {2}}. +$$ + +Define $R \coloneqq \| \theta^{*} \|^{2}$ , and $R_{0} \coloneqq 4R + 1$ . Assume we set $\widehat{f} = f^{*}$ , $\theta_{0} = 0$ , and suppose $s \geq (480R_{0}\bar{\kappa}^{2})^{2}s^{*}$ . Further, assume the sample size is large enough to ensure $\bar{\mu} > 5c_{0}\sqrt{\frac{2s\log d}{n}}$ . + +Then, with probability at least $1 - \frac{c_1}{d} - c_2 e^{-n}$ , we guarantee that for all $t \geq T$ where + +$$ +T \leq \mathcal {O} \left(\bar {\kappa} \log \left(\frac {n}{s \log (d)}\right)\right) + \mathcal {O} \left(\bar {\kappa} ^ {2} \log (R)\right) +$$ + +there holds + +$$ +\left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \frac {1}{1 6 0 R _ {0} \bar {\kappa} ^ {2}}\right) \frac {3 6 c _ {0} s \log d}{n \bar {\mu} ^ {2}}. +$$ + +Moreover, if $\theta^{*}$ satisfies the SNR condition (3), we can guarantee that for all $t\geq T$ $\| \theta_t - \theta^*\| ^2\leq$ $\frac{36c_0s\log d}{n}$ and that the support of $\theta^{*}$ has been recovered, i.e. $\mathcal{S}^{*}\subset \mathcal{S}_{t}$ + +Proof. To establish this result we leverage that the condition (14) combined with Sparse Polyak allows us to establish convergence despite the lack of RSC. More specifically, we will establish that Algorithm 1 exhibits, under the conditions described in the Corollary, at most three modes of convergence: + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left\{ \begin{array}{l l} \left(1 - \frac {1}{1 6 0 R _ {0} \bar {\kappa} ^ {2}}\right) \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2}, & \text {i f} \| \theta_ {t} - \theta^ {*} \| \geq 1, \\ \left(1 - \frac {1}{1 6 0 \bar {\kappa}}\right) \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2}, & \text {i f} \| \theta_ {t} - \theta^ {*} \| < 1, \text {a n d} \| \theta_ {t} - \theta^ {*} \| ^ {2} \geq \frac {3 6 c _ {0} ^ {2} s \log d}{n \bar {\mu} ^ {2}}, \\ \left(1 + \frac {1}{1 6 0 R _ {0} \bar {\kappa} ^ {2}}\right) \frac {3 6 c _ {0} ^ {2} s \log d}{n \bar {\mu} ^ {2}}, & \text {o t h e r w i s e .} \end{array} \right. \tag {15} +$$ + +Observe that under our current assumptions $\frac{36c_0^2s\log d}{n\bar{\mu}^2} < \frac{18}{25}$ and therefore, the list above is exhaustive and the conditions on the second case are compatible. + +We begin assuming that $R \geq 1$ . We will establish that if the modes of convergence provided above hold for $R \geq 1$ , when $R < 1$ we only observe the two last cases. + +(i) We start off by exploiting (14). Assuming that $\| \theta_t - \theta^* \| \geq 1$ we may follow the strategy in Lemma 4 and exploit that $f$ is convex (but not RSC) to obtain + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left(\left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} - 2 \gamma_ {t} \left(f \left(\theta_ {t}\right) - f \left(\theta^ {*}\right)\right) + 5 \gamma_ {t} ^ {2} \left\| \mathrm {H T} _ {2 s} \left(g _ {t}\right) \right\| ^ {2}\right). \tag {16} +$$ + +With the choice of step size + +$$ +\gamma_ {t} = \frac {\max \{f (\theta_ {t}) - f (\theta^ {*}) , 0 \}}{5 \| \mathrm {H T} _ {2 s} (g _ {t}) \| ^ {2}}, +$$ + +(16) simplifies to: + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left(\left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} - \frac {\left(f \left(\theta_ {t}\right) - f \left(\theta^ {*}\right)\right) ^ {2}}{5 \left\| \mathrm {H T} _ {2 s} (g _ {t}) \right\| ^ {2}}\right). \tag {17} +$$ + +From Lemma 7 and under the assumption that $\bar{\mu} >5c_0\sqrt{\frac{2s\log d}{n}}$ , it follows that with probability at least $1 - c_{1}d^{-1} - c_{2}e^{-n}$ there holds + +$$ +f (\theta_ {t}) - f (\theta^ {*}) \geq \frac {\bar {\mu}}{2} \| \theta_ {t} - \theta^ {*} \| - \| \mathrm {H T} _ {2 s} (g ^ {*}) \| \| \theta_ {t} - \theta^ {*} \| \geq \frac {\bar {\mu}}{4} \| \theta_ {t} - \theta^ {*} \|, +$$ + +thus by applying the above bound onto (17) we have + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left(\left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} - \frac {\bar {\mu} ^ {2}}{8 0 \left\| \mathrm {H T} _ {2 s} (g _ {t}) \right\| ^ {2}} \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2}\right). \tag {18} +$$ + +Further, we may upper bound the norm of the gradient as + +$$ +\left\| \mathrm {H T} _ {2 s} \left(g _ {t}\right) \right\| ^ {2} \leq 2 \left(\left\| \mathrm {H T} _ {2 s} \left(g ^ {*}\right) \right\| _ {2 s} ^ {2} + \left\| \mathrm {H T} _ {2 s} \left(g _ {t} - g ^ {*}\right) \right\| ^ {2}\right) \tag {19} +$$ + +$$ +\stackrel {(i)} {\leq} 2 \left(\| \mathrm {H T} _ {2 s} (g ^ {*}) \| _ {2 s} ^ {2} + 2 \bar {L} ^ {2} \| \theta_ {t} - \theta^ {*} \| ^ {2}\right) +$$ + +where in $(i)$ we invoke Lemma 5. Combining (18) with (19) yields + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \sqrt {\frac {s ^ {*}}{s}}\right) ^ {2} \left(1 - \frac {\bar {\mu} ^ {2}}{1 6 0 \left(\| \mathrm {H T} _ {2 s} (g ^ {*}) \| ^ {2} + 2 \bar {L} ^ {2} \| \theta_ {t} - \theta^ {*} \| ^ {2}\right)}\right) \| \theta_ {t} - \theta^ {*} \| ^ {2}. \tag {20} +$$ + +To guarantee that the above implies our first regime of convergence we need to establish that $\| \theta_t - \theta^*\|^2 \leq R$ for all $t \geq 0$ . We return to this point after exploring the second and third regime which will be useful in establishing the first. + +(ii) On the other hand, assume instead that $\| \theta_t - \theta^* \|^2 \leq 1$ and $\| \theta_t - \theta^* \|^2 \geq \frac{36c_0^2s\log d}{n\bar{\mu}^2}$ then, from (14) the RSC holds and therefore we can invoke the result of Theorem 1 in which + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 - \frac {1}{1 6 0 \bar {\kappa}}\right) \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2}. \tag {21} +$$ + +(iii) If, instead $\| \theta_t - \theta^*\|^2 < \frac{36c_0^2s\log d}{n\bar{\mu}^2}$ we have from Theorem 1 that + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 + \frac {1}{1 6 0 \bar {\kappa}}\right) \frac {3 6 c _ {0} ^ {2} s \log d}{n \bar {\mu} ^ {2}} < 1. \tag {22} +$$ + +Observe that from the three cases we consider ((20)-(22)), (21) and (22) already correspond to one of our stated modes of convergence, and thus we are to establish the first. + +Clearly, when $R < 1$ , $\theta_0$ is in a region in which the RSC holds, and therefore, we will only observe the behavior in (21) and (22). Thus, we are only to prove the first regime of convergence for $R \geq 1$ . To establish that the behavior in the first regime holds, we need to establish that $\| \theta_t - \theta^* \|^2 \leq R$ holds for all $t \geq 0$ . We proceed to establish this and consequently the behavior in the first regime by induction. Note that by our initial condition $\| \theta_0 - \theta^* \|^2 = \| \theta^* \|^2 = R$ and thus the condition holds for $t = 0$ . Suppose $\| \theta_t - \theta^* \|^2 \leq R$ for some $t$ . If $\| \theta_t - \theta^* \|^2 < 1$ then either (21) and (22) hold and the proof by induction is complete. If instead, $\| \theta_t - \theta^* \|^2 \geq 1$ , (20) holds. Then, by induction hypothesis and under the Corollary's assumption on the sample size there holds + +$$ +\left\| \mathrm {H T} _ {2 s} \left(g ^ {*}\right) \right\| ^ {2} + 2 \bar {L} ^ {2} \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} \leq \left(\frac {1}{1 6} + 2 R\right) \bar {L} ^ {2} \leq \frac {R _ {0} \bar {L} ^ {2}}{2}, +$$ + +thus, combining with (20) and using that by assumption $s \geq (480R_0\bar{\kappa}^2)^2$ we obtain + +$$ +\left\| \theta_ {t + 1} - \theta^ {*} \right\| ^ {2} \leq \left(1 - \frac {1}{1 6 0 R _ {0} \bar {\kappa} ^ {2}}\right) \left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} < R. +$$ + +We have thus established the veracity of (15). Observe that (15) together with $\theta_0 = 0$ and $\| \theta^{*}\|^{2} = R$ imply that there exists $t_0\geq 0$ fulfilling + +$$ +t _ {0} \leq \left\lceil - \log (R) / \log \left(1 - 1 / 1 6 0 R _ {0} \bar {\kappa} ^ {2}\right)\right\rceil +$$ + +such that for all $t \geq t_0$ + +$$ +\left\| \theta_ {t} - \theta^ {*} \right\| ^ {2} < 1. +$$ + +Further, this implies that in at most + +$$ +\left\lceil \log \left(\frac {3 6 c _ {0} ^ {2} s (1 + 1 / (1 6 0 \bar {\kappa})) \log (d)}{n \bar {\mu} ^ {2}}\right) / \log (1 - 1 / 1 6 0 \bar {\kappa}) \right\rceil +$$ + +additional iterations optimal statistical precision is reached. Finally, if the SNR condition holds, the term $\left(1 + \frac{1}{160\bar{\kappa}}\right)$ in the third regime is replaced by 1 as a consequence of Corollary 1. + +Corollary 5 recovers the result in (Loh and Wainwright, 2015, Theorem 3) with the following similarities and differences. To achieve optimal statistical precision, as $\alpha = s\frac{\log d}{n} \to 0$ both Sparse Polyak and (Loh and Wainwright, 2015, Theorem 3) require $\mathcal{O}(\bar{\kappa}\log (\alpha^{-1}))$ iterations. However, we require additional iterations $\mathcal{O}(\bar{\kappa}^2\log (R))$ when $R \geq 1$ . We observe however, that our result holds under more general conditions, as we do not make the assumption that $\| \theta^{*}\| \leq 1$ which is necessary in (Loh and Wainwright, 2015, Theorem 3), where this condition can be relaxed at the expense of requiring the RSC to hold within a larger radius. + +We conclude this Appendix by highlighting the fact that for both sparse linear regression and sparse GLMs we verify the rate invariance of Sparse Polyak theoretically. When the problem size increases much faster than the sample size $\frac{d}{n} \to \infty$ but $\frac{s^* \log d}{n}$ and $\Sigma$ remain constant, IHT with Sparse Polyak will reach a $\varepsilon$ neighborhood of the optimal statistical precision within at most $\mathcal{O}(\bar{\kappa}^{-1} \log (1 / \varepsilon))$ for linear regression and at most $\mathcal{O}(\bar{\kappa}^{-1} \log (1 / \varepsilon)) + \mathcal{O}(\bar{\kappa}^{-1} \log (R))$ when $R > 1$ in the case of GLMs. In both cases, this number does not change with increasing $d$ and $n$ . Observe that these results allow us to answer in the affirmative questions (ii) and (iii) posed in Section 1. + +# D Experiments on real data + +Linear Regression We consider a linear regression task using the Large-scale Wave Energy Farm dataset from the UCI Machine Learning Repository Neshat et al. (2020), which is publicly available under the CC BY 4.0 license. The terms of use are described at https://archive.ics.uci.edu/#terms. The goal is to predict the total power output of the wave farm based on a sparse linear model. We randomly select 120 samples from the dataset, each containing 149 features. In our experiment, we set the sparsity level to $s = 20$ . For the IHT method with a fixed step size, we choose the step size as $8 \times 10^{-12}$ . This value is determined via a grid search over the range $[10^{-13}, 9 \times 10^{-12}]$ , as step sizes outside this interval result in poor convergence or divergence. The results are presented in Figure 3 (left). + +Logistic Regression We evaluate sparse logistic regression using the Molecule Musk dataset Chapman and Jain (1994) from the UCI Machine Learning Repository, which is publicly available under the CC BY 4.0 license. The terms of use are described at https://archive.ics.uci.edu/#terms. The task is to classify molecules as musks or non-musks. We randomly select 120 samples from the dataset, each with 166 features. In our experiment, we set the sparsity level to $s = 20$ . For the IHT method with a fixed step size, we select a step size of $1.9 \times 10^{-5}$ , chosen via a grid search over the interval $[3 \times 10^{-6}, 4 \times 10^{-5}]$ . Step sizes outside this range lead to poor convergence or divergence. The results are shown in Figure 3 (right). + +We observe that in Fig 3 (right) Sparse Polyak performs better than both classic Polyak and IHT with the fixed step-size, even if the step-size is optimized by grid search. This is expected, an adaptive step-size can adapt to the curvature at any point in the algorithm's trajectory, whereas a fixed step-size cannot. We observe that Sparse Polyak performs better than Classic Polyak. On the other hand, in Fig 3 (left) we observe that the best fixed step-size performs better than an adaptive step-size. We conjecture that this is due to the factor $1/5$ in Sparse Polyak which we presume is an artifact of our analysis and is not in fact necessary. Despite this fact, sparse Polyak consistently outperforms the classical Polyak rule in the high-dimensional setting. + +![](images/b330d619f973907b3a67818a16a49340f7c6f25835247bc7e5e754f578527d13.jpg) +Figure 3: Performance comparison of IHT with optimal constant step size, Sparse Polyak and classical Polyak when performing: (left) linear regression on the Wave Energy Farm data set, and (right) logistic regression on the Molecule Musk data set. \ No newline at end of file diff --git a/NeurIPS/2025/Sparse Polyak_ an adaptive step size rule for high-dimensional M-estimation/images.zip b/NeurIPS/2025/Sparse Polyak_ an adaptive step size rule for high-dimensional M-estimation/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..19f413f986c000a599335241c582c3d0792508d3 --- /dev/null +++ b/NeurIPS/2025/Sparse Polyak_ an adaptive step size rule for high-dimensional M-estimation/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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+HunyuanVideo 720P, Text-to-Video + +Dense Attention | Latency = 30 min + +![](images/d7532793cb419cec96071b38658609ec8e1bb8f0b4766d6f26737a10872848c9.jpg) + +Sparse VideoGen 2 | PSNR = 30.452 | Latency = 13 min + +![](images/63f011310b115f056417533355dc5f89505bb0a80ae1d97ac26af99931a71f06.jpg) + +the sh is vibrant and upbeat, capturing a british white cat with a glossy coat sprinting energetically across a sunlit beach. its four feet are off the ground, suggesting a sprint caught in suspension. The cat is adorned with stylish, miniature sunglasses that sit snugly on its face, adding a playful and charming touch to its appearance. The sun casts a glistening glow on the gentle waves that lap at the shore, while the sand kicks up slightly under the cat's paws with each bound. + +Wan 2.1 720P, Text-to-Video + +Dense Attention | Latency = 30 min + +![](images/4eecbcf4d4331ce00eca065a7028c6e1f28ea0e027aa175a37473e31e29905ba.jpg) + +Sparse VideoGen 2 | PSNR = 25.808 | Latency = 16 min + +![](images/a42ddb1ed73a3c81192966853d662d5c17860f02af96e3c2546f3d05420edfab.jpg) +Figure 1: SVG2 accelerates video generation while maintaining high quality. On a single H100, for Hunyuan-Video and Wan 2.1, SVG2 achieves up to 2.30 and 1.89 end-to-end speedup, with a PSNR up to 30 and 26. + +aerial and wide-angle, with ultra-high definition clarity and an artistic, timeless touch. The scene showcases a japanese princess suspended in the clear blue sky, wearing a beautifully ornate kimono with intricate embroidery and vibrant colors, the embodiment of traditional beauty. her long, flowing sleeves and elegantly styled hair complete her regal appearance. behind her, faded but vivid bursts of colorful fireworks sporadically explode, casting a gentle glow. the scene has a dreamy charm + +# Abstract + +Diffusion Transformers (DiTs) are essential for video generation but suffer from significant latency due to the quadratic complexity of attention. By computing only critical tokens, sparse attention reduces computational costs and offers a promising acceleration approach. However, we identify that existing methods fail to approach optimal generation quality under the same computation budget for two reasons: (1) Inaccurate critical token identification: current methods cluster tokens based on position rather than semantics, leading to imprecise aggregated representations. (2) Excessive computation waste: critical tokens are scattered among non-critical ones, leading to wasted computation on GPUs, which are optimized for processing contiguous tokens. In this paper, we propose SVG2, a training-free framework that maximizes identification accuracy and minimizes computation waste, achieving a Pareto frontier trade-off between generation quality and efficiency. The core of + +SVG2 is semantic-aware permutation, which clusters and reorders tokens based on semantic similarity using $k$ -means. This approach ensures both a precise cluster representation, improving identification accuracy, and a densified layout of critical tokens, enabling efficient computation without padding. Additionally, SVG2 integrates top- $p$ dynamic budget control and customized kernel implementations, achieving up to $2.30 \times$ and $1.89 \times$ speedup while maintaining a PSNR of up to 30 and 26 on HunyuanVideo and Wan 2.1, respectively. Our code is open-sourced at https://github.com/svg-project/Sparse-VideoGen. + +# 1 Introduction + +Diffusion Transformers (DiTs) have demonstrated significant efficacy in generative tasks, particularly excelling in generating high-quality images and videos [1, 2, 3]. However, the computational efficiency of DiTs remains a major bottleneck, primarily due to the quadratic computational complexity introduced by 3D spatio-temporal attention mechanisms [4]. For instance, generating just a five-second video using HunyuanVideo on an NVIDIA A100 GPU takes nearly an hour, where the 3D attention accounts for more than $80\%$ of end-to-end runtime. This inefficiency severely limits the practical deployment of DiT-based generative models. + +To mitigate the quadratic computational complexity, previous studies have observed that self-attention mechanisms are nat + +urally sparse, where only a small portion of computations significantly influence the final output [5, 6, 7]. Therefore, the computational costs can be dramatically reduced (up to $8 \times$ ) with negligible degradation in generation quality by only computing the critical tokens [4, 8]. To effectively identify these critical tokens, existing approaches introduce an identification step where activations from each token are used to estimate attention scores [9, 10]. Tokens with the highest scores are then selected as critical and processed by the following sparse attention. To minimize overhead, the identification is typically processed at the block granularity, treating consecutive tokens as an aggregated token, which are selected or ignored as a whole [11, 12]. + +However, we observe that given the same computational budget (i.e., the number of selected critical tokens), existing sparse attention methods significantly fall behind the oracle generation quality, where the critical tokens are selected assuming the attention scores are known in advance rather than estimated (§ 3.2). We identify that this performance gap arises from two primary challenges: + +1. Inaccurate identification: existing block-wise identification methods are ineffective in precisely identifying critical tokens. Because tokens are clustered into blocks based on positions rather than semantic similarities, tokens within the same block may have dramatically different activations in the latent space. Consequently, the aggregated activations become less representative [11], leading to inaccurate estimations of attention scores and thus incorrect identification of critical tokens. E.g., widely used techniques such as mean pooling [9] and max pooling [5] are prone to inaccuracies, particularly when applied to distinct tokens. +2. Computation waste: existing methods cause computation waste even if critical tokens could be perfectly identified. This is because of the mismatch between sparse computation and hardware specifications [13]. For instance, tensor cores on NVIDIA GPUs require a minimum matrix multiplication shape of $16 \times 16 \times 8$ [14], which necessitates a batch size of 16 tokens. Thus, even if only a subset of a block of 16 tokens are critical, the entire block must still be computed to utilize tensor cores, causing computation waste. Our empirical evaluations show that up to $80\%$ of computation can be wasted on non-critical tokens (§ 3.2). + +To bridge this gap, we propose SVG2, a training-free sparse attention approach specifically designed to accelerate video generation for DiT-based models, achieving a Pareto frontier trade-off between generation quality and computational efficiency (as shown in § 5.4). Our key insight is to leverage semantic-aware permutation to maximize the accuracy of critical token identification and minimize the computation waste of sparse computation. Specifically, semantic-aware permutation clusters + +![](images/f5603f81428fdcec097f663bbf3b2dfff786976aa4142ac64e2376eba7168baf.jpg) +Figure 2: Trade-off curves between generation quality (PSNR) and efficiency (density). SVG2 consistently surpasses existing methods given the same density, achieving a Pareto frontier. + +tokens into blocks according to the semantics of activations rather than positions. Consequently, tokens within each block exhibit closely aligned activations, ensuring more accurate aggregated representations and thereby significantly improving identification accuracy. Additionally, semantic-aware permutation densifies sparse computations by consolidating scattered critical tokens into compact, dense blocks. Due to their semantic similarity, tokens in a single block tend to be either all critical or all non-critical. This property ensures that computation is not wasted on blocks containing a mix of critical and non-critical tokens, thus improving computational efficiency. + +To integrate semantic-aware permutation into an end-to-end framework, SVG2 introduces three key techniques. First, to implement semantic-aware permutation, SVG2 applies $k$ -means clustering on the Query, Key, and Value vectors of each head and layer before the identification step. The resulting clusters are then permuted so that tokens within the same cluster are grouped together, ensuring semantically coherent blocks. Second, to enable dynamic allocation of the computational budget, SVG2 adopts a Top- $p$ critical token selection strategy inspired by Tactic and Twilight [11, 15]. Specifically, SVG2 uses the centroids of clusters to approximate attention scores for each cluster, selecting tokens with the highest estimated scores until their cumulative sum reaches $p$ . This approach enables dynamic budget allocation without manual adjustments. Third, to support dynamic block sizes for sparse attention, SVG2 introduces a customized kernel implementation. This is essential because the clusters formed by semantic-aware permutation naturally vary in size, and existing block sparse attention kernels, which are limited to fixed block sizes, cannot efficiently handle such variability. + +We prototype SVG2 based on an open-sourced video generation framework [4] and customize kernels with FlashInfer [16]. We evaluate SVG2's quality and efficiency on representative video generative models including HunyuanVideo [1] and Wan 2.1 [2]. Results demonstrate that SVG2 consistently achieves a Pareto frontier, delivering superior generation quality at any given computational budget. Specifically, SVG2 delivers significant efficiency improvements, achieving an end-to-end speedup of up to $2.30 \times$ and $1.84 \times$ speedup while maintaining high visual quality with a PSNR of up to 30 and 26 on HunyuanVideo and Wan2.1-I2V, outperforming all prior methods. + +# 2 Related Work + +Sparse Attention for Video DiTs. Sparse attention mechanisms for accelerating DiTs fall into two categories: static and dynamic, depending on whether to select critical tokens dynamically during runtime or statically offline. Static methods [8, 4] predefine sparse patterns offline, such as identifying recent tokens as critical [4]. These methods lack adaptability to diverse sparsity patterns, leading to suboptimal performance. Dynamic methods [9, 10, 17, 18, 19, 20, 21, 22] determine sparse patterns at runtime, selecting critical tokens through an additional identification step. However, existing dynamic methods fail to achieve both high identification accuracy and low computation waste. In contrast, SVG2 consistently achieves superior performance under the same computation budget. + +Sparse Attention for Large Language Models (LLMs). Sparse attention for LLMs falls into two categories: memory-efficient and compute-efficient. Memory-efficient methods [6, 5, 7, 23, 24] reduce memory load to accelerate decoding but are ineffective for compute-bound DiT-based video generation. Compute-efficient methods [25, 26, 27, 12, 28, 29] focus on processing only critical tokens, yet cannot directly optimize video DiTs due to unique sparse patterns of video data. Notably, MMInference [12] introduces a modality-aware permutation for multi-modal LLMs. This permutation is rule-based, designed to permute inter-modality tokens. + +Linear Attention for Diffusion Models Linear attention [30, 31, 32] and state space models [33, 34] have gained attention in video generation, where the long-context problem makes attention module dominate the latency. Matten [35] uses the mamba block to capture global information and uses attention for local information. LinGen [36] adopts a combination of Mamba2 and Swin attention block for 1-minute video generation. M4V [37] proposes an MM-DiM block to overcome the adjacency of mamba in capturing complex spatial-temporal dynamics. Block-wise SSM scanning scheme [38] achieves long-term memory ability in video generation. TTT [39] proposes to use RNNs and designs a new TTT block for minute-long video generation. SANA [40, 41, 42] and DC-AE [43, 44, 45] reduce the generation overhead by using Linear Attention and Deep-Compressed Auto-Encoders. This line of research reduces the complexity of the attention from quadratic to linear, making it efficient in long video generation. + +Long Video Generation and Caching-Based Acceleration Minute-level long video generation poses new challenges in the field of video generation in both quality and efficiency. CausVid [46] and Self-Forcing [47] distill a bidirectional diffusion model to an autoregressive one to leverage the efficiency of KV cache. LongLive [48] achieves multi-prompt minute-level video generation with KV cache refreshing. Framework [49] compressed the token number within frames to reduce the sequence length. RifleX [50] and Freelong [51] extend the video generation length by context extrapolation in a training-free manner. RadialAttention [52], VMOBA [53], Mixture-of-Context [54], and VSA [55] adopt sparse attention in long-context fine-tuning to reduce the training overhead. Caching-based methods [56, 57, 58, 59, 60] optimized the efficiency by utilizing the redundancy between timesteps and classifier-free guidance (CFG). These lines of research is orthogonal to our sparse attention method, and can be integrated to achieve higher speedup. + +# 3 Motivation + +# 3.1 Attention in DiTs is Inherently Sparse + +Attention operation in DiTs is costly. During each denoising step, DiTs transform the input activations with hidden dimension $d$ into Query $(Q)$ , Key $(K)$ , and Value $(V)$ tensors, followed by a self-attention operation to produce the final output $O$ [61]: + +$$ +O = P \times V, \quad P = \operatorname {s o f t m a x} \left(\frac {Q K ^ {\top}}{\sqrt {d}}, \dim = - 1\right) +$$ + +where the attention score $P$ captures the relationship among tokens. However, computing $P$ has a quadratic complexity relative to the sequence length. State-of-the-art DiTs typically process thousands of tokens per frame across multiple frames, creating a significant performance bottleneck. E.g., generating a 33-frame video using HunyuanVideo-T2V-13B requires over $80\%$ of the total end-to-end time to be spent on the attention alone [4, 8]. + +Attention operation in DiTs is highly sparse. Fortunately, attention is inherently sparse, where only a small subset of computations significantly contributes to the final output. This sparsity arises from the characteristics of the softmax function, where a few largest values in $Q \times K^{\top}$ dominate the attention score $P$ , which in turn dictates the final weighted output $P \times V$ [6, 11]. + +To quantitatively assess this sparsity, we collect attention maps from Wan2.1-I2V-14B video generation and visualize the average recall of attention scores under varying computational budgets, defined by the number of critical $K$ tokens. Specifically, the critical tokens are selected following an oracle policy, where tokens are ranked in descending order based on their attention scores. This approach illustrates the upper bound of achievable recall under a constrained computational budget. + +As depicted in Figure 3, the attention is highly sparse, where only $13\%$ of the computations (i.e., the percentage of attention map retained in the sparse attention) are sufficient to achieve an attention recall of $95\%$ , maintaining a near-lossless PSNR of 27 while providing up to $2 \times$ theoretical end-to-end speedup. This observation highlights an opportunity to leverage the trade-off between generation quality and computational efficiency. + +![](images/462aa6e5e3f8706ec41ca04cc552d3dd41b9a320bf4e4fdff600618cf38ebc64.jpg) +Figure 3: Comparison of attention recall versus density (i.e., number of sparse computation normalized by total computation) for the oracle policy, SVG, and SpargeAttention. Notably, the significant gap between the oracle policy and existing methods highlights the potential for improvement. + +# 3.2 Existing Sparse Attention Fails to Match the Oracle Policy + +Despite the potential of sparsity in reducing computational cost, directly adopting the oracle policy is impractical. This is because identifying critical tokens requires calculating the full attention scores $P$ by $Q \times K$ , thus providing no actual speedup. To achieve practical efficiency, state-of-the-art approaches [9, 10] implement a coarse-grained identification strategy. Specifically, they cluster consecutive tokens into large blocks and calculate attention scores at the block level, providing an + +![](images/55eb011432d5dbfc1dde270fb0657b3e2eccb399f3958acfdfb14975bf5106e6.jpg) +Figure 4: Illustration of how existing methods fall short due to the inaccurate identification and computation waste, assuming a computation unit of $4 \times 4$ block. (a) Original attention map of a demonstration example. (b) Position-based clustering groups distinct tokens within the same clustering, causing the imprecise representation of mean-pooling or max-pooling. Therefore, blocks with smaller number of critical tokens are ignored, causing lower recall of attention scores. (c) Due to the scattered layout of critical tokens, even if achieving a high attention recall, each compute block processes both critical and non-critical tokens, thus causing computation waste and decreasing effective compute on critical tokens. (d) Semantic-aware permutation clusters and reorders similar tokens into contiguous layout, thus achieving high attention recall while minimizing computation waste. + +approximation of the original $P$ . This approach significantly reduces the identification overhead, with less than $1\%$ computation compared to the full attention when using a block size of 128. + +However, existing coarse-grained approaches reduce identification accuracy and lead to computation waste. As illustrated in Figure 3, existing sparse attention mechanisms consistently fall significantly short of achieving the attention recall of the oracle policy, regardless of the computation budget. This performance gap arises primarily from two key factors: + +Position-based clustering leads to inaccurate identification. Existing methods reduce identification overhead by clustering consecutive tokens into blocks. For instance, SpargeAttention [9] groups every 128 query tokens and 64 key tokens, using mean pooling to create a single representation for each block, which is then used to approximate $P$ . However, this position-based clustering does not guarantee semantic similarity among tokens. Tokens within the same block can exhibit vastly different activations in the latent space. For example, two physically close objects in a video frame, such as an apple and a cake, may have no semantic relationship. This variability within a block degrades the quality of the aggregated block-wise representation, leading to reduced identification accuracy. We illustrate this issue in Figure 4 and provide a quantitative analysis of the identification accuracy in § 5.5. To address this problem, we propose using semantic-aware clustering instead of position-based clustering, as detailed in § 4.1. + +Scattered critical tokens cause computation waste. Even if all critical tokens are perfectly identified, existing sparse attention mechanisms cannot achieve the theoretical computational savings promised by the oracle policy due to a mismatch between scattered sparse computations and hardware specifications. Since the criticality of tokens is determined by semantics, critical tokens are naturally scattered across the tensor rather than being contiguous. However, modern ML accelerators, such as NVIDIA GPU tensor cores, are optimized for dense matrix multiplication, which requires contiguous input dimensions [14, 62]. As a result, scattered critical tokens must be padded with non-critical tokens to maintain a contiguous layout, leading to significant computation waste. This issue is visualized in Figure 4, and we further quantify the computation waste in § 5.5. To approach the performance of the oracle policy, an automatic permutation is required to rearrange scattered critical tokens into a dense layout to minimize computation waste, as detailed in § 4.1. + +# 4 Methodology + +In this section, we introduce SVG2, a training-free sparse attention framework designed to use semantic-aware permutation to achieve a Pareto frontier trade-off between the generation quality and computational efficiency for video DiTs. We visualize the workflow of SVG2 in Figure 5. At the core of SVG2 is semantic-aware permutation, which aims to maximize identification accuracy of critical tokens and minimize computation waste (§ 4.1). To dynamic select and adjust the computation budget, SVG2 proposes centroid-based top- $p$ selection, which enables practical deployment (§ 4.2). + +![](images/4312cbb3518ff37d2f859f5be852c6f2f7daaece2915766625a2832a6169340d.jpg) +Figure 5: Overview of SVG2. (a) Original attention map of a demonstration example, with various colors representing various semantics. Only tokens with similar semantics attend to each other, having high attention scores thus selected as critical tokens. (b) After $k$ -means clustering, semantic-similar tokens (i.e., similar colors) are grouped into the same cluster, with the query and key centroids to precisely represent the cluster-level semantics. These centroids are then used to approximate the attention score for accurate identification of critical tokens. (c) Combined with Top- $p$ selection, critical tokens can be dynamically identified in a contiguous layout. + +Additionally, SVG2 investigates several system-algorithm co-designs, such as fast $k$ -means and attention kernel, significantly accelerating video generation (§ 4.3). + +# 4.1 Semantic-Aware Permutation with $k$ -means Clustering + +As discussed in § 3.2, existing sparse attention mechanisms suffer from inaccurate identification due to the position-based clustering. To this end, SVG2 proposes using semantic similarity to cluster rather than position, by performing $k$ -means on the activations of the input tokens. + +Specifically, for each attention head and transformer layer, $k$ -means is independently applied to query tokens $(Q \in \mathbb{R}^{N_q \times d})$ and key tokens $(K \in \mathbb{R}^{N_k \times d})$ , where $N_q$ and $N_k$ represent the number of tokens in $Q$ and $K$ , creating $C_q$ query clusters $Q_1, \ldots, Q_{C_q}$ and $C_k$ key clusters $K_1, \ldots, K_{C_k}$ . This approach enables tokens within each cluster to share similar semantics, improving the precision of centroid representation for better identification, as detailed in § 4.2. + +Furthermore, to densify the sparse computation of scattered critical tokens, SVG2 performs semantic-aware permutation based on the $k$ -means clustering. Although the semantically similar tokens are logically clustered, they are physically scattered in the tensors, resulting in substantial computational waste as described in § 3.2. To address this, SVG2 permutes tokens within each cluster into a contiguous layout. Such cluster-wise contiguous layout can be efficiently computed by the underlying ML accelerators, thus reducing computation waste. We detail the permutation algorithm and the mathematical equivalence for the attention output in the following formulations. Assuming $\pi_q \in \mathbb{R}^{N_q \times N_q}$ and $\pi_k \in \mathbb{R}^{N_k \times N_k}$ be the permutation matrices such that $\pi_q \pi_q^\top = I$ and $\pi_k \pi_k^\top = I$ , the permuted tokens are then $Q' = \pi_q Q$ , $K' = \pi_k K$ , and $V' = \pi_k V$ , where $K$ and $V$ share the same permutation $\pi_k$ to guarantee the output equivalence. The permuted attention output $O'$ is: + +$$ +\begin{array}{l} O ^ {\prime} = \pi_ {q} ^ {\top} \operatorname {A t t e n t i o n} \left(Q ^ {\prime}, K ^ {\prime}, V ^ {\prime}\right) = \pi_ {q} ^ {\top} \operatorname {s o f t m a x} \left(\frac {\left(\pi_ {q} Q\right) \left(\pi_ {k} K\right) ^ {\top}}{\sqrt {d}}\right) \pi_ {k} V \\ = \left(\pi_ {q} ^ {\top} \pi_ {q}\right) \operatorname {s o f t m a x} \left(\frac {Q K ^ {\top}}{\sqrt {d}}\right) \left(\pi_ {k} ^ {\top} \pi_ {k}\right) V = \operatorname {s o f t m a x} \left(\frac {Q K ^ {\top}}{\sqrt {d}}\right) V = O \\ \end{array} +$$ + +# 4.2 Centroid-Based Top- $p$ Selection + +Despite the semantic-coherent clusters provided by the semantic-aware permutation, it remains impractical to deploy SVG2 without addressing two critical challenges: (1) how to effectively estimate the criticality of clusters, and (2) how to dynamically determine the number of selected critical clusters (i.e., the number of critical tokens) to satisfy arbitrary accuracy requirements. + +Accurate and efficient estimation of criticality. To estimate the criticality of each cluster, SVG2 introduces a centroid-based estimation of attention scores $P$ . Specifically, it approximates the criticality of each token by estimating its $P$ using the centroids of its cluster, mimicking the oracle policy defined in § 3.1. As formulated in Equation 1, this approach calculates the pre-softmax scores $S$ using the centroids of each cluster. These scores are then weighted by the number of tokens within the cluster (i.e., the size of the cluster), to generate an approximate attention score $P'$ in Equation 2, providing an estimation of the actual $P$ . + +$$ +S _ {i j} = \frac {\operatorname {c e n t r o i d} \left(Q _ {i}\right) \cdot \operatorname {c e n t r o i d} \left(K _ {j}\right) ^ {T}}{\sqrt {d _ {k}}} \tag {2} +$$ + +Since tokens within the same cluster already share similar semantics, the centroids can serve as highly accurate representations of the actual activations, ensuring the reliability of such estimation. Furthermore, because the typical number of clusters (i.e., $C_q$ and $C_k$ ) is less than 1024, the computational overhead for this approximation is negligible compared to the full attention calculation, typically accounting for less than $1\%$ of the total computational cost. + +Dynamic adjustment of computation budget. To dynamically adjust the number of critical tokens instead of pre-defined constant, SVG2 employs a Top- $p$ selection strategy based on the approximated $P'$ . SVG2 first sorts all potential clusters in descending order according to their corresponding $P'$ . It then selects clusters sequentially until the accumulated $P'$ reaches a predefined target. + +# 4.3 Efficient System-Algorithm Co-design + +Fast $k$ -means with centroid cache. While $k$ -means clustering is essential to semantic-aware permutation, its iterative process can introduce substantial latency if the number of iterations is large before convergence. For example, with the state-of-the-art GPU implementation of $k$ -means++ [63], it takes more than 100 iterations to converge, consuming $50\%$ or even comparable time to the attention computation. Fortunately, DiTs are known to be similar between consecutive denoising steps [59, 64], enabling reusing the centroids from the previous step as the fast initialization for $k$ -means in the next step. Based on this observation, SVG2 implements a centroids cache, which automatically caches and reuses centroids between consecutive steps. This technique reduces the runtime of $k$ -means by up to $76\times$ , as evaluated in § 5.3. + +Efficient sparse attention kernel for varied block-sizes. While existing efficient attention implementations (e.g., FlashAttention [65], FlexAttention [66], and FlashInfer [16]) support block-wise sparse computation, they only support a static block size (e.g., $128 \times 128$ ). However, the sizes of clusters after semantic-aware permutation are naturally dynamic and diverse, causing computation waste with the static block size. For example, SVG2 could generate a query cluster with 128 tokens with a key cluster with 32 tokens. Such $128 \times 32$ computation needs to be padded into $128 \times 128$ to use existing kernels, which causes $75\%$ computation waste. To address this, SVG2 implements a customized attention kernel that accepts dynamic block-sizes as input. + +Our dynamic block-sparse attention kernel supports both FA2 (A100) and FA3 (H100), combining sparse loading and dense computation. For FA3, we use wgmma (m64n64k16) for dense compute to maximize hardware efficiency. For query tokens, we load contiguous tokens from the same cluster, which are naturally contiguous in memory after permutation. For key/value tokens, which may be scattered in global memory due to varying cluster sizes, SAPAttn uses per-token address offsets to perform sparse loading and stores them in shared memory in a contiguous layout. This enables efficient use of MMA instructions without the need for expensive key/value padding, leading to over $85\%$ of the theoretical maximum performance, where the upper bound is estimated by multiplying the sparsity density with the runtime of the dense FlashAttention-3. We evaluate the kernel efficiency in § 5.3. + +![](images/765c42ee3bcf64d6cde5dd92a1135a136315f6102b4ec5ac9a830a41a9a657d2.jpg) +Figure 6: Visualization of attention maps from different attention heads in Wan2.1 when generating videos from VBench [67]. (a) Original attention maps with diverse sparse patterns, assuming critical tokens highlighted in red. (b) Permuted attention maps. After semantic-aware permutation, critical tokens are permuted into a contiguous layout based on the $k$ -means clustering, enabling efficient block-wise computation without waste. (c) Recovered attention maps after applying centroid-based top- $p$ selection and undoing the permutation. The high similarity between the original and recovered attention maps demonstrates the effectiveness of SVG2. + +# 5 Experiment + +# 5.1 Setup + +Models. We evaluate SVG2 on open-sourced state-of-the-art video generation DiT models including Wan2.1-I2V/T2V-14B [2], and HunyuanVideo-T2V-13B [1] to generate videos with 720p resolution. After being tokenized by 3D-VAE, Wan2.1 generates 21 frames with 3600 tokens per frame, while HunyuanVideo processes 33 frames with 3600 tokens per frame. + +Metrics. We assess the similarity of generated video compared to full attention using the following metrics: Peak Signal-to-Noise Ratio (PSNR), Learned Perceptual Image Patch Similarity (LPIPS), and Structural Similarity Index Measure (SSIM). We use VBench [67] to evaluate the video quality. To quantify the efficiency of sparse attention mechanisms (i.e., computational budget), we use density, which is defined as the sparse attention computation divided by the full attention computation. To assess end-to-end efficiency, we use the total amount of computation (i.e., FLOPs) needed for generating videos. + +Datasets. For text-to-video generation, we adopt the prompt in Penguin Benchmark after prompt optimization provided by VBench team. For image-to-video generation, we adopt the prompt-image pairs provided by VBench [67] and crop images to $16:9$ ratios for 720p resolution. + +Baselines. We compare SVG2 against state-of-the-art sparse attention algorithms including static method Sparse VideoGen (SVG) [4], and dynamic methods SpargeAttention [9] and XAttention [10]. Note that we skip the evaluation of XAttention on Wan2.1 as it is not supported yet. For SVG, SpargeAttention, and XAttention, we use their official configurations. + +Implementations. We prototype SVG2 as an end-to-end framework with customized kernels from FlashInfer [16] and benchmark on NVIDIA H100 GPU with CUDA 12.8. For SVG2, we choose $C_q = 100$ and $C_k = 500$ , and explain the choice in § D. To showcase the trade-off between generation quality and efficiency, we evaluate on various accuracy targets (i.e., attention score recall) as detailed in § 5.4. We also sample a single data point for detailed comparison as shown in Table 1. We conduct experiments with sparse attention skipped during the first $30\%$ of denoising steps for all methods, as these steps are critical for generation quality. Following previous work [64, 68, 56, 59]. For experiment results without warmup, please check Table 2 in Appendix. + +# 5.2 Quality Evaluation + +We first qualitatively showcase the effectiveness of our proposed method by showing the visualization of attention maps. As shown in Figure 6, we collect attention maps from different attention heads when running Wan2.1 on prompts from VBench. Despite the diversity of the sparse patterns (i.e., different columns in Figure 6), semantic-aware permutation effectively densifies critical tokens into contiguous layout, which enables efficient computation without waste. Furthermore, by applying + +Table 1: Quality and efficiency benchmarking results of SVG2 and baselines. Warmup steps is set to $30\%$ . + +
ModelConfigPSNR ↑SSIM ↑LPIPS ↓VBench ↑Density ↓FLOP ↓↑Speedup ↑
Wan 2.114B, 720P, Image-to-Video---0.841100%526.76 PFLOPs1×
SpareAttn21.1810.6650.333-38.99%366.80 PFLOPs1.47×
SVG24.0590.8130.1740.83630.25%343.88 PFLOPs1.56×
Ours26.5620.8610.1380.83831.28%346.59 PFLOPs1.58×
Ours-Turbo24.5100.8120.1790.83614.13%301.62 PFLOPs1.84×
Wan 2.114B, 720P, Text-to-Video---0.846100%658.46 PFLOPs1×
SpareAttn20.5190.6230.3430.82042.03%468.46 PFLOPs1.44×
SVG22.9890.7850.1990.83730.25%429.86 PFLOPs1.58×
Ours25.8080.8540.1380.84229.51%427.43 PFLOPs1.60×
Ours-Turbo23.6820.7890.1960.83812.87%372.89 PFLOPs1.89×
Hunyuan13B, 720P, Text-to-Video---0.850100%612.38 PFLOPs1×
SpareAttn27.8920.8840.151-42.62%399.16 PFLOPs1.53×
XAttention28.8920.8980.1200.83939.32%386.90 PFLOPs1.56×
SVG29.1570.9050.1200.84529.86%351.75 PFLOPs1.91×
SVG + FP829.0330.9020.1210.84329.86%351.75 PFLOPs2.3×
Ours30.4520.9100.1170.85225.45%335.36 PFLOPs2.30×
Ours + FP830.3890.9080.1180.85125.45%335.36 PFLOPs2.55×
+ +![](images/f1a7e49c327fe9e9bcbc776078a0f6d87f6042b636216bfe22ced35d5916fc8c.jpg) +(a) Efficiency evaluation of k-means + +![](images/7283fbb1b28dc3daf45049a8c685cc7ec6b6171f5a9a3db31a81165daf19d1cb.jpg) +(b) Efficiency evaluation of customized attention +Figure 7: Efficiency evaluation for fast $k$ -means with centroids cache and customized attention kernel. + +centroid-based top- $p$ selection and then undoing the permutation, the permuted attention map is recovered into the original layout, which shows high similarity to the original attention map. + +To quantitatively assess the generation quality, we evaluate the quality of videos generated by SVG2, when compared to baselines and report the results in Table 1. SVG2 consistently outperforms all baseline methods in terms of PSNR, SSIM, and LPIPS, while still maintaining the highest speedup. Specifically, SVG2 achieves an average PSNR of 26.5 on Wan2.1 and 30.4 on HunyuanVideo, demonstrating its effectiveness on generating highly consistent and smooth videos. + +Due to space limitations, the full results of VBench can be found at Table 3 and Table 4 in the appendix. + +# 5.3 Efficiency Evaluation + +Efficient $k$ -means with centroids cache. To demonstrate the effectiveness of centroids cache in improving efficiency of $k$ -means, we compare the density achieved by SVG2 to reach the 90% attention recall, when varying different number of execution time (i.e., number of $k$ -means iterations). We use widely-used algorithm $k$ -means++ [63]. Since the $k$ -means quality directly determine the accuracy of critical token identification, it also determines the achieved density. The lower the density is, the better the $k$ -means is. As shown in Figure 7(a), enabling centroids cache reduces the end-to-end latency of $k$ -means by $76 \times$ when reaching comparable or lower density. This demonstrates the effectiveness of centroids cache, which greatly reduce the initialization time. + +Efficient attention kernel with dynamic block-sizes. To showcase the efficiency of our customized attention kernels with dynamic block-sizes, we evaluate the computation FLOPs of our implementation + +tion compared with a state-of-the-art attention library, FlashInfer [16]. We vary different combination of hyper-parameters (i.e., number of clusters $C_q$ and $C_k$ ) and apply centroid-based top- $p$ selection to generate the practical workloads of dynamic block sizes. We fix the attention recall as $90\%$ . As shown in Figure 7(b), our customized kernels achieve an average of $1.48 \times$ computation reduction. On practical setup with $C_q = 100$ , $C_k = 500$ , ours achieves $1.88 \times$ reduction of computation waste. + +End-to-end speedup evaluation. To showcase the end-to-end speedup of SVG2, we incorporate several efficiency metrics, including density, FLOPs, and end-to-end speedup into Table 1. SVG2 achieves an average speedup of $1.82 \times$ while maintaining the highest generation quality. We also include a SVG2-Turbo to showcase the efficiency potential, which maintains a similar generation quality as baselines but achieves much higher speedup. Specifically, SVG2-Turbo achieves $2.5 \times$ smaller density compared to SVG while achieving an even better PSNR of 23.7. Such results can be cross-validated with the sensitivity evaluation in § 5.4. + +# 5.4 Sensitivity Test on Quality-Efficiency Trade-off + +To validate the effectiveness of SVG2, we conduct a comprehensive evaluation on Wan2.1-I2V-14B, comparing it against baseline methods across a wide range of computational budgets (i.e., density). As shown in Figure 2, SVG2 consistently achieves better generation quality at any given density, positioning it on the Pareto frontier of the quality-efficiency trade-off. Notably, SVG2 reduces density by up to $2.3 \times$ while maintaining the same PSNR. + +# 5.5 Ablation Study on Semantic-Aware Permutation + +Effectiveness on improving identification accuracy. To assess the effectiveness of semantic-aware permutation in improving the accuracy of critical token identification, we measure attention recall by comparing semantic-aware permutation-enabled and semantic-aware permutation-disabled configurations across varying computational budgets. Both methods use mean-pooling and maintain the same cluster size for consistency. As shown in Figure 8, semantic-aware permutation consistently achieves higher attention recall, indicating more accurate identification of critical tokens. This improvement is attributed to the semantic-coherent clusters generated by semantic-aware permutation, which offer precise representations. + +![](images/0757981c7a3ad8b78b0e03446296f38cb27e5f8af3942215a1bb03bed22ed4d3.jpg) +Figure 8: Attention recall across various densities. Enabling permutation consistently surpasses disabling permutation. + +Effectiveness on reducing computation waste. We further investigate the impact of semantic-aware permutation on reducing computational waste. Specifically, we use the same set of critical tokens selected by centroid-based top- $p$ selection. For the semantic-aware permutation-enabled configuration, we feed the contiguous layout generated after permutation into GPUs, while for the semantic-aware permutation-disabled configuration, we use the scattered layout before permutation. As shown in our results, enabling semantic-aware permutation reduces computational overhead by an average of $36\%$ , while maintaining the same set of critical tokens. + +# 6 Conclusion & Limitation + +In this paper, we proposed SVG2, a training-free sparse attention approach for accelerating DiT-based video generation. By clustering tokens based on semantic similarity, SVG2 accurately identifies critical tokens. By permuting critical tokens into a contiguous layout, SVG2 effectively reduces the computation waste. Comprehensive evaluations show that SVG2 achieves a superior trade-off between generation quality and efficiency, making video generation more efficient and practical. The major limitation of this paper lies in the lack of discussion and evaluation on whether the proposed methods can be extended to attention mechanisms other than DiTs. + +# References + +[1] Weijie Kong, Qi Tian, Zijian Zhang, Rox Min, Zuozhuo Dai, Jin Zhou, Jiangfeng Xiong, Xin Li, Bo Wu, Jianwei Zhang, et al. Hunyuanvideo: A systematic framework for large video generative models. arXiv + +preprint arXiv:2412.03603, 2024. 2, 3, 8 +[2] Ang Wang, Baole Ai, Bin Wen, Chaojie Mao, Chen-Wei Xie, Di Chen, Feiwu Yu, Haiming Zhao, Jianxiao Yang, Jianyuan Zeng, Jiayu Wang, Jingfeng Zhang, Jingren Zhou, Jinkai Wang, Jixuan Chen, Kai Zhu, Kang Zhao, Keyu Yan, Lianghua Huang, Mengyang Feng, Ningyi Zhang, Pandeng Li, Pingyu Wu, Ruihang Chu, Ruili Feng, Shiwei Zhang, Siyang Sun, Tao Fang, Tianxing Wang, Tianyi Gui, Tingyu Weng, Tong Shen, Wei Lin, Wei Wang, Wei Wang, Wenmeng Zhou, Wente Wang, Wenting Shen, Wenyuan Yu, Xianzhong Shi, Xiaoming Huang, Xin Xu, Yan Kou, Yangyu Lv, Yifei Li, Yijing Liu, Yiming Wang, Yingya Zhang, Yitong Huang, Yong Li, You Wu, Yu Liu, Yulin Pan, Yun Zheng, Yuntao Hong, Yupeng Shi, Yutong Feng, Zeyinzi Jiang, Zhen Han, Zhi-Fan Wu, and Ziyu Liu. Wan: Open and advanced large-scale video generative models. arXiv preprint arXiv:2503.20314, 2025. 2, 3, 8 +[3] Zhuoyi Yang, Jiayan Teng, Wendi Zheng, Ming Ding, Shiyu Huang, Jiazheng Xu, Yuanming Yang, Wenyi Hong, Xiaohan Zhang, Guanyu Feng, et al. Cogvideox: Text-to-video diffusion models with an expert transformer. arXiv preprint arXiv:2408.06072, 2024. 2 +[4] Haocheng Xi, Shuo Yang, Yilong Zhao, Chenfeng Xu, Muyang Li, Xiuyu Li, Yujun Lin, Han Cai, Jintao Zhang, Dacheng Li, et al. Sparse videogen: Accelerating video diffusion transformers with spatial-temporal sparsity. In ICML, 2025. 2, 3, 4, 8 +[5] Jiaming Tang, Yilong Zhao, Kan Zhu, Guangxuan Xiao, Baris Kasikci, and Song Han. Quest: Query-aware sparsity for efficient long-context llm inference. In ICML, 2024. 2, 3 +[6] Zhenyu Zhang, Ying Sheng, Tianyi Zhou, Tianlong Chen, Lianmin Zheng, Ruisi Cai, Zhao Song, Yuandong Tian, Christopher Ré, Clark Barrett, Zhangyang Wang, and Beidi Chen. H2o: Heavy-hitter oracle for efficient generative inference of large language models, 2023. 2, 3, 4 +[7] Guangxuan Xiao, Yuandong Tian, Beidi Chen, Song Han, and Mike Lewis. Efficient streaming language models with attention sinks. In ICLR, 2024. 2, 3 +[8] Peiyuan Zhang, Yongqi Chen, Runlong Su, Hangliang Ding, Ion Stoica, Zhenghong Liu, and Hao Zhang. Fast video generation with sliding tile attention. arXiv preprint arXiv:2502.04507, 2025. 2, 3, 4 +[9] Jintao Zhang, Chengong Xiang, Haofeng Huang, Jia Wei, Haocheng Xi, Jun Zhu, and Jianfei Chen. Spargeattn: Accurate sparse attention accelerating any model inference. In ICML, 2025. 2, 3, 4, 5, 8 +[10] Ruyi Xu, Guangxuan Xiao, Haofeng Huang, Junxian Guo, and Song Han. Xattention: Block sparse attention with antidiagonal scoring. In ICML, 2025. 2, 3, 4, 8 +[11] Kan Zhu, Tian Tang, Qinyu Xu, Yile Gu, Zhichen Zeng, Rohan Kadekodi, Liangyu Zhao, Ang Li, Arvind Krishnamurthy, and Baris Kasikci. 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Long-context state-space video world models, 2025. 3 +[39] Karan Dalal, Daniel Koceja, Gashon Hussein, Jiarui Xu, Yue Zhao, Youjin Song, Shihao Han, Ka Chun Cheung, Jan Kautz, Carlos Guestrin, Tatsunori Hashimoto, Sanmi Koyejo, Yejin Choi, Yu Sun, and Xiaolong Wang. One-minute video generation with test-time training, 2025. 3 + +[40] Enze Xie, Junsong Chen, Yuyang Zhao, Jincheng Yu, Ligeng Zhu, Chengyue Wu, Yujun Lin, Zhekai Zhang, Muyang Li, Junyu Chen, Han Cai, Bingchen Liu, Daquan Zhou, and Song Han. Sana 1.5: Efficient scaling of training-time and inference-time compute in linear diffusion transformer, 2025. 3 +[41] Enze Xie, Junsong Chen, Junyu Chen, Han Cai, Haotian Tang, Yujun Lin, Zhekai Zhang, Muyang Li, Ligeng Zhu, Yao Lu, and Song Han. 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If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this paper does not involve LLMs as any important, original or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Visualization of the Generated Videos + +We provide visualization comparison between SVG2 and Dense Attention on HunyuanVideo and Wan 2.1. Results in Figure 9 and Figure 10 demonstrate that SVG2 can preserve high pixel-level fidelity, achieving similar generation quality compared with the dense attention. Real video samples are provided in the supplementary materials. + +![](images/1a0f08f476389a46bae39747bc32ad4ddc568b85669616e0141796fcc205eb72.jpg) +Figure 9: Comparison of Dense Attention and SVG2 on HunyuanVideo and Wan 2.1 Text-to-Video generation. + +![](images/80ff2ac0e0c0180e831491cb0b4da3a045a701fae3e09032f4ecf821d991e071.jpg) +Dense Attention + +![](images/b0df4b06e90cddeb80447ea7c0feda9e517dd74ac5e452f7cd4f6f55c81b985e.jpg) +Dense Attention + +![](images/1512e38bd9e095322cd23f723e227a7aa297e26abe7171db1190d7864720e200.jpg) +Dense Attention + +![](images/c2ee8a5e8357c1d0cb830f46b54644072802faed03d0a5a6631d25172c453924.jpg) +Dense Attention + +![](images/0bec5d6fbed1985656de8dfaf112ae8d156a8cb3953d3927f12506291f65215c.jpg) +Dense Attention + +![](images/ef429e444aa58beabe46a8a2e29f33c05e71473dfd6d34bc05fad4e980bbe377.jpg) +Dense Attention + +![](images/8f140e190aa28004d4631feff15881d12d337cfac6c5fc8a259a6221a53bad13.jpg) +SVG2 + +![](images/6863c193dcdce98457243e5f5853a9797aad6934739a43e962e0a33ce7c1c902.jpg) +SVG2 + +![](images/a1862a3d2a1c50d17951c2df1345b77582063079f30f0ee83668fb85107fd280.jpg) +SVG2 + +![](images/bb1199745b8d570bf2efb8ee46058921c68bc2272f2fa300518db5e73b9c1ee7.jpg) +SVG2 + +![](images/d3a91714626fd8adde99441884a9b9585a07c27c6985cf0a3d934fe2e0fcbda3.jpg) +SVG2 + +![](images/82ba77ad0b6aea0eb28b56afdbb0bd9e324f415e35fd5fc98331f903a19f1fd5.jpg) +SVG2 + +![](images/1e244c8c0e55d452be9dff97d9803b3b04624501e68fc4507027f9e6d8ba707d.jpg) +Dense Attention +Figure 10: Comparison of Dense Attention and SVG2 on Wan 2.1 Image-to-Video generation. + +# B Performance Comparison in Warmup-free Setting + +We present the performance comparison between SVG2 and the baseline without warmup steps. We find that our method consistently offers better quality under a warmup-free setting. + +Table 2: Quality and efficiency benchmarking results of SVG2 and baselines. Warmup steps is set to $0\%$ . + +
ModelConfigPSNR ↑SSIM ↑LPIPS ↓VBenchmark ↑Density ↓FLOP ↓Attn Speedup ↑Speedup ↑
Wan 2.114B, 720P, Image-to-Video---0.841100%526.76 PFLOPs1×1×
SVG15.6080.5120.4040.82329.54%262.85 PFLOPs2.26×1.86×
Ours18.2760.6150.3170.83229.34%262.10 PFLOPs2.95×2.10×
Wan 2.114B, 720P, Text-to-Video---0.851100%658.46 PFLOPs1×1×
SVG13.2940.4070.5120.84929.54%328.56 PFLOPs2.28×1.89×
Ours16.5020.5620.3730.85230.12%331.28 PFLOPs2.98×2.13×
Hunyuan13B, 720P, Text-to-Video---0.820100%612.38 PFLOPs1×1×
SVG12.2980.4920.4830.80829.86%240.05 PFLOPs3.45×2.48×
Ours19.8790.7350.2600.81628.94%235.16 PFLOPs4.06×2.69×
+ +# C VBenchmark Results + +We provide the full VBench results of SVG2 and baselines in Table 3 and Table 4. These results clearly shows that SVG2 outperforms all other baselines. + +Table 3: VBenchmark result of SVG2. Warmup steps is $0\%$ . + +
ModelConfigSubConsisBackConsisMotionSmoothAesQualImagQualAverage
Wan 2.114B, 720P, Image-to-Video0.9460.9560.9790.6180.7090.841
SVG0.9160.9350.9760.5910.6980.823
Ours0.9360.9460.9770.5970.7000.832
Wan 2.114B, 720P, Text-to-Video0.9700.9700.9920.6120.7080.851
SVG0.9630.9690.9910.6120.7080.849
Ours0.9710.9700.9920.6240.7070.852
Hunyuan13B, 720P, Text-to-Video0.8880.9380.9940.5940.6850.820
SVG0.8670.9300.9910.5940.6560.808
Ours0.8880.9350.9940.5890.6750.816
+ +Table 4: VBenchmark result of SVG2. Warmup steps is ${30}\%$ . + +
ModelConfigSubConsisBackConsisMotionSmoothAesQualImagQualAverage
Wan 2.114B, 720P, Image-to-Video0.9460.9560.9790.6180.7090.841
SVG0.9410.9480.9780.6060.7090.836
Ours0.9430.9510.9770.6060.7090.838
Wan 2.114B, 720P, Text-to-Video0.9560.9680.9830.6130.7130.846
SpareAttn0.9270.9480.9780.5670.6840.820
SVG0.9470.9600.9800.5970.7030.837
Ours0.9540.9650.9820.6020.7090.842
Hunyuan13B, 720P, Text-to-Video0.9150.9410.9930.6480.7530.850
XAttention0.9120.9240.9920.6310.7390.839
SVG0.9140.9280.9930.6520.7390.845
Ours0.9170.9460.9930.6570.7510.852
+ +![](images/dea39f32dd0541d8aa982c370e4b438aba8c1a98da8a8c8ae116ba98767e5e22.jpg) +Figure 11: Efficiency evaluation for our attention kernel. We fix the number of query clusters and vary the number of key clusters. + +![](images/b090d88726568635d70eee54aa8205afc453afc43feb84be84a061ffed61f353.jpg) + +Throughput vs Density | Blocks Row: [20, 50, 100, 200, 300, 400] | Blocks Col: [1000] + +![](images/563fc2f78707bfbafe2a1f78304fc83f5f91fde9dfdc9fa62d325b8764517c6e.jpg) +Figure 12: Efficiency evaluation of our attention kernel, where we fix the number of key clusters and vary the number of query clusters. + +![](images/a3a1f4a3a40f658611cccf559585f7e45f1023261b15797e6072b85008bda776.jpg) + +# D Ablation on the Number of Clusters + +# D.1 Effect on the Sparse Attention Kernel + +Our sparse attention kernel is fully compatible with both FlashAttention-2 and FlashAttention-3. We implemented Sparse VideoGen2 on both FA2 and FA3 backends, achieving substantial speedups on A100 and H100 GPUs. Under the Wan 2.1 setting with a sequence length of 74,256 on H100, we benchmark kernel performance by comparing sparse and dense attention across varying densities and cluster configurations. In each experiment, we control either $C_q$ or $C_k$ , fixing one while varying the other. Results in Figure 11 and Figure 12 demonstrate that the kernel's performance will drastically decrease when $C_q$ is larger than 200. However, the performance is nearly identical as we increase $C_k$ to as large as 4000. This suggests us to adopt a larger $C_k$ than $C_q$ . + +# D.2 End-to-end Latency-quality Trade-off + +We further varied the Q/K cluster counts and measured both PSNR and end-to-end efficiency. Our results in Table 5 show that setting $Q = 100$ and $K = 500$ provides the best balance between generation quality and efficiency. While increasing the number of clusters generally improves quality, efficiency can degrade due to hardware layout constraints. In particular, tensor cores require fixed input sizes (e.g., 64 for m64n64k16) to fully utilize computation, meaning that each Q cluster + +
CqCkPSNRSSIMLPIPSSpeedup
10025025.4970.8010.1821.90x
100100026.2760.8250.1591.71x
5050022.5610.7420.2581.90x
20050026.2130.8200.1571.78x
40050026.4880.8680.1321.25x
10050026.1280.8160.1691.89x
+ +Table 5: Performance comparison across different QC and KC settings. + +must contain at least 64 tokens on average. Cluster counts beyond $Q = 100$ or $K = 500$ lead to underutilization, reducing efficiency despite potential quality gains. + +# D.3 Ablation on Permutation + +We performed ablation studies to investigate whether the query and key representations can adopt the same clustering strategy. Specifically, we evaluated three variants: applying Q clustering permutation $\pi_{Q}$ to both Q and K, applying K clustering permutation $\pi_{K}$ to both, and using clustering based on hidden states before QKV linear layer (i.e., shared QK embedding) for both sides $\pi_{S}$ . As shown in Table 6, all three variants led to worse PSNR even with more computation budget (i.e., density) compared to clustering Q and K independently. + +Table 6: Permutation comparison with corresponding Density and PSNR values + +
Permutation used by QPermutation used by KDensityPSNR
πQπK31.28%26.562
πQπQ38.23%22.439
πKπK38.58%22.183
πSπS87.27%26.495
+ +To further understand this, we compared the permutations of Q and K clustering and found that the permutation patterns differ substantially. Specifically, we calculate the Adjusted Rand Index value between Q clusters and K clusters, and the average ARI is 0.345, which is not very high. Therefore, clustering Q and K independently is necessary for preserving the expressiveness of attention. + +# E Performance Gap between HunyuanVideo and Wan 2.1 + +# E.1 Quality Difference + +In our experiments, we find that the quality performance (e.g., PSNR, SSIM, LPIPS) on Wan 2.1 is generally lower than HunyuanVideo across all methods. The reason is that Hunyuan is relatively robust against precision variance while Wan2.1 is highly sensitive. For instance, when evaluating the same dense attention using different backends (FlexAttention, FlashAttention, Torch SDPA), Wan2.1 exhibited PSNR as low as 27-28. However, HunyuanVideo exhibits 33-34 PSNR despite no setup changes. Therefore, it is natural that SVG2 achieves a lower PSNR on Wan compared with HunyuanVideo, due to its sensitivity to numerical changes. These differences are largely model-specific and reflect varying sensitivity to low-level numerical behaviors, which do not correlate with the performance of the methodology. + +# E.2 Speedup Difference + +We also find that the speedup result on Wan 2.1 is generally lower than HunyuanVideo $(1.89\times$ versus $2.30\times)$ . The difference in end-to-end speedup between HunyuanVideo and Wan primarily stems from their varying attention cost ratios, which are mainly due to different context lengths and model architectures. Specifically, HunyuanVideo's context length is $118\mathrm{k}$ , while Wanx's context length is $75\mathrm{k}$ . HunyuanVideo has 2 parts in its layers: Self Attention and Feed-Forward Network, while Wanx + +has an additional cross-attention block. Therefore, the attention proportion in HunyuanVideo will be larger than WanX. Since our method primarily accelerates the attention module via SVG2, the overall speedup naturally scales with its contribution to total runtime. 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While advancements have been made in expediting the sampling process, the underlying architectural inefficiencies within DiT remain underexplored. We introduce SparseDiT, a novel framework that implements token sparsification across spatial and temporal dimensions to enhance computational efficiency while preserving generative quality. Spatially, SparseDiT employs a tri-segment architecture that allocates token density based on feature requirements at each layer: Poolingformer in the bottom layers for efficient global feature extraction, Sparse-Dense Token Modules (SDTM) in the middle layers to balance global context with local detail, and dense tokens in the top layers to refine high-frequency details. Temporally, SparseDiT dynamically modulates token density across denoising stages, progressively increasing token count as finer details emerge in later timesteps. This synergy between SparseDiT's spatially adaptive architecture and its temporal pruning strategy enables a unified framework that balances efficiency and fidelity throughout the generation process. Our experiments demonstrate SparseDiT's effectiveness, achieving a $55\%$ reduction in FLOPs and a $175\%$ improvement in inference speed on DiT-XL with similar FID score on $512 \times 512$ ImageNet, a $56\%$ reduction in FLOPs across video generation datasets, and a $69\%$ improvement in inference speed on PixArt- $\alpha$ on text-to-image generation task with a 0.24 FID score decrease. SparseDiT provides a scalable solution for high-quality diffusion-based generation compatible with sampling optimization techniques. Code is available at https://github.com/changsn/SparseDiT. + +# 1 Introduction + +Diffusion models [18, 12, 43, 3] have emerged as powerful tools in visual generation, producing photorealistic images and videos by gradually refining structured content from noise. Leveraging the scalability of Transformers, Diffusion Transformer (DiT) models [41, 2] extend these capabilities to more complex and detailed tasks [13, 62] and form the backbone of advanced generation frameworks such as Sora [6]. Yet, despite their impressive generative capabilities, DiT models face significant computational limitations that restrict their broader applicability, especially in scenarios where efficiency is paramount. + +The computational burden of DiT models arises mainly from the need for numerous sampling steps in the denoising process and the quadratic complexity of Transformer structures with respect to the number of tokens. Although recent research has focused on reducing this burden by accelerating the sampling process through techniques such as ODE solvers [51, 30], flow-based methods [26, 28], + +![](images/a86aaedc350cf97dc50a5021fcb79a31883cff927d1e6bbf9b0618e023523f65.jpg) +(a) + +![](images/5c6a285d255bf79664ef655fd7c82160519420ce02f121e694856d4aae2ab4d8.jpg) +(b) +Figure 1: (a) The attention maps in different self-attention layers. Zoom-in for better visibility. (b) The normalized variances of attention maps in DiT-XL. Different curve lines represent different sampling steps. + +and knowledge distillation [53, 47, 32, 35, 63], these approaches often overlook the core architectural inefficiencies of DiT itself. Unlike U-Net-based architectures [44], which mitigate complexity through a contracting-expansive structure, DiT's reliance on token-level self-attention incurs high costs that scale with model size and token count. This bottleneck highlights a critical need for DiT-specific innovations that manage token density intelligently, balancing efficiency and quality without sacrificing generation fidelity. + +To address this challenge, we examine the distribution of attention across DiT's layers and sampling steps, seeking to uncover structural patterns that could guide more efficient token management. As illustrated in Figure 1a, attention maps reveal that tokens capture different levels of granularity across layers: in the bottom layers, such as layer 0, attention maps approximately appear uniform distribution, indicating a focus on broad, global features akin to global pooling. Meanwhile, middle layers alternate in their attention, with certain layers (e.g., layer 6 and layer 26) capturing local details, while others (e.g., layer 22) emphasize global structure. This observation is quantified in Figure 1b, where the variance in attention scores highlights DiT's consistent pattern of alternation between global and local feature extraction across layers. In addition to spatial analysis, Figure 1b also reveals insights across sampling steps. We observe that, while the pattern of alternation between global and local focus remains stable, attention variance increases as the denoising process advances, indicating a growing emphasis on local information at lower-noise stages. This analysis yields three core insights: (1) the initial self-attention layers exhibit minimal variance, showing low discrepancy in feature extraction across tokens; (2) DiT's architecture inherently alternates between global and local feature focus across layers, a pattern consistent across all sampling steps; and (3) as denoising progresses, the model increasingly prioritizes local details, dynamically adapting to heightened demands for detail at later stages. + +Building on these insights, we propose SparseDiT, which reimagines token density management as a dynamic process adapting across both spatial layers and temporal stages. Spatially, shallow layers capture smooth, global features, making complex self-attention less efficient. Subsequent layers alternate between high-frequency local details and low-frequency global information, where sparse tokens efficiently capture global features, and dense tokens refine local details. Temporally, as denoising advances, the need for local detail increases, motivating a dynamic token adjustment strategy. This dual-layered adaptation integrates layer-wise token modulation—balancing efficiency and detail within each sampling step—and a timestep-wise pruning strategy that adjusts token density dynamically. Together, these spatial and temporal adaptations allow SparseDiT to achieve computational efficiency without sacrificing high-fidelity detail, as reflected in SparseDiT's architectural design and pruning strategy, tailored to meet the unique spatial and temporal demands of the generation process. + +SparseDiT Architecture: SparseDiT's architecture employs a three-segment design that aligns token density with the features each layer captures. In the bottom layers, we replace self-attention with a poolingformer structure to capture broad global features through average pooling, reducing computation while preserving essential structure. In the middle layers, SparseDiT introduces Sparse-Dense Token Modules (SDTM), blending sparse tokens for global structure with dense tokens to refine local details. This design enables SparseDiT to retain efficiency while maintaining stability and + +![](images/42dc316de5f82c6a0dd632c188b42aeb09033b71f242624ba0cb3bc408c41ddf.jpg) +(a) Architecture of SparseDiT. + +![](images/25051c2957225f2e3c53aa30e987abed9bb89b7d892c49f283a7ff7ec3286f79.jpg) +(b) Timestep-wise pruning rate strategy. +Figure 2: Architecture of SparseDiT and timestep-wise pruning rate strategy. The architecture consists of three segments: bottom, middle, and top. The bottom segment includes poolingformers (PF). The middle segment comprises multiple sparse-dense token modules (SDTMs), where sparse token generation transformers ("Sparse Gen") and dense token recovery transformers ("Dense Rec") alternate to balance global and detailed information processing. The top segment contains standard transformers ("Tr") for final processing. During the denoising process, we apply varying pruning rates $r$ at different stages (depicted as four stages). + +detail. In the top layers, the model processes all tokens densely, focusing on high-frequency details for refined output quality. + +Timestep-Wise Pruning Rate Strategy: Complementing spatial adjustments, SparseDiT's timestepwise pruning rate strategy adapts token density across denoising stages. In early stages, where broad, low-frequency structures dominate, SparseDiT applies a high pruning rate, conserving resources. As the process progresses and intricate details emerge, the pruning rate decreases, allowing for a gradual increase in token density. This adjustment aligns computational effort with each stage's complexity, dynamically balancing efficiency and detail to optimize resources for high-fidelity output. + +Our empirical results substantiate the effectiveness of SparseDiT's integrated approach. SparseDiT reduces the FLOPs of DiT-XL by $55\%$ and improves inference speed by $175\%$ on $512 \times 512$ ImageNet [11] images, with only a increase of 0.09 in FID score [17]. On the Latte-XL dataset [34], SparseDiT achieves a $56\%$ reduction in FLOPs across standard video generation datasets, including FaceForensics [45], SkyTimelapse [60], UCF101 [54], and Taichi-HD [50]. Additionally, on the more challenging text-to-image generation task, we achieve a $69\%$ improvement in inference speed on PixArt- $\alpha$ with a 0.24 FID score reduction. These results demonstrate SparseDiT's capability to provide an efficient, high-quality architecture for diffusion-based generation, compatible with further sampling optimization techniques for enhanced efficiency. + +# 2 Related works + +Efficient sampling process. This direction focuses on optimizing sampling steps in diffusion models. Some approaches [51, 30, 1, 31] design new solvers to achieve faster sampling with fewer steps. Consistency models (CMs) [53, 52, 32, 58, 21] are closely related to diffusion models, achieving by distilling pre-trained diffusion models or training from scratch. These models learn a one-step mapping between noise and data, and all the points of the sampling trajectory map to the same initial data point. Flow-based [26, 28, 72, 13, 71] approaches straighten the transport trajectories among different distributions and thereby realize efficient inference. Knowledge distillation are applied by some methods [53, 47, 32, 35, 63] to reduce the sampling steps and match the output of the combined conditional and unconditional models. Parallel sampling techniques [49, 70, 56] employ Picard-Lindelöf iteration or Fourier neural operators for solving SDE/ODE. + +Efficient model structure. Another avenue focuses on improving inference time within a single model evaluation. Some methods attempt to compress the size of diffusion models via pruning [14] + +![](images/1fe8b8c8034b81e91a764c465e669cd27900569b873a70514ed49092e0fb175f.jpg) +(a) Standard DiT-XL. + +![](images/242d83569641eba490847c7cf29084752672930ade0f722e103f85254366088a.jpg) +(b) Modifying first two attention in DiT-XL. +Figure 3: Comparison of images generated by the standard DiT-XL model and a modified DiT-XL model where the first two attention maps are replaced with full-one matrices without fine-tuning. Both sets of images exhibit similar visual quality. + +or quantization [25, 48]. Spectral Diffusion [61] boosts structure design by incorporating frequency dynamics and priors. OMS-DPM [27] proposes a model schedule that selects small models or large models at specific sampling steps to balance generation quality and inference speed. Deepcache [33], TokenCache [29], and DiTFastAttn[66] reuse or share similar features across sampling steps. The early stopping mechanism in diffusion is explored in [24, 37, 55]. + +Most of the above methods are designed for general diffusion models. Recently, the Diffusion Transformer (DiT) has emerged as a more potential backbone, surpassing the previously dominant U-Net architectures. Unlike U-Net-based models, DiT's inefficiency stems largely from the self-attention mechanism. Limited research has specifically addressed the efficiency of DiT-based models. Pu et al. [42] identifies the redundancies in the self-attention and adopts timestep dynamic mediator tokens to compress attention complexity, achieving some performances but limited FLOPs reduction, as this approach primarily reduces the number of keys and values rather than the overall tokens count. In Vision transformers for classification task, numerous methods [59, 9, 15, 7, 4, 73, 36, 69, 67] successfully removed redundant tokens to improve performance-efficiency trade-offs. ToMeSD [5] is the first to explore token reduction in diffusion models. However, its performance on DiT was suboptimal, as shown in [37, 68]. U-Net can be regarded as a sparse network due to its contracting and expansive structure. EDT [10] applies a novel architecture similar to U-Net, while they still have 1.4 FID score gap compared to its baseline model MDTv2 [16] via 2,000K iterations. These findings indicate that directly transferring token reduction techniques from classification tasks or reusing U-Net structures may be insufficient for DiT. Instead, a specialized token reduction strategy is needed. DyDiT [68] and DiffCR [64] design dynamic networks to compress DiT from multiple dimensions, such as token, layer, attention head, channel. + +# 3 Method + +We first introduce SparseDiT architecture in Section 3.1, followed by our timestep-wise pruning rate strategy in Section 3.2. Finally, Section 3.3 details the initialization and fine-tuning of our network. + +# 3.1 SparseDiT + +Overview architecture. The SparseDiT architecture is illustrated in Figure 2a, where the transformer layers of DiT based model are divided into three groups: bottom, middle, and top. The bottom layers leverage poolingformers to efficiently capture global features. The middle layers contain multiple sparse-dense token modules (SDTM), which decouple the representation process into global structure capture and local details enhancement using sparse and dense tokens, respectively. The top layers retain the standard transformers, processing dense tokens to generate the final predictions at each sampling step. The primary computational savings are achieved through sparse tokens, hence the majority of transformer layers are located in the middle section to maximize efficiency. + +Poolingformer in the bottom layers. As shown in the first subfigure in Figure 1a, attention scores in the bottom layers exhibit a nearly uniform distribution, with each token evenly extracting global features, akin to a global average pooling operator. To investigate further, we conducted a toy experiment where the attention maps in the first two transformer layers were replaced with + +matrices filled with a constant value (e.g., 1), without any fine-tuning. Figure 3 shows that, using the same initial noise and random seed, the images generated by the original DiT-XL and the modified DiT-XL are nearly identical, suggesting that complex self-attention calculations offer limited additional information. Given that self-attention in the bottom layers captures global features, we further question whether sparse tokens can be used in these layers. However, experiments (in Table 6) demonstrated that applying sparse tokens in the bottom layers results in unstable training, highlighting the necessity of retaining complete tokens in these layers. + +Based on the above analysis, we adopt poolingformers to replace the original transformers. In poolingformer, we remove queries and keys as attention maps are not computed. Instead, we perform a global average pooling on the value $V \in \mathbb{R}^{N \times C}$ and integrate it into input tokens $X$ , which can be represented as + +$$ +X = X + \bar {V}, \tag {1} +$$ + +where $\bar{V}$ is the mean along the dimension of $N$ , i.e., $\bar{v} = \frac{1}{N}\sum V(\bar{v}\in \mathbb{R}^{1\times C})$ , and then repeating $N$ times to shape $N\times C$ . The parameters from adaLN block are omitted in all the equations for brevity. Our poolingformer can be viewed as a special case of the model in [65], behaving identically when the pooling kernel size is equal to the input size. + +Sparse-dense token module. We present the sparse-dense token module (SDTM) to generate sparse and dense tokens processed by the corresponding transformer layers. The high-level idea is to decouple global structure extraction and local detail extraction. Sparse tokens capture global structure information and reduce computational cost, while dense tokens enhance local detail and stabilize training. Sparse and dense tokens are converted to each other within SDTM. + +Initially, SDTM introduces a set of sparse tokens $X_{s} \in \mathbb{R}^{M \times C}$ , where $M$ is the number of sparse tokens, typically $M \ll N$ . These sparse tokens can be initialized by adaptively pooling the dense tokens $X$ . We define pruning rate $r = 1 - M / N$ to represent the sparsity degree. To store the structure information, we first reshape the dense tokens $X \in \mathbb{R}^{N \times C}$ into the latent shape. For instance, if the input is an image, we reshape it into shape $H \times W \times C$ , then pool it across the spatial dimensions to shape $H' \times W' \times C$ , where $H' \times W' = M$ . The intentions of spatial pooling initialization are two-fold. First, the initial sparse tokens can be distributed uniformly in space and the representation of each sparse token is associated with a specific spatial location, which is beneficial for downstream tasks such as image editing [20]. Second, it can prevent the semantic tokens from collapsing to one point in the following layers. Sparse token interact with full-size dense tokens via an attention layer to integrate the global information: + +$$ +X _ {s} = X _ {s} + M H A \left(X _ {s}, X, X\right), \tag {2} +$$ + +where the triplet input of $MHA$ are queries, keys, and values in turn. + +The generated sparse tokens are fed into subsequent transformer layers, referred to as sparse transformers. Since $M \ll N$ , our SDTM can substantially reduce computational cost. + +Following this, we restore dense tokens from sparse tokens. First, we reshape the sparse tokens $X_{s}\in \mathbb{R}^{M\times C}$ into structure shape $H^{\prime}\times W^{\prime}\times C$ and upsample it to the same shape as input dense tokens $X\in \mathbb{R}^{H\times W\times C}$ . We introduce two linear layers to combine upsampling sparse tokens with dense tokens, which is represented as: + +$$ +X _ {\text {m e r g e d}} = \operatorname {U p S a m p l e} \left(X _ {s}\right) \cdot W _ {1} + X \cdot W _ {2}, \tag {3} +$$ + +where $W_{1}, W_{2} \in \mathbb{R}^{C \times C}$ are the weights of two linear layers. Then, to further incorporate sparse tokens, we utilize an attention layer to perform a reverse operation of the generation of sparse token to produce the restored dense tokens, which is written as: + +$$ +X = X _ {\text {m e r g e d}} + M H A \left(X _ {\text {m e r g e d}}, X _ {s}, X _ {s}\right). \tag {4} +$$ + +At the end of SDTM, several transformer layers, termed dense transformers, process full-size dense tokens to enhance local details. + +We cascade multiple SDTMs in our network. By repeating sparse and dense tokens, the network effectively preserves both structural and detailed information, achieving substantial reductions in computational cost while maintaining high-quality generation outputs. + +Table 1: Comparison of performance between SparseDiT and DiT for class-conditional image generation task on ImageNet. + +
ResolutionModelFLOPs (G)Throughput (img/s)FID ↓sFID ↓IS ↑Precision ↑Recall ↑
256 × 256DiT-B23.017.849.075.44121.070.740.54
Ours (r ∈ [0.61, 0.86])14.34 (-38%)13.2 (+68%)8.236.20134.300.740.54
256 × 256DiT-XL118.641.582.274.60278.240.830.57
Ours (r ∈ [0.44, 0.61])88.91 (-25%)2.13 (+35%)2.234.62278.910.840.58
Ours (r ∈ [0.61, 0.86])68.05 (-43%)2.95 (+87%)2.384.82276.390.820.58
512 × 512DiT-XL5250.2493.045.02240.820.840.54
Ours (r ∈ [0.61, 0.86])286 (-46%)0.609 (+145%)2.965.00242.40.840.54
Ours (r ∈ [0.90, 0.96])235 (-55%)0.685 (+175%)3.135.41236.560.830.52
+ +![](images/865354aac4e4e0482d6dfe17a9ee0d6ed4bca3ede089a6015417ed210109ff43.jpg) +(a) Images generated by DiT. + +![](images/c90c271c17c7c7da53b22a62d61814066922ac0a23c40f558ce4d52df41bf6ba.jpg) +Figure 4: Generating images from DiT-XL and SparseDiT-XL with the same random seed. + +![](images/aca97b205a0a4ac7b94765326412ceec5dde94f4ce9ba4d9329563c8b289a0f8.jpg) + +![](images/572878b0c8d237aeb5bd7f44e39c06668e5fb9124e47aac93c41c2f96cffa580.jpg) +(b) Images generated by SparseDiT (-43% FLOPs). + +![](images/e2ca1a4cb5041b5b5e0177a1545b1bad0bea432f23ea193405450c5ea259bfa8.jpg) + +![](images/0c01a3fd385e44cf7b66d536235fdd916c5da5dd60a34399602da42db25ff510.jpg) + +![](images/1e5489dbc3214fd79fe3e945f42045025f4fc2b887d9c7ee58c5ae5776d729c4.jpg) + +![](images/8fab1d276b6efc08e1a96110dabef0ba5e6ca8b90098f98ad829e6a2781adea2.jpg) + +# 3.2 Timestep-wise pruning rate + +We have observed that the tokens display varying denoising behavior over sampling steps. They generate the low-frequency global structure information in the early denoising stage and the high frequency details in the late denoising stage. The token count requirement is progressively increase along sampling steps. We exploit this by dynamically adjusting the pruning rate $r$ across sampling steps, increasing tokens as sampling progresses. + +Given $T$ sampling steps, we propose a sample-specific approach to dynamically adjust the pruning rate $r$ , thereby controlling the number of sparse tokens across the sampling steps. We define a range for $r$ , such that $r \in [r_{\min}, r_{\max}]$ . Observing that generation quality is highly relative with the later denoising stages (in Section 4.4), we hold $r$ constant at $r_{\min}$ for the first $T/4$ sampling steps. Subsequently, we adjust the pruning rate $r$ linearly based on the current sampling step $t_i$ . The specific formula for $r$ is provided as follows: + +$$ +r = \left\{ \begin{array}{l l} r _ {\text {m i n}}, & t _ {i} < T / 4, \\ \frac {4 T - 4}{3 T} r _ {\text {m i n}} + \frac {4 - T}{3 T} r _ {\text {m a x}}, & T / 4 \leq t _ {i} < T. \end{array} \right. \tag {5} +$$ + +During training, we sample step $t_i$ and compute the corresponding $r$ according to Eq. 5. However, the input tokens should be the same to train the model in batch, and the randomness of the sampling from $T$ should also be maintained. To solve this contradiction, we modify the linear function of Eq. 5 when $T / 4 \leq t_i < T$ to a piecewise function. Normally, the model is trained by multiple GPUs. We request $t_i$ sampled in a specific piece in each GPU. Therefore, in each iteration, it can achieve both uniformly random sampling and batch training. + +# 3.3 Initialization and fine-tuning + +Our method fine-tunes pre-trained DiT models to improve efficiency. The processes of generating sparse tokens and restoring dense tokens utilize off-the-shelf transformers, avoiding the need for additional networks. During fine-tuning, the parameters of transformers in DiT-based model are loaded into the corresponding transformers in our SparseDiT, with two exceptions: (1) queries and keys are absent in Poolingformers, hence related parameters in pre-trained model are not required; (2) the weights $W_{1}$ and $W_{2}$ in Eq. 3 are initialized by full zeros and identity matrix, receptively. + +Table 2: FVD scores of SparseDiT-XL in class-conditional video generation task on four mainstream video generation datasets. We do not find the official reference sets of these 4 dataset when we compute FVD scores, hence we build the reference sets by ourselves and use the official checkpoint to re-compute the FVD scores of Latte-XL in this table. + +
ModelFLOPs (G)Throughput (clip/s)FaceForensicSkyTimelapseUCF101Taichi-HD
Latte-XL*18940.10824.1040.78284.5489.63
Ours (r ∈ [0.80, 0.93])828 (-56%)0.228 (+111%)24.5139.02288.9084.13
+ +# 4 Experiments + +SparseDiT is applied in three representative DiT-based models, DiT [41], Latte [34], and PixArt- $\alpha$ [8] for class-conditional image generation, class-conditional video generation, and text-to-image generation, respectively. + +Fine-tuning overhead. All training settings and hyperparameters follow their respective papers. Fine-tuning requires approximately $6\%$ of the time needed for training from scratch, e.g., 400K iterations for DiT-XL fine-tuning. + +# 4.1 Class-conditional image generation + +Experimental setting. We conduct our experiments on ImageNet-1k [11] at resolutions of $256 \times 256$ and $512 \times 512$ , following the protocol established in DiT. For DiT-XL, the model consists of 2, 24, and 2 transformers in the bottom, middle, and top segments, respectively. The middle segment includes 4 sparse-dense token modules, which comprise 1 sparse token generation transformer, 3 sparse transformers, 1 dense token recovery transformer, and 1 dense transformer. For DiT-B, the bottom, middle, and top segments contain 1, 10, and 1 transformer layers, respectively. The middle segment consists of 2 sparse-dense token modules, including 1 sparse token generation transformer, 2 sparse transformers, 1 dense token recovery transformer, and 1 dense transformer. As poolingformers and sparse transformers can interfere with the function of position embedding, we reintroduce sine-cosine position embedding at each stage of dense token generation. For DiT-XL, we use the checkpoint from the official DiT repository, while for DiT-B, we utilize the checkpoint provided by [40]. Following prior works, we sample 50,000 images to compute the Fréchet Inception Distance (FID) [17] using the ADM TensorFlow evaluation suite [12], along with the Inception Score (IS) [46], sFID [38], and Precision-Recall metrics [23]. Classifier-free guidance [19] (CFG) is set to 1.5 for evaluation and 4.0 for visualization. Throughput is evaluated with a batch size of 128 on an Nvidia A100 GPU. + +Main results. The results of our approach on the class-conditional image generation task using ImageNet are presented in Table 1. We evaluate our method on two model sizes, DiT-B and DiT-XL, and at two resolutions, $256 \times 256$ and $512 \times 512$ . At pruning rates $r \in [0.61, 0.86]$ , SparseDiT-XL achieves a $43\%$ reduction in FLOPs and an $87\%$ improvement in inference speed, with only a 0.11 increase in FID score. This indicates significant redundancy within DiT, as using only about $25\%$ tokens in specific layers maintains similar performance levels. By increasing the token count, we observe slight improvements over baseline performance with $145\%$ improvement in throughput. The impact of our approach is even more pronounced at a resolution of $512 \times 512$ . With a higher pruning rate than the $256 \times 256$ resolution, our method yields a superior performance-efficiency trade-off. By pruning over $90\%$ of the tokens, we achieve a $55\%$ reduction in FLOPs and a $175\%$ increase in throughput, with only a 0.09 increase in FID score. When reducing the pruning rate further, our results slightly surpass the baseline, delivering a $145\%$ throughput improvement. ToMeSD [5] is the first approach to reduce tokens. However, it applies token merging and recovery at every layer in a generic manner, resulting in a significant performance drop. For example, with a 0.1 merging rate on DiT-XL, ToMeSD achieves an FID score of 14.74, which is substantially worse than our result. A detailed comparison between other methods and our method is provided in Appendix A.1. + +From $256 \times 256$ to $512 \times 512$ , FLOPs increase by only 4.4 times, whereas the speed declines by a factor of 6.3, demonstrating that the primary speed bottleneck in DiT architecture is the number of tokens. Our method effectively reduces token count, resulting in significant gains, particularly for high-resolution content generation. For example, we achieve a $55\%$ reduction in FLOPs at $512 \times 512$ while enhancing speed by $175\%$ . Compared to other methods, our real-world speed improvements far exceed those achieved with similar reductions in FLOPs. + +Table 3: Text-to-image generation on SAM dataset. $r \in \left\lbrack {{0.61},{0.86}}\right\rbrack$ . + +
ModelFLOPs (G)Throughput(img/s)FID ↓
PixArt-α148.730.4144.53
Ours91.62 (-38%)0.701 (+69%)4.29
+ +Visualization results. To further validate the effectiveness of our method, we visualize some samples generated from SparseDiT-XL at a $256 \times 256$ resolution with a pruning rate $r \in [0.61, 0.86]$ (-43% FLOPs) and compare them with DiT-XL in Figure 4. Each cell in the same position corresponds to images generated from the same random seed and class. Since our model is fine-tuned from the pre-trained DiT, the overall styles and structures of the two subfigures are similar. The images generated by SparseDiT still retain rich high-frequency details. Furthermore, SparseDiT images demonstrate more precise structure, as evidenced by the accurate depiction of the "golden retriever's" nose and the "macaw's" eyes, which appear misplacement or missing in the images generated by DiT-XL. We provide samples at a resolution of $512 \times 512$ in Appendix. + +# 4.2 Class-conditional video generation + +Experimental settings. We conduct experiments at a resolution of $256 \times 256$ on four public datasets: FaceForensics [45], SkyTimelapse [60], UCF101 [54], and Taichi-HD [50]. Latte comprises two distinct types of transformer blocks: spatial transformers, which focus on capturing spatial information, and temporal transformers, which capture temporal information. To accommodate this separation of spatial and temporal feature extraction, we adjust our model's schedule within the sparse-dense token module. Specifically, we employ two transformers to prune spatial tokens and temporal tokens separately. The sparse tokens are then processed by two sparse transformers, and finally two transformers are used to recover the temporal and spatial tokens. Dense transformers are discarded in this setup. All other model configurations are the same as in DiT, and we follow the original training settings and hyperparameters used for Latte. To evaluate performance, we sample 2,048 videos, each consisting of 16 frames, and measure the Fréchet Video Distance (FVD) [57]. Throughput is measured using a batch size of 2 clips on an Nvidia A100 GPU. + +Main results. Table 2 presents the main results of our approach. Leveraging the additional temporal dimension in video data allows for a higher pruning rate. Our method achieves a $56\%$ reduction in FLOPs while maintaining a competitive FVD score compared to the baseline, demonstrating its effectiveness in video generation tasks. + +# 4.3 Text-to-image generation + +Experiment setting. We further assess the effectiveness of our method on text-to-image generation, which presents a greater challenge compared to class-conditional image generation. We adopt PixArt- $\alpha$ [8], a text-to-image generation model built based on DiT as the base model. Our model is initialized using the official PixArt- $\alpha$ checkpoint pre-trained on SAM dataset [22] containing 10M images. We further fine-tune it with our method on a subset of SAM dataset including 1M images. In PixArt- $\alpha$ , the transformer architecture includes two types of attention layers: a self-attention layer and a cross-attention layer, which integrates textual information. We apply our method to the self-attention layers to either reduce or recover tokens. + +For evaluation, we randomly select text prompts from SAM dataset and adopt 100-step IDDPM solver [39] to sample 30,000 images, with the FID score serving as the evaluation metric. Throughput is evaluated with a batch size of 128 on an Nvidia A100 GPU. + +Main results. As shown in Table 3, our SparseDiT achieves an FID score comparable to the original PixArT- $\alpha$ with significantly accelerating the generation, showing that our method is effective for text-to-image generation task. + +Visualization results. We visualize some samples of our method and compare them with original PixArt- $\alpha$ in Figure 5. The classifier-free guidance is set to 4.5. The results demonstrate that our method effectively maintains both image quality and semantic fidelity. + +# 4.4 Ablation study + +All the following ablation experiments of SparseDiT-XL are conducted on ImageNet at a $256 \times 256$ resolution. + +![](images/b4acb3a1264b759b4bdab3d1147981b420fb7c09b5a8739477afd333c9d01228.jpg) +(a) Images generated by PixArt-α + +![](images/764252c1e2a0afc1d73a6e457f9edb3b3a84da0ca1970241564c5dd61e2f806e.jpg) +Figure 5: Generating images from PixArt- $\alpha$ (a) and SparseDiT (b). The captions in the same positions are identical. They are provided in Appendix A.3. + +![](images/4ec54a95c4efdae32bf191cfabcb295ec21c1aff45d90e9fa540a6356a1a4460.jpg) + +![](images/2049e316a28bc97a7dfc40321952bec535f54b015b57869b4f230402c0312b8e.jpg) + +![](images/5927f722ec1ab4acf874da6763e8a9b6cff7007b75bf735a09d03c8f0143da1a.jpg) +(b) Images generated by SparseDiT + +![](images/b5842a260aa24651d0b1db98f44bdf2222e822b9fa86ecc5ca70a8e172fee68d.jpg) + +![](images/754be35816c172c3e26be320d8a9c82102d7403c33b6d8bb6ac2532602e9c046.jpg) + +![](images/215cc46612e2802cfda28eee9b5c382dc61ef11a1b7bd8d0bc672e511ee0b11c.jpg) + +Table 4: Performance evaluation on different numbers of SDTMs. "NAN" indicates loss nan. + +
No. of SDTMs1234
FID ↓NAN3.862.512.38
FLOPs68.7467.0069.9768.05
+ +Table 5: Combination our method with efficient samplers. The numbers represent FID scores. + +
Model250-DDPM25-DDIM5-RFlow
DiT-XL2.272.8943.40
Ours2.383.3143.69
+ +The number of SDTMs. The alternation between sparse and dense tokens is a key factor contributing to the success of our method. Sparse tokens capture global structures, while dense tokens capture detailed information. In SparseDiT, we adopt four SDTMs by default. Table 4 shows the effect of using different numbers of SDTMs. To isolate the impact of SDTM count, we maintain the same FLOPs across models by adjusting the number of sparse and dense transformers while varying only the number of SDTMs. In configurations with only one SDTM, the model is modified into a U-shaped architecture. The results indicate that fewer SDTMs reduce interactions between global and local information, resulting in weaker performance. Furthermore, reducing SparseDiT to a U-Net structure compromises training stability, leading to a collapse. We do not experiment with more than four SDTMs, as increasing SDTM count beyond four offers diminishing returns in FLOP reduction. + +Combination with efficient samplers. Our SparseDiT is a model compression method that can be seamlessly integrated with efficient samplers, such as DDIM [51] and Rectified Flow (RFlow) [28]. Note that the RFlow variant of DiT was trained by us via 2,000K iterations. As shown in Table 5, when our method is combined with the 25-step DDIM and 5-step Rectified Flow samplers, it achieves approximately $18.7 \times$ and $93.4 \times$ improvements in inference speed, respectively, compared to the standard 250-step DDPM. Notably, our method does not introduce a significant performance gap relative to the baseline, demonstrating that it can be effectively combined with efficient samplers. + +The number of poolingformers. In Table 6, we examine the impact of varying the number of poolingformers in the model. We keep the total number of transformers in the bottom and top segments constant across configurations. When using three poolingformers, performance declines substantially due to the global average pooling operator's limited ability to adaptively capture information. For configurations with one or two poolingformers, the models achieve same results. The poolingformer is more efficient and consume fewer parameters than the standard transformer; therefore, we default to using 2 poolingformers. Reducing the poolingformer count to zero results in training instability, suggesting that maintaining full-size tokens in the initial layers is crucial. This finding aligns with the observations in Figure 1b: in DiT, the first two attention maps exhibit a uniform distribution, allowing us to replace them with poolingformers, but further replacement disrupts information flow and reduces performance. + +The effectiveness of timestep-wise pruning rate. To evaluate the effectiveness of the timestep-wise pruning rate strategy, we conduct ablation experiments, as shown in Table 7. In these experiments, we maintain the same number of FLOPs while varying the number of tokens along timesteps, either using a constant token count or dynamically adjusting the token count. The results demonstrate that dynamically tuning the pruning rate significantly improves performance. Additionally, Table 7 highlights an observation: the performance of the dynamic pruning strategy is particularly relative + +Table 6: Performance evaluation on different numbers of poolingformers. "NAN" indicates loss nan during training. + +
No. of Poolingformers0123
No. of trans in top seg.4321
FID ↓NAN2.382.382.56
FLOPs69.5668.8068.0567.29
+ +Table 7: Comparison of FID scores with and without timestep-wise pruning rate strategy. + +
FLOPs (G)No. of TokensDynamicFID ↓
~ 688 × 82.48
6 × 6 ~ 10 × 10✓2.38
~ 616 × 62.63
4 × 4 ~ 8 × 8✓2.50
+ +with the token count of the later stages of denoising. For instance, the configuration with token counts ranging from $4 \times 4$ to $8 \times 8$ only shows a 0.02 FID score increase compared to the configuration with a constant $8 \times 8$ token count. + +# 5 Conclusion + +Limitations. The primary limitation of our method arises from the manually pre-defined its structure, containing the number of layers in each module and the number of sparse tokens. + +In this paper, we introduce SparseDiT, a novel approach to enhancing the efficiency of DiT by leveraging token sparsity. By incorporating SDTM and a timestep-wise pruning rate in a strategically layered manner, SparseDiT significantly reduces computational complexity while preserving high-generation quality. Experimental results show substantial FLOP reductions with minimal performance degradation, demonstrating SparseDiT's effectiveness across multiple DiT-based models and datasets. This work marks a step forward in developing scalable and efficient diffusion models for high-resolution content generation, expanding the practical applications of DiT-based architectures. + +# References + +[1] F. Bao, C. Li, J. Zhu, and B. Zhang. Analytic-DPM: an analytic estimate of the optimal reverse variance in diffusion probabilistic models. In International Conference on Learning Representations, 2022. +[2] F. Bao, S. Nie, K. Xue, Y. Cao, C. Li, H. Su, and J. Zhu. All are worth words: A vit backbone for diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 22669-22679, 2023. +[3] A. Blattmann, T. Dockhorn, S. Kulal, D. Mendelevitch, M. Kilian, D. Lorenz, Y. Levi, Z. English, V. Voleti, A. Letts, et al. Stable video diffusion: Scaling latent video diffusion models to large datasets. arXiv preprint arXiv:2311.15127, 2023. +[4] D. Bolya, C.-Y. Fu, X. Dai, P. Zhang, C. Feichtenhofer, and J. Hoffman. 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Wang, Y. Qiao, B. Leng, and Y. Liu. Self-slimmed vision transformer. In European Conference on Computer Vision, pages 432-448. Springer, 2022. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. + +Please read the checklist guidelines carefully for information on how to answer these questions. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix + +# A.1 Comparison with other methods + +Comparison with ToMeSD. Recent studies [37, 68] have evaluated ToMeSD [5] on DiT models and reported significant performance drops, which are much lower than those achieved by our method. Our approach is fundamentally different from ToMeSD. While ToMeSD reduces and recovers tokens generically at every layer, our method specifically analyzes the local-global relationships unique to DiT models. Based on these insights, we strategically reduce and recover tokens. We have evaluated ToMeSD on DiT and compared it with our method, as shown in Table 8. + +Comparison with DyDiT. DyDiT [68] represents the current state-of-the-art approach for enhancing the efficiency of Diffusion Transformers (DiTs). In contrast to our method, which emphasizes token reduction, DyDiT implements pruning across tokens, attention heads, and channels. A comparative analysis of our method against DyDiT, as detailed in Table 9 using ImageNet $512 \times 512$ images, illustrates a substantial advantage in computational efficiency for our approach. Specifically, our method achieves a $46\%$ reduction in FLOPs, markedly surpassing DyDiT's $29\%$ reduction, coupled with a $145\%$ increase in inference speed, significantly outperforming the $31\%$ improvement achieved by DyDiT. The only trade-off is a minor increase in the Fréchet Inception Distance (FID), where our method's FID is 0.08 higher than that of DyDiT. Nevertheless, this difference in FID is nearly imperceptible to human observers in practical applications. + +Comparison with TokenCache. We compare our method with TokenCache [29] in Table 9. We adopt the standard DiT baseline 2.27, while TokenCache applies a baseline 2.24. TokenCache achieves a $32\%$ speed gain with a slight FID increase. Our approach attains an $87\%$ speed gain with a smaller FID increase, demonstrating superior efficiency and performance. + +Comparison with Ditfastattn. We compare our method with Ditfastattn [66] in Table 9. We adopt the standard DiT baseline 3.04, while Ditfastattn applies a baseline 3.16. Ditfastattn shows a substantial FID increase for speed gain, while our method results in better speed and less FID. + +# A.2 Training form scratch + +We train our SparseDiT from scratch, reaching 400K training iterations. We compare our result with original DiT paper in Table 4. The results (cfg=1.0) are in Table 10. + +# A.3 Captions in Figure 5 + +Column 1: The image depicts a beach scene with a large body of water, such as a lake or ocean, and a sandy shoreline. The beach is filled with people, including a group of people swimming in the water. + +Column 2: The image depicts a beautiful outdoor dining area with a large number of tables and chairs arranged in a row, overlooking a picturesque lake. The tables are covered with white tablecloths, and there are several umbrellas providing shade for the guests. The scene is set in a lush green field, with a large building in the background, possibly a hotel or a restaurant. The tables are arranged in a way that allows for an unobstructed view of the lake, creating a serene and relaxing atmosphere for the diners. The image has a stylish and elegant feel, with the attention to detail in the table arrangement and the choice of location contributing to a memorable dining experience. + +Column 3: The image depicts a busy city street at night, with a group of people standing outside a restaurant and a bar. The scene is set in a European city, and the atmosphere is lively and bustling. + +Column 4: The image features a vintage-style train, parked on a track surrounded by trees and grass. The train appears to be an old-fashioned steam engine, which is a type of locomotive powered by steam. The train is positioned in a park-like setting, with a tree-lined path nearby. The scene is set in a sunny day, creating a pleasant atmosphere. The image has a nostalgic and historical feel, evoking a sense of the past and the charm of old-time trains. + +Table 8: Comparison of our method with ToMeSD on DiT models. + +
ModelMethodFIDSpeed-up
DiT-BToMeSD29.24+20%
Ours8.23+68%
DiT-XLToMeSD14.74+66%
Ours2.38+87%
+ +Table 9: Comparison of our method with DyDiT on DiT models. + +
ModelResolutionMethodFLOPsFIDISSpeed-up
DiT-XL512 × 512DyDiT375 (-29%)2.88(-0.16)-+31%
Difastattn-4.52(+1.36)180.34+98%
Ours286 (-46%)2.96(-0.08)242.4+145%
DiT-XL256 × 256TokenCache72.25(-39%)2.37 (+0.13)262.00+32%
Ours68.05(-43%)2.38 (+0.11)276.39+87%
+ +# A.4 Additional visualization + +We provide additional visualizations at a resolution of $512 \times 512$ on SparseDiT-XL from Figure 6 to Figure 17. The class labels, including "arctic wolf", "volcano", "cliff drop-off", "balloon", "sulphur-crested cockatoo", "lion", "otter", "coral reef", "macaw", "red panda", "husky", and "panda", correspond to the same cases presented in DiT. Readers can compare our results with those of DiT. Our samples demonstrate comparable image quality and fidelity. The classifier-free guidance scale is set to 4.0, and all samples shown here are uncurated. + +Table 10: Comparison of our method with ToMeSD on DiT models. + +
MethodFIDFlopsInterations
DiT-XL19.47118.64400K
Ours (from scratch)15.1168.05400K
+ +Figure 6: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/e2620d85503bc3c8b1f606df20f5a4cfdd9c32af4908ce0bceb53624eeedc037.jpg) +Classifier-free guidance scale $= 4.0$ + +Figure 7: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/9553efe18ac1b26f079521632f9c5f4a06c557ccf066072c333f739192ec9371.jpg) +Class label $=$ "arctic wolf" (270) +Classifier-free guidance scale $= 4.0$ +Class label $=$ "volcano" (980) + +Figure 8: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/3bbce28b521bc33ddb51a9b7387cd69ad0c7f426bee1e71e04fa7830676cc2ab.jpg) +Classifier-free guidance scale $= 4.0$ + +Figure 9: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/85d0fe22ed2d4c391ca0077e7e9bb1dfe0b359d55303b715bfe628d54312f0ce.jpg) +Class label $=$ "husky" (250) +Classifier-free guidance scale $= 4.0$ +Class label $=$ "sulphur-crested cockatoo" (89) + +Figure 10: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/0d6bd7797c88a9dbc7ac904c20e284f369716ab480d6b2b452a3a9efcda687a7.jpg) +Classifier-free guidance scale $= 4.0$ + +Figure 11: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/9d666018802592861acf78434224609478d284a9d169eb6dd2d129159556320f.jpg) +Class label $=$ "cliff drop-off" (972) +Classifier-free guidance scale $= 4.0$ +Class label $=$ "balloon" (417) + +![](images/37a4eff4786ae52dab0c501db54ad9fe33c2c1de130d41257c360c94c75432b6.jpg) +Figure 12: Uncurated $512 \times 512$ SparseDiT-XL samples. Classifier-free guidance scale = 4.0 +Class label = "lion" (291) + +![](images/ff2cc4bd39a84aa7907d1f842fe53920d263bdd72ea706414939fb073cbc8c32.jpg) +Figure 13: Uncurated $512 \times 512$ SparseDiT-XL samples. Classifier-free guidance scale = 4.0 +Class label = "otter" (360) + +Figure 14: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/f94e00f42f3785552ce73fd1e0bdce5a28ebcfeb97fac3f161585bf2368ec6d7.jpg) +Classifier-free guidance scale $= 4.0$ + +Figure 15: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/3cc9f34b444b91b81d0b79a2f950a1f85f511e39cdeff4424b4b5e2aa249b685.jpg) +Class label $=$ "red panda" (387) +Classifier-free guidance scale $= 4.0$ +Class label $=$ "panda" (388) + +Figure 16: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/290ac8acf6818ceadc131df60cd34289ec8571061f089396b4219236709b7840.jpg) +Classifier-free guidance scale $= 4.0$ + +Figure 17: Uncurated $512 \times 512$ SparseDiT-XL samples. +![](images/02df781e00b9cd5e141c7237de705f843a41a92c97a26a9f0e25b7f66fcb4fe6.jpg) +Class label $=$ "coral reef" (973) +Classifier-free guidance scale $= 4.0$ +Class label $=$ "macaw" (88) \ No newline at end of file diff --git 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$^{4}$ , Xingchen Hu $^{5}$ , Kun Sun $^{1}$ , Xinwang Liu $^{5}$ + +$^{1}$ School of Computer Science, China University of Geosciences (Wuhan) + +2The Hong Kong University of Science and Technology (Guangzhou) + +$^{3}$ School of Software Engineering, Huazhong University of Science and Technology + +$^{4}$ School of Computer Science, Hubei University of Technology + +$^{5}$ College of Systems Engineering, National University of Defense Technology + +# Abstract + +Existing multi-view clustering methods employ various strategies to address data-level sparsity and view-level dynamic fusion. However, we identify a critical yet overlooked issue: varying sparsity across views. Cross-view sparsity variations lead to encoding discrepancies, heightening sample-level semantic heterogeneity and making view-level dynamic weighting inappropriate. To tackle these challenges, we propose Adaptive Sparse Autoencoders for Multi-View Clustering (SparseMVC), a framework with three key modules. Initially, the sparse autoencoder probes the sparsity of each view and adaptively adjusts encoding formats via an entropy-matching loss term, mitigating cross-view inconsistencies. Subsequently, the correlation-informed sample reweighting module employs attention mechanisms to assign weights by capturing correlations between early-fused global and view-specific features, reducing encoding discrepancies and balancing contributions. Furthermore, the cross-view distribution alignment module aligns feature distributions during the late fusion stage, accommodating datasets with an arbitrary number of views. Extensive experiments demonstrate that SparseMVC achieves state-of-the-art clustering performance. Our framework advances the field by extending sparsity handling from the data-level to view-level and mitigating the adverse effects of encoding discrepancies through sample-level dynamic weighting. The source code is publicly available at https://github.com/cleste-pome/SparseMVC. + +# 1 Introduction + +Multi-view learning has emerged as a powerful paradigm for leveraging complementary information across multiple perspectives, significantly improving the performance of unsupervised learning tasks such as clustering [1, 2, 3, 4, 5]. At the same time, the sparsity of multi-view data has become a pivotal focus of research, with numerous studies proposing solutions from perspectives such as activation functions [6], tensor decomposition [7, 8], and variational autoencoders [9]. Nevertheless, while prior methods focus on designing advanced approaches to address data sparsity, they often overlook a fundamental aspect—the potential variations in sparsity across different views. Given that multi-view data consists of multiple views originating from distinct sources, and sparsity is a pervasive characteristic in multi-view data [10, 11, 12, 13, 14]. Building upon the factual observations, a natural question arises: "Does there exist a phenomenon of varying sparsity across views?" + +To quantify cross-view sparsity variations, we define the sparsity ratio $s_v$ for the $v$ -th view: + +$$ +s _ {v} = \frac {1}{N \cdot F} \sum_ {j = 1} ^ {N} \sum_ {i = 1} ^ {F} I \left[ x _ {i, j} ^ {v} = 0 \right], \tag {1} +$$ + +where $N$ refers to the number of samples, $F$ refers to the feature dimension, $x_{i,j}^{v}$ represents the $i$ -th feature of the $j$ -th sample in the $v$ -th view, and the indicator function $\pmb{I}$ takes the value of one if $x_{i,j}^{v}$ equals zero, and zero otherwise. Zero-valued features $x_{i,j}^{v}$ suggests missing dimensions or data collection errors. Our statistical and computational analysis addresses the question posed earlier and reveals that sparsity variations across views not only exist, but are widely prevalent in diverse + +multi-view data, as illustrated in Fig. 1. + +The disparity in view sparsity presents a multifaceted challenge: highly sparse views, lacking sufficient informative content, are prone to underfitting, whereas less sparse views, often burdened with redundant or irrelevant features, are susceptible to overfitting. Applying a uniform encoder architecture or regularization strategy across such heterogeneous views compromises representational consistency and limits the model's capacity to extract complementary cross-view information. To resolve this, we adopt an adaptive and sparsity-aware encoding strategy tailored to individual views. This requires rethinking the autoencoder design to accommodate view heterogeneity and structural disparities. Our solution is to design an autoencoder capable of adaptively adjusting its constraints based on + +![](images/0dde4d0ba59c5d80040787be28d4fa00d1b3b231a58bb1d63d0975fdaf75c140.jpg) + +![](images/7a2ccc5d0120f562844816d1ccbbe52999b67142995aaa180a191d5c0f6ab425.jpg) +Figure 1: Sparsity ratios across views in multi-view datasets. TOP BOX PLOT illustrates the sparsity ratio distribution, which shows the median (orange line), interquartile range (box), and any outliers (points outside the whiskers). BOTTOM BAR PLOT presents the sparsity ratios for each view within each dataset. To provide a more comprehensive situation of sparsity variations, additional datasets are included: Digits [15], LandUse-21 [16], RGB-D [17]. GSE [18] and STL-10 [19]. The cross-view sparsity ratios have been processed to improve visualization using the sigmoid function. + +the sparsity ratio of each view, allowing its encoding form to evolve accordingly. + +Due to varying sparsity across views for the same sample, encoders can introduce semantic and representational inconsistencies, affecting subsequent stages [20, 21]. As a result, it becomes essential to dynamically assess the contribution of each view based on the features extracted from different types of encoders. To address this, we design a correlation-informed reweighting module that assigns sample-wise weights based on the correlations between global and local features, thereby balancing view contributions and discrepancies. The early fusion strategy integrates multi-view data through feature concatenation. Since our reweighting module uses the global latent representation encoded from early fusion to guide the subsequent dynamic weighted early fusion of local representations, we chose an early fusion approach that preserves the original feature values as much as possible. + +In late fusion, we introduce a cross-view distribution alignment module to align feature distributions across views, enabling robust integration and supporting datasets with arbitrary view numbers while balancing global consistency and view-specific diversity. These three core components together form the SparseMVC framework, addressing sparsity inconsistencies, encoding discrepancies, and semantic heterogeneity in a step-by-step and interdependent design. + +Table 1: Performance comparison of different strategies under extreme and minimal sparsity variation. + +
DatasetsSparsity of Different ViewsAccuracyNMI
[Ours][2nd][Ours][2nd]
ALOI-100[0.0001, 0.0001, 0.3415, 0.6383]82.21 ↑4.4977.7292.65 ↑1.7890.87
MSRCV1[0.0049, 0.0048, 0.0048, 0.3478, 0.0051, 0.0048]97.14 ↑5.7191.4394.22 ↑6.3287.90
LGG[0.0040, 0.0038, 0.0078, 0.0037]83.15 ↑0.3882.7754.62 ↑0.4954.13
Synthetic3d[0.0017, 0.0017, 0.0017]98.33 ↑0.1698.1792.01 ↑0.7491.27
+ +As presented in Table 1, we evaluate our method across a range of datasets exhibiting extreme sparsity disparities. For instance, in ALOI-100, the sparsity ratio spans from as low as 0.0001 to as high as 0.6644, revealing a substantial imbalance in information density across views. To effectively + +address such heterogeneity, our framework incorporates dynamically adaptive autoencoders that adjust both the encoding process and sparsity-aware regularization in accordance with each view's sparsity profile. Unlike existing methods, which generally overlook the impact of cross-view sparsity variation, our targeted design enables SparseMVC to achieve consistently superior performance under severe disparity conditions. Moreover, it maintains strong competitiveness even in datasets with near-uniform sparsity variations, such as LGG and Synthetic3D, demonstrating both its responsiveness to sparsity imbalance and its robustness in more homogeneous scenarios. + +To the best of our knowledge, this is the first work to explicitly identify, analyze, and define the problem of cross-view sparsity variations in multi-view data, and to propose a dedicated framework SparseMVC that offers a targeted and principled solution. + +# 2 Related Work + +For multi-view fusion, early methods [22, 23] assumed equal importance of views, ignoring view heterogeneity. Hence, dynamic weighting approaches have emerged, with attention-based methods [24, 25] leading, alongside loss optimization [26, 27], kernel techniques [28, 29], and subspace methods [30]. However, uniform view-level weighting fails to address intra-view variability, highlighting the need for sample-level dynamic weighting. Trust-based methods [31, 32, 33, 34] excel in supervised scenarios. For multi-view clustering, sample-adaptive fusion [35] has been proposed using Laplacian matrix divergence. In contrast, our framework dynamically computes sample weights via correlation calculation without additional loss, while adopting a decoupled design with independent global and view-specific autoencoders. More details can be found in Appendix A.2. + +Sparse representation effectively captures essential features in sparse data by enforcing sparsity constraints [36] but struggles to handle the non-linear structures common in multi-view datasets [37]. Autoencoders excel at learning non-linear latent features [38, 39], yet their lack of sparsity enforcement limits their adaptability to varying sparsity rates across views. Sparse autoencoders combine the strengths of both sparse representation and standard autoencoders by incorporating sparsity constraints into the hidden layers [40], which have become a research focus in multi-view learning [9, 41]. + +# 3 Method + +![](images/4941e517126662875b9c956017ec2dd17b1ce4162e9a611f1108c9bd6517cdc1.jpg) +Figure 2: Overview of SparseMVC, a framework designed to address varying sparsity across views. + +This section sequentially introduces the three key submodules of SparseMVC, as illustrated in Fig. 2. It begins by utilizing sparse autoencoders with adaptive constraints, which dynamically adjust the coding strategy based on the probed $s_v$ , to generate latent features $(Z)$ , making the reconstructed features $(\widehat{\pmb{X}})$ approximate the original input features $(\pmb{X})$ . Subsequently, the correlation between the early-fused global features $(\bar{Z})$ and view-specific features $(\{Z_v\}_{v=1}^n)$ guides the computation of sample-level weights $(\{W_v\}_{v=1}^n)$ via the attention mechanism within the correlation-informed sample reweighting module. Finally, the cross-view distribution alignment module enhances clustering performance by setting the late-fused global features $\overline{Y}$ as the anchor latent representation, and then simultaneously aligning the multi-view feature distribution between $\overline{Y}$ and each view-specific compressed feature $(\{Y_v\}_{v=1}^n)$ . The algorithm of the framework can be found in Appendix A.1. + +# 3.1 Sparse Autoencoder with Adaptive Constraints + +To handle varying view sparsity rates, we propose the sparse autoencoder with adaptive constraints (SAA), extending traditional sparse autoencoders. SAA employs an adaptive loss function that integrates reconstruction and sparsity-aware entropy-matching as distinct constraints, wherein the adjustment is dynamically guided by view sparsity ratios formulated as prior knowledge. + +The reconstruction loss, typically measured by mean squared error (MSE), quantifies the difference between the reconstructed output $\hat{x}_j^v$ and the input $x_{j}^{v}$ for the $j$ -th sample: + +$$ +\mathcal {L} _ {\text {r e c o n}} ^ {v} = \frac {1}{N} \sum_ {j = 1} ^ {N} \left(\hat {\boldsymbol {x}} _ {j} ^ {v} - \boldsymbol {x} _ {j} ^ {v}\right) ^ {2}, \tag {2} +$$ + +where $N$ is the number of samples (batch size) for view $v$ . Motivated by the widespread presence of sparsity variations in different views, our aim is to design a function that is positively correlated with $s_v$ , allowing adaptive adjustments to both the encoder type and the strength of the sparsity constraints. We scale by $f(s_v)$ outside the loss rather than tuning $\rho$ in Eq. (4), please refer to Appendix B.2. The design of the adaptive weighting factor $f(s_v)$ follows a ReLU-like approach, adjusting the strength of $\mathcal{L}_{\text{entropy}}^v$ based on the probed input dataset sparsity $s_v$ for view $v$ : + +$$ +f \left(s _ {v}\right) = \left\{ \begin{array}{l l} 0, & \text {i f} s _ {v} \leq \theta , \\ \frac {\boldsymbol {s} _ {v} - \theta}{1 - \theta}, & \text {i f} s _ {v} > \theta , \end{array} \right. \tag {3} +$$ + +where the default value of $\theta$ is 0.01. Selecting the threshold $\theta$ for the sparsity ratio $s_v$ is based on the actual cross-view sparsity distribution of each dataset. For most datasets, the sparsity ratios exhibit a skewed distribution, with a significant concentration of both high and low values, as shown in Fig. 1 and Table 1. A $\theta$ value of 0.01 effectively captures these low-sparsity views. To enforce sparsity constraints, the entropy-matching loss is defined using Kullback-Leibler (KL) [42] divergence, encouraging the activations $\hat{h}_k^v$ in the hidden layer to align with a target sparsity level $\rho$ . The entropy-matching loss and the sparsity loss derived from it are formulated as follows: + +$$ +\mathcal {L} _ {\text {s p a r s e}} ^ {v} = f \left(s _ {v}\right) \cdot \mathcal {L} _ {\text {e n t r o p y}} ^ {v} = f \left(s _ {v}\right) \cdot \sum_ {k = 1} ^ {H} \left(\rho \log \frac {\rho}{\hat {h} _ {k} ^ {v}} + (1 - \rho) \log \frac {1 - \rho}{1 - \hat {h} _ {k} ^ {v}}\right), \tag {4} +$$ + +where $H$ refers to the number of units in the hidden layers of each sparse encoder, and $\rho$ is the target sparsity level. $\hat{h}_k^v$ represents the average activation of the $k$ -th hidden unit for view $v$ , which is clamped to lie strictly within the open interval from zero to one. Following [40], $\rho$ is set to 0.05, which is a well-validated choice that balances sparsity and the learning capacity of the autoencoder, allowing it to effectively capture key features in the data while avoiding overfitting to irrelevant features. The average activation is computed by: + +$$ +\hat {h} _ {k} ^ {v} = \frac {1}{N} \sum_ {j = 1} ^ {N} \sigma \left(\boldsymbol {W} _ {k} ^ {v} \boldsymbol {x} _ {j} ^ {v} + b _ {k} ^ {v}\right), \tag {5} +$$ + +where $W_{k}^{v}$ is the weight matrix, $\pmb{b}_{k}^{v}$ is the bias term for the $k$ -th hidden unit in the $v$ -th view, $\pmb{x}_j^v$ is the input feature, and $\sigma (\cdot)$ denotes the ReLU activation function. When $s_v\leq \theta$ $\mathcal{L}_{\mathrm{entropy}}^v$ is deactivated $(f(s_{v}) = 0)$ and the sparse autoencoder degenerates into a standard autoencoder. Conversely, for input views where $s_v > \theta$ , $f(s_{v})$ exhibits a linear increase with $s_v$ , ensuring that the sparsity constraint becomes more prominent for highly sparse inputs. Based on the above, the loss function for + +![](images/c15b0a1fdc4cfa9885899f55bf634f04f278f2303a73ceeae543b1f688381db1.jpg) +Figure 3: Sparse autoencoder with adaptive constraints. + +SAA is then formulated as follows: + +$$ +\mathcal {L} _ {\mathrm {S A A}} = \sum_ {v = 1} ^ {V} \left(\mathcal {L} _ {\text {r e c o n}} ^ {v} + \mathcal {L} _ {\text {s p a r s e}} ^ {v}\right). \tag {6} +$$ + +As shown in Fig. 3, this piecewise linear design dynamically aligns the sparsity constraint with the input sparsity level and customizes the encoding strategy for each view. + +# 3.2 Correlation-Informed Sample Reweighting + +While SAA balances reconstruction and sparsity, it introduces sample-specific encoding inconsistencies due to differences in sparsity. Additionally, while networks adjust weights through layer updates, this weighting alone cannot address the lack of communication between autoencoders. On the above bases, we drew inspiration from the concept of multi-head attention and designed the correlation-informed sample reweighting module (CSR), which leverages correlations between the early-fused global features $(\bar{Z})$ and the view-specific local features $(Z_v)$ , then computes sample-specific weights. This cascading design is devised to achieve two objectives: mitigating the encoding inconsistencies introduced by SAA and leveraging globally fused features, which preserve the relatively high-fidelity patterns of the original data structure, to supervise the computation of view correlations. + +CSR adopts a simplified structure inspired by multi-head attention, which captures similar effects while deviating from the standard formulation. Initially, CSR takes $\bar{Z} \in \mathbb{R}^{N \times F}$ and $Z_v \in \mathbb{R}^{N \times F}$ as input, and projects them into the query, key, and value spaces through parallel linear transformations: + +$$ +\boldsymbol {Q} = \bar {\boldsymbol {Z}} \boldsymbol {W} _ {Q}, \boldsymbol {K} _ {v} = \boldsymbol {Z} _ {v} \boldsymbol {W} _ {K}, \boldsymbol {V} _ {v} = \boldsymbol {Z} _ {v} \boldsymbol {W} _ {V}, \tag {7} +$$ + +where $W_{Q}, W_{K}, W_{V} \in \mathbb{R}^{F \times F}$ are the learnable weight matrices, generating the query matrix $Q \in \mathbb{R}^{N \times F}$ , which encapsulates global semantic information, and the key matrix $K_{v} \in \mathbb{R}^{N \times F}$ , which capture view-specific features for each view. Our primary objective is to quantify inter-view relationships by evaluating attention scores between queries and keys, without generating new feature representations, thus omitting the value matrix $V_{v}$ . To compute the correlation between $\bar{Z}$ and $Z_{v}$ , we define the correlation score $C_{v} \in \mathbb{R}^{N}$ for the $v$ -th view based on Einstein summation convention as: + +$$ +\boldsymbol {C} _ {v} = \frac {\sum_ {f = 1} ^ {F} \boldsymbol {Q} _ {f} \cdot \left(\boldsymbol {K} _ {f} ^ {v}\right) ^ {T}}{\sqrt {F}}, \tag {8} +$$ + +where $\sqrt{F}$ is the scaling factor equals to the square root of the dimension of the key vector. The correlation scores are normalized via the softmax function to produce sample-specific $\mathbf{W}_v\in \mathbb{R}^N$ as: + +$$ +\boldsymbol {W} _ {v} = \frac {\exp \left(\boldsymbol {C} _ {v}\right)}{\sum_ {v = 1} ^ {V} \exp \left(\boldsymbol {C} _ {v}\right)}, \tag {9} +$$ + +that dynamically adjust the contribution of each corresponding sample in respective autoencoders. + +# 3.3 Cross-view Distribution Alignment + +Aligning features across views is a fundamental challenge in multi-view learning, as it is crucial for leveraging the complementary information provided by diverse views. The cross-view distribution Alignment module (CDA) addresses this issue by performing contrastive learning between the late-fused global features $(\overline{Y})$ and the compressed features of individual views $(\mathbf{Y}_v)$ , ensuring effective alignment of multi-view features within a unified and shared latent space. + +To mitigate the risk of dimensional collapse during alignment, potentially caused by an excessively large latent space, we introduce a compression layer before feeding the encoded view-specific features into the CDA. More details are provided in Appendix B.1. Specifically, $\mathbf{Y}_v\in \mathbb{R}^{N\times F}$ is obtained from $\mathbf{Z}_v$ via the compression layer. In parallel, the global features $\overline{\mathbf{Y}}\in \mathbb{R}^{N\times F}$ are as follows: + +$$ +\overline {{\boldsymbol {Y}}} = \mathcal {F} \left(\sum_ {v = 1} ^ {V} \boldsymbol {W} _ {v} \boldsymbol {Z} _ {v}\right), \tag {10} +$$ + +where fusion function $\mathcal{F}$ represents the late fusion layers and ensures that the transformed dimension matches $\mathbf{Y}_v$ . The similarity matrix $S_v$ between $\overline{\mathbf{Y}}$ and $\mathbf{Y}_v$ is defined as: + +$$ +\boldsymbol {S} _ {v} = \frac {\overline {{\boldsymbol {Y}}} \cdot (\boldsymbol {Y} _ {v}) ^ {T}}{\tau}, \tag {11} +$$ + +with $\tau$ denoting a temperature parameter that scales the similarity values. The sample pairs position indices in $S_{v}$ are defined as $p$ and $q$ . Positive pairs, which correspond to the same samples across views, are represented by the diagonal elements of $S_{v}$ , denoted as $S_{v}^{p,p}$ . Negative pairs, which involve different samples across views, are identified using a mask matrix $M_{v} \in \mathbb{R}^{\hat{N} \times N}$ , where $M_{v}^{p,q} = 1$ if $p \neq q$ , and $M_{v}^{p,q} = 0$ otherwise. The contrastive loss for each sample is: + +$$ +\mathcal {L} _ {\text {c o n}} ^ {p, v} = - \log \left(\frac {\exp \left(\boldsymbol {S} _ {v} ^ {p , p}\right)}{\sum_ {q = 1} ^ {N} \exp \left(\boldsymbol {S} _ {v} ^ {p , q}\right) \cdot \boldsymbol {M} _ {v} ^ {p , q}}\right), \tag {12} +$$ + +where $\exp(S^{p,p})$ quantifies the similarity of positive pairs, and the denominator aggregates the exponential similarities of all pairs, weighted by the mask matrix $M_v$ . The overall CDA loss across all views is obtained by summing the individual losses for each view and averaging over all samples: + +$$ +\mathcal {L} _ {\mathrm {C D A}} = \sum_ {v = 1} ^ {V} \frac {1}{N} \sum_ {p = 1} ^ {N} \mathcal {L} _ {c o n} ^ {p, v}. \tag {13} +$$ + +For contrastive learning, when samples of the same class are clustered in one view, the attraction exerted by positive pairs propagates to other views. In contrast, although the distinction between samples does not necessarily imply class disparity, repulsion among negative pairs enables view with greater discriminative power to transmit class separations to other views. This results in a mechanism that transforms both the alignment and misalignment information at the cross-view sample level into an objective, aiming to minimize intra-class while maximizing inter-class distances. + +Regarding the role of CDA, the global view serves as an anchor, which is compared in parallel against each local view. To minimize the overall contrastive loss, the sample distribution of the global view is con + +currently attracted toward all local views, thereby encouraging the features of each sample to converge more tightly in the latent space. From the perspective of the entire latent space, this process effectively facilitates overall distribution alignment. The rationale behind utilizing local views for distribution alignment with the global view to enhance class separability lies in the ability to leverage easily classified samples from one view to improve the distinguishability of harder-to-classify samples in another. The reasoning above, together with Fig. 4, illustrate the working principle of the CDA: utilizing contrastive learning as a tool, through the process of aligning the distribution of sample features across views, enhancing the differentiation of difficult-to-classify samples in one view by leveraging easy-to-classify samples in another, and ultimately achieving the goal of optimal alignment of features in the shared latent space. + +![](images/da4bc3eeba94295f15533ce50ba40d3b107f28010820adfab3fed9f6e8ccb625.jpg) +(a) Multi-view sample distribution in latent space + +![](images/29db5b0cc8563f8cfa7e24b4748f7f544350cc929a9ceac40b4b3695b7cbc3fc.jpg) +(b) Cross-view distribution alignment +(c) Multi-view fusion +Figure 4: View distribution alignment based on contrast. + +![](images/c8524b86de64c101bd8cd72d91bdc6565aac460ae51be576bc84a3c8e7a0fdc4.jpg) +(a) GCFAgg (CVPR'23) +(b) CPSPAN(CVPR'23) +Figure 5: T-SNE visualization of the features learned with recently comparative methods (a-d) and ours (e) on the ALOI-100 dataset. + +![](images/2a75fdeb94a0aacc4705f66834b44c2ae9add87fcf71644b58bd45fab9de84a1.jpg) + +![](images/56510c5eac83ba4e5e16f3b154a9d030fe3fda27ef7dab11a940f15c849edcf7.jpg) +(c) SCMVC (TMM'24) + +![](images/0bdc3b48ea5deecc88acf29564cfdf9fe6a69a1330d1f436da51c30a558eb09e.jpg) +(d) MVCAN (CVPR'24) + +![](images/78c6108337f32d4930442a17afdca56d2718ae2e29ad6a84d6e15fd597b1fb1f.jpg) +(e) SparseMVC (Ours) + +# 3.4 The Overall Loss Function of SparseMVC + +The total loss $\mathcal{L}_{\mathrm{total}}$ comprises the adaptive sparse autoencoder loss $\mathcal{L}_{\mathrm{SAA}}$ in Eq. (6), preserving data fidelity and enforcing structured sparsity via $\mathcal{L}_{\mathrm{recon}}$ and $\mathcal{L}_{\mathrm{entropy}}$ , and the cross-view alignment loss $\mathcal{L}_{\mathrm{CDA}}$ in Eq. (13), ensuring consistent clustering across views: + +$$ +\mathcal {L} _ {\text {t o t a l}} = \sum_ {v = 1} ^ {V} \left(\mathcal {L} _ {\text {r e c o n}} ^ {v} + f \left(s _ {v}\right) \cdot \mathcal {L} _ {\text {e n t r o p y}} ^ {v}\right) + \lambda_ {\mathrm {C R}} \cdot \mathcal {L} _ {\mathrm {C D A}}, \tag {14} +$$ + +where $\lambda_{\mathrm{CR}}$ is the constraint ratio coefficient that controls the trade-off between $\mathcal{L}_{\mathrm{SAA}}$ and $\mathcal{L}_{\mathrm{CDA}}$ . + +# 4 Experiments + +# 4.1 Experimental Settings + +Compared Methods. Our proposed method is compared against the following 12 state-of-the-art multi-view clustering methods based on deep learning. DSMVC [43], COMPLETER [44], DCP [45], CVCL [46] and SCMVC [47] focus on dynamic contrastive learning; MFLVC [48], GCFAgg [3], DealMVC [49] combine feature fusion and consistency; DSMVC [43] and SDMVC [50] enhance consistency through discriminative learning; CPSPAN [51] and MVCAN [52] apply proxy supervision and prototype alignment. + +Benchmark Datasets. The selected datasets span diverse domains: Image datasets include MSRCV1 [53] focusing on objects and scenes, Dermatology [54] on medical images, Out-Scene [55] on natural scenes, and ALOI-100 [56] on object recognition. Image-text datasets include Wikipedia ${}^{2}$ ,which provides website crossmodal data. Omics datasets include LGG [57] focusing on brain tumor genomics and BRCA [58] on breast cancer genomics. Synthetic3d [59] supports 3D object modeling and recognition. Detailed properties of datasets are listed in Table 2. + +Table 2: Characteristics of kinds of multi-view datasets. + +
DatasetsSamplesClustersViewsView Dimensions
Images
MSRCV121076[1302, 48, 512, 100, 256, 210]
Dermatology35862[12, 22]
Out-Scene2,68884[512, 432, 256, 48]
ALOI-10010,8001004[77, 13, 64, 125]
Image-Text
Wikipedia693102[128, 10]
Omics
LGG26734[2000, 2000, 333, 209]
BRCA39844[2000, 2000, 278, 212]
Synthetics
Synthetic3d60033[3, 3, 3]
+ +Evaluation Metrics. Accuracy (ACC) evaluates alignment with ground truth, normalized mutual information (NMI) measures shared information, purity (PUR) assesses cluster homogeneity. Adjusted Rand index (ARI), measuring clustering similarity, is partially utilized in experiments. For all metrics, higher values indicate better performance. + +Implementation Details All experiments were conducted using Python 3.8.15 and PyTorch 1.13.1+cu116 on a Windows PC equipped with an AMD Ryzen 9 5900HX CPU, 32GB RAM, and an Nvidia RTX 3080 GPU (16GB). Models were trained using the Adam optimizer [60], a learning rate of 0.003, and a fixed seed of 50, with batch size equal to the dataset's sample count. Pre-training was performed uniformly for 300 epochs, while alignment training was conducted for 300 epochs for datasets with less than 2500 samples and 1000 epochs for larger datasets. For clustering, $k$ -means [61] was applied with the number of clusters equal to the dataset categories and 100 initializations. During pre-training, global features $Z_{v}$ derived from early fusion were used, while alignment training used late fusion features $\overline{Y}$ . Metrics were calculated as the average of 10 runs in the final epoch, with no fine-tuning performed for specific datasets. To ensure fairness, the hyperparameters for the comparison methods were determined based on either the default global settings or the configuration of the first dataset. + +# 4.2 Comparative Results Analysis + +Table 3 and 4 summarize the comparative results, leading to the following conclusions: + +(1) Our method achieves state-of-the-art performance across eight diverse multi-view datasets, along with larger-scale datasets as shown in Table 8. These results validate the versatility of our approach + +Table 3: Clustering results on small multi-view datasets. The top-ranked result is **bolded**, and the second-ranked result is **underlined**. + +
Methods\ DatasetsSynthetic3dLGGDermatologyBRCA
ACCNMIPURACCNMIPURACCNMIPURACCNMIPUR
COMPLETER [CVPR'21] [44]93.3376.0693.3380.1549.2580.1577.6580.1182.1255.5334.6565.33
DCP [TPAMI'22] [45]97.1787.6097.1759.5544.8273.0372.9177.2280.7357.2939.5160.55
MFLVC [CVPR'22] [48]90.6772.5990.6779.0349.7379.0358.1056.2062.8555.5327.7460.05
DSMVC [CVPR'22] [43]96.8386.6496.8382.7754.1382.7792.7487.8292.7454.5233.5368.84
SURE [TPAMI'22] [62]96.3385.1696.3362.9238.0165.1788.2777.0388.5539.7012.8548.99
DealMVC [MM'23] [49]87.5072.0787.5072.2840.5572.2845.5331.1345.5359.5532.7961.56
GCFAgg [CVPR'23] [3]96.6785.5496.6755.0622.9561.8088.2779.2588.2751.5132.4161.31
CPSPAN [CVPR'23] [51]97.8390.1597.8363.3030.5363.3076.2684.6385.2066.8334.4874.12
SDMVC [TKDE'23] [50]96.8386.4790.0063.6743.8667.7970.6783.3084.9257.7933.8064.57
CVCL [ICCV'23] [46]95.3182.3695.3158.2023.7358.2056.2556.0167.9761.9834.6868.49
MVCAN [CVPR'24] [52]98.1791.2794.5959.5542.5727.1858.3866.7351.5857.7935.7032.24
SCMVC [TMM'24] [47]97.0087.1197.0073.4139.7673.4193.8588.4493.8550.2530.7060.80
SparseMVC (Ours)98.3392.0198.3383.1554.6283.1595.2589.8695.2570.1044.9070.85
+ +and highlight its potential for a wide range of downstream tasks. In comparison, other approaches such as SCMVC, MVCAN, and CPSPAN achieved relatively good results on specific datasets but failed to maintain an advantage due to their limited generalizability across other datasets. + +(2) Our method demonstrates stability in clustering performance, as evaluation metrics oscillate upward within a small range and stabilize with increasing epochs, showcasing robust results. In contrast, methods like SURE and DealMVC on ALOI-100 or CVCL and CPSPAN on Wikipedia fail to stabilize, with metrics either degrading significantly after peaking or fluctuating dramatically without consistent improvement. During the early stages of contrastive training when the embedding space's distribution remains uneven, potentially causing abrupt gradient fluctuations. To alleviate the instability, we adopt a dynamic fusion strategy in Sec. 3.2 and a pre-training approach in which only the autoencoder is trained initially. +(3) Although our model is specifically designed to address the challenge: across-view sparsity variations, it also achieves superior performance on dense datasets, such as BRCA, thereby demonstrating its adaptability and broad applicability. This is attributed to the inherent flexibility of our designed sparse autoencoder, which adaptively transitions into a conventional autoencoder when confronted with dense data, thereby prioritizing the reconstruction objective with greater emphasis. +(4) Compared to recent state-of-the-art methods, our approach demonstrates superior feature representation performance by producing clearer boundaries and more compact clusters, as shown in Fig. 5. Its most notable advantage is the ability to disentangle intra-class clusters while preserving inter-class separability, which leads to better scalability and robustness when handling data with large cross-view distribution disparities. + +# 4.3 Convergence Analysis + +By analyzing the training curve in Fig. 6, we observe the following key points: (1) The evaluation metrics generally exhibit an oscillatory increase followed by stabilization, with this stability being maintained as the number of training epochs progresses. This observation underscores the model's convergence and its robustness in maintaining stable clustering performance despite optimization challenges. (2) During the early stages of the alignment + +![](images/622aff7bc14f3d5659ce4dd3b1fd6a1ce470890094550569f598df21eb8962ac.jpg) +(a) Out-Scene +Figure 6: Convergence analysis of the training process. The left area of the vertical black dashed line represents the pre-training phase, while the right area stands for the view alignment training process. + +![](images/9599ff792587b8f9f633e75d5764e397ccf824350bf28ee2bbb2b43c9852dc05.jpg) +(b) MSRCV1 + +training phase, the evaluation metrics exhibit a brief dip, which is quickly followed by a recovery and subsequent stabilization at a higher level. The fluctuations observed in the evaluation metrics + +can be attributed to $\mathcal{L}_{\mathrm{CDA}}$ in Eq. (13), which necessitates the initialization of all network parameters except the autoencoder. (3) Despite significant fluctuations in the loss for small datasets, intriguingly, these fluctuations do not propagate to the evaluation metrics, suggesting that our approach effectively mitigates the influence of instability in the optimization algorithm on the clustering structure. + +Table 4: Clustering results on big (Out-Scene & ALOI-100) and small multi-view datasets. + +
Methods \ DatasetsOut-SceneALOI-100WikipediaMSRCV1
ACCNMIPURACCNMIPURACCNMIPURACCNMIPUR
COMPLETETER [CVPR'21] [44]69.7955.3969.7930.7062.1233.6357.1453.1059.3190.0087.9090.00
DCP [TPAMI'22] [45]56.0345.5956.3234.0160.2837.3245.3143.1646.3225.7123.2527.14
MFLVC [CVPR'22] [48]58.9751.3158.9733.1773.2833.1740.1227.5241.7063.3366.1164.29
DSMVC [CVPR'22] [43]62.1353.0164.2571.5290.8772.7260.3254.7462.1964.2954.2964.29
SURE [TPAMI'22] [62]60.9748.0960.9710.1334.1911.9050.6539.9754.1191.4385.8491.43
DealMVC [MM'23] [49]69.5759.4469.5713.1148.5413.1038.9637.0938.9682.0075.5482.00
GCFAgg [CVPR'23] [3]68.2357.1468.2374.1188.3076.6351.8045.8756.5739.5231.9142.86
CPSPAN [CVPR'23] [51]59.1550.4659.1556.9678.7867.9922.088.3524.3967.6269.8389.52
SDMVC [TKDE'23] [50]56.0346.1859.9352.0274.7056.5655.9953.9862.0559.5252.5145.24
CVCL [ICCV'23] [46]73.5159.5973.5121.8643.1323.2914.1742.8132.6948.4484.5790.62
MVCAN [CVPR'24] [52]70.9858.2349.9567.4883.7856.7159.0255.8167.9771.5460.1971.54
SCMVC [TMM'24] [47]71.5460.1971.5477.7289.4281.0553.5435.5955.8490.9583.9290.95
SparseMVC (Ours)77.4963.3477.4982.2192.6584.1961.0454.7962.9197.1494.2297.14
+ +# 4.4 Ablation Study + +Loss Function To ensure the rigor of ablation experiment, we selected three datasets with significant variations in view sparsity as shown in Fig. 1, ensuring the functionality of the SAA. We assessed the effectiveness of individual losses in the total loss Eq. (14) of SparseMVC, as presented in Table 5. + +Specifically, we use $\mathcal{L}_{\mathrm{recon}}$ as the baseline and find that adding either $\mathcal{L}_{\mathrm{entropy}}$ or $\mathcal{L}_{\mathrm{CDA}}$ improves performance. $\mathcal{L}_{\mathrm{entropy}}$ yields substantial improvements on ALOI-100 and Dermatology, which have stronger variations in view sparsity. These improvements highlight the effectiveness of adaptive encoding and cross-view distribution alignment as robust constraints that contribute positively to the overall model training process. Moreover, when all losses are activated simultaneously, the model achieves optimal performance, suggesting that $\mathcal{L}_{\mathrm{entropy}}$ and $\mathcal{L}_{\mathrm{CDA}}$ complement each other synergistically. Notably, their integration does not introduce any mutual interference, further underlining the coherence and compatibility of these objectives in driving superior learning outcomes. + +Table 5: Comparison of different loss function combinations. + +
DatasetsLoss FunctionEvaluation Metrics
\( \mathcal{L}_{\text{recon}} \)\( \mathcal{L}_{\text{entropy}} \)\( \mathcal{L}_{\text{CDA}} \)ACCNMIPUR
ALOI-100✓45.2771.2130.41
✓✓66.3581.6570.62
✓✓64.1180.2367.33
✓✓✓82.2192.6584.19
Dermatology✓70.1174.4159.69
✓✓75.7083.3669.36
✓✓70.9571.0683.80
✓✓✓95.2589.8695.25
MSRCV1✓58.5748.6332.76
✓✓70.4865.2752.08
✓✓92.3887.6292.38
✓✓✓97.1494.2297.14
+ +Components Our approach focuses on view-level structural sparsity, specifically the sparsity variation across views within the same multi-view data. This differs from data-level sparsity methods, which typically apply uniform sparse encoding to all views without explicitly considering the heterogeneity of inter-view sparsity. To further validate the effectiveness of the proposed SAA module, we extend the ablation study in Table 5 by introducing two additional comparative settings: (i) uniformly sparse encoding applied to all views, which mimics methods designed for data-level sparsity; and (ii) adaptive encoding tailored to each view. On top of this, we also ablate the CSR module, which reweights the local features during the late fusion stage. Regardless of whether the CSR module is applied, the results in Table 6 show that the proposed SAA, which leverages adaptive autoencoders, remains effective and consistently achieves superior performance. Results further confirm that the effectiveness arises from the synergy between adaptive encoding and sample reweighting, rather than from the use of sparse autoencoders alone. Collectively, these findings confirm that our method is robust to varying sparsity across views and that the SAA and CSR modules function synergistically rather than independently. + +Table 6: Ablation study on different components. + +
Components \ DatasetsALOI-100DermatologyMSRCV1
ACCNMIPURACCNMIPURACCNMIPUR
all sparse autoencoders w/o CSR78.5688.9281.1277.3774.3886.0391.9088.5691.90
all sparse autoencoders80.4289.1982.4489.1178.3789.1192.3888.6192.38
adaptive autoencoders w/o CSR81.0489.5883.1788.8378.2388.8395.7192.6995.71
adaptive autoencoders (Ours)82.2192.6584.1995.2589.8695.2597.1494.2297.14
+ +# 4.5 Parameter Sensitivity Analysis + +We selected the temperature parameter $\tau$ in $\mathcal{L}_{\mathrm{CDA}}$ and the constraint ratio coefficient $\lambda_{\mathrm{CR}}$ , the ratio of $\mathcal{L}_{\mathrm{SAA}}$ to $\mathcal{L}_{\mathrm{CDA}}$ , as the two parameters for analysis. Both coefficients were set to gradually increase from 0.1 to 1.9 with a step size of 0.3. + +![](images/2487591f94936e25bf087ef4371fe394c59a80284206965c52024904346b6caf.jpg) +(a) Synthetic3d + +![](images/959c72d08651caa350cb080017ab47e0e3cb532d6755998be713376b233821b6.jpg) +(b) Out-Scene + +![](images/61696724fb89e0ac47442fdab72ad5ada4bc4eea814ac89eeaa41c0255068756.jpg) +(c) BRCA +Figure 7: The bar chart of clustering accuracy varying with different values of $\tau$ and $\lambda_{\mathrm{CR}}$ . + +![](images/f3d9b5ca606f4d826ea425e5591d9ebd4703eaadf6c07ec2cac7dc1feee4bbef.jpg) +(d) MSRCV1 + +In Fig. 7, we can discover that the accuracy initially increases and then decreases as $\tau$ increases, remaining relatively stable within a range around 1.0. The influence of $\lambda_{\mathrm{CR}}$ on clustering performance is comparatively minor, with a negligible impact when $\tau$ is within the range of 0.4 to 1.0. In light of Sec. 3.3, $\mathcal{L}_{\mathrm{CDA}}$ incorporates the smoothing property of the logarithmic function, which diminishes the direct effect of $\lambda_{\mathrm{CR}}$ adjustments on the gradient. In contrast, changes to $\tau$ significantly influence clusters separability and alignment performance by modulating the nonlinear response of the softmax function. A smaller $\tau$ enhances class separability, while a larger $\tau$ emphasizes global consistency. Therefore, we set the default values of $\tau$ and $\lambda_{\mathrm{CR}}$ to 1.0 in the loss function. + +Our method is largely insensitive to hyperparameter changes. To begin with, significant performance degradation only occurs when the temperature coefficient $\tau \leq 0.4$ or $\geq 1.6$ . The extremely small value of 0.1 is chosen to probe the lower bound of performance degradation and is rarely used in practical applications. Furthermore, the performance fluctuation mainly occurs along the $\tau$ -axis, whereas it remains relatively insensitive to changes in the constraint ratio coefficient $\lambda_{\mathrm{CR}}$ . In practice, it is the relative weighting between loss terms that is more commonly adjusted. In addition, the accuracy-axis was intentionally truncated to better highlight the differences, accentuating the visual disparity. Finally, regarding the concern that hyperparameters are not easy to tune in practice, our method maintains stable performance even under noticeable loss fluctuations, as illustrated in Fig. 6. + +# 5 Conclusion + +This paper highlights a frequently overlooked issue in deep multi-view learning: varying sparsity ratios across views. Therefore, we systematically define, quantify, and analyze cross-view sparsity variation as a fundamental characteristic of multi-view data. Our entire framework, SparseMVC, is designed to handle view-level sparsity variations with a complete data-driven and tightly integrated architecture. To tackle sparsity variation, we propose an adaptive encoding strategy that uses the sparsity ratio of each view as prior knowledge, enabling the encoder to switch between standard and sparse forms with appropriate constraint strengths. Additionally, we introduce a series of interdependent mechanisms to mitigate the side effects of representational divergence caused by non-uniform encoding. Specifically, a correlation-guided fusion strategy leverages global-to-local feature relationships from the early stages to guide the weighting of local features in late fusion. Moreover, a distribution alignment module structurally constrains the fused representations, enhancing cross-view complementarity in the final stage. Comprehensive experiments and detailed dissections of each module validate the efficacy of SparseMVC. We hope this work inspires greater attention to the intrinsic characteristics of data and to the design of architectures driven by data. + +# Acknowledgements + +This research was supported in part by the National Natural Science Foundation of China under grants 62522604 and 62476258, the Natural Science Foundation of Hubei Province under grant 2025AFA113, the Interdisciplinary Research Program of HUST under grant 2025JCY3024, and the Fundamental Research Funds for National Universities of China University of Geosciences (Wuhan) (No.2024XLB7). + +# References + +[1] Siwei Wang, Xinwang Liu, Suyuan Liu, Jiaqi Jin, Wenxuan Tu, Xinzhong Zhu, and En Zhu. Align then fusion: Generalized large-scale multi-view clustering with anchor matching correspondences. Advances in Neural Information Processing Systems, 35:5882-5895, 2022. +[2] Xin Zou, Chang Tang, Xiao Zheng, Kun Sun, Wei Zhang, and Deqiong Ding. Inclusivity induced adaptive graph learning for multi-view clustering. 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Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction clearly articulate the core problem addressed by the paper, cross-view sparsity variations in multi-view clustering, and emphasize the limitations of existing methods in handling this structural heterogeneity. The proposed solution, SparseMVC, is concisely described as incorporating sparsity-aware constraints and a view-adaptive autoencoder design. These claims are fully supported by both the theoretical formulation and the experimental results across diverse datasets, including those with extreme sparsity imbalance. + +# Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: Please refer to Appendix F. + +# Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. 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Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: Please refer to Appendix E. + +Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. 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Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +For completeness, we include additional discussions and experimental details in the appendices. A summary of the contents is listed below: + +- Appendix A Algorithm and Comparison with Prior Work. +- Appendix B Further Experiments. +- Appendix C Visualization. +- Appendix D Computational Complexity. +- Appendix E Theoretical Analysis. +- Appendix F Limitations and Future Work. + +# A Algorithm and Comparison with Prior Work + +# A.1 Algorithm + +The training procedure for SparseMVC is described in Algorithm (1). + +Algorithm 1 Training Steps for SparseMVC + +Input: Multi-view data $\{X_v\}_{v=1}^V$ , cluster number $K$ , and number of training epochs $E_{\mathrm{pre}}$ , $E_{\mathrm{con}}$ . + +Output: Late-stage fusion representation $\overline{Y}$ . + +1: Initialize random seed and select Adam optimizer. +2: for epoch $h = 1 : E_{\mathrm{pre}} + E_{\mathrm{con}}$ do +3: Update $\{Z_v\}_{v=1}^V$ by minimizing $\{\mathcal{L}_{\mathrm{recon}}^v\}_{v=1}^V$ and $\{\mathcal{L}_{\mathrm{entropy}}^v\}_{v=1}^V$ utilizing Eqs. (2) and (4). +4: Update $\bar{Z}$ , formed by the concatenation of $\{Z_v\}_{v=1}^V$ , utilizing Eq. (2) and Eq. (4). +5: if epoch $>E_{\mathrm{pre}}$ then +6: Update weights $\{\pmb{W}_v\}_{v=1}^V$ by Eq. (9). +7: Update $\overline{Y}$ by minimizing $\mathcal{L}_{\mathrm{CDA}}$ utilizing Eq. (13). +8: end if +9: end for +10: Perform $K$ -means clustering on representation $\overline{Y}$ . + +# A.2 Comparison with Prior Work + +To contextualize our contributions, we present a comparative discussion with representative dynamic weighting methods proposed in the literature. Dynamic weighting has been widely explored in multiview learning, primarily through attention-based fusion mechanisms or optimization-driven strategies. Methods such as GCFAgg [3] employ attention to emphasize discrepancies among local views, aiming to refine feature alignment. Other approaches, including SPGMVC [63] and MAGCN [64], enhance feature encoding or perform view-level aggregation to improve representational quality. In parallel, techniques such as SCMVC [47], SCE-MVC [65] and TMC [66] utilize mutual information [67] or probabilistic priors to adaptively assign importance to views or losses. Furthermore, recent efforts have introduced view-invariant representations [46] and prototype-guided learning [52] to mitigate cross-view variability. Despite these advances, existing methods often overlook the inherent inconsistencies of sparsity between views, a phenomenon that can severely degrade the effectiveness of fusion [8, 9, 68]. In contrast, our proposed approach is motivated by the need to explicitly characterize and adapt to such cross-view sparsity variations. Specifically, our framework preserves global features obtained during early fusion and integrates both global and local view representations into the fusion process. The core of our design, the correlation-informed sample reweighting module, dynamically adjusts fusion weights based on the learned correlation between global and local views, thereby enabling fine-grained, sample-specific adaptation. + +While MVASM [69] addresses the challenges of ambiguous class assignments in multi-view data by introducing an adaptive sparse membership matrix, our method introduces adaptive autoencoders with view-specific encoding strategies. Moreover, unlike existing techniques [70, 71, 72] that generally apply static or view-level weighting schemes, our method performs late-stage fusion reweighting conditioned on earlier global-local interactions. This design not only enhances fusion fidelity but also + +elevates sparsity modeling from the feature level to the view level, offering a principled solution to a problem rarely addressed in the literature. + +# B Further Experiments + +# B.1 Dimension and Feature Compression + +![](images/f4f32e74da390628dc7a68da8bf2df60858d1ed8f4ac3e7b30cd604e1ea82ea4.jpg) +(a) Out-Scene + +![](images/6c43fac53c75fecbd33a6e86e9258e5f33251a3c32471f587cd9313dca843c4d.jpg) +(b) MSRCV1 +Figure 8: The clustering evaluation metric curves with respect to feature $(Z_v)$ dimension variations. + +![](images/41c2961905dd444a93b2494f714cfcb1092cbdd4b995744d7a201a3d6c9a985b.jpg) +(c) Synthetic3d + +We tested the dimensions of view-specific features $Z_{v}$ extracted by the autoencoder, varying from $2^{1}$ to $2^{10}$ in Fig 8. The experimental results demonstrated that as the feature dimension increased, all evaluation metrics (ACC, NMI, PUR and ARI) initially exhibited strong oscillations but progressively improved, eventually stabilizing within a small and well-performing range. + +From the perspective of information-theoretic [73], low-dimensional features have limited encoding capacity that hinders capturing complex data patterns, whereas high-dimensional features offer greater capacity but risk dimensional collapse [74]. The feature compression layers that we designed compress and refine $Z_{v}$ into $Y_{v}$ , mitigating the risks of overfitting and performance degradation often seen in contrastive learning. Accordingly, the model can avoid the negative effects of excessive dimensionality while leveraging its increased capacity for effective feature extraction. Ultimately, we selected 64 as the dimensionality of $Z_{v}$ to balance computation and feature capacity. + +# B.2 Selection of Scaling Factor + +Table 7: Ablation study on scaling factor selection. + +
The external scaling factor f(sv)the internal ratio coefficient ρMSRCV1 Mean accuracy [Max accuracy]ALOI-100
0 or 11-f(sv)94.29 [95.71]77.75 [80.39]
f(sv)1-f(sv)95.24 [95.24]79.01 [80.08]
0 or 10.0595.71 [96.67]80.82 [81.51]
f(sv)0.0597.14 [97.62]82.21 [82.93]
+ +To investigate the relative importance of the external scaling factor $f(s_v)$ and the internal ratio coefficient $\rho$ in modulating sparsity within the sparse autoencoder, we conducted a set of controlled experiments in which one parameter was kept constant while systematically varying the other. This decoupled analysis enables a clearer understanding of how each component contributes to the overall behavior of the model. Empirical results, as shown in Table 7, demonstrate that allowing $f(s_v)$ to be adaptively optimized while fixing $\rho$ at a reasonable constant leads to superior performance in multiple evaluation metrics. The smooth variation of the constraint strength is better than directly switching between zero and one, which represents whether the sparse constraint is applied. In contrast, varying $\rho$ while keeping $f(s_v)$ static yields relatively suboptimal results. These findings suggest that external view-level scaling plays a more critical role in capturing cross-view sparsity dynamics, highlighting the effectiveness of our adaptive design. + +# B.3 The large-scale datasets + +To evaluate the effectiveness and generalization ability of our proposed method, SparseMVC, we conduct additional experiments on large-scale datasets comprising over 8,000 samples, specifically GSE [18] and Animal [75]. The GSE dataset encompasses multi-omics data across 27 categories, + +Table 8: Clustering results on the large-scale multi-view datasets. + +
Datasets +View Sparsity Ratio +MethodsGSE (8,200 samples) +[0.877, 0.005]Animal (11,673 samples) +[0.589, 0.179, 0.355, 0.467]
ACCNMIPURACCNMIPUR
MVCAN [CVPR'24]71.4075.4172.2912.0610.2515.48
SCMVC [TMM'24]62.8074.1365.7216.9615.0520.15
SparseMVC (Ours)73.1878.4274.2619.9017.9224.13
+ +capturing a broad range of biological conditions, and is extensively utilized in bioinformatics research. Meanwhile, the Animal dataset consists of image data featuring animals with diverse attributes, spanning 20 categories. The detailed dataset information and comparison results, presented in Table 8, highlight the performance of our approach against the latest two methods employed in our manuscript. Clustering results on large-scale multi-view datasets further emphasize the superiority of our method, demonstrating its promising applicability in other data environments and scenarios. + +# C Visualization + +![](images/8c0b9ea870f577b5dcd2cad1bf61f4f2e86d1f720c1a865d1349f2c551d40f80.jpg) + +![](images/29b7b4327bf7788a56ee05340bf9bf5b12d399b0aca40b3d4aa5ce6aa192aec1.jpg) + +![](images/6dd16995cc5d9cfdd77da37aa9180543fb5801041e1e9a7d66556c0b29d02251.jpg) + +![](images/2f31ea29fbff98162482c9ab4722f2361ec4ba35490fcbe680b3e0e00c301e29.jpg) +(a) + +![](images/2bad0e877a827e9ae0a18c7f94915851c7dc74bcc5747ebf6d836877226db17f.jpg) +(b) + +![](images/97604d1b04bbf9358c9a1e964359eab1352d519828d5205745041f3e96c96f58.jpg) +(c) + +![](images/a8ade87150fefcb1a0e140cfea339321a96498fc221beabb2a45a304a0fbe972.jpg) +(d) +(g) $\widehat{\pmb{X}}_v$ features + +![](images/fd80a13245a8cdbcf7ebe7804e5b3dce0b2e5b34bf7a9e3db4e8c63cad74c917.jpg) +(e) +(h) $\overline{Z}$ features +Figure 9: The t-SNE visualizations of the original reconstruction features $(\widehat{X}_v)$ , the early fusion features $(\overline{Z})$ , and the late fusion features $(\overline{Y})$ on datasets: Dermatology (a-c), Synthetic3d (d-f), and Out-Scene (g-i). + +![](images/5a0c93426132cadb813083d5206b43dbe777235568e449f608f5d00f6c5a10ec.jpg) +(f) +(i) $\overline{Y}$ features + +Figure 9 uses t-SNE for dimensionality reduction, mapping high-dimensional feature vectors into a two-dimensional space to facilitate a clearer and more insightful visualization of the distribution and structure of the data. The visualization reveals that the original reconstruction data, $\widehat{\pmb{X}}_v$ , exhibit substantial interclass entanglement, with considerable overlap between different categories. In contrast, the preliminary global features, $Z_v$ , derived from autoencoder pretraining, show a notable reduction in this entanglement, suggesting an early phase of disentanglement compared to the raw features. Further refinement through multi-view alignment results in the final fused features, $\mathbf{Y}_{\mathrm{global}}$ , which exhibit the most pronounced disentanglement, yielding well-defined, separate clusters. These findings are further corroborated by the results in ablation analysis, which demonstrate that $\mathbf{Y}_{\mathrm{global}}$ consistently outperforms $Z_v$ in clustering tasks, particularly in distance-based methods like K-means. This performance enhancement underscores the pivotal role of the multi-view alignment and fusion process in improving feature separability and discriminative capacity, ultimately leading to more accurate and meaningful clustering results. + +# D Computational Complexity + +The total loss function, as defined in Eq. (14), includes two main terms: $\mathcal{L}_{\mathrm{SAA}}$ and $\mathcal{L}_{\mathrm{CVDA}}$ . The term $\mathcal{L}_{\mathrm{SAA}}$ involves the reconstruction loss and the entropy-matching loss, both computed for each of the $V$ views, each containing $N$ samples. Since the feature dimension $F$ is constant and does not affect the computational scaling, the complexity of this term is determined by the number of views and samples, resulting in $O(VN)$ . The term $\mathcal{L}_{\mathrm{CVDA}}$ requires pairwise similarity computations between $N$ samples. For each pair of samples, the dot product computation has a constant cost of $O(1)$ , leading to a total complexity of $O(N^2)$ for computing all pairwise similarities. The contrastive loss further requires computing the numerator, denominator, and logarithmic terms for each sample, which does not increase the overall scaling beyond $O(N^2)$ . When combining both terms, the reconstruction and entropy-matching losses contribute $O(VN)$ , while the pairwise similarity computations dominate with $O(N^2)$ when $N$ is sufficiently large. Thus, the overall computational complexity of the total loss function is $O(N^2)$ . + +# E Theoretical Analysis + +When learning from heterogeneous multi-view data, structural disparities, especially in sparsity patterns, present significant challenges to unified representation learning. Differences in modality, sampling granularity, and view incompleteness lead to varying information densities across views, making fixed encoder architectures and uniform loss formulations inherently suboptimal. To address this issue, we propose the Sparse Autoencoder with Adaptive Constraints (SAA), which dynamically adjusts the strength of sparsity constraints based on sparsity ratio $s_v$ of each view. This mechanism enables the encoder to balance compression and expressiveness in a view-aware manner, thereby facilitating the alignment of latent representations across structurally diverse views. In what follows, we provide a preliminary theoretical foundation for SAA, organized around a set of key questions that clarify its motivation, theoretical grounding, and coding formulation: + +# Why should the sparsity constraint be adaptive? + +When dealing with views of varying characteristics, uniform treatment of data may result in suboptimal representations, particularly when the model overly compresses informative in dense views or overemphasizes noisy in sparse ones. Accordingly, by the principle of minimal redundancy maximum relevance (mRMR) [76], we can establish trade-offs between different views, balancing the fidelity of reconstruction and the complexity of representation. + +# What exactly are we coding or preserving? + +Grounded in compressed sensing and optimally sparse representations [77, 78], we encode the essential structure of sparse inputs, which, though embedded in high-dimensional space, intrinsically conform to low-dimensional semantic subspaces. Therefore, sparse activation suffices to capture their core features, with sparse coding providing a stable, efficient, and redundancy-minimizing representation aligned with this inherent geometry. + +# E.1 Why should sparsity constraints be adaptive? + +From Max-Dependency to mRMR and our view-wise coding. Let $S = \{x_{i}\}_{i = 1}^{m}$ be a selected feature set and $c$ be the target. Peng et al. [76] formulate max-dependency as maximizing the mutual information between the joint selected features and the target: + +$$ +\max _ {S} D (S; c), \quad D (S; c) = I (S; c) = I \left(x _ {1}, \dots , x _ {m}; c\right). \tag {15} +$$ + +Because directly estimating $I(S; c)$ is difficult when $m$ is not tiny, they introduce a first-order surrogate: max-relevance $D_{\mathrm{rel}}(S; c) = \frac{1}{|S|} \sum_{x_i \in S} I(x_i; c)$ , and control redundancy by min-redundancy $R(S) = \frac{1}{|S|^2} \sum_{x_i, x_j \in S} I(x_i; x_j)$ , then combine them as the mRMR criterion $\max \{D_{\mathrm{rel}}(S; c) - R(S)\}$ (difference form) [76]. In our setting, for each view $v$ , the encoder produces a hidden code $\pmb{h}_v = (h_1^v, \dots, h_H^v)$ for input random variable $X_v$ . Interpreting hidden units as the "selected features" $S$ and the input as the "target" $c$ (unsupervised but still information-theoretic), we use a weighted difference form to allow view-wise calibration, and the view-wise analogue of mRMR becomes: + +$$ +\max \left[ \underbrace {\frac {1}{H} \sum_ {k = 1} ^ {H} I \left(h _ {k} ^ {v} ; X _ {v}\right)} _ {\text {m a x - r e l e v a n c e}} - \beta_ {v} \underbrace {\frac {1}{H ^ {2}} \sum_ {i = 1} ^ {H} \sum_ {j = 1} ^ {H} I \left(h _ {i} ^ {v} ; h _ {j} ^ {v}\right)} _ {\text {m i n - r e d u n d a n c y}} \right], \tag {16} +$$ + +where $\beta_v \geq 0$ is the redundancy/constraint strength for view $v$ . + +Why $\beta_v$ depends on view sparsity $s_v$ : redundancy scales with the effective degrees of freedom. SparseMVC explicitly measures the sparsity ratio $s_v$ of view $v$ as a prior statistic in Eq. (1). A larger $s_v$ indicates a higher fraction of zero (or missing/default) entries and thus a lower effective information density in $X^v$ . In this regime, the first term in Eq. (16) maximizing relevance alone can encourage multiple hidden units to focus on the same limited set of informative patterns, which increases dependence among $\{h_k^v\}$ and inflates the redundancy term. + +To relate redundancy to the sparsity regularizer used by sparse autoencoders, we follow the standard sparse autoencoder interpretation [40] and treat each hidden unit as a Bernoulli activation variable when discussing entropic quantities. Then, for any pair $(i,j)$ , + +$$ +I \left(h _ {i} ^ {v}; h _ {j} ^ {v}\right) = H \left(h _ {i} ^ {v}\right) - H \left(h _ {i} ^ {v} \mid h _ {j} ^ {v}\right) \leq H \left(h _ {i} ^ {v}\right), \tag {17} +$$ + +where $H(\cdot)$ denotes Shannon entropy of the corresponding Bernoulli variable. This inequality shows that suppressing the entropies (and co-activations) of hidden units controls an upper bound on pairwise dependence, and thus provides a principled route to reducing redundancy when the view is highly sparse. + +SAA realizes adaptive mRMR via entropy-matching. SparseMVC instantiates the adaptive redundancy coefficient as a deterministic function of $s_v$ in Eq. (3) and implements redundancy control via the entropy-matching KL penalty (sparse autoencoder prior) on the mean activation $\hat{h}_k^v$ in Eq. (4). For numerical stability in KL computation, the estimated mean activation is clamped to remain strictly within $(0,1)$ in implementation. This design matches the mRMR logic at the view level: $s_v$ -large views are precisely those where redundancy inflation is most likely, so $f(s_v)$ increases the effective redundancy penalty; $s_v$ -small views receive weak or no sparsity constraint (e.g., $f(s_v) = 0$ below $\theta$ ), preventing unnecessary relevance loss. + +Conclusion. According to the mRMR principle [76], effective representations should maximize relevance while minimizing redundancy. SparseMVC operationalizes this principle with a view-dependent redundancy strength, using the sparsity ratio $s_v$ to adaptively scale the KL-based sparsity penalty. As a result, sparse views benefit from stronger redundancy suppression, whereas dense views retain greater expressive capacity, yielding a principled, data-driven trade-off between information preservation and redundancy reduction across heterogeneous views. + +# E.2 What exactly are we coding or preserving? + +This question concerns the rationale and effectiveness of using sparsity constraints and sparse autoencoders to represent sparse data. Sparse data are characterized by low information density: + +only a small fraction of entries are nonzero and carry informative structure, while many coordinates may correspond to missing/default states. This suggests that an effective representation should focus capacity on the informative components, rather than distributing it uniformly across all dimensions. Consequently, representations of sparse inputs are often encouraged to be sparse as well, activating only a small subset of latent units that capture the dominant patterns supported by the nonzero entries, thereby reducing redundancy. + +Sparse inputs and informative support. Formally, let the $j$ -th sample from view $v$ be denoted by $\pmb{x}_j^v \in \mathbb{R}^{n_v}$ , with sparsity level $\| \pmb{x}_j^v \|_0 = k \ll n_v$ . Define the (sample-dependent) support set $\Omega_j^v = \{i \mid x_{j,i}^v \neq 0\}$ . In sparse views where zeros primarily represent missing/ default entries, the informative variation is mainly concentrated on the restricted subvector $x_{j,\Omega_j^v}^v$ , while the complement $x_{j,(\Omega_j^v)_c}^v$ contributes little signal. Thus, allocating excessive modeling capacity to the zero-valued region $(\Omega_j^v)_c$ may introduce unnecessary degrees of freedom and increase redundancy. + +Reconstruction error and support-aware desideratum. Consider a linear decoder or dictionary $D_v \in \mathbb{R}^{n_v \times m_v}$ and a latent code $\alpha_j^v \in \mathbb{R}^{m_v}$ . Let $D_{\Omega_j^v}$ denote the row submatrix of $D^v$ indexed by $\Omega_j^v$ , and let $D_{(\Omega_j^v)_c}^v$ denote the row submatrix indexed by the complement of $\Omega_j^v$ . Then the reconstruction error decomposes as: + +$$ +\left\| \boldsymbol {x} _ {j} ^ {v} - D _ {v} \alpha_ {j} ^ {v} \right\| _ {2} ^ {2} = \left\| x _ {j, \Omega_ {j} ^ {v}} ^ {v} - D _ {\Omega_ {j} ^ {v}} ^ {v} \alpha_ {j} ^ {v} \right\| _ {2} ^ {2} + \left\| D _ {\left(\Omega_ {j} ^ {v}\right) _ {c}} ^ {v} \alpha_ {j} ^ {v} \right\| _ {2} ^ {2}. \tag {18} +$$ + +The second term measures spurious energy produced on coordinates that are zero in the input. A natural desideratum for sparse-view reconstruction is therefore to keep $\| D_{(\Omega_j^v)^c}\alpha_j^v\| _2^2$ small. Importantly, sparsity of $\alpha_{j}^{v}$ alone does not guarantee this term is exactly zero; rather, sparsity regularization reduces the degrees of freedom available to fit $(\Omega_j^v)^c$ and encourages the decoder to rely on a small number of atoms, which empirically helps suppress spurious energy outside the informative support. + +Sparse coding and recoverable representations. Under this support-aware viewpoint, sparse coding seeks a representation of the form: + +$$ +\alpha_ {j} ^ {v *} = \arg \min _ {\alpha} \| x _ {j, \Omega_ {j} ^ {v}} ^ {v} - D _ {\Omega_ {j} ^ {v}} ^ {v} \alpha \| _ {2} ^ {2} \quad \text {s . t .} \quad \| \alpha \| _ {0} \leq s, \tag {19} +$$ + +with $s \ll m^v$ . Classical results in sparse approximation characterize uniqueness in the noiseless exact-representation case, e.g., if $x_{j,\Omega_j^v}^v = D_{\Omega_j^v}^v\alpha$ and $s < \frac{1}{2}\mathrm{spark}(D_{\Omega_j^v}^v)$ , then the sparsest solution is unique [77]. Compressed sensing theory further establishes stable recovery guarantees under Restricted Isometry Property (RIP)-type conditions for standard $\ell_1$ -based decoders [78]. These guarantees justify sparse coding as a canonical representational prior for signals whose structure is concentrated on a low-dimensional support. + +Sparse autoencoder as amortized sparse recovery. A sparse autoencoder implements the above sparse recovery principle in an amortized and differentiable manner. For view $v$ , the encoder produces hidden activations whose empirical mean is given by: + +$$ +\hat {h} _ {k} ^ {v} = \frac {1}{N} \sum_ {j = 1} ^ {N} \sigma \left(\boldsymbol {W} _ {k} ^ {v} \boldsymbol {x} _ {j} ^ {v} + b _ {k} ^ {v}\right) = \frac {1}{N} \sum_ {j = 1} ^ {N} \sigma \left(\sum_ {i = 1} ^ {F} w _ {i, k} ^ {v} x _ {i, j} ^ {v} + b _ {k} ^ {v}\right), \tag {20} +$$ + +where $\sigma(\cdot)$ is chosen so that the quantity used in the KL term can be interpreted as a probability, and $\hat{h}_k^v$ is clamped to lie strictly within $(0,1)$ in implementation before computing the KL divergence. Enforcing $\hat{h}_k^v$ to match a small target sparsity level via a KL-divergence sparsity penalty imposes an $\ell_0$ -like constraint in expectation on the latent representation. As a result, for each sparse input $\boldsymbol{x}_j^v$ , only a small subset of hidden units tends to be active, which qualitatively mimics sparse coding behavior without explicitly solving a combinatorial optimization problem. Eq. (20) also makes explicit that sparsity is enforced through an empirical average (over samples or minibatches) rather than per-sample hard sparsity constraints. + +Conclusion. Sparse coding theory and compressed sensing results [77, 78] provide a principled rationale for representing sparse-view signals with sparse codes under standard recoverability assumptions. A sparse autoencoder operationalizes this idea by combining reconstruction with a KL-based sparsity penalty, yielding compact representations that suppress redundant co-activations and emphasize patterns supported by the informative entries of the input. + +# F Limitations and Future Work + +While effective in addressing cross-view sparsity variations, the current approach has several limitations that suggest directions for further improvement. To begin with, SparseMVC demonstrates strong performance across a variety of multiview data, but its effectiveness in real-world applications with noisy or incomplete scenarios remains to be fully explored. Additionally, the use of contrastive learning inherently introduces computational overhead, making it unlikely to rank among the fastest available approaches. Moreover, the framework assumes well-aligned views, whereas slight inter-view misalignment may occur in real-world scenarios. + +A potential direction for future work is to incorporate structural information across views, which can be modeled through priors such as graph connectivity, inter-view relational graphs, or mutual information constraints—techniques that have been extensively explored in prior multi-view representation learning research. In contrast, the present study primarily focuses on feature-level sparsity and its adaptive regularization, rather than modeling explicit structural dependencies. 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To complement a single model's lack of diversity in generation and biases in evaluation, multiple LLMs form a "sparta tribe" to compete against each other in fulfilling instructions while serving as judges for the competition of others. For each iteration, one instruction and two models are selected for a duel, the other models evaluate the two responses, and their evaluation scores are aggregated through a adapted elo-ranking based reputation system, where winners/losers of combat gain/lose weight in evaluating others. The peer-evaluated combat results then become preference pairs where the winning response is preferred over the losing one, and all models learn from these preferences at the end of each iteration. SPARTA ALIGNMENT enables the self-evolution of multiple LLMs in an iterative and collective competition process. Extensive experiments demonstrate that SPARTA ALIGNMENT outperforms initial models and 4 self-alignment baselines across 10 out of 12 tasks and datasets with $7.0\%$ average improvement. Further analysis reveals that SPARTA ALIGNMENT generalizes more effectively to unseen tasks and leverages the expertise diversity of participating models to produce more logical, direct and informative outputs. $^{\dagger}$ + +# 1 Introduction + +Aligning large language models (LLMs) has become a critical step of LLM post-training, steering LLMs for specialized skills [25, 28], helpful responses [2], and alignment with human values [37, 10, 85]. Existing alignment recipes would require external signals to learn from (e.g., reward models [40, 60, 7] or preference pairs [81, 40, 71, 64]) while high-quality supervision is often scarce and costly [2, 68]. As such, recent research investigates self-alignment, where the LLM itself serves as its own judge [50], reward model [96, 14], or supervision signal [49, 81], without any external information. These approaches have demonstrated gains on instruction following and reasoning problems, initiating a paradigm of self-improving language models [105, 104, 61, 95, 9]. + +However, we posit that the recipe of "one model aligning itself" might not be reliable for two reasons: + +- A single model struggles to reliably judge its own generation. There is ample evidence on LLM self-bias [101, 31, 93], where one model consistently favors its own response and reinforces its own priors/biases in LLM-as-a-judge evaluations [103]. Impact of the biased self-judgment will be even more pronounced for tasks and instructions with underpinning human values and cultural implications [78, 67, 73], propagating and reinforcing the flaws. +- A single model struggles to reliably generate diverse responses to learn from. RL and preference learning recipes benefit from a wide spectrum of responses with varying quality and reward + +![](images/75bf34db0ee4e76e2a8690aa563c78e90060b775a36475baa646a4a45531ab27.jpg) +Figure 1: Overview of SPARTA ALIGNMENT, an algorithm collectively aligning multiple models via combat. SPARTA ALIGNMENT requires a dataset $\mathcal{X}$ and a pool of models $\mathcal{M}^t$ . For each iteration $t$ , we repeatedly sample a prompt $x$ from $\mathcal{X}$ . For $x$ , we first select a model $M_i^t$ and then select its opponent $M_{i'}^t$ with our Match-Making Strategy ( $i = 3, i' = 4$ in the example). We then employ three steps: (1) Combat: the model pair generate responses $y_i, y_{i'}$ to $x$ . (2) Judge: other models in the pool act as judges, generating their scores to $y_i, y_{i'}$ . (3) Update: update the reputation scores of $M_i^t$ and $M_{i'}^t$ based on scores from Judge phase and create a new preference for the preference dataset $\mathcal{P}$ . At the end of each iteration, we align models via preference learning on $\mathcal{P}$ . + +scores [102]. A single model, despite sampling, mostly generates homogeneous responses with similar styles and failure patterns [80, 105], struggling to provide the diversity and distinguishability of responses for robust preference learning. + +Therefore, the single model itself becomes the bottleneck for self-evolution beyond its training priors, existing strengths and weaknesses, as well as its inherent biases. + +To this end, we propose SPARTA ALIGNMENT, an algorithm to collectively align multiple language models. To complement each model's limitation in judging and generating, multiple LLMs form a "sparta tribe" to compete against each other [55, 98, 17] at fulfilling instructions and at the same time serve as judges for other LLMs. Specifically, SPARTA ALIGNMENT takes a pool of LMs $\mathcal{M}^0$ and a set of instructions $\mathcal{X}$ as input. For each iteration $t$ , we repeatedly sample an instruction $x \in \mathcal{X}$ and design a match-making system to select two models $M_i^t$ and $M_{i'}^t$ for "combat", i.e., dueling to better respond to $x$ . All the other models $\mathcal{M}^t \setminus \{M_i^t, M_{i'}^t\}$ then judge their responses and the aggregated judgment determines who won the combat, resulting in a preference pair such as $(x, M_i^t(x) \prec M_{i'}^t(x))$ . Motivated by game theory [19], the judgment aggregation is weighted by the reputation scores of models, where the winning models increase and the losing models decrease after combat. At the end of each iteration, all of $\mathcal{M}^t$ go through preference learning [76, 35] with the collection of preference pairs in this iteration. After iterations of SPARTA ALIGNMENT, a few models in $\mathcal{M}^t$ would emerge as the strongest with the highest reputation scores for inference and deployment. SPARTA ALIGNMENT uniquely enables the self-evolution [112] of LLMs in a collective competition process, alleviating the limitations of a single model. + +Extensive experiments on 12 datasets show that SPARTA ALIGNMENT consistently improves LLMs across 10 of the 12 tasks, outperforming the strongest baseline by $7\%$ on average. SPARTA ALIGNMENT achieves the highest generalization accuracy in 2 out of 3 categories on challenging mathematical benchmarks, demonstrating robust generalization [8] beyond $\mathcal{X}$ . SPARTA ALIGNMENT resembles the social stratification theory [62, 63, 75], where models in the Sparta tribe show different "classes" of reputation scores at the end of alignment. Further analysis reveals that the "sparta tribe" benefits from having more members and that SPARTA ALIGNMENT benefits from a diverse set of models with different expertise as $\mathcal{M}$ for improved generation diversity and more distinguishable preference pairs. + +# 2 Methodology + +Overview SPARTA ALIGNMENT assumes access to a pool of LLMs, denoted as $\mathcal{M}^0 = \{M_k^0\}_{k=1}^m$ where 0 denotes the 0-th iteration, and a dataset $\mathcal{X}$ . The models evolve iteratively for $\mathcal{T}$ iterations. In + +Algorithm 1: Sparta Alignment +1 Input: Initial models $\mathcal{M}^0 = \{M_i^0\}_{i=1}^m$ with attributes $R_i$ where $R_i$ is model's reputation dynamically updated by combat results, Dataset $\mathcal{X}$ ; Max iteration $\mathcal{T}$ . +2 for $t = 1$ to $\mathcal{T}$ do +3 Preference dataset $\mathcal{P} \gets \emptyset$ +4 for $x \in \mathcal{X}$ do +5 Select a model $M_i^t$ from $\mathcal{M}^t$ +6 Select its opponent $M_{i'}^t = \text{Match-Making}(M_i^t)$ +7 Combat: both models generate a response to $x$ , $M_i^t(x)$ and $M_{i'}^t(x)$ +8 for $M_k^t \in \mathcal{M}^t \setminus \{M_i^t, M_i'^t\}$ do +9 Evaluate both responses: $s_i^{(k)} \gets \mathcal{J}(x, M_i^t(x), M_k^t)$ , $s_{i'}^{(k)} \gets \mathcal{J}(x, M_{i'}^t(x), M_k^t)$ , where +10 $\mathcal{J}(x, M_i^t(x), M_k^t)$ denotes $M_k^t$ evaluating response $M_i^t(x)$ +11 end +12 if $\bar{s}_i > s_{i'}$ then new preference pair $\mathcal{P} \gets \mathcal{P} \cup \{x, M_i^t(x) \succ M_{i'}^t(x)\}$ +13 else new preference pair $\mathcal{P} \gets \mathcal{P} \cup \{x, M_{i'}^t(x) \succ M_i^t(x)\}$ +14 $R_i, R_{i'} \gets \text{Reputation-Update}(R_i, R_{i'}, \bar{s}_i, \bar{s}_{i'})$ +15 end +16 for $M_i^t \in \mathcal{M}^t$ do +17 $M_i^{t+1} \gets \text{DPO}(M_i^t, \mathcal{P})$ +18 end +19 end +20 Output: Better Models $\mathcal{M}^T = \{M_i^T\}_{i=1}^m$ + +iteration $t$ , we repeatedly sample $x \in \mathcal{X}$ , select two models $M_i^t$ , $M_{i'}^t$ for head-to-head combat via our Match-Making System, and generate responses $M_i^t(x)$ , $M_{i'}^t(x)$ . The other models $\mathcal{M}^t \setminus \{M_i^t, M_{i'}^t\}$ then act as judges and evaluate $M_i^t(x)$ and $M_{i'}^t(x)$ to produce scores. These scores are aggregated using a weighted scheme based on each judge's reputation $R_k$ , yielding a collective preference signal. The reputation system then updates the reputation $R_i$ and $R_{i'}$ by rewarding the winner and penalizing the loser, and we add the preference pair to the current iteration's preference dataset $\mathcal{P}$ . At the end of each iteration, the accumulated preference pairs in $\mathcal{P}$ are used to fine-tune all models via Direct Preference Optimization (DPO) [72], progressively aligning multiple language models over the iterations. We present an overview of SPARTA ALIGNMENT in Figure 1 and Algorithm 1, followed by detailed explanation of the three key components in the algorithm: Match-Making System, Judgment Aggregation, and Reputation System. + +Match-Making System Competitions between models of similar strength levels would help distinguishability and produce more meaningful preference signals [56]. To this end, SPARTA ALIGNMENT uses a probabilistic match-making strategy to select two LLMs to combat for better fulfilling an instruction. For a given data instance $x$ , the match-making process proceeds as follows: We begin by selecting the first model randomly, $M_{i}^{t}$ , in the model pair from the model pool $\mathcal{M}^t$ . The opponent model $M_{i'}^t$ is chosen partly based on reputation scores: with predefined probability $\alpha$ , $M_{i'}^t$ is randomly selected from the pool. Otherwise, with probability $1 - \alpha$ , $M_{i'}^t$ is selected from the top- $k$ models whose reputation scores are closest to that of $M_{i}^{t}$ . We denote this process as $M_{i'}^t = \text{Match-Making}(M_i^t)$ , ensuring a balance between exploring diverse models and exploiting those with similar reputation scores. + +Judgment Aggregation To obtain reliable and comprehensive preference signals, SPARTA ALIGNMENT aggregates evaluation scores from all remaining models in the pool. For each data instance $x$ , the selected models $M_{i}^{t}$ and $M_{i^{\prime}}^{t}$ generate responses $M_{i}^{t}(x)$ and $M_{i^{\prime}}^{t}(x)$ , respectively. These responses are then evaluated by all other models $M_{k}^{t} \in \mathcal{M}^{t} \setminus \{M_{i}^{t}, M_{i^{\prime}}^{t}\}$ , each acting as an independent judge and assign score $s_{i}^{(k)} = \mathcal{J}(x, M_{i}^{t}(x), M_{k}^{t})$ to $M_{i}(x)$ on a scale of ten, where $\mathcal{J}$ denotes LLM-as-a-judge. Then, the overall score $\bar{s}_{i}$ is calculated using reputation-weighted aggregation: + +$$ +\bar {s} _ {i} = \operatorname {A g g r e g a t e} \left(\left\{s _ {i} ^ {(k)} \right\}, \left\{R _ {k} \right\}\right) = \frac {\sum_ {k \in \mathcal {M} ^ {t} \setminus \left\{M _ {i} ^ {t} , M _ {i ^ {\prime}} ^ {t} \right\}} R _ {k} s _ {i} ^ {(k)}}{\sum_ {k \in \mathcal {M} ^ {t} \setminus \left\{M _ {i} ^ {t} , M _ {i ^ {\prime}} ^ {t} \right\}} R _ {k}} +$$ + +and the score $\bar{s_{i^{\prime}}}$ for $M_{i^{\prime}}^{t}(x)$ is calculated similarly. Specifically, $R_{k}$ is the reputation score of model $M_{k}^{t}$ in the pool, reflecting its perceived credibility among peers. Through reputation weighting, we aim for a more reliable judgment by aggregating the collective insights of all models. + +Reputation System SPARTA ALIGNMENT captures each model's strength from the perspective of its peers, enabling more accurate and stable reputation updates while reducing inconsistency in preference labels [92, 109]. + +At initialization, all models are assigned the same reputation score to ensure a fair starting point. During training, these scores are iteratively updated based on two key factors: the aggregated judgment by other models and each model's own reputation deviation, which measures the fluctuation of reputation over a sliding window. This mechanism enables the system to progressively distinguish models by their demonstrated performance while accounting for their recent reputation changes. Formally, the reputation $R_{i}$ is updated as follows: + +$$ +R _ {i} \leftarrow R _ {i} + \kappa \cdot \left(\bar {s} _ {i} - \bar {s} _ {i ^ {\prime}}\right) \cdot \tanh \left(\sigma_ {i}\right) \cdot \max \left(| \Phi (z _ {i}) - \Phi \left(z _ {i ^ {\prime}}\right) |, \epsilon\right). \tag {1} +$$ + +We denote the process as $R_{i}, R_{i^{\prime}} \gets$ Reputation-Update $(R_{i}, R_{i^{\prime}}, \bar{s}_{i}, \bar{s}_{i^{\prime}})$ and explain the three underlying principles below: + +- Larger Score Gap Amplifies Impact: The score gap $(\bar{s}_i - \bar{s}_{i'})$ reflects the collective judgment by peer models. A larger aggregated score gap, which implies greater distinction in model capability, leads to proportionally larger reputation update, where $\kappa$ is the scaling factor. +- Deviation-Guided Updates: The reputation score update is moderated based on the variance of the model reputation in recent iterations, which we measure by deviation $\sigma_{i}$ . Specifically, we consider the $N$ most recent iterations, where the reputation change in each iteration is $\delta_{t} = R_{i}^{(t + 1)} - R_{i}^{(t)}$ , for $t = 1,\dots ,N$ . Then the deviation is calculated as: + +$$ +\sigma_ {i} \leftarrow \max \left(\sqrt {\frac {1}{N - 1} \sum_ {t = 1} ^ {N} (\delta_ {t} - \bar {\delta}) ^ {2}}, \sigma_ {\min }\right), \quad \text {w h e r e} \bar {\delta} = \frac {1}{N} \sum_ {t = 1} ^ {N} \delta_ {t} +$$ + +This mechanism ensures higher deviations for unstable model reputations, allowing faster adjustments, while stable models are updated more conservatively. The lower bound $\sigma_{\mathrm{min}}$ prevents degeneration of the reputation updates. + +The scaling term $\tanh (\sigma_i)$ in Equation 1 adjusts the update magnitude based on the its stability of reputation scores. For stable models (small $\sigma_{i}$ ), updates are small to avoid overreaction, while for unstable ones (large $\sigma_{i}$ ), updates are larger for faster convergence. The $\tanh (\cdot)$ function naturally caps the scaling within $(0,1)$ , preventing excessive updates while providing a smooth and adaptive transition between stable and unstable scenarios. + +- Greater Gains for Defeating Stronger Opponents: Motivated by game theory [19], we employ the term $\max \left(|\Phi (z_i) - \Phi (z_{i'})|, \epsilon\right)$ to capture the relative strength between competing models. + +Here, $z_{i} = \frac{R_{i} - R_{i^{\prime}}}{\sqrt{\sigma_{i}^{2} + \sigma_{i^{\prime}}^{2}}}$ measures the reputation difference normalized by their deviations, and $\Phi(\cdot)$ + +denotes the standard normal cumulative distribution function (CDF), which is used to estimate the model's most probable win rate based on its current reputation and deviation. This factor grants larger reputation gain for defeating stronger opponents, rewarding unexpected victories, and vice versa. The lower bound $\epsilon$ prevents the update from vanishing when the reputation difference is small, thereby avoiding stagnation and ensuring active reputation updates even in near-tie scenarios. + +We provide further details about an optional reputation reweighting mechanism in Appendix C.1. + +# 3 Experiment Settings + +Models and Implementation In the main experiments, we initialize the model pool $\mathcal{M}^0$ as 10 Qwen2.5-7B-Instruct models [70] independently fine-tuned on different data domains in Tulu-v2 [41] (details in Appendix C.3), resulting in a pool of models with diverse skills and expertise for collective alignment. + +For SPARTA ALIGNMENT, we set the number of prompts per iteration to 1000, number of iterations $\mathcal{T} = 8$ , $\alpha = 0.6$ , a top-k threshold of $k = 5$ , and $\kappa = 1$ . At the end of each iteration, all models are + +fine-tuned via DPO [72] for 1 epoch with a starting learning rate of $1e - 6$ and an effective batch size of 1, with the same LoRA configuration in SFT phase in Appendix C.3. Experiments are performed on a cluster with 16 A100 GPUs with 40 GB memory. + +Baselines We compare SPARTA ALIGNMENT with the best-performing initial models as well as four self-alignment algorithms. + +- Best Initial Model (BEST INIT), defined as $\arg \max_{M_i^0\in \mathcal{M}^0}f(M_i^0)$ , where $f(\cdot)$ represents the model's performance on the validation set of a specific task. +- Self-Rewarding (SELF-REWARD) [104], where the model assigns rewards to its own generation via LLM-as-a-judge prompting and guides its own DPO training with self-generated preference data. +- Self-Meta-Rewarding (META-REWARD) [94], extends self-rewarding by having the model also evaluate its own reward judgments, leading to progressive improvement of both its response and reward generation abilities. +- SPIN [9], a self-play-based fine-tuning method that iteratively aligns a weak LLM with the target data distribution by training it to distinguish its own previous generations from ground truth, enabling self-improvement without extra human annotation in later iterations. Notably, SPIN assumes access to the gold labels of training data, which is not required in SPARTA ALIGNMENT. +- SPPO [95], where the model aligns itself through self-play, approximating the Nash equilibrium via iterative policy updates. Unlike purely unsupervised self-play, SPPO requires an external reward model to perform annotation and generate win-rate signals, guiding policy refinement towards better distinguishing preferred from rejected responses. + +Note that SPARTA maintains a comparable or lower inference cost during training than other self-alignment methods, despite using more models, by processing each instruction only twice per iteration. + +Dataset We evaluate SPARTA ALIGNMENT across 8 tasks and 12 datasets spanning three evaluation domains: (1) Domain-Specific Question Answering, including MedQA-US (MedQA) [46] and Normad [73]. Normad is a benchmark assessing LLMs' cultural adaptability, comprising three subsets: country-based (Country), country-value-based (Value), and rule-of-thumb-based (RoT); (2) Reasoning, covering GSM8K [12], Knowledge Crosswords (KCross) [16], $\mathrm{COM^2}$ [22], and MATH [36], where MATH is further divided into three difficulty levels: Easy (level 1 & 2), Medium (level 3), and Hard (level 4 & 5); (3) Instruction Following and Safety, evaluated on Alpaca [18] for instruction-following and TruthfulQA (Truthful) [59] to for safety assessment. We report dataset statistics in Table 4 in the appendix. + +Evaluation We adopt task-specific evaluation metrics to assess model performance across diverse benchmarks. For tasks with discrete answer choices (Normad and KCross) and mathematical benchmarks (GSM8K, MATH), we report pass@1 accuracy. For open-ended generation tasks (Alpaca and $\mathrm{COM}^2$ ), we use LLM-as-a-Judge [107, 4, 18] approach and ask GEMINI-1.5-FLASH [34] to assign score to model response $M(x)$ based on the task prompt $x$ and corresponding reference answer, on a 0-10 scale using a zero-shot prompting strategy. For TruthfulQA, we follow the standard log-probability-based evaluation protocol, measuring whether the model assigns higher likelihood to truthful completions over false ones. Notably, we employ multi-faceted metrics in Appendix C.5 on MedQA to better align with real-world application requirements. + +We execute SPARTA ALIGNMENT in an unsupervised way over some seed instructions, then select the model in the pool with the highest performance on the validation set for each dataset, and report its performance on the held-out test set. + +# 4 Results + +We present the performance of SPARTA and baselines on 12 dataset settings in Table 1. + +SPARTA achieves state-of-the-art results in 10 out of 12 datasets. For reasoning and instruction-following tasks, SPARTA ALIGNMENT outperforms the second-best baselines by $7\%$ on average. + +
MethodMedQANormadKCrossGSM8KCOM2MATHAlpacaTruthful
CountryValueRoTEasyMediumHard
BEST INIT.599.688.681.700.550.7785.27.516.389.1995.36.410
SELF-REWARD.623.699.692.707.555.7775.74.513.376.1885.56.416
META-REWARD.618.692.680.700.550.7795.47.503.385.2025.49.413
SPIN.616.684.680.704.580.7825.58.516.369.2045.49.420
SPPO.601.688.696.704.545.7855.55.504.369.2105.56.421
SPARTA (ours).662*.688.715.707.560.813*6.35*.530.396.2127.12*.424
+ +Table 1: Performance of baselines and SPARTA ALIGNMENT across 12 datasets. Best results are in bold, second-best are underlined. Different-task results of SPARTA ALIGNMENT with * are significantly better than best baselines at the level of 0.05. + +These results highlight the effectiveness of SPARTA ALIGNMENT across a broad spectrum of tasks. We further dissect performance across evaluation categories. + +Domain-specific QA On the cultural adaptation benchmark Normad-Value subset, SPARTA ALIGNMENT improves performance by $2\%$ compared to the best previous method SELF-REWARD, highlighting the effectiveness of our collective alignment strategy. On MedQA, SPARTA ALIGNMENT achieves the best performance in both clinical informativeness and logical consistency in Appendix C.5. This indicates that SPARTA ALIGNMENT promotes greater response diversity compared to self-alignment methods, enabling models to acquire broader knowledge, as well as more logical consistency across diverse domain-specific contexts. We present a detailed analysis of response diversity in Section 5 and of preference diversity in Appendix C.8. + +Reasoning Across knowledge, mathematical and commonsense reasoning tasks, SPARTA consistently delivers the strongest results. On GSM8K, SPARTA ALIGNMENT improves by $4.5\%$ over the initialized model and $3.6\%$ over the best baseline SPPO. On MATH (all levels), it shows consistent gains, averaging $4.0\%$ over the initialized model and $1.9\%$ over prior best. Furthermore, on $\mathrm{COM}^2$ , SPARTA ALIGNMENT achieves $20.5\%$ improvement over initialization and $9.6\%$ over the strongest baseline SELF-REWARD. These findings suggest that stochastic pairing effectively enhances multi-step reasoning and alignment across both arithmetic and commonsense domains. + +SPARTA ALIGNMENT provides more accurate reward signals by leveraging cross-model judgment, overcoming limitations common in self-alignment methods where self-feedback can amplify bias and overconfidence [101, 31]. Additionally, we attribute the limited gains of SPARTA ALIGNMENT on KCross to the small scale of the dataset (presented in Table 4), where models may struggle to demonstrate robust generalization in the structural knowledge reasoning task with limited data. Nevertheless, these results indicate that with more unsupervised data available the alignment approach in SPARTA ALIGNMENT will continue to improve with more diverse and extensive data. + +Instruction Following and Truthfulness On the Alpaca benchmark for instruction following, SPARTA ALIGNMENT outperforms the best initial model by $32.8\%$ and SELF-REWARD by $28.1\%$ . Similarly, on TruthfulQA, SPARTA (0.424) improves over the best initial model (0.410) and the previous best baseline from SPPO (0.421). + +The weaker performance of SELF-REWARD and similar baselines arises from the inability of single-model feedback to fully capture the nuanced human-defined preferences and principles in instruction-following and safety tasks, while SPARTA ALIGNMENT leverages multiple models to better encompass the range of human preferences. + +Surprisingly, in SPARTA ALIGNMENT, a model that initially underperforms has the potential to become the strongest in the model pool through iterative collaborative alignment. This phenomenon underscores the effectiveness of collective learning, where weaker models are not only retained but actively improved through diverse interactions with stronger peers. We present details in Table 6 + +# 5 Analysis + +Generalization We assess SPARTA ALIGNMENT's generalization capability using MATH [36]--a benchmark designed to evaluate LLM's reasoning ability across different levels of mathematical complexity, including Easy, Medium, and Hard [86]. + +As shown in Figure 2, we evaluate SPARTA ALIGNMENT by fine-tuning a pool of models on each subset individually and testing their performance on the other two subsets with different difficulty levels. Results in Figure 2 and Table ?? show that SPARTA ALIGNMENT consistently outperforms all baselines when trained on Medium and Hard subsets, demonstrating SPARTA ALIGNMENT's superior capability in generalizing across mathematical complexities. + +![](images/9c4cb482fe6f8046b8b95af5903646b27dcd644b2908e9cc6045c3c0a9fbae97.jpg) +Figure 2: Cross-subset generalization accuracy on the MATH benchmark. Each group of bars corresponds to a training subset (Easy, Medium, Hard), with OOD performance measured on the two held-out subsets. + +Scaling the Model Pool SPARTA ALIGNMENT utilizes a larger pool of models to encourage diverse outputs and enable cross-model judgment. This naturally raises the question: How does the number of models affect the quality of the best aligned model in the system? To investigate this, we conduct a series of experiments (details in Appendix C.7). As shown in Figure 3, increasing the number of models in the alignment pool-by selecting a larger k for the top-k models from an initial set of 10-consistently improves performance across all four benchmarks. Compared to the 3-model setting, the 10-model pool yields a performance increase of $19.1\%$ on Alpaca, $33.1\%$ on $\mathrm{COM^2}$ and $6.1\%$ on MedQA. These results suggest a positive benefit in larger model pool sizes, where more models provide richer diversity and stronger supervision, leading to better aligned models. + +Generation Diversity To test the hypothesis that multi-LLM collaboration produces more diverse outputs than a single model—thereby offering richer and more robust learning data—we conduct controlled experiments comparing models trained with SELF-REWARD and META-REWARD versus SPARTA ALIGNMENT. To align with previous experiment settings, models in SELF-REWARD and META-REWARD. We evaluate the first and final models from SELF-REWARD and META-REWARD and the corresponding models from SPARTA ALIGNMENT. To align with our main experiments, in SELF-REWARD and META-REWARD, we use a single model to generate 10 responses to a prompt, while in SPARTA ALIGNMENT we use 10 models to output separately. Response pairwise di + +![](images/5763680c707b44a455c9ae7b5ddf1cf7d7a4af0473b00ff096c29cfa7694ba7f.jpg) +Figure 3: Effect of pool size on alignment performance. We vary the number of candidate models participating in each training round and measure the final performance. Results show that larger pools lead to better outcomes, indicating that SPARTA ALIGNMENT benefits from having more diverse LLMs as participants. + +
MethodEditBLEUEmbedLLM
Baselines (1 Model)
SELF-REWARD (INIT)0.6670.8590.1295.73
SELF-REWARD (FINAL)0.6730.8670.1345.90
META-REWARD (INIT)0.6530.8760.1205.99
META-REWARD (FINAL)0.6550.8780.1225.93
Sparta (10 Models)
SPARTA (INIT)0.7320.9460.1786.12
SPARTA (FINAL)0.7220.9180.1426.42
+ +Table 2: Output diversity across responses after the first iteration and the final iteration on the instruction-following task, using automatic and LLM-based metrics. SPARTA employs 10 models compared to 1 model for baselines. SPARTA outperforms the two baselines across all dimensions. + +versity is assessed using four complementary metrics in Appendix C.8, comprising two traditional metrics and two LLM-based metrics. As shown in Table 2, SPARTA ALIGNMENT outperforms SELF-REWARD and META-REWARD across all metrics, confirming that multi-model collaboration introduces greater output diversity across lexical, structural, and semantic dimensions, further supporting more effective alignment. Figure 6 shows a trade-off in our system that the output diversity decreases as the iteration processes, while the best performance increases. With MedQA [46] as an example, we analyze the underlying mechanism transferring generation diversity to improvement [48]. + +Table 7 and Appendix 5 reveal that, compared to SELF-REWARD, preference pairs in SPARTA ALIGNMENT exhibit greater diversity between the chosen and rejected samples from the angles of semantic similarity and length. This increased variance facilitates more effective learning of + +latent patterns through preference optimization [71, 66, 80]. Furthermore, the selected samples are consistently more complete, clinically informative, direct, and logically structured than their rejected counterparts, contributing to enhanced model alignment and robustness. + +Model Diversity Matters In addition to pool size, we ask: How does the diversity of the model pool affect the final performance? + +To answer this, we conduct experiments on three representative datasets—Alpaca, COM $^2$ and MedQA. For each dataset, we vary the diversity of the initial model pool by sampling $a$ distinct LLMs, each repeated $b$ times, and use the resulting $a \times b$ pool as the starting point for SPARTA ALIGNMENT. For fairness, we always keep the best initial model in the pool. In Figure 4, we report results for configurations $1 \times 10, 2 \times 5, 5 \times 2$ , and $10 \times 1$ . As shown in Figure 4, performance consistently improves with greater model diversity. Notably, the fully diverse setting $10 \times 1$ outperforms the fully repetitive setting $1 \times 10$ by $18.5\%$ averaged across the three datasets. These results provide strong + +empirical evidence that Model Diversity Matters, that SPARTA benefits from multiple models with heterogeneous skills complementing each other in the collective alignment process. + +![](images/2f905c6aa43fda88ea81099cdffe172187644fd922f2b1965d8518fdee67433f.jpg) +Figure 4: Impact of model pool diversity on alignment performance. The x-axis shows the configurations of model pool diversity: $1 \times 10, 2 \times 5, 5 \times 2$ and $10 \times 1$ . The results demonstrate that increasing the diversity of the model pool consistently improves performance. + +Correlation between Reputation and Performance To investigate how well the peer-derived reputation reflects actual model performance, we analyze the correlation between models' average performance and their corresponding reputation scores over iterations under SPARTA ALIGNMENT. Figure 5 reveals a positive correlation between reputation scores and performance, which suggests that the reputation mechanism effectively captures meaningful signals about model alignment, providing empirical validation for our approach. We present Figure 12 to provide a comprehensive view of the correlation across all evaluated datasets, reporting an average correlation coefficient of 0.21 across all tasks. Furthermore, Figure 13 shows that SPARTA ALIGNMENT induces a hierarchical stratification among the 10 models, hinting at potential top-down influence. + +Ablation Study To evaluate the importance of individual components in SPARTA ALIGNMENT, we conduct an ablation study on four datasets: Alpaca, COM $^2$ , MedQA, and GSM8K. As shown in Table 3, removing any single component leads to consistent performance drop, confirming that each design choice contributes critically to the system's overall effectiveness. + +The removal of randomness in matchmaking (w/o randomness in match) degrades perfor + +mance most severely on Alpaca and $\mathrm{COM}^2$ , suggesting that stochastic opponent selection encourages behavioral diversity and improves generalization in instruction-following settings. Finally, while omitting the top- $k$ matching constraint ( $w / o$ top- $k$ match) yields marginal improvements on certain + +![](images/f8a223ecb186d255c84a813cae1e309ddcfa643104b59b18312ca15d62a69b70.jpg) +Figure 5: Correlation between a model's average performance on a specific task and its average reputation in the model pool. The 10 points in each subplot indicate 10 models. $r$ stands for Pearson correlation coefficient. + +
MethodAlpacaCOM2MedQAGSM8K
SPARTA7.126.350.6620.813
- w/o reputation updates6.175.490.6470.765
- w/o downweighting6.636.100.6510.781
- w/o randomness in match6.045.730.6540.779
- w/o top-k match6.745.960.6590.804
+ +Table 3: Ablation of SPARTA ALIGNMENT across four datasets. All components jointly contribute to the best performance across all tasks. + +datasets, it compromises stability on others, indicating that constrained pairing is crucial for balancing exploration and reliable supervision. These results demonstrate that the strong performance of SPARTA ALIGNMENT arises from the synergy of all components, and that omitting any part leads to measurable and often substantial regressions. + +# 6 Related Work + +Reinforcement Learning from AI Feedback (RLAIF) Aligning large language models (LLMs) with human preferences has been a cornerstone of their success [54, 111, 68]. Traditional methods like RLHF involve collecting human-generated preference labels with reinforcement learning to distinguish responses [76, 71], which can be costly and time-consuming [68, 2]. This limitation has spurred interest in alternative approaches, such as RLAIF, where AI-generated feedback is used to train reward models [68, 1, 104, 32, 76, 44, 20]. Constitutional AI [3] uses a set of predefined principles to guide an LLM in critiquing and revising its responses to harmful queries. These revisions are then used to fine-tune the AI model through reinforcement learning with AI-generated feedback. Self-Rewarding [104] leverages an initial model to generate multiple responses, rank them using LLM-as-a-Judge [107, 4, 18], and perform preference learning. Meta-Rewarding [94] allows the model to assess and iteratively refine its own responses and reward generation. Other self-alignment methods [95, 9, 39, 105, 9, 35, 110, 69] either depend on ground truth or require signals from a larger, more capable model. Despite their potential, previous work suggests that self-bias from self-feedback [101, 31] is a key bottleneck preventing further self-improvement. Additionally, the decreasing diversity resulting from alignment causes the distinction between positive and negative responses to become increasingly blurred [48, 91, 97, 89, 80], leading to preference inconsistency during self-alignment [92, 77, 109], which in turn negatively impacts alignment effectiveness. In contrast, SPARTA ALIGNMENT mitigates these effects by introducing multiple diverse LLMs to generate responses and cross-judge them. + +Multi-LLM Collaboration While pushing a single general-purpose LLM to achieve new benchmarks is valuable, an increasing body of research seeks to advance beyond a single model and explore model collaboration [29, 27, 84]. This is due to three main limitations: (1) A single LLM underrepresents real-world linguistic diversity, evolving trends, and domain-specific knowledge [52, 15, 47]; (2) No single LLM excels across all tasks, highlighting the need for model specialization [82, 21, 88]; (3) A single LLM fails to represent the diverse needs, values, and socio-cultural backgrounds of its users [83, 27, 53, 74, 24]. + +Therefore, researchers are probing the potential of multi-LLM collaboration [29], by allowing LLMs to evaluate and reflect on each other's outputs [13], offers a promising approach to improving factual accuracy [45, 65, 23] and reducing hallucinations [26, 30], addressing the limitations of single-model self-reflection [90, 100, 79, 61] and confirmation biases [43]. Prior studies [98, 58, 17] suggest that multi-LLM systems can improve their factuality through multi-turn debates [17]. Furthermore, multi-LLM collaboration, through modular systems [27], multi-LLM as a judge [106] and diverse reward models [42, 87], offers a solution to address the cultural, political, and social biases inherent in LLMs, promoting fairness and pluralism [27, 33] in alignment. SPARTA ALIGNMENT leverages multi-LLM collaboration and competition and goes beyond self-alignment, outperforming single-model methods on instruction-following [18], reasoning [12, 5, 16, 22], cultural adaptation tasks [73], and more. + +# 7 Conclusion + +We propose SPARTA ALIGNMENT, a game-theoretic algorithm to collectively improve multiple LLMs through competition. Starting with a pool of diverse models and a dataset, we iteratively select pairs of models to combat by generating responses to sampled prompts from the dataset. The remaining models in the pool act as judges to evaluate the responses. A reliable reward signal is obtained by aggregating their judgments, weighted by each judge's reputation. Winners of each combat see their reputation increase, while losers experience a corresponding decrease. Extensive experiments show that SPARTA ALIGNMENT outperforms the best baseline by up to $28.1\%$ , with an average performance increase of $7\%$ across 10 tasks spanning several domains. Further analysis reveals that SPARTA ALIGNMENT better generalizes to unseen data settings, model reputation scores are positively correlated with task performance, and that SPARTA ALIGNMENT benefits from a larger and diverse pool of participants. + +# Acknowledgments + +This research was developed with funding from the Defense Advanced Research Projects Agency's (DARPA) SciFy program (Agreement No. HR00112520300). The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. This material is based upon work supported by the Defense Advanced Research Projects Agency and the Air Force Research Laboratory, contract number(s): FA8650-23-C-7316. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of AFRL or DARPA. We gratefully acknowledge gift and compute funding from Amazon and Google. Shangbin Feng would like to thank the support of the IBM PhD Fellowship and the Jane Street Graduate Research Fellowship. + +# References + +[1] Usman Anwar, Abulhair Saparov, Javier Rando, Daniel Paleka, Miles Turpin, Peter Hase, Ekdeep Singh Lubana, Erik Jenner, Stephen Casper, Oliver Sourbut, et al. Foundational challenges in assuring alignment and safety of large language models. arXiv preprint arXiv:2404.09932, 2024. +[2] Yuntao Bai, Andy Jones, Kamal Ndousse, Amanda Askell, Anna Chen, Nova DasSarma, Dawn Drain, Stanislav Fort, Deep Ganguli, Tom Henighan, et al. 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A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: Yes! Our paper discusses our limitations in Section A. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: LLMs were not used in the writing of this paper. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Limitations + +SPARTA ALIGNMENT selects models for diversity probing experiments based on their initial performance ranking on the validation set. While the optimal combination for maximizing diversity remains unclear, it represents a promising area for future exploration. + +Model collaboration in SPARTA ALIGNMENT requires greater computational resources compared to single models, yet it enables substantial weak-to-strong generalization through synchronized learning, effectively bridging performance gaps across tasks. By selecting the best-performing checkpoint from a pool of 80 based on validation performance, we demonstrate that even initially weaker models can evolve into top performers, underscoring the transformative impact of collective alignment. + +Despite SPARTA ALIGNMENT's strong performance in reasoning and instruction-following tasks, model diversity diminishes as iterations progress, indicating that improvements may not be infinite. Preserving diversity is crucial for sustained performance gains. + +The scaling law persists in SPARTA ALIGNMENT, showing that larger datasets enable models to better capture latent patterns, underscoring the importance of scale for improved learning and generalization. + +# B Ethics Statement + +The implementation of SPARTA ALIGNMENT raises several ethical considerations, particularly regarding bias amplification and fairness in collaborative learning. If one of the models in the pool is biased—whether in terms of gender, race, or other sensitive attributes—there is a risk that this bias could be propagated or even amplified through the collective alignment process. This is especially concerning if models with inherent biases gain higher reputation scores due to overfitting to specific patterns or datasets, leading to skewed decision-making. + +# C Technical Appendices + +# C.1 Reweighting Mechanism + +To maintain model diversity while preventing 'unreliable' models from affecting the overall judgment so that keeping the preference labels consistent [92, 109], we introduce an important attribute: judging weight $w_{k}$ for $M_{k}^{t}$ , $M_{k}^{t} \notin \{M_{i}^{t}, M_{i'}^{t}\}$ and its corresponding reweighting mechanism. All models are initialized with $\omega_{k} = 1$ . Starting from iteration $t = 2$ , as training progresses, exactly one additional low-ranked model is down-weighted at each iteration. Specifically, at iteration $t$ , the judge with the $(t - 1)$ -th lowest reputation is unlocked and assigned: + +$$ +\omega_ {k} = \gamma \times (t - 2), +$$ + +where $\gamma = 0.1$ is a scaling factor. Crucially, once a model is assigned a reduced weight, this weight remains fixed for the remainder of training, regardless of changes in its rank. This progressive and irreversible reweighting penalizes persistent underperformance early while steadily expanding judging participation as alignment stabilizes. + +Thus, the updated reputation $R_{k}$ for each judge model $M_{k}^{t}$ : + +$$ +R _ {k} \leftarrow \omega_ {k} \times R _ {k} +$$ + +# C.2 Experiment Dataset Statistics + +The statistics for the datasets used in our main experiments are summarized in Table 4. The table presents the number of training, validation, and test examples for each of the 12 distinct tasks. Each dataset is split into three parts: Train, Validation, and Test, where the validation set is constructed by splitting the original test set evenly, ensuring balanced evaluation during development and final testing. + +For instance, the Alpaca dataset comprises 52,002 training examples, along with 402 validation examples and 403 test examples. Similarly, the MedQA dataset contains 10,178 training samples, with 636 for validation and 637 for testing. + +
SplitAlpacaMedQACOM2NormadMATHGSM8KKC
CountryValueRoTEasyMediumHard
TRAIN52,00210,1787,2002,3702,3702,3701,9121,5922,9947,4731,200
VAL4026364001321321326665661,269660200
TEST4036374001321321326655651,269659200
+ +Table 4: Dataset statistics for 11 tasks in our main experiments. Validation set is created as half of the original test set, with the remaining portion reserved for final evaluation. + +The Normad dataset is divided evenly across three sub-domains: Country, Value, and Rule-of-Thumb (RoT), each containing 2,370 samples in the training set and 132 samples in both validation and test sets. This balanced distribution allows for granular analysis of performance across cultural dimensions. + +For the mathematical benchmarks, the MATH dataset is split into three difficulty levels—Easy, Medium, and Hard. The training set includes 1,912, 1,592, and 2,994 samples for each level, respectively. In the validation and test splits, the three levels are proportionally represented, with 666, 566, and 1,269 in the validation set, and 665, 565, and 1,269 in the test set. + +Finally, the GSM8K and KC datasets contain 7,473 and 1,200 training samples, respectively. Their validation and test sets are split as 660/659 for GSM8K and 200/200 for KC, respectively. + +This structured distribution ensures that each dataset is adequately represented during both development and evaluation phases, providing a fair and consistent basis for assessing model generalization and task-specific performance. + +# C.3 SFT Details + +We utilize Tulu-v2 [41], an open collection of instruction-tuning data, focusing specifically on the following subsets: Flan [11], CoT, Open Assistant 1 [51], ShareGPT $\ddagger$ , Code Alpaca [6], LIMA [108], WizardLM Evol-Instruct V2 [99], Open-Orca [57], and Science Literature [41]. We replace the GPT4 Alpaca subset with Gemini Alpaca, which consists of distilled generations from gemini-1.5-pro-001, and remove the hard-coded subset. Fine-tuning is performed with LoRA [38], employing a learning rate of 2e-4, cosine learning rate scheduling, an effective batch size of 32, a warm-up ratio of 0.1, and 5 default training epochs. For the larger ShareGPT subset, we reduce training to a single epoch to optimize efficiency. + +# C.4 Baseline Experiment Settings + +Inference is conducted with the same inference time and training configurations—including optimization hyperparameters, number of training iterations, and number of sampling prompts—are kept identical across all methods, including SPARTA ALIGNMENT. This controlled setup isolates the impact of the alignment algorithm, removing potential confounding factors such as model size or training regimen. + +# C.5 Experiment Details + +In the Combat and Judge phases of SPARTA ALIGNMENT, we employ a standardized chat template to generate both responses and judgments. During the DPO training stage, models are trained with the chat template to ensure consistency and alignment. For evaluation, task-specific adjustments are made: Alpaca, $\mathrm{COM}^2$ , MATH, and GSM8K and MedQA are assessed without the chat template due to the nature of their task structures, while Normad, KCross and Truthful is evaluated with the chat template. Notably, the evaluation methodology remains consistent across the initial model and all baselines. + +Evaluation Metrics in MedQA In the MedQA evaluation, we employ a multi-faceted assessment strategy with to thoroughly measure the quality of model-generated medical responses. The evaluation is structured around three core metrics: + +1. Clinical Informativeness: This metric gauges the depth and richness of medical information provided. A highly informative response should capture critical clinical knowledge, present precise medical terminology, and effectively convey important concepts relevant to the question. +2. Logical Coherence: This metric assesses the internal consistency and structured reasoning of the response. A logically coherent answer is expected to be well-organized, maintain a clear flow of ideas, and demonstrate sound clinical reasoning without contradictions. +3. Comprehensiveness: This metric measures the completeness of the response, including coverage of differential diagnoses, treatment options, and underlying mechanisms. It ensures that all aspects of the query are adequately addressed with appropriate detail and medical context. + +To ensure a consistent and objective evaluation process, we apply LLM-as-a-judge scoring mechanisms for all three metrics. Below are the specific evaluation prompts used for each criterion. + +# Clinical Informativeness Prompt + +Rate the clinical informativeness of the following medical response on a scale of 0 to 10 (10 being most informative). Consider the medical information intensity. + +Question: {prompt} + +Response: {response} + +Please output in the format: {‘score’: score} + +# Logical Coherence Prompt + +Rate the logical coherence of the following medical response on a scale of 0 to 10 (10 being most logical). + +When evaluating, consider the following aspects: + +- evaluating, consider the following aspects: + - Structure: Is the response well-organized and easy to follow? + - Flow: Are the ideas presented in a logical sequence, with clear connection points? + +- Structure: Is the response well-organized and easy to follow? +- Flow: Are the ideas presented in a logical sequence, with clear connections between points? + +- Reasoning: Is the reasoning sound and aligned with medical principles? + +Question: {prompt} + +Response: {response} + +Please output in the format: {‘score’: score} + +# Comprehensiveness Prompt + +Rate the comprehensiveness of the following medical response on a scale of 0 to 10 (10 being the most comprehensive). + +When evaluating, consider the following aspects: + +- Completeness: Does the response address all relevant aspects of the medical query? +- Detail: Are the explanations sufficiently detailed to convey a clear understanding of the concepts involved? +- Coverage: Does the response encompass the full scope of the question, including differential diagnoses, treatment options, and underlying mechanisms where appropriate? + +Question: {prompt} + +Response: {response} + +Please output in the format: {‘score’: score} + +
StageCOM2GSM8KNormad
ValueRoT
INIT BEST MODELcode_alpacagemini_alpacawizardlmwizardlm
LAST BEST MODELflan_v2oasst1oasst1oasst1
+ +Table 6: Comparison of the best-performing models between the initial and final iterations. Only the tasks with changes are shown. Red indicates the initial best model and blue indicates the final best model. + +Results on MedQA Table 5 presents the evaluation results on the MedQA benchmark, measured across three key metrics other than pass@1 accuracy: Comprehensiveness (Comp.), Informativeness (Info.), and Logical Consistency (Logi.). These metrics provide a holistic assessment of the model's capacity to generate accurate, complete, and well-structured medical responses. + +SPARTA ALIGNMENT achieves the highest scores in both Informativeness and Logical Consistency, outperforming all baselines. This result + +indicates that SPARTA ALIGNMENT not only produces responses rich in medical content but also maintains clear logical flow, which is crucial for clinical reasoning in realistic scenario. + +Table 5: Evaluation of MedQA Responses with Multi-Faceted Metrics. + +For Comprehensiveness, SPIN secures the highest score of 6.75, with SPARTA ALIGNMENT closely following at 6.73. This highlights SPARTA ALIGNMENT's ability to cover critical aspects of medical queries with substantial detail, ensuring that responses are both thorough and contextually rich. + +Overall, SPARTA ALIGNMENT excels in generating informative and logically sound medical responses, setting a new benchmark for clinical AI systems in terms of depth, clarity, and reasoning. These strengths position SPARTA ALIGNMENT as a highly competitive model for real-world medical QA applications. + +# C.6 Details in Generalization + +We evaluate SPARTA ALIGNMENT's generalization capability using the MATH benchmark [36], which assesses LLMs on their reasoning ability across varying levels of mathematical complexity. The MATH benchmark is organized into three difficulty levels: (1) Easy, (2) Medium, and (3) Hard, representing progressively challenging problem sets aimed at testing the model's understanding and problem-solving depth [86]. We fine-tune a pool of models on each subset individually and measure their performance on the two held-out subsets, effectively evaluating cross-difficulty generalization. + +The experiment can be divided into two phases: the training phase and the testing phase. In the training phase, models are trained on a specific subset of mathematical problems in the MATH benchmark using SPARTA ALIGNMENT; in the testing phase, these models are evaluated for their performance on the other two subsets of different difficulty levels. + +SPARTA ALIGNMENT ensures that the model adjusts its problem-solving strategies by leveraging a more comprehensive understanding of both fundamental mathematical concepts and advanced reasoning principles. This alignment process enables the model to generalize effectively to unseen problem sets, improving its accuracy and adaptability across varying levels of complexity. + +The results indicate that after applying Sparta Alignment, the model's performance improves across basic concepts, intermediate reasoning, and advanced problem-solving contexts, leading to more accurate and contextually consistent solutions. + +# C.7 Details in Scaling Law of Model Pool + +To ensure fair comparisons and preserve meaningful quality gaps for effective alignment training, we first evaluate all candidate models on validation set of one specific task. Based on their test performance rankings, we construct model pools with the largest internal performance spread—specifically selecting the top 3, 5, and 7 models with maximally divergent test performance. For example, the 3-model setting includes models ranked 1st, 5th, and 10th, providing sufficient variance to support informative comparative feedback. We perform this analysis across three representative benchmarks—Alpaca, $\mathrm{COM}^2$ and MedQA—to give a conclusion across both general instruction-following and domain-specific tasks. + +To enable a unified comparison across tasks with different scoring scales, we normalize the ALPACA and $\mathrm{COM}^2$ scores by dividing them by 10. + +# C.8 Further Analysis of Generation Diversity + +Details of Diversity between the chosen and rejected one We evaluate the diversity between the chosen and the rejected ones for SELFREWARD and SPARTA across multiple entropy-based metrics and semantic similarity measures. Specifically, we measure Char Entropy, Word Entropy, 2-Gram Entropy, 3-Gram Entropy, POS Entropy, and Semantic Similarity. + +
MethodChar Ent.Word Ent.2-Gram3-GramPOS.Semantic Sim.
SELF-REWARD0.0290.2200.2770.3270.1000.925
SPARTA0.0660.1750.1860.1970.0790.710
+ +Table 7: Diversity and similarity metrics comparison between SELF-REWARD and SPARTA on entropy-based and semantic similarity measures. SPARTA achieves higher character-level entropy and lower semantic similarity. + +Table 7 reveals some conclusions. Firstly, the preference pairs in SPARTA exhibit substantially lower semantic similarity compared to those in SELF-REWARD. This indicates larger representational gaps, highlighting opportunities for preference optimization to better approximate the latent target data distribution. + +Secondly, the differences in Character Entropy are more pronounced in SPARTA, suggesting that variations between the chosen and rejected samples are partially influenced by disparities in length. Among all entropy-based metrics, Character Entropy is the most sensitive to text length, which amplifies its variance. Moreover, SPARTA consistently demonstrates lower values across other entropy metrics compared to SELF-REWARD, indicating reduced diversity in its generated text. And Further case studies validate this observation, revealing that even with identical extracted answers, the shorter responses are consistently marked as negative. This bias implicitly encourages the model to generate more detailed and comprehensive answers during preference optimization. + +Details of Evaluation Metrics We employ Pairwise Diversity Calculation to quantify cross-response diversity for each prompt. Specifically, generated responses are compared in a pairwise manner across three primary metrics: Edit Distance, BLEU Distance, and Embedding Distance. For each metric, the average pairwise distance is computed to reflect the diversity of responses associated with the prompt. To further enhance the evaluation, we incorporate an LLM-based Judgment Mechanism leveraging the GEMINI-1.5-FLASH model. This model acts as a judge, providing an independent assessment of semantic, stylistic, and strategic differences across generated responses. + +- Edit Distance (Edit): This metric measures the surface-level lexical diversity by calculating the normalized character-level edit distance between pairs of generated responses. Specifically, for two responses $r_1$ and $r_2$ , the edit distance is defined as the minimum number of character-level insertions, deletions, or substitutions required to transform $r_1$ into $r_2$ . The result is normalized by the maximum length of the two responses to ensure comparability across different response lengths. Formally, the edit distance $d_{\mathrm{edit}}$ is given by: + +$$ +d _ {\mathrm {e d i t}} \left(r _ {1}, r _ {2}\right) = \frac {\operatorname {E d i t D i s t a n c e} \left(r _ {1} , r _ {2}\right)}{\max \left(\left| r _ {1} \right| , \left| r _ {2} \right|\right)} +$$ + +Higher values indicate greater surface-level variations, reflecting more diverse lexical choices among the responses. + +- BLEU Distance (BLEU): This metric assesses the structural or phrasal diversity of generated responses by capturing $n$ -gram overlap differences. It is computed as 1 - BLEU, where BLEU + +is the traditional bilingual evaluation understudy score. For two responses, the BLEU score is calculated with equal weight for unigram, bigram, trigram, and four-gram matches. The smoothing function is applied to avoid zero scores for sentences with no overlap. The BLEU distance is formulated as: + +$$ +d _ {\mathrm {B L E U}} \left(r _ {1}, r _ {2}\right) = 1 - \mathrm {B L E U} \left(r _ {1}, r _ {2}\right) +$$ + +A higher BLEU distance suggests more structural or syntactic variations between the responses, enhancing phrasal diversity. + +- **Embedding Distance (Embed):** This metric quantifies the semantic distinctness between responses by leveraging a pre-trained Sentence Transformer model (BAAI/bge-large-en-v1.5) to compute embeddings. For each pair of responses, the cosine similarity is measured between their embeddings, and the embedding distance is derived as: + +$$ +d _ {\mathrm {e m b e d}} (r _ {1}, r _ {2}) = 1 - \cos (\mathbf {e} _ {1}, \mathbf {e} _ {2}) +$$ + +where $\mathbf{e}_1$ and $\mathbf{e}_2$ are the embedding vectors for $r_1$ and $r_2$ , respectively. This value reflects the semantic dissimilarity, with higher scores indicating more substantial differences in meaning. + +- LLM Diversity Score (LLM): This score is calculated by querying the GEMINI-1.5-FLASH model with a series of generated responses and evaluating the diversity using a proprietary evaluation mechanism. The model scores each prompt based on semantic, stylistic, and strategic differences, providing an aggregate diversity score on a scale from 1 to 10. This metric not only captures linguistic diversity but also considers deeper contextual shifts and strategic variations in the generated responses. + +Tracking the Changes of Generation Diversity We employ four key metrics—Edit Pairwise Distance, BLEU Pairwise Distance, Embedding Pairwise Distance,and LLM-as-a-Judge scores—to comprehensively track diversity changes throughout the optimization process. At each iteration, the ten models generate responses to a given prompt, and their pairwise diversity is measured across these four metrics. This evaluation framework allows us to quantify structural, lexical, and semantic variations, as well as perceived quality from a large language model perspective, providing a holistic view of convergence and alignment dynamics. + +The results indicate a clear trade-off between diversity and optimization during the iterative training process of SPARTA ALIGNMENT. The consistent decline in Edit Score, BLEU Score, and Embedding Score suggests that model outputs are converging towards more structured and semantically aligned representations, potentially at the expense of surface-level diversity. This is + +a typical characteristic of optimization algorithms that prioritize correctness and consistency over variability. + +However, the test performance continues to improve across iterations, reflecting that semantic alignment contributes positively to improvement of instruction-following task. Interestingly, the LLM-as-a-Judge score remains relatively stable, which may suggest that LLM-based evaluations are less sensitive to structural optimizations and may rely more heavily on stylistic or high-level contextual markers rather than purely logical coherence. + +These observations highlight the effectiveness of SPARTA ALIGNMENT in enhancing model robustness and generalization for instruction-following tasks while maintaining semantic integrity. The stability of LLM-based judgment also implies that further optimization may require alignment strategies that extend beyond surface-level consistency and address deeper contextual understanding. + +Case Study in MedQA We present representative examples of preference pairs from the MedQA benchmark to illustrate the effectiveness of SPARTA ALIGNMENT. The results demonstrate that SPARTA ALIGNMENT consistently favors responses that are comprehensive, accurate, direct, and rich in clinical information, aligning with real-world expectations for medical decision-making. + +![](images/1ef7d3be3113c1ea04f62fe9ae3cbe51e77b6b6251f00dda6d40cdf8966a32dd.jpg) + +![](images/521e55bee5110bbd4145b2c53e331bd07052075e0c1102da32e69f38a313a98f.jpg) + +![](images/9bcb12a9d45ece92f38220f9c98ab9dbb6d8fc34213b79b151d95f9207e9c08a.jpg) +Figure 6: Evolution of cross-response diversity over processed iterations, measured using four distinct metrics: Edit Score, BLEU Score, Embedding Score and LLM-as-a-Judge Score. A consistent decline in these scores suggests improved structural coherence and semantic alignment in generated responses, reflecting enhanced model optimization. + +![](images/1c62c590fdbc14bf092c8103b0ee00568042a24f266001d0a9be341501e369f1.jpg) + +# Type: more comprehensive + +# Instruction + +An exclusively breast-fed, 5-day-old boy is brought to the physician by his mother for a routine examination. He was born at term and delivery was uncomplicated. He received all standard treatment and testing prior to being discharged from the hospital 3 days ago. Examination shows no abnormalities. Without receiving additional supplementation at this time, this newborn is at greatest risk of developing which of the following conditions? A. Scaly dermatitis, B. Intracranial bleed, C. Microcytic anemia, D. Peripheral neuropathy, E. Rickets + +# Generated Text (Chosen) + +The best answer is C. Microcytic anemia. Here's why: - Breast Milk: Breast milk is considered the ideal food for infants in the first six months of life, providing essential nutrients and antibodies. However, it lacks sufficient iron content to meet the rapidly growing needs of a 5-day-old baby. - Iron Deficiency Anemia: Without any iron supplementation, the baby is at high risk of developing iron deficiency anemia due to the insufficient iron intake from breast milk alone. - Other Options: - Scaly Dermatitis (A): This could be related to a vitamin D deficiency, but it's not the most immediate concern in this scenario. - Intracranial Bleed (B): This is highly unlikely with an uncomplicated birth. - Peripheral Neuropathy (D): This condition is more likely to develop later on due to nerve damage, usually from a lack of vitamin B12 or other factors. - Rickets (E): While possible with vitamin D deficiency, it typically takes longer to develop compared to iron deficiency anemia. + +# Generated Text (Rejected) + +C. Microcytic anemia Newborns who are exclusively breastfed may be at risk for developing iron deficiency anemia due to the low iron content in breast milk. This can lead to microcytic anemia, which can be prevented by adding iron-fortified infant formula or iron supplements to the diet. + +# Annotation + +The chosen answer is more comprehensive in its analysis, detailing the relationship between breastfeeding and iron deficiency anemia. It methodically rules out other differential diagnoses, enhancing the clarity of why microcytic anemia is the most likely condition. Furthermore, it explains the expected timeline and typical risks associated with each incorrect option, which strengthens the clinical reasoning. In contrast, the rejected answer, although correct, is very brief and does not explore the differential options or the reason why microcytic anemia is the most immediate concern for a 5-day-old newborn. It lacks the clinical reasoning and differential analysis, making it less informative and less aligned with a step-by-step diagnostic process. + +Figure 7: Case Study of MedQA: The chosen one is more comprehensive than the rejected one + +# Type: more accurate + +# Instruction + +An exclusively breast-fed, 5-day-old boy is brought to the physician by his mother for a routine examination. He was born at term and delivery was uncomplicated. He received all standard treatment and testing prior to being discharged from the hospital 3 days ago. Examination shows no abnormalities. Without receiving additional supplementation at this time, this newborn is at greatest risk of developing which of the following conditions? A. Scaly dermatitis, B. Intracranial bleed, C. Microcytic anemia, D. Peripheral neuropathy, E. Rickets + +# Generated Text (Chosen) + +The best answer is C. Microcytic anemia. + +- Breast Milk: Breast milk is considered the ideal food for infants during the first six months, providing essential nutrients and antibodies. However, it contains insufficient iron to meet the growing needs of a 5-day-old infant. +- Iron Deficiency Anemia: Without iron supplementation, the infant is at a high risk of developing iron deficiency anemia, leading to microcytic anemia. +- Exclusion of Other Options: - Scaly Dermatitis (A): Typically linked to vitamin D or essential fatty acid deficiencies, but not the most immediate concern. - Intracranial Bleed (B): Unlikely given the uncomplicated birth history and normal examination. - Peripheral Neuropathy (D): This condition generally develops later and is associated with vitamin B12 deficiency or other metabolic issues. - Rickets (E): A consequence of vitamin D deficiency, which takes longer to manifest compared to iron deficiency anemia. + +The comprehensive analysis highlights that iron deficiency is the most immediate risk for an exclusively breast-fed newborn due to the low iron content in breast milk. Thus, the best answer is: C. Microcytic anemia. + +# Generated Text (Rejected) + +# C. Microcytic anemia + +Newborns who are exclusively breastfed may be at risk for developing iron deficiency anemia due to the low iron content in breast milk. This can lead to microcytic anemia, which can be prevented by adding iron-fortified infant formula or iron supplements to the diet. + +# Annotation + +The chosen response is significantly more accurate and informative, providing a clear clinical reasoning path that connects the infant's exclusive breastfeeding to iron deficiency and subsequently to microcytic anemia. It not only explains the pathophysiology but also systematically eliminates the incorrect options based on clinical evidence. + +In contrast, the rejected answer, while correct, is overly simplistic and lacks the differential diagnosis or consideration of clinical presentation. It fails to engage with the full spectrum of possible conditions or justify why microcytic anemia is the most immediate concern, leading to a less informative clinical understanding. + +Figure 8: Case Study of MedQA: The chosen one is more accurate and clinically comprehensive than the rejected one + +# Type: more clinically informative + +# Instruction + +An 18-year-old male in his first year of college presents to the emergency room with a fever and a severe headache. He reports having unprotected sex with several partners over the past few weeks. Upon examination, the male demonstrates nuchal rigidity and photophobia. His past medical history is notable for a lack of vaccinations beginning from infancy due to his parents' belief that vaccinations may cause autism. The bacteria causing these symptoms would most likely demonstrate which of the following? A. Positive quellung reaction, B. Negative quellung reaction, C. Lactose fermentation, D. Urease positive, E. Negative catalase test + +# Generated Text (Chosen) + +The patient's presentation suggests a diagnosis of meningitis, possibly caused by Neisseria meningitidis. + +- Clinical Symptoms: N. meningitidis is known to cause meningococcal meningitis, characterized by fever, headache, nuchal rigidity, and photophobia. +- Positive Quelling Reaction: The bacteria causing meningitis would most likely demonstrate a positive quelling reaction (Option A). The quelling reaction is a test used to identify Neisseria species by their ability to agglutinate with antibodies against their polysaccharide capsule. N. meningitidis has a positive quelling reaction due to its polysaccharide capsule. +- Exclusion of Other Options: - Negative quelling reaction (B) is inconsistent with the capsule structure of N. meningitidis. - Negative catalase test (E) does not apply, as N. meningitidis is catalase positive. - Lactose fermentation (C) and urease positive (D) are not characteristic of N. meningitidis. + +Therefore, the best answer is: A. Positive quelling reaction. + +# Generated Text (Rejected) + +The relevant information to answer the above question is: The bacteria causing these symptoms would most likely be Neisseria meningitidis, a Gram-negative diplococcus that causes meningococcal disease. Therefore, the answer is A. + +# Annotation + +The chosen answer is more comprehensive and clinically informative, detailing the connection between the patient's symptoms and the pathophysiology of Neisseria meningitidis. It explains the diagnostic significance of the positive quellung reaction and systematically rules out differential diagnoses based on microbiological characteristics. Additionally, it justifies the exclusion of options B, C, D, and E with clear reasoning, demonstrating a deeper understanding of bacterial pathogenesis. + +In contrast, the rejected answer, while correct, is oversimplified and does not engage with differential reasoning or the biological mechanisms that support the diagnosis. It lacks the clinical depth necessary for robust medical decision-making and offers minimal justification for the selected answer. + +Figure 9: Case Study of MedQA: The chosen one is more clinically informative than the rejected one + +# Type: more direct + +# Instruction + +Your colleague has been reading the literature on beta-carotene supplementation and the risk of heart disease. She thinks they may share a clinically relevant association and would like to submit an editorial to a top journal. Upon final literature review, she discovers a newly published study that refutes any association between beta-carotene and heart disease. Your colleague is upset; you suggest that she, instead, mathematically pool the results from all of the studies on this topic and publish the findings. What type of study design are you recommending to your colleague? + +A. Randomized control trial, B. Systematic review, C. Case-cohort study, D. Meta-analysis, E. Cross-sectional study + +# Generated Text (Chosen) + +The type of study design you are recommending to your colleague is: D. Meta-analysis. + +Definition: A meta-analysis is a statistical technique used to mathematically pool data from multiple studies to derive a more precise estimate of the overall effect. +- Relevance: In this case, by aggregating the results of all studies on beta-carotene and heart disease, your colleague can obtain a clearer and more robust estimate of the true association. +- Why Meta-analysis? This method not only improves statistical power but also allows the identification of patterns or sources of variation across studies. + +# Generated Text (Rejected) + +The correct answer is D. Meta-analysis. A meta-analysis is a statistical method that combines data from multiple studies to provide a more precise estimate of the effect size or association between variables. In this case, your colleague would use a meta-analysis to mathematically pool the results from all of the studies on the topic of beta-carotene and heart disease. This would allow her to obtain a more accurate and robust estimate of the association between beta-carotene and heart disease than could be obtained from a single study. Additionally, by using a meta-analysis, she could also identify potential sources of heterogeneity among the studies and explore possible reasons for any differences in their results. + +# Annotation + +The chosen answer is more direct and focused, clearly identifying meta-analysis as the recommended study design and succinctly explaining its purpose. It effectively emphasizes the method's statistical advantage in aggregating results for more robust conclusions. + +In contrast, the rejected answer, while correct, is more verbose and less focused. It redundantly describes the statistical benefits and expands on secondary points (like heterogeneity analysis) that, while relevant, dilute the core recommendation. The more concise response is better aligned with clinical decision-making where clarity and precision are essential. + +Figure 10: Case Study of MedQA: The chosen one is more direct than the rejected one + +# Type: more logical reasoning + +# Instruction + +A 32-year-old G2P0A1 woman presents at 36 weeks of gestation for the first time during her pregnancy. The patient has no complaints, currently. However, her past medical history reveals seizure disorder, which is under control with valproic acid and lithium. She has not seen her neurologist during the past 2 years, in the absence of any complaints. She also reports a previous history of elective abortion. The physical examination is insignificant. Her blood pressure is $130/75 \mathrm{mmHg}$ and pulse is $80 / \mathrm{min}$ . The patient is scheduled to undergo regular laboratory tests and abdominal ultrasound. Given her past medical history, which of the following conditions is her fetus most likely going to develop? + +A. Trisomy 21, B. Neural tube defects (NTDs), C. Iron deficiency anemia, D. Intrauterine growth restriction, E. Limb anomalies + +# Original Generated Text (Chosen) + +The patient's past medical history of seizure disorder, specifically the use of valproic acid and lithium, increases the risk of neural tube defects (NTDs) in her fetus. Therefore, the correct answer is: B. Neural tube defects (NTDs). + +- Mechanism of Action: Valproic acid is known to inhibit folate metabolism, which is critical for neural tube closure during embryonic development. Its teratogenic effects are well-documented, with a marked increase in the risk of spina bifida and other NTDs. +- Clinical Correlation: Lithium, while primarily associated with cardiac anomalies, is often used alongside valproic acid in mood stabilization. Its usage further necessitates regular screening during pregnancy. +- Exclusion of Other Options: - Trisomy 21 (A): No strong correlation with valproic acid or lithium usage. - Iron deficiency anemia (C): Not indicated by her medical history. - Intrauterine growth restriction (D): More commonly linked to placental issues or maternal hypertension, neither of which is present. - Limb anomalies (E): Not typically associated with these medications. + +# Generated Text (Rejected) + +The best answer is B. Neural tube defects (NTDs). + +- Valproic Acid: This medication is known to increase the risk of neural tube defects (NTDs) in developing fetuses. NTDs are congenital malformations that occur when the neural tube fails to close properly, resulting in conditions like spina bifida or anencephaly. +- Past History of Abortion: While not definitive proof of NTDs, this history may be relevant given the association between valproic acid and increased risk of NTDs. + +Important Note: This is a generalization based on the information provided. Each pregnancy is unique, and factors beyond medication use can contribute to pregnancy outcomes. The patient should discuss her complete medical history and any concerns with her healthcare provider. + +# Annotation + +The chosen answer presents a clear and logical pathway from the patient's medication history to the increased risk of Neural Tube Defects (NTDs). It systematically explains the teratogenic mechanism of valproic acid, links lithium's relevance, and rules out other differential diagnoses with clear clinical reasoning. This structured approach not only justifies the answer but also demonstrates a deeper understanding of pharmacological risks in pregnancy. + +In contrast, the rejected answer, although correct, is less structured and introduces non-specific associations, like abortion history, without strong clinical evidence. It lacks a step-by-step breakdown of the differential process, making the reasoning less robust and less logical compared to the chosen response. + +Figure 11: Case Study of MedQA: The chosen one is more logically structured than the rejected one + +![](images/f88cbff0829a0985842c39f94543e052703b1aaf4e6cb3fe8522f1f095c93492.jpg) + +![](images/848307d39184a8952b51bbf064a5f08cab5fe5ba666ecf18baa08c48b66d92fa.jpg) + +![](images/dd6f47703fc4d4251e4c5d6d9cb847f86b09a969bab3ec6d209aa19b4c3b4240.jpg) + +![](images/5b0f5440275bbd8ae2ca7fae93b7a1409d9f78264448664a394b82517f7dae0f.jpg) + +![](images/e08a9e6fd10511fc55ba0df40ef53d48b6fb1e7caedf841b21faccd7a5296e66.jpg) + +![](images/1e9a203622715deba44b64833b59e3cc4f18d39b5045c7e23f61a1feb51cb1b2.jpg) + +![](images/476fe5d0b53a753ae019b2b1a18ca3445ed672fe92de8ba4ccfa0dff739b461a.jpg) + +![](images/bbd40bf8ee4c521c313f073d86c66b8f22996511d4b8d0b8c59fad6b899cc01d.jpg) + +![](images/5c82c2871d4841044d41f99f62c63c5b0a6486bf6f0cc42eda0cdd08b63f4d52.jpg) +Figure 12: Correlation between the model's average performance on a specific task and its average reputation in the model pool. Each subplot contains 10 points representing 10 models. $r$ denotes the Pearson correlation coefficient. + +![](images/2b8a7aac72ed595fbd1ca5cd0002ceae3e44d4f18ad6b3ad7755f14c64c91e87.jpg) +Average Performance + +![](images/ee10312e467070e981e36d70d5b967c7ebdf0982262fec6f9f8559d62568ac6e.jpg) + +![](images/570c7b08d825d85b2dbbe205a08fa9c1a37312063ad9d6a4b941dabe020f940c.jpg) + +# C.9 Details in Correlation + +Figure 12 visualizes the correlation between models' average performance on specific evaluation benchmarks (x-axis) and their average reputation within the model pool (y-axis). Each subplot represents a distinct benchmark, including MedQA, Normad (Country), Normad (Value), Normad (RoT), KCross, GSM8K, COM $^2$ , Math (Easy), Math (Medium), Math (Hard), Alpaca, and TruthfulQA. In each subplot, ten colored points correspond to ten different models, and the dashed line represents the linear regression fit. The Pearson correlation coefficient $r$ is annotated in the bottom-right corner, quantifying the linear relationship strength between performance and reputation. + +Several benchmarks exhibit strong positive correlations. For instance, KCross ( $r = 0.52$ ), Math (Medium) ( $r = 0.4$ ), and Math (Hard) ( $r = 0.5$ ) demonstrate clear trends where higher task performance is generally associated with higher average reputation. This alignment suggests that model evaluations on these tasks are well-reflected in their reputation scores, indicating consistency between task-specific competence and collective judgment in the model pool. + +Class Stratification Figure 13 demonstrates that SPARTA ALIGNMENT naturally induces hierarchical stratification among the 10 models, forming distinct performance tiers. Some models consistently occupy the upper strata, while several other models converge towards lower tiers as iterations process, suggesting that reputation-driven clustering effectively differentiates model reliability. This emergent hierarchy hints at potential top-down influence, where models with high reputation scores may guide ones with low reputation, highlighting an interesting direction for understanding social dynamics in multi-LLM alignment: Does bottom-up effect exist in multi-LLM fine-tuning system?. + +Interestingly, if models with suboptimal performance were to consistently occupy the upper tiers, it would indicate potential misalignment in the reputation assessment mechanism. Such scenarios could lead to cascading errors, where less capable models disproportionately influence collective decisions, ultimately degrading the align + +![](images/d28514e2bf205304dd1cf88326f528f5379095a413ef10184ec4a6ddbffe8086.jpg) +Figure 13: Correlation between model's performance on a specific task and its reputation in every iteration in the model pool. 10 types of 80 points mean 10 models in 8 iterations. $r$ stands for Pearson correlation coefficient. + +ment process. This underscores the importance of accurate reputation signals that faithfully represent model competence, ensuring that performance-based stratification genuinely mirrors underlying capabilities. + +Looking forward, enhancing the precision of reputation assignments remains a crucial direction. Implementing more granular evaluation metrics that capture edge cases and task-specific nuances could help refine the hierarchical structure, preventing misjudged models from dominating higher tiers. Moreover, the current top-down influence observed in the hierarchy hints at an intriguing question: Does a bottom-up effect exist in multi-LLM fine-tuning systems? Understanding whether lower-tier models can influence upper-tier strategies could reveal new pathways for improving collective learning dynamics, potentially making the alignment process more robust and equitable. + +# C.10 Case Study + +# Instruction Following - Alpaca Analysis + +# Instruction + +What breed dog is smallest? + +# Original Generated Text + +What breed is the largest? Name three breeds that are in the middle. + +The smallest dog breed is the Chihuahua. The largest dog breed is the Mastiff. Three breeds that are in the middle are the Miniature Schnauzer, Boxer, and Beagle. + +# Generated Text + +The smallest breed of dog is the Chihuahua. Chihuahuas typically weigh between 2 and 6 pounds and stand about 6 to 9 inches tall at the shoulder. Other small breeds include the Toy Poodle, Toy Terrier, and Toy Pomeranian. However, the Chihuahua is considered the smallest breed by weight and size. It is important to note that individual dogs within a breed can vary in size, and some Chihuahuas may be slightly smaller or larger than the average. Additionally, there are some non-breed dogs that are even smaller, such as the Italian Greyhound and the Silky Terrier. However, Chihuahua remains the smallest recognized breed of dog by the American Kennel Club and other major kennel clubs. + +# Annotation + +The instruction specifically asks for the smallest breed of dog. The generated output focuses accurately on the Chihuahua, providing details about its size and weight, and comparing it with other small breeds. It also clarifies the basis of recognition by major kennel clubs, making the answer thorough, informative, and aligned with the question. + +Figure 14: Case Study of Instruction Following in Alpaca + +# Commonsense Reasoning - Com² Analysis + +# Prompt + +As an expert in commonsense reasoning, your task is to provide a concise response to a question based on the given context. The question focuses on studying the causes, effects, or attributes of personas related to the given context. + +# Context + +Violet lives the dream. + +# Question + +What does Violet need to do before Violet lives the dream? + +# Ground Truth + +To make plans. + +# Original Generated Text + +Violet needs to be born. + +# Generated Text + +Violet needs to identify her goals and aspirations, take steps towards achieving them, and work hard to make her dreams a reality. + +# Annotation + +The context indicates that Violet is living her dream, so the question asks what Violet needs to do before living her dream. Based on the given context, Violet needs to set goals, take action towards them, and work hard to achieve her dreams. This answer is derived from the context and aligns with the question's focus on Violet's actions leading up to living her dream. + +Figure 15: Case Study of Commonsense Reasoning $\mathrm{COM}^2$ + +# Medical Question Answer - MedQA Analysis + +# Question + +Two weeks after undergoing an emergency cardiac catheterization with stenting for unstable angina pectoris, a 61-year-old man has decreased urinary output and malaise. He has type 2 diabetes mellitus and osteoarthritis of the hips. Prior to admission, his medications were insulin and naproxen. He was also started on aspirin, clopidogrel, and metoprolol after the coronary intervention. His temperature is $38^{\circ}\mathrm{C}$ $(100.4^{\circ}\mathrm{F})$ , pulse is $93 / \mathrm{min}$ , and blood pressure is $125 / 85\mathrm{mmHg}$ . Examination shows mottled, reticulated purplish discoloration of the feet. Laboratory studies show: + +- Hemoglobin count: $14\mathrm{g / dL}$ +- Leukocyte count: $16,400 / \mathrm{mm}^3$ +- Segmented neutrophils: $56\%$ +Eosinophils: $11\%$ +Lymphocytes: $31\%$ +- Monocytes: $2\%$ +- Platelet count: $260,000 / \mathrm{mm}^3$ +- Erythrocyte sedimentation rate: $68~\mathrm{mm / h}$ + +# Serum + +- Urea nitrogen: $25\mathrm{mg / dL}$ +- Creatinine: $4.2\mathrm{mg / dL}$ + +Renal biopsy shows intravascular spindle-shaped vacuoles. Which of the following is the most likely cause of this patient's symptoms? + +# Options: + +A. Renal papillary necrosis +B. Allergic interstitial nephritis +C. Cholesterol embolization +D. Eosinophilic granulomatosis with polyangiitis +E. Polyarteritis nodosa + +# Ground Truth + +Cholesterol embolization (C) + +# Original Generated Text + +F. Acute tubular necrosis +G. Acute glomerulonephritis +H. Acute pyelonephritis +I. Acute pancreatitis +J. Acute hepatic failure +Remember to answer the question with your own reasoning, but don't hesitate to check for the answer to the question, as there might be more than one answer. + +# Generated Text + +Assistant: C. Cholesterol embolization + +Cholesterol embolization can occur in patients with coronary artery disease, especially those who have undergone cardiac catheterization. It typically presents with acute kidney injury and skin changes such as livedo reticularis. The intravascular spindle-shaped vacuoles seen on renal biopsy are characteristic of cholesterol embolization. Other causes of acute kidney injury in this patient, such as allergic interstitial nephritis or polyarteritis nodosa, are less likely given the patient's history and clinical presentation. + +# Annotation + +The assistant correctly identifies "Cholesterol embolization" as the answer, recognizing the typical presentation of renal failure, reticular skin changes, and biopsy findings. The reasoning is aligned with the clinical scenario and laboratory findings, indicating a solid understanding of the pathophysiology. + +Figure 16: Case Study of MedQA \ No newline at end of file diff --git a/NeurIPS/2025/Sparta Alignment_ Collectively Aligning Multiple Language Models through Combat/images.zip b/NeurIPS/2025/Sparta Alignment_ Collectively Aligning Multiple Language Models through Combat/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..e163a44ae36fa0d2a7b56db4a370ca24b397bed3 --- /dev/null +++ b/NeurIPS/2025/Sparta Alignment_ Collectively Aligning Multiple Language Models through Combat/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eee26bbfd068bef011bf20faa264e2f25bf4f280419a10225f893aa4f88865c4 +size 504363 diff --git a/NeurIPS/2025/Sparta Alignment_ Collectively Aligning Multiple Language Models through Combat/layout.json b/NeurIPS/2025/Sparta Alignment_ Collectively Aligning Multiple Language Models through Combat/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..3ae331e8919b01b650af5c21ecccc8c77d0ac2e3 --- /dev/null +++ b/NeurIPS/2025/Sparta Alignment_ Collectively Aligning Multiple Language Models through Combat/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:38c063551ec6b3ce5162ccd879d51c97d3261c8e50b6278d7a3ca661aea9f0be +size 1222855 diff --git a/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_content_list.json b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..51f74ac606bdc1c856f4afc4d760ae1b49fd1519 --- /dev/null +++ b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:199c0f9b331abc028b78aa7c8f58fb26211917f293807d853d73ea1246c31b6a +size 162461 diff --git a/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_model.json b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_model.json new file mode 100644 index 0000000000000000000000000000000000000000..a445307fa21b9fb01dfcaf6587417f728ff86252 --- /dev/null +++ b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:69980fc84daef00e9ea502b7709c16088f126a31cdf1d48bd94256d61af87eee +size 209508 diff --git a/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_origin.pdf b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1f518eb4fd802ba63217943154cdb40abb8e4d49 --- /dev/null +++ b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/12d95042-7358-4b5e-ac55-87d45da7eb41_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2b411ece5bfa59554e6f3bb41ec884ab3f27ecdd19de3d1e65aefc158a676a1e +size 2113983 diff --git a/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/full.md b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/full.md new file mode 100644 index 0000000000000000000000000000000000000000..c5834537e474efd99ec7f0f4639f8153f0ac21e2 --- /dev/null +++ b/NeurIPS/2025/Spatial Understanding from Videos_ Structured Prompts Meet Simulation Data/full.md @@ -0,0 +1,765 @@ +# Spatial Understanding from Videos: Structured Prompts Meet Simulation Data + +Haoyu Zhang $^{1,2}$ , Meng Liu $^{3,4*}$ , Zaijing Li $^{1,2}$ , Haokun Wen $^{1}$ , Weili Guan $^{1}$ , Yaowei Wang $^{1,2}$ , Liqiang Nie $^{1*}$ + +$^{1}$ Harbin Institute of Technology (Shenzhen) $^{2}$ Pengcheng Laboratory + +$^{3}$ Shandong Jianzhu University $^{4}$ Zhongguancun Academy + +# Abstract + +Visual-spatial understanding, the ability to infer object relationships and layouts from visual input, is fundamental to downstream tasks such as robotic navigation and embodied interaction. However, existing methods face spatial uncertainty and data scarcity, limiting the 3D spatial reasoning capability of pre-trained vision-language models (VLMs). To address these challenges, we present a unified framework for enhancing 3D spatial reasoning in pre-trained VLMs without modifying their architecture. This framework combines SpatialMind, a structured prompting strategy that decomposes complex scenes and questions into interpretable reasoning steps, with ScanForgeQA, a scalable question-answering dataset built from diverse 3D simulation scenes through an automated construction process designed for fine-tuning. Extensive experiments across multiple benchmarks demonstrate the individual and combined effectiveness of our prompting and fine-tuning strategies, and yield insights that may inspire future research on visual-spatial understanding. + +# 1 Introduction + +Visual-spatial understanding, the ability to infer spatial relationships and the layout of objects from visual input, is a core component of human perception [1, 2, 3]. From a single image, human observers can intuitively estimate distances, relative sizes, and even infer occluded structures. As intelligent systems become increasingly embedded in real-world applications such as autonomous driving [4, 5, 6], robotic navigation [7, 8, 9], and augmented reality [10, 11, 12], it becomes crucial to endow models with similar spatial reasoning capabilities for robust perception and interaction. + +Unfortunately, a single image is inherently limited in capturing the complexity of real-world 3D scenes, constraining its utility in practical scenarios [13, 14, 15]. To address this, point clouds have become a mainstream representation for 3D scene understanding due to their ability to encode rich geometric information [16, 17]. Yet, generating high-quality point clouds typically requires expensive sensors and incurs significant computational overhead, limiting scalability and accessibility. + +These limitations motivate the pursuit of vision-only solutions that operate on scanning videos or multi-view images of scenes. Such approaches offer a more human-like and scalable pathway to spatial understanding [18]. However, performing 3D spatial reasoning from scanning videos presents two significant challenges: (1) Spatial Uncertainty. In the absence of explicit depth information, models must infer 3D structure from inherently limited 2D observations. This process is further complicated by occlusions, perspective distortions, and texture ambiguities, all of which introduce significant spatial uncertainty. Effectively addressing this challenge demands multi-step logical reasoning across frames to reconstruct coherent spatial layouts. (2) Data Scarcity. Existing datasets for this task are limited in both scale and diversity, restricting the ability of vision-language models + +(VLMs) to acquire robust spatial knowledge and perceptual capabilities. Moreover, these datasets involve scans of real-world scenes, which leads to poor scalability. This highlights the need for scalable and extensible data sources to support effective spatial reasoning in VLMs. + +To address these challenges, we propose a dual approach for enhancing 3D spatial reasoning in pre-trained VLMs, without modifying their underlying architecture. First, we introduce SpatialMind, a structured Chain-of-Thought (CoT) prompting strategy that guides VLMs through step-by-step reasoning over spatial relationships. Second, we present ScanForgeQA, a large-scale synthetic question-answering (QA) dataset constructed from diverse 3D simulation scenes using an automated generation pipeline. Fine-tuning VLMs on this dataset equips them with spatial commonsense knowledge, significantly improving their generalization to unseen spatial layouts. We have validated our approach through extensive experiments across multiple benchmarks. Results demonstrate the individual and combined effectiveness of our prompting and fine-tuning strategies, and yield insights that may inspire future research on visual-spatial understanding. + +Our contributions are summarized as follows: + +- We introduce SpatialMind, a spatial prompting strategy that decomposes spatial reasoning into structured steps, enabling pre-trained VLMs to perform multi-step inference over spatial relationships from visual input alone. +- We develop a scalable dataset generation pipeline to construct ScanForgeQA, a synthetic spatial question-answering dataset that enables VLMs to acquire spatial commonsense through fine-tuning. +- Experimental results validate the effectiveness and generalizability of both SpatialMind and ScanForgeQA, with their combination achieving further gains and providing valuable insights for future research. + +# 2 Related Work + +2D Image Spatial Understanding focuses on modeling spatial relationships among objects within the 2D image. Most existing models are trained on 2D images paired with textual descriptions, which offer limited cues about 3D structure. Consequently, their capacity for spatial reasoning remains constrained. To mitigate this, several approaches, such as SpatialVLM [13], SpatialRGPT [14], and SpatialBot [15], have been proposed. These methods enhance the spatial understanding by fine-tuning models on datasets specifically designed for spatially grounded QA tasks. To enable more comprehensive evaluation, recent studies [19, 20, 21, 22, 23] have introduced hierarchical benchmarks that assess models across varying levels of spatial reasoning complexity. Parallel efforts have explored more explicit forms of spatial interaction [24, 25]. For example, point-based methods [26, 27] interpret spatial instructions by predicting specific target points. Building on this trend, SpatialCoT [28] proposes a two-stage strategy that aligns multimodal inputs with spatial coordinates and incorporates CoT reasoning to better address complex embodied tasks. Despite these advancements, model performance often degrades in complex real-world 3D environments, highlighting the limitations of 2D-based approaches in representing complex 3D scenes. + +3D Indoor Spatial Understanding focuses on enabling intelligent agents to identify object positions and infer their spatial relationships within enclosed environments, thereby supporting both object manipulation and interactive scene comprehension. Early 3D models are trained on standard indoor datasets [29, 30, 31, 32, 33, 34, 35] using point clouds to facilitate downstream tasks like 3D object detection and instance segmentation [36, 37, 38, 39] and primarily focus on object-level geometry and appearance features [40, 41, 42, 43]. More recent work extends this focus to complex indoor scenes, emphasizing inter-object spatial relationships and holistic scene-level understanding. To address challenges such as geometric complexity and annotation sparsity, many of these models employ cross-modal strategies that combine point cloud data with auxiliary multi-view 2D images [16, 17, 44]. Inspired by the way humans perceive spatial layouts through vision alone, emerging research [1, 45, 18] has begun to explore purely vision-based approaches to 3D spatial understanding. These methods rely solely on visual inputs, such as scanning videos, without requiring explicit 3D priors like point clouds. This line of work offers a more practical and scalable alternative for real-world deployment. In this context, we further investigate whether purely vision-based inputs can provide a more effective solution for indoor scene understanding. + +![](images/94e91adc9b0ef3ded6def00974d52bcf3801f645a5526ddbd7c5ba018fe4d18f.jpg) +Figure 1: Illustration of our SpatailMind prompting strategy. + +# 3 SpatialMind Prompting Strategy + +As shown in Figure 1, our SpatialMind prompting strategy consists of two main components: 1) Scene Decomposition, where the 3D scene depicted in the video is transformed into multiple different representations; and 2) Question Decomposition, in which the question is broken down into a sequence of fine-grained reasoning steps. Further details can be found in Appendix D. + +# 3.1 Scene Decomposition + +The scene decomposition process includes three sequential steps: local modeling, coordinate mapping, and cognition generation. + +Local Modeling. The first step processes scanning video frames to extract object instances and their relative spatial configurations within localized coordinate systems. To handle scene complexity and reduce the search space, we leverage GPT-4o² to identify all objects mentioned across the questions associated with a given scene, using them as candidate targets. For each frame $i$ , we prompt VLMs to detect a subset of objects $\{c_{ij}\}$ from the candidate targets and estimate their positions $\mathbf{p}_{ij}^{\mathrm{local}} \in \mathbb{R}^3$ . These positions are defined relative to a randomly selected reference object (i.e., origin) within the same frame, forming a local 3D map: + +$$ +\mathcal {L} _ {i} = \left\{\left(c _ {i j}, \mathbf {p} _ {i j} ^ {\text {l o c a l}}\right) \mid j = 1, \dots , n _ {i} \right\}, \tag {1} +$$ + +where $n_i$ denotes the number of objects in frame $i$ . Because each video frame captures only a limited field of view, the same object may appear across multiple frames from different perspectives. Thus, this step focuses on accurate per-frame object detection and spatial localization, laying the foundation for subsequent alignment in a global coordinate system. + +Coordinate Mapping. To integrate spatial information across video frames, this step transforms all locally detected object positions into a unified global coordinate system. The global origin is defined by selecting the reference object in the first frame. To estimate motion between frames, we prompt the VLM to infer the relative rotation and translation between adjacent frames. These relative transformations are accumulated sequentially to compute each frame's transformation $\mathbf{T}_i$ with respect to the global coordinate system: + +$$ +\mathbf {T} _ {i} = \prod_ {k = 1} ^ {i} \left[ \begin{array}{c c} \mathbf {R} _ {k, k - 1} & \mathbf {t} _ {k, k - 1} \\ \mathbf {0} & 1 \end{array} \right], \tag {2} +$$ + +where $\mathbf{R}_{k,k-1}$ and $\mathbf{t}_{k,k-1}$ denote the relative rotation and translation from frame $k-1$ to frame $k$ , respectively. This accumulated approach provides more stable and accurate alignment than directly estimating each frame's absolute pose. Using these transformations, each object's local coordinates are converted into global coordinates via homogeneous transformation: + +$$ +\left[ \begin{array}{c} \mathbf {p} _ {i j} ^ {\text {g l o b a l}} \\ 1 \end{array} \right] = \mathbf {T} _ {i} \cdot \left[ \begin{array}{c} \mathbf {p} _ {i j} ^ {\text {l o c a l}} \\ 1 \end{array} \right], \tag {3} +$$ + +where $\mathbf{p}_{ij}^{\mathrm{global}}$ denotes the global coordinates of the object $j$ in the frame $i$ . This step ensures that all detected objects across frames are positioned consistently within the same 3D space. Since objects may appear in multiple frames under different perspectives, we merge duplicate detections based on spatial proximity and semantic consistency via prompting. The result is a global 3D map of the scene: + +$$ +\mathcal {G} = \left\{\left(c _ {k}, \mathbf {p} _ {k} ^ {\text {g l o b a l}}\right) \right\} _ {k = 1} ^ {N}, \tag {4} +$$ + +where $N$ is the total number of all object instances in the entire scene. This map serves as a unified spatial abstraction that captures the overall layout from egocentric scanning videos. + +Cognition Generation. Beyond constructing a 3D map, we explore two additional formats for representing scene structure: a 2D spatial grid and natural language descriptions. We define a regular 2D grid over the global scene, typically aligned with the $XY$ -plane. Each grid cell corresponds to a fixed real-world area (e.g., 1 meter per cell, denoted by cell size $s$ ). Each object $c_k$ is mapped to a discrete grid location $(i_k, j_k)$ : + +$$ +\left(i _ {k}, j _ {k}\right) = \left(\left\lfloor \frac {x _ {k}}{s} \right\rfloor , \left\lfloor \frac {y _ {k}}{s} \right\rfloor\right), \tag {5} +$$ + +where $(x_{k},y_{k})$ are the horizontal components of the object's global position $\mathbf{p}_k^{\mathrm{global}}$ . In parallel, we generate natural language descriptions of object locations relative to a designated reference point. Using prompting, the model produces statements such as {"monitor": "locate 1 meter to the left of the reference point)}. These descriptions serve as a human-interpretable form of spatial cognition, bridging visual perception and symbolic reasoning. + +# 3.2 Question Decomposition + +Different types of spatial questions require distinct reasoning strategies [46]. To accommodate this diversity, we first categorize questions into several types (e.g., object size, relative distance, and relative direction). For each category, we design a dedicated reasoning procedure using GPT-40, followed by human verification to ensure correctness and interpretability. For instance, consider a question from the "relative distance" category: Among the refrigerator, window, and microwave, which object is closest to the door? The reasoning process for this type follows four structured steps: 1) Identify all mentioned objects, 2) Estimate the spatial coordinates of all relevant objects, 3) Compute the pairwise distances between the door and each candidate object, and 4) Select the object with the minimum distance as the answer. During inference, the system correspondingly selects the appropriate reasoning procedure based on the identified question type. + +To perform 3D spatial reasoning, we feed the VLMs with the input scanning video, one form of scene representation (e.g., 3D map, 2D grid, or textual position descriptions), the question, and the corresponding step-by-step reasoning plan. To assess which scene representation format is most interpretable for VLMs, we have conducted comparative experiments, as shown in Figure 3. + +# 4 ScanForgeQA Dataset Construction + +The construction of the ScanForgeQA dataset involves a three-stage pipeline, illustrated in Figure 2. These stages are: 1) Scene Construction, where single-room 3D environments are created; 2) Scan Creation, in which egocentric videos are simulated by scanning through the constructed scenes; and 3) QA Generation, where textual question-answering pairs are automatically generated based on object annotations and the spatial layout of each scene. + +# 4.1 Scene Construction + +To ensure diversity and richness in single-room scene collection, we adopt two parallel strategies: + +Separation. We modify existing scene datasets to leverage available resources effectively. Specifically, we utilize the 3D-FRONT dataset [47], which contains 6,813 multi-room scenes furnished with diverse 3D objects and annotated with detailed layout semantics and high-quality textures. Since our focus is on single-room environments, we disassemble each multi-room scene into individual rooms. For each scene, we isolate and load one room at a time, along with its corresponding ceiling and walls, and save it as an independent instance. This disassembly process yields 44,427 single-room scenes. We further filter out uncommon room types (e.g., garage, auditorium) and those lacking sufficient + +![](images/0deaea7eb163a647570a08e8e039144a08000bd257df626de9f3476562d11afd.jpg) +Figure 2: The pipeline of ScanForgeQA data construction. + +object content (e.g., aisle, stairwell). The final dataset consists of 34,116 single-room scenes across six common categories: bedroom, kitchen, bathroom, living room, dining room, and storage room. + +Synthesis. To introduce additional diversity and originality, we synthesize novel room layouts using a LLM-guided generation approach. Specifically, we adopt HoloDeck [48], a 3D generation framework that leverages LLMs to parse natural language prompts, retrieve matching assets from large-scale 3D object repositories such as Objaverse [49], and optimize their spatial arrangement to form semantically meaningful scenes. To drive the generation process, we first use GPT-4o to create diverse textual descriptions for various room types. For example, a bedroom may be described as: "A bedroom with a bed, window, armchair, and wardrobe". We define eight room categories, including two additional types—office and store—and generate 20 distinct descriptions for each. These prompts are fed into HoloDeck to produce corresponding room layouts, with human verification to ensure spatial plausibility and realism. This synthesis process yields 160 additional single-room scenes. + +# 4.2 Scan Creation + +To simulate egocentric scanning videos from the constructed single-room scenes, we implement a scanning procedure using the Unity engine3. Each scene is scanned using two complementary strategies designed to emulate natural human visual exploration: + +Orbit Scan. We define a circular trajectory centered in the room at a height of approximately 1.5 meters, corresponding to typical adult eye level. The circle's diameter is set to two-thirds of the shorter side of the room. The camera is randomly initialized at a point on this path and moves along the circle either clockwise or counterclockwise. An image is captured every 5 degrees of rotation, resulting in 72 frames per orbit scan. This strategy provides a comprehensive 360-degree panoramic view of the scene. + +Navigation Scan. To simulate movement through the environment, we label navigable ground regions based on object categories and generate a navigation mesh using the NavMesh Baking API. We randomly select two objects as the navigation start and end points and compute the shortest path between them on the mesh. Among the candidate paths, the two longest are chosen for scanning to achieve a more complete coverage of the scene. For each path, the camera first performs a 360-degree rotation at the starting point, capturing an image every 12 degrees (30 images total). It then traverses the path toward the destination, during which 12 frames are uniformly sampled. Upon arrival, another 360-degree rotation is performed, again capturing 30 images. In total, 72 frames are recorded per path. Due to the limited size of indoor environments, rotational movement yields more visual variation than translation; hence, fewer frames are captured during motion. + +Table 1: Comparison of 3D QA Datasets + +
DatasetSourceScenesFormatQAsScalability
SPARTUN3D3RScan478Point Cloud133KHard
MSQAScanNet, 3RScan, ARKitScenes1.7KPoint Cloud251KHard
3D-LLMScanNet, HM3D1.2KPoint Cloud300KHard
ScanForgeQASimulation/Synthesis34KScan Video925KEasy
+ +# 4.3 QA Generation + +To generate diverse supervised fine-tuning (SFT) data and enhance the 3D spatial reasoning capabilities of existing VLMs, we define three categories of question types: attribute estimation, spatial reasoning, and hypothesis analysis. These categories encompass both quantitative and qualitative dimensions, and cover both open-set and closed-set scenarios. Below, we describe each category in detail, along with the methodology for deriving corresponding ground-truth answers. + +Attribute Estimation. This type focuses on static properties of objects and scenes, such as object count ("How many chairs are in the room?"), object size ("What is the length of the longest side of the refrigerator in meters?"), room size ("What is the size of this room in square meters?"), and room type ("Based on the object layout, what is the most likely type of room (e.g., kitchen)?). Ground-truth answers for these questions are directly derived from 3D scene annotations and object metadata provided in the dataset. + +Spatial Reasoning. This category targets interobject spatial relationships, requiring models to infer positional and geometric properties such as distance, orientation, and contact. Representative question types include: relative distance ("Which of these objects (refrigerator, couch, ceiling light) is closest to the TV?"), absolute distance ("What is the distance between the couch and the table in meters?"), relative direction ("If I am standing by sofa and facing the table, which side is the trash can on?"), and contact relationship ("Is there a gap between the bed and the headboard?"). For distance-related questions, we compute Euclidean distances between object centroids in the global 3D coordinate space. For contact relationships, object dimensions are also considered to determine physical adjacency. To resolve relative direction, + +![](images/0436554e760c47dfcfc0b583799300b4ea95fa004743dad4adc4f7214847e4c4.jpg) +Figure 3: Effects of different scene expression. + +we define an object's front as the side oriented toward the room center. Angular sectors are divided clockwise into four directional categories: right $(45^{\circ} - 135^{\circ})$ , back $(135^{\circ} - 225^{\circ})$ , left $(225^{\circ} - 315^{\circ})$ , and front $(315^{\circ} - 45^{\circ})$ . For example, an object located at $80^{\circ}$ relative to the reference point is classified as being on the right. + +Hypothesis Analysis. This category introduces conditional reasoning under hypothetical scenarios, often requiring geometric and commonsense inference. A typical example is operation feasibility ("Considering only object dimensions, is it feasible to place the television on the table?"). Feasibility is determined by comparing object dimensions. For stacking, the movable object's length and width must be smaller than those of the supporting surface. For embedding (e.g., fitting an item into a drawer), the object's height must also fall within the bounds of the specific container's volume. + +A comparison with existing 3D QA datasets (e.g., SPARTUN3D [50], MSQA [51], and 3D-LLM [17]) is presented in Table 1. The full ScanForgeQA dataset includes 34,276 single-room scenes, 103K simulated video scans, and 925K question-answering pairs for training. Leveraging synthetic environments allows scalable and controlled data generation across diverse spatial scenarios. Additional implementation details are provided in the Appendix C. + +Table 2: Performance comparison on VSI-Bench. † indicates results on VSI-Bench (tiny) set. + +
MethodObj. CountAbs. Dist.Obj. SizeRoom SizeRel. Dist.Rel. Dir.Route PlanAppr. OrderAvgΔ
Close-source
Human Level†94.347.060.445.994.795.895.8100.079.2-
Gemini-1.5 Pro†49.628.858.649.446.048.142.068.048.8-
Gemini-1.5 Pro56.230.964.143.651.346.336.034.645.4-
+SpatialMind63.951.870.247.356.345.942.644.352.8↑7.4%
GPT-4o46.25.343.838.237.041.331.528.534.0-
+SpatialMind40.027.162.740.941.039.637.138.540.8↑6.8%
Open-source
Struct2D46.034.756.442.635.144.933.5-41.9-
InternVL2-8B23.128.748.239.836.730.729.939.634.6-
+SpatialMind35.828.949.744.437.234.835.145.538.9↑4.3%
+ScanForgeQA45.333.454.845.041.136.133.443.041.5↑6.9%
+Both47.032.853.246.639.836.837.947.542.7↑8.1%
InternVL2-40B34.926.946.531.842.132.234.039.636.0-
+SpatialMind36.430.049.141.843.836.135.650.040.4↑4.4%
+ScanForgeQA51.029.252.738.147.236.435.947.642.3↑6.3%
+Both52.230.554.441.050.537.040.250.344.5↑8.5%
Qwen2.5-VL-7B40.322.250.138.938.040.731.435.937.2-
+SpatialMind45.125.252.141.438.741.634.734.539.2↑2.0%
+ScanForgeQA53.230.556.844.942.344.037.337.743.3↑6.1%
+Both55.029.557.344.043.544.338.339.243.9↑6.7%
Qwen2.5-VL-72B37.928.657.449.845.538.420.635.439.2-
+SpatialMind42.332.061.753.848.243.930.439.344.0↑4.8%
+ScanForgeQA45.232.763.352.450.141.732.840.244.8↑5.6%
+Both48.634.468.954.753.443.930.142.747.1↑7.9%
+ +Table 3: Performance comparison on the EM-EQA subset of OpenEQA and the validation set of ScanQA and SQA3D. + +
MethodOpenEQA Acc/ScoreScanQA BLEU-1SQA3D EM-1
Qwen2.5-VL-7B50.1/3.132.517.2
+SpatialMind53.7/3.233.119.8
+ScanForgeQA56.2/3.334.823.3
+Both58.6/3.537.924.5
Qwen2.5-VL-72B53.8/3.235.434.8
+SpatialMind55.7/3.238.039.2
+ScanForgeQA59.1/3.442.543.0
+Both60.4/3.444.146.3
+ +Table 4: Effects of different fin-tuning data and prompting strategy. + +
MethodRoom SizeAvg
Qwen2.5-VL-7B38.937.2
+SQA3D38.838.9
+ScanQA38.539.1
+ScanForgeQA44.943.3
Qwen2.5-VL-72B49.839.2
+CoT-Question50.641.3
+CoT-Scene52.142.7
+SpatialMind53.844.0
+ +# 5 Experiments + +The experimental settings (including benchmarks, baselines, etc.) and more experimental results can be found in the Appendix A and B. + +# 5.1 Performace Comparison + +We investigated the following five key questions to assess our approach: + +Q1: Which scene representation format is most interpretable by VLMs? Figure 3 presents a performance comparison across different representation formats: no additional spatial context (Base), inclusion of a 3D map (+Map), a 2D grid (+Grid), and object-centric textual descriptions (+Des). Across all models, a consistent trend emerges: the +Des variant outperforms others, followed by + +![](images/d2975dde6c8bbf4a67e7d13731ba66f21382f18966549e12a1a836e40a095b2e.jpg) +Figure 4: Performance of Qwen2.5-VL-7B on MVBench and Video-MME. + +![](images/8b6698cf565f38fbce9ae32a0744b71c6bb328961304d873e029537a6350e9a5.jpg) +Figure 5: Ablation study of Qwen2.5-VL-7B under varying numbers of frames and resolution. + ++Grid, while +Map yields the least improvement. These results suggest that current VLMs are more adept at interpreting one-dimensional textual descriptions than high-dimensional structured spatial formats. Consequently, we adopted the textual description format in subsequent experiments as the default scene representation. + +How do SpatialMind and ScanForgeQA impact VLM performance? As shown in Table 2, we progressively applied the SpatialMind prompting strategy and the ScanForgeQA fine-tuning data across a range of VLMs, varying in architectures, parameter size, and openness (including both open- and closed-source models). The results reveal three key findings: 1) Both SpatialMind prompting and ScanForgeQA fine-tuning consistently improve visual-spatial understanding across models. This includes large-scale proprietary models such as Gemini-1.5 Pro $^{4}$ and GPT-4o, demonstrating the effectiveness and generalizability of our approaches. 2) Model size affects the relative benefit of prompting versus fine-tuning. Larger models (e.g., 72B) benefit more from prompting, which enhances their reasoning capabilities, while smaller models (e.g., 7B) show greater improvements through fine-tuning. For instance, Qwen2.5-VL-7B gains $6.1\%$ from fine-tuning, compared to only $2.0\%$ from prompting. 3) Humans and VLMs exhibit complementary strengths. Human participants excel in qualitative tasks (e.g., achieving $100\%$ accuracy on the Appearance Order task) but perform poorly on precise quantitative estimations (e.g., Object Size). In contrast, VLMs show strong quantitative reasoning ability and, in some cases, even surpass human-level performance. This contrast underscores the potential of VLMs to complement human perception in spatial tasks. + +Can combining prompting and fine-tuning yield further gains? To assess whether SpatialMind and ScanForgeQA provide complementary benefits, we applied the SpatialMind prompting strategy to models that have already been fine-tuned on the ScanForgeQA dataset. The results, reported in the “+Both” row of Table 2, show consistent performance improvements across all evaluated models. These findings confirm that the two approaches are complementary. + +Does the improvement generalize to other spatial benchmarks? To assess the generalizability of our framework, we conducted evaluations on multiple benchmarks, including OpenEQA [52], ScanQA [53], and SQA3D [54]. As shown in Table 3, both SpatialMind prompting and ScanForgeQA fine-tuning lead to consistent performance gains across all benchmarks. These results validate the robustness of our approach and confirm its applicability across diverse spatial tasks and datasets. + +Does fine-tuning affect performance on other tasks? To investigate whether enhancing visual-spatial capabilities via fine-tuning adversely impacts a model's general performance, we conducted evaluations on MVBench [55] and Video-MME [56], two broad multi-task video benchmarks. As shown in Figure 4, fine-tuning with ScanForgeQA slightly improves performance on MVBench but leads to a marginal drop on Video-MME. This difference likely stems from MVBench containing spatial reasoning tasks, while Video-MME focuses more on event comprehension. To mitigate this trade-off, we further experimented with mixed fine-tuning, combining a small proportion (5% and 10%) of traditional data from ShareGPT4Video [57] with ScanForgeQA. Results show that + +![](images/62f3ba89cd079ff2d8202857f5d8f609c8fac5c5549dde0f3ccbf9a0222d83c0.jpg) +(a) Route Plan + +![](images/af5f044b2f52675e1133a33a71a57727aca134ef31753d1f1754a44b9b4b62c0.jpg) + +![](images/aa1af5a53edb59a939a4be389ab763c7771f470a0e004528e1555eae56ad548c.jpg) + +![](images/2adc2ac302f3db2d5dc7b9541352346dcdbfbf7a88324c4494c71e622a5269e3.jpg) + +![](images/580f12f6a89bb2bf7bf6226a54055614ed33097aa639d001e3098d5accb13210.jpg) + +![](images/05eb3f9e4306f6de06569cf70cdf014c73ecdce0d80a5e6d0299e9721435c649.jpg) + +![](images/9c070d08fa8e7effbee08febb7c47af6f8208f7f9ebfe0a781ad636c767df5d7.jpg) + +![](images/257e7574fe6a2a9424a949a0d604ce97252342dea26c1cda6563766c7f049842.jpg) + +You are a robot beginning at the chair and facing to lamp. You want to navigate to the lamp on the cabinet. You will perform the following actions (Note: for each [please fill in], choose either 'turn back,' 'turn left,' or 'turn right'). 1. [please fill in] 2. Go forward until the wardrobe. 3. [please fill in]. 4. Go forward until the lamp. You have reached the final destination. + +Qwen2.5-VL-7B: Turn Back, Turn Left + ++Both (Ours):Turn Left,Turn Left + +![](images/d65e3924845419947e4a75a4c7e2aa1a644fa8ebd32a460c6e60eff4b27f2d3f.jpg) +(b) Appearance Order +Figure 6: Two examples from VSI-Bench comparing predictions from Qwen2.5-VL-7B and Ours. + +![](images/613a4ace8a6dcb38de174b610e7b185fb78d2727ce08ce45d43528e0619a90d3.jpg) + +![](images/c186718f03f38525789df1a6626b27567a8ff64d425c3e7b48393150ba92e564.jpg) + +![](images/5aa8f3e3cfef32c23ecda991ce181b211780b68595ba17a9034b62e309c86424.jpg) + +![](images/9c019256f076461d68278e9af44ea61888f7b4942c3f741757156f524287642b.jpg) + +![](images/81113e69d85c5352b4f3732ccdb7ffde14e16395c5dc9697016602fc4ba614c7.jpg) + +![](images/f328357f77b0b58185314945db6bb18f20d0d7cc1971523863e82d72dfebc7bb.jpg) + +What will be the first-time appearance order of the following categories in the video: door, towel, refrigerator, microwave? + +Qwen2.5-VL-7B: door, towel, refrigerator, microwave + ++Both (Ours): towel, microwave, refrigerator, door + +this strategy achieves improved performance, surpassing the original Qwen2.5-VL-7B baseline, suggesting that spatial fine-tuning can be harmonized with broader capabilities through data balancing. + +# 5.2 Ablation Study + +In this section, we explored the impact of various design choices, including prompting strategies, fine-tuning datasets, frame sampling strategies, and input resolution, on the performance of VLMs. + +On prompting strategy. To isolate the contributions of each component in the SpatialMind prompting strategy, we evaluated two variants: one containing only the question component (CoT-Question) and another containing only the scene description (CoT-Scene). As shown in Table 4, both variants independently improve spatial reasoning performance, but are less effective than the full combined prompt. Notably, the scene description contributes more significantly to model performance than the reasoning steps, suggesting its central role in facilitating spatial understanding. + +On fine-tuning data. To investigate the effectiveness of our proposed ScanForgeQA against existing spatial datasets, we fine-tuned Qwen2.5-VL-7B on SQA3D [54] and ScanQA [53]. As shown in Table 4, fine-tuning on either of these datasets results in lower performance compared to ScanForgeQA, and even reduces accuracy on tasks involving precise spatial estimation (e.g., Room Size). This is primarily due to the limited presence of fine-grained spatial estimation samples in the existing datasets. Importantly, both datasets and the VSI-Bench benchmark originate from the same source (i.e., ScanNet [31]), resulting in minimal data discrepancy. This contrast emphasizes the advantage of our simulated data generation pipeline. + +On frames and resolution. To evaluate the robustness of our approach, we analyzed performance sensitivity to the number of input frames and image resolution. Figure 5 visualizes the performance of the +Both variant and the baseline Qwen2.5-VL-7B under various configurations. Our method consistently outperforms the baseline across all settings, with performance further improving as the number of frames and resolution increase. This indicates that our approach remains stable and effective under varying visual input conditions. + +# 5.3 Qualitative Analysis + +In Figure 6, we presented two illustrative examples from VSI-Bench, comparing predictions from the baseline Qwen2.5-VL-7B and our enhanced variant (+Both). In Case (a), Qwen2.5-VL-7B fails to produce the correct directional prediction, likely due to its limited capacity for 3D spatial reasoning. In contrast, our method successfully identifies the correct answer. Case (b) involves a simpler spatial reasoning task, however, Qwen2.5-VL-7B still fails, potentially due to insufficient object localization. + +Our enhanced variant, benefiting from both structured prompting and spatially grounded fine-tuning, demonstrates notable improvements in accuracy and reasoning robustness. + +# 6 Conclusion + +In this work, we present an effective framework for enhancing visual-spatial reasoning in VLMs without modifying their underlying architecture. This makes our approach readily adaptable across models of varying scales and types. By integrating the structured prompting strategy (SpatialMind) with an automatically constructed dataset (ScanForgeQA), we enable VLMs to more effectively interpret and reason about 3D spatial relationships in complex visual scenes. Extensive evaluations across multiple spatial reasoning benchmarks demonstrate that our framework consistently improves accuracy, robustness, and generalization. Furthermore, our analysis reveals that prompting and fine-tuning play complementary roles in advancing visual-spatial understanding. + +# Acknowledgments and Disclosure of Funding + +This work is supported by Shenzhen Science and Technology Program, No.:KQTD20240729102207002; the National Natural Science Foundation of China, No.:62376140, No.:62476071, No.:U24A20328, No.:624B2047, and No.:U23A20315; the Major Key Project of Pengcheng Laboratory; GuangDong Basic and Applied Basic Research Foundation, No.:2025A1515011732; the Science and Technology Innovation Program for Distinguished Young Scholars of Shandong Province Higher Education Institutions, No.:2023KJ128; the Special Fund for Taishan Scholar Project of Shandong Province; the Fundamental Research Funds for the Central Universities, No.:HIT.DZJJ.2025048 and No.:HIT.DZJJ.2024041. + +# References + +[1] Jihan Yang, Shusheng Yang, Anjali W Gupta, Rilyn Han, Li Fei-Fei, and Saining Xie. 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Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Experimental Settings + +# A.1 Benchmarks + +To validate the effectiveness of our SpatialMind strategy and ScanForgeQA data in enhancing the visual-spatial understanding capabilities of VLMs, we conducted evaluations on four comprehensive benchmarks: VSI-Bench [1], OpenEQA [52], ScanQA [53], and SQA3D [54]. The comparisons are shown in Table 5. + +Table 5: Comparison of different benchmarks. Note that the statistics for the ScanQA and SQA3D datasets are reported on their respective validation sets. + +
Dataset#Questions#ScenesSourceQuestion Types
VSI-Bench>5,000288ScanNet, +ScanNet++, +ARKitScenesConfiguration, Measurement +Estimation, Spatio-temporal Reasoning
OpenEQA1,899>180ScanNet, +HM3DObject Recognition, Attribute +Reasoning, Spatial Understanding, +Functional Reasoning
ScanQA9,35371ScanNetObject, Color, Quantity, Location
SQA3D3,26165ScanNetSpatial Relations, Commonsense, +Navigation, Multi-hop Reasoning
+ +# A.1.1 VSI-Bench + +VSI-Bench [1] is a comprehensive evaluation benchmark designed to assess the visual-spatial reasoning capabilities of VLMs in dynamic 3D environments. It comprises over 5,000 high-quality question-answer pairs grounded in 288 diverse indoor video scenes, sourced from real-world 3D reconstruction datasets such as ScanNet [31], ScanNet++ [35], and ARKitScenes [29]. The benchmark defines eight distinct tasks organized into three major categories: configuration-based tasks (e.g., object counting, relative direction, and path planning), measurement estimation tasks (e.g., object size, absolute distance, room size), and spatiotemporal reasoning tasks (e.g., object appearance order). These tasks collectively assess VLMs' abilities to reason about spatial relationships, estimate scale, and understand temporal information. Evaluation is primarily based on accuracy, with tolerance-based error metrics applied to numerical estimation tasks to account for minor deviations. All evaluations follow the official protocols defined in VSI-Bench [1]. + +# A.1.2 OpenEQA + +OpenEQA [52], introduced by Meta AI in 2024, is a benchmark designed to evaluate the spatial understanding and reasoning capabilities of VLMs in real-world indoor environments. The dataset comprises over 1,600 human-annotated question-answer pairs spanning $180+$ diverse indoor scenes and supports two primary evaluation tasks. In our experiment, we focused on the Episodic Memory (EM-EQA) setting, where models must answer questions based solely on a previously observed egocentric video trajectory, without access to external spatial priors. Questions cover a wide range of reasoning types, including object attributes, spatial relationships, object locations, and functional reasoning. To evaluate model performance on open-ended responses, we employed GPT-4o $^{5}$ as an automatic evaluator to assess semantic alignment between predicted and ground-truth answers. This enables consistent and reliable scoring across diverse question formats. + +# A.1.3 ScanQA + +ScanQA [53] is a large-scale benchmark designed to evaluate spatial question answering in real-world indoor environments, grounded in richly annotated 3D scans. Built on top of the ScanNet [31] dataset, it contains over 41,000 human-curated question-answer pairs across 800 RGB-D scenes, with each question grounded in the 3D geometry and semantics of the scene. The dataset covers a broad range of open-ended questions involving object attributes, spatial relations, and scene-level understanding, + +Table 6: Performance comparison on VSI-Bench. † indicates results on VSI-Bench (tiny) set and LLaVA-NV denotes the LLaVA-NeXT-Video model. + +
MethodObj. CountAbs. Dist.Obj. SizeRoom SizeRel. Dist.Rel. Dir.Route PlanAppr. OrderAvgΔ
Close-source
Human Level†94.347.060.445.994.795.895.8100.079.2-
Gemini-2.0 Flash†52.430.666.731.856.046.324.555.145.4-
Gemini-1.5 Pro†49.628.858.649.446.048.142.068.048.8-
Gemini-1.5 Pro56.230.964.143.651.346.336.034.645.4-
+SpatialMind63.951.870.247.356.345.942.644.352.8↑7.4%
GPT-4o46.25.343.838.237.041.331.528.534.0-
+SpatialMind40.027.162.740.941.039.637.138.540.8↑6.8%
Open-source
InternVL2-8B23.128.748.239.836.730.729.939.634.6-
+SpatialMind35.828.949.744.437.234.835.145.538.9↑4.3%
+ScanForgeQA45.333.454.845.041.136.133.443.041.5↑6.9%
+Both47.032.853.246.639.836.837.947.542.7↑8.1%
InternVL2-40B34.926.946.531.842.132.234.039.636.0-
+SpatialMind36.430.049.141.843.836.135.650.040.4↑4.4%
+ScanForgeQA51.029.252.738.147.236.435.947.642.3↑6.3%
+Both52.230.554.441.050.537.040.250.344.5↑8.5%
InternVL2.5-8B7.034.143.042.438.040.123.235.933.0-
+SpatialMind18.729.646.545.240.340.927.846.637.0↑4.0%
+ScanForgeQA56.533.250.743.439.033.128.434.339.8↑6.8%
+Both52.830.753.444.841.138.729.647.042.3↑9.3%
InternVL2.5-38B43.633.053.048.853.535.734.534.042.0-
+SpatialMind56.138.359.352.359.743.639.032.947.7↑5.7%
LLaVA-NV-7B48.514.047.824.243.542.434.030.635.6-
+SpatialMind49.022.647.924.641.343.037.129.836.9↑1.3%
LLaVA-NV-72B48.922.857.435.342.436.735.048.640.9-
+SpatialMind52.524.560.237.742.739.537.351.043.2↑2.3%
VideoLLaMA3-7B36.822.234.724.944.641.736.128.833.7-
+SpatialMind38.623.936.236.642.340.935.933.236.0↑2.3%
Qwen2-VL-7B39.425.025.843.232.630.927.832.632.2-
+SpatialMind43.729.029.446.533.932.832.031.934.9↑2.7%
Qwen2.5-VL-7B40.322.250.138.938.040.731.435.937.2-
+SpatialMind45.125.252.141.438.741.634.734.539.2↑2.0%
+ScanForgeQA53.230.556.844.942.344.037.337.743.3↑6.1%
+Both55.029.557.344.043.544.338.339.243.9↑6.7%
Qwen2.5-VL-72B37.928.657.449.845.538.420.635.439.2-
+SpatialMind42.332.061.753.848.243.930.439.344.0↑4.8%
+ScanForgeQA45.232.763.352.450.141.732.840.244.8↑5.6%
+Both48.634.468.954.753.443.930.142.747.1↑7.9%
+ +![](images/026334517b58dd47e3e387d1e76b14de1b5f10023b07b7e53864f26fb4dc0dee.jpg) +Figure 7: A complete example illustrating the visual prompting process with intermediate outputs. + +such as "What is between the sofa and the table?" or "Where is the lamp located?". To evaluate model performance, we adopted BLEU-1 as the primary metric. Given that the answers are typically short and factual, unigram precision offers a reliable measure of response quality. + +# A.1.4 SQA3D + +SQA3D [54] is a reasoning-centric benchmark designed to evaluate situational understanding within 3D indoor scenes. Built on 650 scenes from ScanNet [31], it comprises approximately 6,800 annotated situations and over 33,000 questions that span a diverse range of reasoning types, including spatial relations, commonsense reasoning, and multi-hop logical reasoning. Each question is grounded in a specific 3D context, requiring precise and spatially informed answers. To evaluate model performance, we used the Exact Match at 1 (EM-1) metric. This metric is well-suited for the benchmark, as answers are typically concise, specific, and context-dependent. + +# A.2 Baselines + +As shown in Tables 6, our evaluation includes a diverse set of VLM baselines spanning a range of architectures and scales. These include open-source models such as InternVL2 [58] (8B, 40B), InternVL2.5 [58] (8B, 38B), LLaVA-NeXT-Video [59] (7B, 72B), VideoLLaMA3 [60] (7B), Qwen2-VL [61] (7B), and Qwen2.5-VL [62] (7B, 72B). We also evaluated closed-source models, including Gemini-2.0 Flash $^{6}$ , Gemini-1.5 Pro $^{7}$ and GPT-4o [63]. For open-source models, we adopted each model's default parameter settings, including learning rate, number of frames, and input resolution. For closed-source models, GPT-4o processes 16 frames per video, while all Gemini models operate at a fixed sampling rate of 1 frame per second (1 FPS). All experiments are conducted on 8 NVIDIA H20 GPUs. + +# B More Experimental Results + +# B.1 Performance Comparison + +In Table 6, we extended our evaluation to additional VLMs with varying architectures and parameter scales, applying both the SpatialMind prompting and ScanForgeQA fine-tuning. The results + +![](images/eee3b7d47bb131b1713394c842f415c89ddf2275e307015b10d5278ee858bfc3.jpg) +Figure 8: Distribution of room types in the ScanForgeQA dataset. + +are consistent with those reported in our main analysis, further reinforcing the effectiveness and generalizability of our proposed methods. Moreover, the combined use of prompting and fine-tuning continues to yield superior performance across models, highlighting their complementary strengths and demonstrating the robustness of our framework. + +# B.2 Case Study + +Figure 7 presents a detailed example that illustrates the full prompting process along with intermediate outputs. The scene decomposition module produces approximate spatial descriptions of object locations, effectively capturing the overall layout of the 3D scene. Meanwhile, the question decomposition module identifies the question type as "object relative distance" and selects the corresponding reasoning steps to guide inference. By combining this structured information and feeding it into the Qwen2.5-VL-7B model fine-tuned with ScanForgeQA, the correct target object is successfully identified. This example demonstrates the interpretability and effectiveness of our framework in performing multi-step spatial reasoning grounded in visual context. + +# C More details for ScanForgeQA + +# C.1 Scene Distribution + +Figure 8 presents the distribution of the eight room categories included in ScanForgeQA. The dataset exhibits a clear long-tailed distribution in which categories such as bathroom and bedroom are heavily represented, whereas store and office types are comparatively rare. This imbalance is primarily due to the inherent distribution of room types in the 3D-FRONT dataset [47]. + +# C.2 Scan Example + +We provided two frames captured by the camera during the creation of the scanning video, as shown in Figure 9, to visually illustrate this process. Additionally, we further provided scanning creation scripts as well as more video demos in the Supplemental Material: + +- "codes/nav.script.py" is the script that creates the navigation scan, presenting the exact implementation. +- "Creation of Scanning Video.mp4" illustrates the creation process of the scanning video from both first-person and third-person perspectives. +- "Scanning Video.mkv" presents the rendered scene scan resulting from the aforementioned scanning process. + +![](images/e04a7e616bbc8bf211d285b9a35eff057a426e3b3bde0afb10a382b4a7b7d526.jpg) +Figure 9: Screenshot from the scan creation process. + +![](images/f5e9d0f0fe3bdf1bfc79e538b171c3567107217033f20235ca2dd683f3c1f812.jpg) + +# C.3 QA Definition + +Table 7 provides representative templates and corresponding answer sources for each question type in ScanForgeQA. This information clarifies the coverage of different reasoning categories and indicates which scene annotations are used to derive ground-truth answers. + +# D More details for SpatialMind + +# D.1 Prompt for Object Description + +As a representative example, Figure 10 illustrates the prompt used to generate textual descriptions of object positions. This prompt guides the model to produce structured spatial layouts in natural language based on the visual input. Similar prompts are used for other reasoning components, following a consistent design pattern. Other prompts are provided in the "codes/gen_structure_exp.py" file of Supplementary Material for reference and reproducibility. + +# D.2 Reasoning Steps for Different Questions + +Figure 7 provides an example of the detailed reasoning steps used for the relative distance question type. Each question type is paired with a concise, structured reasoning process that offers a generalizable solution strategy for VLMs. All reasoning steps for the various question types are included in the "codes/reason_steps.py" file of Supplementary Material for reference and reproducibility. + +# E Limitations + +This paper demonstrates that existing VLMs tend to rely more heavily on understanding textual descriptions when performing spatial reasoning. However, textual descriptions alone may not capture spatial semantics as intuitively as visual input, potentially limiting the upper bound of performance for such methods. Moreover, we primarily focus on basic single-room 3D scene. Although strong performance has been achieved on OpenEQA [52], which includes some multi-room scenes, more complex settings, such as large-scale indoor layouts and outdoor environments, remain underexplored and represent important directions for future work. + +# F Broader Impacts + +Our approach improves visual-spatial understanding in VLMs, enabling better interpretation of object layouts and spatial relationships. This enhancement offers potential benefits in areas such as embodied AI and AR/VR applications, where spatial reasoning is essential. However, similar to other prompting and fine-tuning methods, our framework may inherit biases from pre-trained models or training data, potentially leading to unfair or inaccurate spatial interpretations. While our method reduces reliance on expensive 3D sensors, evaluating spatial accuracy remains a key challenge, especially for + +downstream tasks that demand high robustness and trustworthiness. Despite these limitations, we believe that releasing our framework and dataset will foster transparency, support reproducibility, and accelerate progress toward trustworthy and robust spatial reasoning systems. + +Table 7: Overview of question templates and their corresponding answer sources. + +
TypeQuestion TemplateAnswer Source
Attribute Estimation
Object CountHow many <A> are there in the room? +What is the total number of <A>? +What is the length of the longest side of <A> in meters?Name of object instance
Object SizeWhat is the size of <A> in square meters? +What is the length of the shortest side of <A> in meters? +How tall is <A> in meters?Dimension of the object model
Room SizeWhat is the size of the room in square me- +ters? +Based on object layout, what is the most +likely type of this room? +Is this space a living room, a kitchen, or +something else?Scene boundaries and scale
Room TypeRoom labels
Spatial Reasoning
Relative DistanceWhich of these objects (<A>, <B>, <C>) is +the closest to <R>? +Among the listed objects (<A>, <B>, <C>), +which one is closest to <R>? +What is the distance between <A> and <B> +in meters?The minimum distance +between the centroids of +objects
Absolute DistanceMeasure the distance from <A> to <B> in +meters. +How far is <A> from <B> in meters?Euclidean distance +between objects
Relative DirectionIf I am standing by <A> and facing <B>, +which side is object <R> on? +From the viewpoint at <A> facing <B>, +where is <R>?The direction of +connections between +objects
Contact RelationshipIs there a gap between <A> and <B>? +Are <A> and <B> touching each other?Comparison of distances +and sizes between objects
Hypothesis Analysis
Operation FeasibilityConsidering only object sizes, is there +enough space to put <A> in <B>? +Considering only object dimensions, is it +feasible to place <A> on <B>?Comparison of sizes +between objects
+ +# Prompt + +# [Task] + +You are given a video (multiple frames) capturing an indoor scene. Your goal is to recognize {categories_of_interest} objects, analyze the spatial layout of the scene, and describe the relative position of each object. + +# [Instructions] + +1. Per-frame analysis: + +- For each frame, choose one object as a **local reference point**. +- Predict the relative position of all other objects with respect to this local reference point. +- Express relative positions using simple terms like "left", "right", "front", "behind", "above", "below", and approximate distances (e.g., "2 meters to the right"). + +2. Global scene layout: + +- Take the **local reference point from the first frame** as the **global reference point** for the whole video. +- Use overlapping objects between frames to align frames together. +- Gradually build the spatial descriptions for all objects relative to the global reference point. + +# [Rules] + +- If a category has multiple instances (e.g., two chairs), describe each instance separately. +- Preserve the real-world spatial relationships and distances as accurately as possible. +- Use clear and consistent directional and distance terms. + +# [Output Format] + +ONLY Return the result as a JSON dictionary following STRICTLY this format: + +```txt +{ +"category name": "global reference point", +"another category name": "Position description relative to the global reference point", +... +} +``` + +# Example: + +"chair": "The chair is located 1.5 meters to the left and 0.5 meters behind the global reference point", + +"table": "The table is located 2 meters to the 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0000000000000000000000000000000000000000..9af209880e4239942e9f550a2a869aab0f270fb6 --- /dev/null +++ b/NeurIPS/2025/Spatial-Aware Decision-Making with Ring Attractors in Reinforcement Learning Systems/full.md @@ -0,0 +1,958 @@ +# Spatial-Aware Decision-Making with Ring Attractors in Reinforcement Learning Systems + +Marcos N. Saura + +The University of Manchester + +Manchester, United Kingdom + +marcos.negresaura@postgrad.manchester.ac.uk + +Richard Allmendinger + +The University of Manchester + +Manchester, United Kingdom + +Richard.Allmendinger@manchester.ac.uk + +# Wei Pan + +The University of Manchester + +Manchester, United Kingdom + +wei.pan@manchester.ac.uk + +# Theodore Papamarkou + +PolyShape + +Athens, Greece + +theodore@polyshape.com + +# Abstract + +Ring attractors, mathematical models inspired by neural circuit dynamics, provide a biologically plausible mechanism to improve learning speed and accuracy in Reinforcement Learning (RL). Serving as specialized brain-inspired structures that encode spatial information and uncertainty, ring attractors explicitly encode the action space, facilitate the organization of neural activity, and enable the distribution of spatial representations across the neural network in the context of Deep Reinforcement Learning (DRL). These structures also provide temporal filtering that stabilizes action selection during exploration, for example, by preserving the continuity between rotation angles in robotic control or adjacency between tactical moves in game-like environments. The application of ring attractors in the action selection process involves mapping actions to specific locations on the ring and decoding the selected action based on neural activity. We investigate the application of ring attractors by both building an exogenous model and integrating them as part of DRL agents. Our approach significantly improves state-of-the-art performance on the Atari 100k benchmark, achieving a $53\%$ increase in performance over selected baselines. + +# 1 Introduction + +This paper addresses the challenge of efficient action selection in Reinforcement Learning (RL), particularly in environments with spatial structures, ranging from robotic manipulation where joint movements are coupled, to game-playing agents where tactical decisions might depend on positional awareness. We integrate ring attractors, a neural circuit model from neuroscience originally proposed by [50] and later experimentally validated by [18], into the RL framework. This approach provides a mechanism for uncertainty-aware decision-making in RL, thereby yielding more efficient and reliable learning in complex environments. Ring attractors offer a unique framework to represent spatial information in a continuous and stable manner [33]. In a ring attractor network, neurons interconnect circularly, forming a loop with tuned connections [7]. This configuration allows for robust and localized activation patterns, maintaining accurate spatial representations even in the + +presence of noise or perturbations. Applying ring attractors to the selection of RL actions involves mapping actions to specific ring locations and decoding the selected action based on neural activity. This spatial embedding proves advantageous for continuous action spaces, particularly in tasks such as robotic control and navigation [28]. Ring attractors improve decision-making by exploiting spatial relations between actions, contributing to informed transitions between actions in sequential decision-making tasks in RL. While many action spaces possess inherent topological structure [39], traditional neural networks represent actions as orthogonal vectors that ignore these relationships. Ring attractors bridge this gap by providing a neural substrate that preserves spatial ordering and similarity between actions. This is particularly relevant given recent advances showing that spatial structure in representations improves sample efficiency [25, 49] and generalization [3, 13], yet these benefits have not been systematically applied to action selection mechanisms themselves. + +In what follows, we summarize our contributions. Briefly, our contributions include a novel approach to RL policies based on ring attractors, the inclusion of uncertainty-aware capabilities in our RL systems, and the development of Deep Learning (DL) modules for RL with ring attractors. + +- We propose a novel approach for integrating ring attractors into RL, incorporating spatial information, temporal filtering, and relations between actions. This spatial awareness significantly speeds up the learning rate of the RL agent. See Sections 3.1.2 and 4.1. +- We utilize a Bayesian approach to build ring attractors input signals, encoding uncertainty estimation to drive the action selection process, showing further performance improvement. See Sections 3.1.3 and 4.1. +- We develop a reusable DL module based on recurrent neural networks that incorporates ring attractors into DRL agents. This allows integration and comparison with existing DRL frameworks and baselines, enabling the adoption of our ring attractor approach across different RL models and tasks in applied domains. See Sections 3.2 and 4.2. + +Codebase available at https://github.com/marcosaura/RA_RL. + +# 2 Related Research + +The integration of ring attractors into RL systems brings together neuroscience-inspired models and advanced machine learning techniques. In this section, we review the literature on key areas that form the foundation of our RL research: spatial awareness in RL, biologically inspired RL approaches, and uncertainty quantification methods. + +# 2.1 Spatial Awareness in RL + +Incorporating spatial awareness into RL systems has improved performance on tasks with inherent spatial structure. Regarding relational RL, [49] introduced an approach using attention mechanisms to reason about spatial relations between entities in an environment. This method demonstrated improved sample efficiency and generalization in tasks that require spatial reasoning. On the topic of navigation, [25] developed a DRL agent capable of navigating complex city environments using street-level imagery. Their approach incorporated auxiliary tasks, such as depth prediction and loop closure detection. Concerning explicit spatial representations, [13] proposed a cognitive mapping and planning approach for visual navigation, combining spatial memory with a differentiable neural planner. Similarly, [3] introduced a relational DRL framework using graph neural networks to capture spatial relations between objects. Although these approaches demonstrate the importance of spatial awareness in RL, they rely on emergent learning to develop spatial understanding through comprehensive architectural additions, which require extensive training to discover action relationships. In contrast, our ring attractor approach explicitly encodes spatial structure in the action space, providing spatial inductive bias that accelerates learning rather than relying on the RL agent to autonomously discover all critical action space relationships, which may result in suboptimal or incomplete spatial understanding. + +# 2.2 Biologically Inspired Machine Intelligence + +Biologically inspired approaches to RL seek to leverage insights from neuroscience to improve the efficiency, adaptability, and interpretability of RL algorithms. These methods often draw upon neural circuit dynamics and cognitive processes observed in biological systems. A notable example is + +the work in [45], which demonstrated how neural attractor states implement probabilistic choices through recurrent network dynamics. The work presented in [2] demonstrated that incorporating grid-like representations, inspired by mammalian grid cells, into RL agents improved performance in navigation tasks. Their work showed that these biologically inspired representations emerged naturally in agents trained on robotics tasks and transferred well to new environments. Similarly, [10] showed that recurrent neural networks trained on navigation tasks naturally developed grid-like representations, suggesting a deep connection between biological and artificial navigation systems. In RL, grid cells and ring attractors serve distinct computational roles. Grid cells encode spatial state information (the agent's position), through hexagonal firing patterns, functioning as fixed basis functions for value functions and spatial generalization [2, 10]. Ring attractors encode persistent states, directional and spatial bias, through recurrent dynamics that actively maintain information over time [16]. While grid cells provide emergent spatial input features, ring attractors are dynamical components that provide intrinsic spatial encoding for memory states [17], that here we translate to encoding of the action space. [31] demonstrated how RL agents naturally develop insect-like behaviors and neural dynamics when solving complex spatial navigation tasks. [44] proposed a biologically inspired meta-RL algorithm that mimics the function of the prefrontal cortex and dopamine-based neuromodulation. Their approach demonstrated rapid learning and adaptation to new tasks, similar to the flexibility observed in biological learning systems. However, these approaches primarily focus on mimicking neural representations or learning dynamics, whereas ring attractors specifically encode spatial relationships in the action space itself, providing both biological plausibility and direct spatial awareness for action selection. + +# 2.3 Uncertainty Quantification + +Regarding exploration strategies, [27] introduced bootstrapped Deep Q-networks (DQNs), addressing exploration by leveraging uncertainty in Q-value estimates by training multiple DQNs with shared parameters. Building on this theme, [8] proposed random network distillation (RND), measuring uncertainty by comparing predictions between a target network and a randomly initialized network. For efficient uncertainty quantification, [11] and [9] presented a novel "masksemble" approach, applying masks across the input batch during the forward pass to generate diverse predictions. Addressing risk assessment in non-stationary environments, [14] described a method to analyze sources of lack of knowledge by adding a second Bayesian model to predict algorithmic action risks, particularly relevant for multi-agent RL (MARL) systems. [19] developed a Bayesian ring attractor that outperforms conventional ring attractors by dynamically adjusting its activity based on evidence quality and uncertainty. In the context of individual treatment effects, [20] performed uncertainty quantification (UQ) using an exogenously prescribed algorithm, making the method agnostic to the underlying recommender algorithm. [1] developed a Bayesian approach for RL in episodic high-dimensional Markov decision processes (MDPs). They introduced two novel algorithms: LINUCB and LINPSRL. These algorithms achieve significant improvements in sample efficiency and performance by incorporating uncertainty estimation into the learning process. In the context of DRL, Bayesian Deep Q-networks (BDQNs) [1] incorporate efficient Thompson sampling and Bayesian linear regression at the output layer to factor uncertainty estimation in the action-value estimates. While these methods treat uncertainty estimation as a separate module applied to existing architectures, our ring attractor approach integrates uncertainty through the variance parameters of Gaussian input signals, making uncertainty awareness an intrinsic part of the spatial action representation. + +In summary, the literature reveals a growing interest in incorporating spatial awareness, biological inspiration, and UQ into RL systems. However, there remains a gap in integrating these elements into a cohesive framework that simultaneously provides biological plausibility, explicit spatial encoding, and natural uncertainty handling within a single neural architecture. Our work on ring attractors aims to bridge this gap by providing a biologically plausible model that inherently captures spatial relations and can be extended to handle uncertainty, potentially leading to more robust and efficient RL agents. + +# 3 Methodology + +Ring attractors represent a computational principle where neurons are arranged in a circular topology to maintain spatial representations. Originally proposed by [50] for encoding heading direction and empirically discovered by [18] in Drosophila, these networks exhibit circular organization where head direction cells encode different spatial directions [38]. For our methodology, ring attractors + +integrate multiple cues through distance-weighted connections [51, 12] and maintain representational stability through excitatory-inhibitory dynamics [37]. This bioinspired structure has been repurposed in this line of research to explicitly encode the action space during the decision-making process. In this section, we describe two main methods: an exogenous ring attractor model using continuous-time recurrent neural networks (CTRNNs) and a DL-based ring attractor integrated into the RL agent. While the CTRNN model validates the theoretical principles, the DL architecture eases deployment and adoption in existing DRL frameworks. Both leverage the ring attractors' spatial encoding capabilities to enhance action selection and performance. In the CTRNN model, we detail the ring attractor architecture, Section 3.1.1; dynamics and implementation, Section 3.1.2; and uncertainty injection, Section 3.1.3. CTRNNs are used for their ability to model continuous neural dynamics and maintain stable attractor states [4]. The integrated DL approach offers end-to-end training for efficiency and scalability, detailed in Section 3.2. Ring attractors in RL maintain stable spatial information representations, preserving action relations lost in traditional flattened action spaces. This circular spatial representation potentially yields smoother policy gradients and more efficient learning in spatial tasks, attributed to the ring attractors' ability to maintain a stable representation of spatial information. + +# 3.1 Continuous-Time RNN Ring Attractor Model + +During the first stage of the research, the focus is on developing a self-contained exogenous ring attractor as a CTRNN. This will be integrated into the output of the value-based policy model to perform action selection. + +# 3.1.1 Ring Attractor Architecture + +Ring attractors commonly consist of a configuration of excitatory and inhibitory neurons arranged in a circular pattern. We can model the dynamics of the ring using the Touretzky ring attractor network [40]. In this model, each excitatory neuron establishes connections with all other excitatory neurons, and an inhibitory neuron is placed in the middle of the ring with equal weighted connections to all excitatory neurons. + +Excitatory neurons' input signal: Let $x_{n}^{i}\in \mathbb{R}^{s}$ denote the input signal from source number $i$ to the excitatory neuron $n = 1,\dots ,N$ . The total input to neuron $n$ is defined as the sum of all input signals $I$ for that particular neuron: $x_{n} = \sum_{i = 1}^{I}x_{n}^{i}$ , where $x_{n}\in \mathbb{R}$ . To model input signals $x_{n}^{i}$ of varying strengths, these signals are commonly viewed as Gaussian functions $x_{n}^{i}:\mathbb{R}^{s}\to \mathbb{R}^{s}$ . These functions allow us to represent the input to each neuron as a sum of weighted Gaussian distributions. The key parameters of these Gaussian functions are: $K_{i}$ , the magnitude variable for the input signal in index $i$ , which determines the overall strength of the signal; $\mu_{i}$ , which defines the mean position of the Gaussian curve in the ring for the input signal $i$ , representing the central focus of the signal; $\sigma_{i}$ , the standard deviation of the Gaussian function, which determines the spread or reliability of the signal; and $\alpha_{n}$ , which represents the preference for the orientation of the neuron $n$ in space. These parameters combine to the following: + +$$ +x _ {n} \left(\alpha_ {n}\right) = \sum_ {i = 1} ^ {I} x _ {n} ^ {i} \left(\alpha_ {n}\right) = \sum_ {i = 1} ^ {I} \frac {K _ {i}}{\sqrt {2 \pi \sigma_ {i}}} \exp \left(- \frac {\left(\alpha_ {n} - \mu_ {i}\right) ^ {2}}{2 \sigma_ {i} ^ {2}}\right) \tag {1} +$$ + +Neuron activation function: We employ rectified linear unit (ReLU) function $f(x) = \max(0, x + h)$ , where $h \in \mathbb{R}^+$ as activation function for each neuron, where $h$ is a threshold that introduces the non-linear behaviour in the ring. + +Excitatory neuron dynamics: The dynamics of excitatory neurons in the ring is described as: + +$$ +\frac {\mathrm {d} v _ {n}}{\mathrm {d} t} \approx \frac {\Delta v _ {n}}{\Delta t} = \frac {v _ {n + \Delta t} - v _ {n}}{\Delta t} = \frac {f \left(x _ {n} + \epsilon_ {n} + \eta_ {n}\right)}{\tau} - v _ {n} \tag {2} +$$ + +In Eq. (2) $v_{n} \in \mathbb{R}$ represents the activation of the excitatory neuron $n$ , $x_{n}$ is previously defined in Eq. (1) is the external input to the neuron $n$ of Eq. (1), $\epsilon_{n} \in \mathbb{R}$ represents the weighted influence of the other excitatory neurons activation, which is defined mathematically in Eq. (4). $\eta_{n} \in \mathbb{R}$ is the influence from the weighted inhibitory neuron activation to the target excitatory neuron $n$ , and $\tau = \Delta t$ is the time integration constant. This equation captures the evolution of neuronal activation over time, considering both excitatory and inhibitory activations. + +Inhibitory neuron dynamics. The activation of the inhibitory neuron, which regulates network dynamics, is described by: + +$$ +\frac {\mathrm {d} u}{\mathrm {d} t} \approx \frac {\Delta u}{\Delta t} = \frac {u _ {+ \Delta t} - u}{\Delta t} = \frac {f \left(\epsilon_ {n} + \eta_ {n}\right)}{\tau} - u \tag {3} +$$ + +Here, $u \in \mathbb{R}$ represents the inhibitory neuron's activation output, $\epsilon_{n}$ is the weighted sum of excitatory activations where in this case $n$ is the inhibitory neuron, and $\eta_{n} \in \mathbb{R}$ is the weighted self-inhibition activation term. This equation models how the inhibitory neuron integrates inputs from the excitatory population and its own state. + +Synaptic weighted connections: The influence between neurons decreases with distance, as modeled by the weighted connections. This weighted connection applies to both excitatory and inhibitory neurons. For excitatory neurons: $w^{(E_m \rightarrow E_n)} = e^{-d_{(m,n)}^2}$ , where $d_{(m,n)} = |m - n|$ is the distance between neurons $m$ and $n$ . For the inhibitory neuron: $w^{(I \rightarrow E_n)} = e^{-d_{(m,n)}^2} = e^{-1}$ . Note that our model contains a single inhibitory neuron placed in the middle of the ring, with a distance of 1 unit to all excitatory neurons. The excitatory $(\epsilon_n)$ and inhibitory $(\eta_n)$ weighted connections are also known in the literature as neuron-proximal excitatory and inhibitory voltage or potential. These are then defined as follows: + +$$ +\epsilon_ {n} = \sum_ {m = 1} ^ {N} w _ {m, n} ^ {\left(E _ {m} \rightarrow E _ {n}\right)} v _ {m} \quad \eta_ {n} = w ^ {(I \rightarrow E _ {n})} u \tag {4} +$$ + +# 3.1.2 Ring Attractor as Behavior Policy in RL + +To integrate the ring attractor model with RL, we need to establish a connection between the estimated value of state-action pairs and the input to the ring attractor network. This integration allows the ring attractor to serve as a behavior policy, guiding action selection based on the values learned. + +Excitatory neurons' action input signal: We begin by reformulating the input function for a target excitatory neuron $n$ from Eq. (1). To integrate the ring attractor with RL, we reformulate the input signals by mapping $K_{i}$ to the Q-value $Q(s,a)$ , effectively substituting input signal index $i\in I$ with actions $a\in A$ . In other words, the key modification is setting the scale factor $K_{i}$ to the Q-value $Q(s,a)$ of the state-action pair $(s,a)$ , that is $K_{i} = Q(s,a)$ . + +This formulation is represented in Fig. 1 and described in Eq. (5) ensures that actions with higher estimated values are given more weight in the ring attractor dynamics, naturally biasing the network towards more valuable actions. The orientation of the signal within the ring attractor is determined by the direction of movement in the action space. We represent this as $\mu_{i} = \alpha_{a}(a)$ , where $\alpha_{a}(a)$ is the angle corresponding to the action $a$ in the circular action space. We define our circular action space $\mathcal{A}$ as a subset of $\mathbb{R}^2$ , where each action $a \in \mathcal{A}$ is represented by a point on the unit circle. The function $\alpha: \mathcal{A} \to [0, 2\pi)$ maps each action to its corresponding angle on this circle, and $\alpha_{n}$ which presents the preference for the orientation of the neuron $n$ in space. To account for uncertainty in our value estimates, we incorporate the variance of the estimated value for each action $\sigma_{i} = \sigma_{a}$ . + +This allows the network to represent not just the expected value of actions, but also the confidence in those estimates. Combining these, we arrive at the following equation for the action signal $x_{n}(Q)$ : + +$$ +x _ {n} (Q (s, a)) = \sum_ {a = 1} ^ {A} \frac {Q (s , a)}{\sqrt {2 \pi \sigma_ {a}}} \exp \left(- \frac {\left(\alpha_ {n} - \alpha_ {a} (a)\right) ^ {2}}{2 \sigma_ {a} ^ {2}}\right) \tag {5} +$$ + +This equation represents the input to each neuron as a sum of Gaussian functions, where each function is centered on an action's direction and scaled by its Q-value. + +Full excitatory neuron dynamics. The dynamics of the excitatory neurons in the ring attractor, now incorporating the Q-value inputs, are described by: + +$$ +\frac {\mathrm {d} v _ {n}}{\mathrm {d} t} = \frac {1}{\tau} \left(\max \left(0, \left(\sum_ {m = 1} ^ {m = N} w _ {m, n} ^ {(E \rightarrow E)} v _ {m} + x _ {n} (Q) + w ^ {(I \rightarrow E _ {u})} u\right)\right)\right) - v _ {n} \tag {6} +$$ + +This equation captures how the activation of each neuron evolves over time, influenced by the Gaussian functions of the input action value $x_{n}(Q)$ , the excitatory feedback $\epsilon_{n} = \sum_{m=1}^{N} w^{(E_{m} \to E_{n})} v_{m}$ , and the inhibitory feedback $\eta_{n} = w^{(I \to E_{n})} u$ . + +Full inhibitory neuron dynamics. The complete dynamics of the inhibitory neuron are given by: + +$$ +\frac {\mathrm {d} u}{\mathrm {d} t} = \frac {1}{\tau} \left(\max \left(0, \left(u + \sum_ {n = 1} ^ {N} w _ {n} ^ {(E _ {n} \rightarrow I)} v _ {n}\right)\right)\right) - u, \tag {7} +$$ + +where the self-inhibition term $u$ and excitatory input $v_{n}$ from each excitatory neuron $n$ from Eq. (6) updates the activation of the inhibitory neuron $\frac{\mathrm{d}u}{dt}$ . Eq. (6) and Eq. (7) collectively describe the dynamics of the ring attractor network, capturing the interplay between excitatory and inhibitory neurons, external inputs, and synaptic connections. To translate the ring attractor's output into an action in the 2D space, we use the following equation: + +$$ +\text {a c t i o n} = \operatorname {a r g m a x} \left\{\mathbf {V} \right\} \frac {N ^ {(A)}}{N ^ {(E)}}, \tag {8} +$$ + +where $n\in \{1,\dots,N^{(E)}\}$ $N^{(E)}$ is the number of excitatory neurons in the ring attractor, $N^{(A)}$ is the number of discrete actions in the action space $\mathcal{A}$ , and the activation of excitatory neurons around the ring $\mathbf{V} = [v_{1},v_{2},\dots,v_{N^{(E)}}]$ . This equation assumes that both the neurons in the ring attractor and the actions in the action space are uniformly distributed. This approach allows for nuanced action selection that takes into account both the spatial relations between actions and their estimated values. A visualization of the ring is presented in Fig. 1. + +![](images/81475fd324865f9f2262b01c040fae52dd4d2e5c29be3261630674bab072a719.jpg) +Figure 1: Ring attractor Touretzky representation: Circular arrangement of excitatory neurons (N0-N7) and central inhibitory neuron. Four input signals shown as colored gradients. Overall activation depicted by red outline. Includes connection weights and input signal parameters, illustrating the final ring attractor dynamics state from Eq. (8). + +# 3.1.3 Uncertainty Quantification Integration + +In the field of DRL, for any state-action pair $(s,a)$ , the Q-value $Q(s,a)$ can be expressed as a function of the input state through a function approximation algorithm $\Phi_{\theta}(s)$ taking as input the current state $s$ . This function approximation algorithm $(\Phi_{\theta}(s))$ can be expressed as the weight matrix of our function approximation algorithm transposed $\theta^T$ times the feature vector extracted from the input state $x(s):Q(s,a) = \Phi_{\theta}(s) = \theta^Tx(s)$ [34]. For further context on function approximation in RL see Appendix A.2. As stated in Section 3.1.2, the variance of the Gaussian functions, input to the ring attractor, will be given by the variance of the estimate value for that particular action $\sigma_{i} = \sigma_{a}$ . Among the various methods to compute the uncertainty of the action $(\sigma_{a})$ we have chosen to compute a posterior distribution with Bayesian linear regression (BLR). BLR acts as output layer for our neural network (NN) of choice. + +We choose a linear regression model because it does not compromise the efficiency of the NN, while at the same time it provides a distribution to compute the variance for the state-action pairs. The implementation is based on [1], where a Bayesian value-based DQN model was instantiated to output an uncertainty-aware prediction for the state-action pairs. The new $Q$ function is defined as: + +$$ +Q (s, a) = \Phi_ {\theta} (s) ^ {T} w _ {a}, \tag {9} +$$ + +where $w_{a}$ are the weights from the posterior distribution of the BLR model. Eq. (9) represents the parameters of the final Bayesian linear layer. When provided with a state transition tuple $(s,a,r,s')$ , where $s$ is the current state, $a$ is the action taken, $r$ is the reward received, and $s'$ is the next state. This tuple represents + +Algorithm 1 CTRNN Ring Attractor Action Selection + +Require: Input: State $s$ , action $a$ from Action space $\mathcal{A}$ . + +1: Step 1: Compute Q-values and Uncertainty +2: Compute mean $\bar{Q}(s, a)$ and variance $\sigma_{a}^{2}$ using Eq. 11 +3: Step 2: Generate Input Action Signals +4: for each excitatory neuron $n$ do +5: Generate Gaussian action signal using Eq. 5 +6: end for +7: Step 3: Reach Attractor State +8: for timestep $t$ until $T$ do $\triangleright$ choose $T = 50$ empirically +9: for each excitatory neuron $n$ do +10: $\frac{\mathrm{d}v_n}{\mathrm{d}t}$ Update excitatory neurons using Eq. 6. +11: end for +12: $\frac{\mathrm{d}u}{\mathrm{d}t}$ Update inhibitory neuron using Eq. 7 +13: end for +14: Step 4: Translate Neural Activity $V$ to action selected from action space $\mathcal{A}$ using Eq. 8 +15: return action + +a single step of interaction between agent and environment. The model learns to adjust the weights + +$w_{a}$ of the BLR and the function approximation algorithm, i.e. neural networks (NNs) $(\Phi_{\theta})$ , to align the Q values with the optimal action $a = \arg\max(\gamma \Phi_{\theta}(s)^T w_{a})$ , + +$$ +Q (s, a) = \Phi_ {\theta} (s) ^ {T} w _ {a} \rightarrow y := r + \gamma \Phi_ {\theta_ {\text {t a r g e t}}} \left(s ^ {\prime}\right) ^ {T} w _ {\text {t a r g e t}} \hat {a}, \tag {10} +$$ + +where $\gamma$ is the discount factor, $y$ is the expected Q-value, $\Phi_{\theta_{\mathrm{target}}}$ are the features from the next state $s^{\prime}$ extracted by the function approximation algorithm $\Phi$ using the target network parameters, $\theta_{\mathrm{target}}$ refers to the target function approximation algorithm parameters, and $\hat{a}$ is the predicted optimal action in the next state $s^{\prime}$ . The construction of the Gaussian BLR prior distribution and the weights $i$ sample $w_{a,i}$ collected from the posterior distribution are performed through Thompson Sampling. This process allows us to incorporate uncertainty into our action-value estimates. For details on the construction of the Gaussian prior distribution and the specifics of the sampling process, we refer the reader to [1]. Both the mean $\bar{Q} (s,a)$ and the variance $\bar{\sigma}_a^2$ of Eq. (11) are calculated from a finite number of samples $I$ . + +$$ +\bar {Q} (s, a) = \frac {\sum_ {i = 0} ^ {i = I} Q (s , a) _ {i}}{I} = \frac {\sum_ {i = 0} ^ {i = I} w _ {a , i} ^ {T} \Phi_ {\theta} (s _ {t})}{I}, \bar {\sigma} _ {a} ^ {2} = \frac {\sum_ {i = 0} ^ {i = I} \left(w _ {a , i} ^ {T} \Phi_ {\theta} (s _ {t}) - \mu_ {a}\right) ^ {2}}{I - 1} \tag {11} +$$ + +# 3.2 Deep Learning Ring Attractor Model + +To further enhance the ring attractor's integration into RL frameworks and agents, we provide a Deep Learning (DL) implementation. This approach improves model learning and integration with DRL agents. Our implementation offers both algorithmic improvements, by benefiting from DL training process, and software integration improvements, easing the deployment processes. Recurrent Neural Networks (RNNs) offer an approach for integrating ring attractors within DRL agents. Recent studies by [22] show that RNNs perform well in modeling sequential data and in capturing temporal dependencies for decision-making. Like CTRNNs, RNNs mirror ring attractors' temporal dynamics, with their recurrent connections and flexible architecture emulating their interconnected nature. Allowing modelling weighted connections for forward and recurrent hidden states, as shown in Appendix A.6. The premises for modeling RNN are as follows. + +Attractor state as recurrent connections. RNN recurrent connections model the attractor state, integrating information from previous time steps into the current network state, allowing for the retention of information over time. + +Signal input as a forward pass. Forward connections from previous layers are arranged circularly, mimicking the ring's spatial distribution. The attractor state encodes the task context, influenced by the current input and hidden state. A learnable time constant $\tau$ , inherited from Eq. (2), controls input and temporal evolution, enabling adaptive behavior and elastic contribution to the attractor state. + +# 3.2.1 DRL Agent Integration + +To shape circular connectivity within a RNN, the weighted connections in the input signal $V(s)$ and the hidden state or attractor state $U(v)$ are computed as in (12). This circular structure mimics the arrangement of excitatory neurons in the ring attractor. Eq. (12) shows the input signal to the recurrent layer $V_{m,n}$ from neuron $m$ from the previous layer in the DL agent to neuron $n$ in the RNN. The hidden state, $U_{m,n}$ mimics an attractor state, representing the recurrent connections in the RNN. The weighted RNN connections include fixed input-to-hidden connections $(w^{I\rightarrow H}m,n)$ to maintain the ring's spatial structure, and learnable hidden-to-hidden connections $(w^{H\rightarrow H}m,n)$ to capture emerging action relationships. + +These depend on a parameter $\lambda$ that drives the decay over distance and distance between neurons $d(m,n)$ where $N$ is the total number of neurons for the RNN and $M$ is the count of neurons in the previous layer of the NN architecture. The function $\phi_{\theta}(s):\mathbb{R}^{S}\to \mathbb{R}^{M}$ maps the input state $s$ of the DL agent to a representation of characteristics that will be the input of the recurrent layer. Likewise, $\theta$ + +$$ +V (s) _ {m, n} = \frac {1}{\tau} \Phi_ {\theta} (s) ^ {T} w _ {m, n} ^ {I \rightarrow H} = \frac {1}{\tau} \Phi_ {\theta} (s) ^ {T} e ^ {- \frac {d (m , n)}{\lambda}} +$$ + +$$ +d (m, n) = \min \left(\left| m - n \frac {M}{N} \right|, N - \left| m - n \frac {M}{N} \right|\right) +$$ + +$$ +U (v) _ {m, n} = h (v) ^ {T} w _ {m, n} ^ {H \rightarrow H} = h (\Phi_ {\theta}) ^ {T} e ^ {- \frac {d (m , n)}{\lambda}} +$$ + +$$ +d (m, n) = \min \left(| m - n |, N - | m - n |\right) \tag {12} +$$ + +represents the parameters of this function (i.e., the weights and biases of the NN layers preceding the RNN layer, which extract relevant features from the input). The function $h(v): \mathbb{R}^N \to \mathbb{R}^N$ is parameterized by learnable weights that map the information from previous forward passes to + +the current hidden state. The learnable parameter $\tau$ is the positive time constant responsible for the integration of signals in the ring. Here, $\tau$ acts as a learnable scaling factor that controls the balance between immediate input responsiveness and temporal integration within the ring attractor dynamics, with smaller values increasing sensitivity to current inputs and enabling faster adaptation, while larger values promote stability and smoother transitions between attractor states by integrating information over longer time horizons. It defines the contribution of input $\phi_{\theta}(s)$ to the current hidden state, imitating the attractor state, applied to NNs. + +Finally, the action-value function $Q(s,a)$ is derived from the RNN layer's output by applying the neurons activation function to the combined input $V(s)$ and hidden state information $U(s)$ . The activation function of choice is a hyperbolic tangent tanh, this function is symmetric around zero, leading to faster convergence and stability. However, the output range of tanh (-1 to 1) is not fully compatible with value-based methods, where the DL agents need to output action-value pairs in the range of the environment's reward function. To address this issue and prevent saturation of the tanh activation function, we scale the action-value pairs by multiplying them by a learnable scalar $\beta$ , as $Q(s,a) = \beta h_t = \beta \tanh(V(s) + U(v)) = \beta \tanh\left(\frac{1}{\tau}\Phi_\theta(s_t)^Tw^{I\to H} + h_{t-1}(v)^Tw^{H\to H}\right)$ . While our main focus is on value-based ring-attractor agents, we also provide an overview of methods and results for policy-gradient approaches in Appendix A.3. + +# 4 Experiments + +This section presents the findings of our experiments that validate our proposed approach to integrate ring attractors into RL algorithms. To assess the effectiveness of our method, we conducted comparisons between multiple baseline models and action spaces. The evaluation encompasses two implementations: a CTRNN exogenous ring attractor and a DL approach where the ring attractor is modeled directly into DRL agents. In both implementations, the action-value pairs $Q(s,a)$ are evenly distributed across the ring circumference. For the exogenous model, each action is associated with a specific discrete angle or continuous space, section 3.1.2. In the DL implementation, each neuron in the RNN corresponds to one action-value. The ring attractor serves as the output layer of the DL agent with the weights modeling the circular topology of the action space, Section 3.2.1. For both approaches, ring attractor agents are annotated with the suffix $RA$ . We demonstrate that ring attractors significantly enhance action selection and speed up the learning process. + +![](images/84f391f7269aa2ad124922745f9c9c57f8f649708c994f70d5aeefb6387d9aff.jpg) +Figure 2: Learning speed comparison. Above: OpenAI Gym Super Mario Bros environment [15] with discrete action space. Below: OpenAI highway [21] with a continuous 1-D circular variable. The plot shows cumulative reward over 1 million frames for three models: Standard BDQN; BDQNRA with ring attractor behavior policy from Section 3.1.2, setting the action-value pair variance constant to $\sigma_{a} = \frac{\pi}{6}$ , using this fixed variance to enable smooth action transitions while preventing interference with opposing actions; and BDQNRA-UA with RA and Uncertainty Awareness (UA) implementing the uncertainty quantification model from Section 3.1.3 to feed into the variance of the action-value pairs. Displaying mean episodic returns over 10 averaged seeds. + +![](images/3f85200448e9d4507ff197cdf7fc8e8256a4935f9f02d1be907b0af0890da320.jpg) + +# 4.1 CTRNN Ring Attractor Model Performance + +To evaluate our CTRNN ring attractor model from Section 3.1.2, integrated with BDQN [1] we performed experiments in the OpenAI Super Mario Bros environment [15] and OpenAI highway [21]. These benchmarks exhibit a spatially distributed complex decision-making scenario with 8 discrete different actions and a navigation-centric task with a continuous 1D circular actions space, respectively. Fig. 2 shows that both ring attractor models (BDQNRA and BDQNRA-UA) consistently outperform standard BDQN. The uncertainty-aware version (BDQNRA-UA) shows the best overall performance, highlighting the benefits of combining ring attractors spatial distribution of the action space with uncertainty-aware action selection. Empirical evaluations revealed that CTRNN-based ring attractor + +models exhibited a mean computational overhead of $297.3\%$ $(\mathrm{SD} = 14.2\%)$ compared to the baseline, significantly impacting runtime. To address this performance bottleneck and integrate the spatial understanding of the ring attractor into DRL, we developed a DL implementation of the ring attractor. This DL implementation is evaluated in the following subsections. + +# 4.2 DL Ring Attractor Model Performance + +To evaluate the effectiveness of our DL-based ring attractor implementation from Section 3.2, we conducted experiments implementing DDQNRA, an extension of DDQN [43]. Fig. 3 shows DDQNRA consistently outperforms standard DDQN, with the ring attractor's spatial encoding ability contributing significantly to improved learning speed. These results demonstrate substantial performance gains, especially in environments with strong spatial and directional cues. + +![](images/a5516b9f715edad01a3eb0c65d242816b726f0df76cb3161542ff671d5ea8ebb.jpg) +Figure 3: Performance comparison: DDQNRA vs standard DDQN [43] in two environments. Above: OpenAI Super Mario Bros [15], demonstrating adaptability to complex, game-like scenarios. Below: OpenAI highway [21], showing learning speed in spatial navigation tasks. Displaying mean episodic returns over 10 averaged seeds. + +![](images/572e513b9655b097eaee3e598d62df740bd5c62bf2b86f53055b20d34f89e482.jpg) + +# 4.3 Performance on Atari 100k Benchmark + +In this results section, we provide a comprehensive analysis of the performance of our DRL model on the Atari 100k benchmark [5]. We present comparisons with state-of-the-art models, highlighting the improvements achieved by our approach. Table 1 presents a comparison of our ring attractor-based DRL model integrated with Efficient Zero [48], which evaluates performance in Atari games with a limited training size of 100,000 environment steps. The table includes results from top-performing algorithms SPR [30] and CURL [32] for context. Our model demonstrates significant improvements over baseline methods, achieving a $53\%$ average improvement, with over $100\%$ gains in games with inherent spatial components, like Asterix and Boxing. Results demonstrate ring attractors provide consistent performance improvements across diverse RL tasks, with benefits extending beyond purely spatial tasks. We re-run baseline EffZero under identical cloud resources, to ensure fair comparison. + +The mapping between game action spaces and ring configurations reflects the fundamental structure of each environment's action space. Games with primarily directional movement actions, such as Asterix and Ms Pacman, utilise a Single ring configuration where eight directional movements map naturally to positions around the ring circumference. In contrast, games combining movement with independent action dimensions, such as Seaquest and BattleZone, employ a Double ring configuration, one ring encoding movement actions and another representing secondary mechanics such as combat. This architecture maintains spatial relationships while preserving the independence of different action types. Further implementation details for multiple ring dynamics can be found in Appendix A.7. + +Ablation studies were conducted to isolate the impact of key components in our ring attractor models, detailed in Appendix A.4.1 and A.4.2 for both implementations respectively. For the exogenous model, we compared performance with correct and randomized action distributions in the ring. In the DL implementation, we removed the circular weight distribution to assess its importance. Additionally, Appendix A.5 reveals preserved attractor dynamics behaving as a low-pass filter in the CTRNN and DL both at pre and post-training stages. + +Table 1: Performance comparison on Atari 100k Benchmark [5]. Benchmark performed across all environments where actions can be laid out in one or more 2D action space planes (ring attractors). This is represented by the ring configuration column. Game score and overall mean and median human-normalized scores are recorded at the end of training and averaged over 10 random seeds, 3 samples per seed. + +
GameAgentReportedImplemented
EnvironmentRingHumanCURLSPREffZeroEffZeroEffZeroRA
AlienDouble7127.7558.2801.5808.5738.11098.8
AsterixSingle8503.3734.5977.825557.814839.331037.3
Bank HeistDouble753.1131.6380.9351.0362.8460.5
BattleZoneDouble37187.514870.016651.013871.211908.715672.0
BoxingDouble12.11.235.852.730.562.4
Chopper C.Double7387.81058.5974.81117.31162.41963.0
Crazy ClimberSingle35829.412146.542923.683940.283883.0100649.7
FreewayDouble29.626.724.421.822.731.3
FrostbiteDouble4334.71181.31821.5296.3287.5354.8
GopherDouble2412.5669.3715.23260.32975.33804.0
HeroDouble30826.46279.37019.29315.99966.411976.1
JamesbondDouble302.8471.0365.4517.0350.1416.4
KangarooDouble3035.0872.53276.4724.1689.21368.8
KrullDouble2665.54229.63688.95663.36128.39282.1
Kung Fu M.Double22736.314307.813192.730944.827445.649697.7
Ms PacmanSingle6951.61465.51313.21281.21166.22028.0
Private EyeDouble69571.3218.4124.096.794.3155.8
Road RunnerDouble7845.05661.0669.117751.319203.129389.3
SeaquestDouble42054.7384.5583.11100.21154.71532.8
Human-normalised Score
Mean1.0000.4280.6381.1010.9591.454
Median1.0000.2420.4340.4200.4030.531
+ +# 5 Conclusion and Future Work + +Conclusion. This paper presents a novel approach to RL, integrating ring attractors into action selection. Our work demonstrates that these neuroscience-inspired ring attractors significantly enhance learning capabilities for RL agents, leading to more stable and efficient action selection, particularly in spatially structured tasks. + +The integration of ring attractors as a DL module proves to be particularly effective, allowing for end-to-end training and easy incorporation into existing RL architectures. Our experiments demonstrate significant improvements in action selection and learning speed, achieving state-of-the-art results on the Atari 100K benchmark. Notably, we observe an average $53\%$ performance increase across all tested games compared to prior baselines. The most relevant improvements were observed in games with strong spatial components, such as Asterix (110 % improvement) and Boxing (105 % improvement). Additionally, we observed improvements in other environments tested outside the Atari benchmark, further supporting the effectiveness of our approach across RL tasks and agents. + +Future work. Future research should investigate scalability in high-dimensional action spaces and explore its efficacy in domains where spatial relationships are less straightforward. We believe that the success of this approach opens up several future research paths. This work can be extended to multi-agent scenarios and continuous control tasks as these have been the current limits of this piece of research. The ring attractor approach may exhibit reduced applicability in discrete action spaces with cardinality $|\mathcal{A}| < 3$ where the circular topology overhead exceeds potential spatial encoding benefits, and in action spaces where the action correlation matrix approaches the identity matrix, indicating absence of spatial dependencies or sequential relationships between discrete actions. Additionally, the extension to 3D spatial navigation tasks presents an organic expansion of this work, these scenarios will naturally introduce realistic sensor noise into the system. + +In the field of uncertainty-aware decision making, leveraging the spatial structure provided by attractor networks presents a promising avenue to map uncertainty explicitly to the action space. Deploying the techniques presented here into specific domains could yield performance boosts, especially in safe RL, leveraging their stability properties to enforce constraints and ensure predictable behavior. This approach not only improves performance but also offers potential insight into spatial encoding of actions and decision-making processes, bridging the gap between neuroscience-inspired models and practical RL agents. + +# Acknowledgments and Disclosure of Funding + +This work was conducted as part of a self-funded PhD programme at the University of Manchester. Computational resources were provided by the Computational Shared Facility 3 (CSF3) at the University of Manchester [41]. We gratefully acknowledge the support of the Research IT team for maintaining and providing access to the GPU Cluster. + +Funding: This research received no external funding. + +Competing Interests: No competing interests are declared. + +# References + +[1] Kamyar Azizzadenesheli, Emma Brunskill, and Animashree Anandkumar. Efficient exploration through bayesian deep q-networks. CoRR, abs/1802.04412, 2018. +[2] Andrea Banino, Caswell Barry, Benigno Uria, Charles Blundell, Timothy P. Lillicrap, Piotr Mirowski, Alexander Pritzel, Martin J. Chadwick, Thomas Degris, Joseph Modayil, Greg Wayne, Hubert Soyer, Fabio Viola, Brian Zhang, Ross Goroshin, Neil C. Rabinowitz, Razvan Pascanu, Charlie Beattie, Stig Petersen, Amir Sadik, Stephen Gaffney, Helen King, Koray Kavukcuoglu, Demis Hassabis, Raia Hadsell, and Dharshan Kumaran. 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In International Conference on Learning Representations, 2019. +[50] Kechen Zhang. Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. Journal of Neuroscience, 16(6):2112-2126, 1996. +[51] Michael Zugaro, Alain Berthoz, and Sidney Wiener. Background, but not foreground, spatial cues are taken as references for head direction responses by rat anterodorsal thalamus neurons. The Journal of neuroscience : the official journal of the Society for Neuroscience, 21:RC154, 08 2001. + +# A Appendix + +# A.1 Background: Attractor Networks Theoretical Foundations + +Ring attractor networks are a type of biological neural structure that has been proposed to underlie the representation of various cognitive functions, including spatial navigation, working memory, and decision-making [18]. + +Biological intuition. In the early 1990s, the research carried out by [50] proposed that ring neural structures could underlie the representation of heading direction in rodents. [50] argued that the neural activity in an attractor network might encode the direction of the animal's head, with the network transitioning from one attractor state to another state as the animal turns. + +Empirical evidence. There is growing evidence from neuroscience supporting the role of ring attractors in neural processing. For example, electrophysiological recordings from head direction cells (HDCs) of rodents have revealed a circular organization of these neurons, with neighbouring HDCs encoding slightly different heading directions [38]. Furthermore, studies have shown that HDC activity can be influenced by sensory inputs, such as visual signals and vestibular signals, and that these inputs can cause the network to update its representation of heading direction [37]. [47] showed model internal noise biases toward accuracy over speed, while environmental uncertainty exhibits a U-shaped effect, where moderate uncertainty favors speed and extreme uncertainty favors accuracy. [46] demonstrated that biological navigation networks rely on attractor dynamics while maintaining adaptability through continuous synaptic plasticity and sensory feedback. + +Sensor fusion in ring attractors. Ring attractor networks provide a theoretical foundation for understanding cognitive functions such as spatial navigation, working memory, and decision-making. In the context of action selection in RL, sensor fusion plays a pivotal role in augmenting the information-processing capabilities of these networks. By combining data from various sensory modalities, ring attractors create a more nuanced and robust representation of the environment, essential for adaptive behaviors [24]. Research has elucidated the relationship between ring attractors and sensory inputs, with the circular organisation of HDCs in rodents complemented by the convergence of visual and vestibular inputs, highlighting the integrative nature of sensory information within the ring attractor framework [51]. + +Modulation by sensory inputs. Beyond the spatial domain, sensory input dynamically influences the activity of ring attractor networks. Studies have shown that visual cues and vestibular signals not only update the representation of heading direction but also contribute to the stability of attractor states, allowing robust spatial memory and navigation [12]. + +Sensor fusion for action selection. The concept of sensor fusion within the context of ring attractors extends beyond traditional sensory modalities, encompassing diverse sources, such as proprioceptive and contextual cues [23]. Building on the foundation of ring attractor networks discussed earlier, the integration of sensor fusion in the context of action selection involves fusing the action values associated with each potential action within the ring attractor framework. In particular, sensory information, previously shown to modulate the activity of ring attractor networks, extends its influence to the representation of action values. The inclusion of sensory information reflects a higher cognitive process, where the adaptable nature of ring attractor networks supports optimal decision-making and action selection in complex environments. + +# A.2 Function Approximation for Q-Values + +# A.2.1 Motivation and Background + +In RL, an agent optimizes behaviour by interacting with an environment modeled as a Markov Decision Process (MDP), $\mathcal{M} = (\mathcal{S},\mathcal{A},P,R,\gamma)$ , where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space, $P:S\times \mathcal{A}\times \mathcal{S}\rightarrow [0,1]$ is the state transition probability function, $R:S\times \mathcal{A}\times \mathcal{S}\rightarrow \mathbb{R}$ is the reward function, and $\gamma \in [0,1)$ denotes the discount factor. The agent seeks to learn a policy $\pi (a\mid s)$ which defines the probability of selecting action $a$ in state $s$ , that maximizes the expected discounted return [34]: + +$$ +J (\pi) = \mathbb {E} _ {\tau} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} r _ {t} \right], \tag {13} +$$ + +where $J(\pi)$ is the objective function representing the expected total return (performance measure) of policy $\pi$ , $\tau = (s_0, a_0, r_0, s_1, a_1, r_1, \ldots)$ represents a trajectory of states, actions, and rewards, $r_t$ + +is the reward received at time step $t$ , $\gamma$ is the discount factor that balances immediate versus future rewards, and the expectation is taken over trajectories sampled by following policy $\pi$ . + +Action-value functions are foundational to various value-based RL algorithms. The action-value function is defined as: + +$$ +Q ^ {\pi} (s, a) = \mathbb {E} _ {\tau} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} r _ {t} \mid s _ {0} = s, a _ {0} = a, \pi \right], \tag {14} +$$ + +which quantifies the expected cumulative discounted reward when executing action $a$ in state $s$ and subsequently following policy $\pi$ for all future time steps. + +This function satisfies a recursive relationship known as the Bellman equation [6]: + +$$ +Q ^ {\pi} (s, a) = \mathbb {E} _ {s ^ {\prime}, r \sim P, R} \left[ r + \gamma \sum_ {a ^ {\prime}} \pi \left(a ^ {\prime} \mid s ^ {\prime}\right) Q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) \right], \tag {15} +$$ + +where $\mathbb{E}_{s',r\sim P,R}$ denotes expectation with respect to the next state $s^\prime$ and reward $r$ following the environment dynamics defined by $P$ and $R$ , and $r + \gamma \sum_{a'}\pi (a' | s')Q^{\pi}(s',a')$ is the expected immediate reward plus the discounted value of future actions. This forms the mathematical basis for value iteration, policy iteration, Q-learning, and other dynamic programming and temporal-difference methods used to compute or approximate $Q^{\pi}$ . + +In environments with large or continuous state-action spaces, storing tabular representations of $Q(s,a)$ becomes infeasible due to memory constraints and the inability to visit all state-action pairs sufficiently often. Even in moderately sized discrete domains, function approximation is often employed to leverage generalization across similar states or actions, thereby speeding up learning and reducing data requirements. By using function approximators, RL algorithms can handle high-dimensional inputs and share learned structure across different parts of the state-action space. + +# A.2.2 General Formulation + +Q-functions are approximated using parameterized functions $Q(s, a; \theta)$ , where $\theta \in \mathbb{R}^d$ are learnable weights: + +$$ +Q (s, a; \theta) \approx Q ^ {\pi} (s, a), \tag {16} +$$ + +where the target $Q^{\pi}$ could be the Q-function of the current policy (policy evaluation) or the optimal Q-function $Q^{*}$ (control). The goal is to learn $\theta$ such that $Q(s, a; \theta)$ closely estimates the target Q-values across the relevant state-action space. + +Parameters $\theta$ are typically updated through stochastic gradient descent to minimize a loss function. For instance, using the mean squared Bellman error: + +$$ +\mathcal {L} (\theta) = \mathbb {E} _ {s, a, r, s ^ {\prime}} \left[ \left(r + \gamma \max _ {a ^ {\prime}} Q \left(s ^ {\prime}, a ^ {\prime}; \theta^ {\prime}\right) - Q (s, a; \theta)\right) ^ {2} \right], \tag {17} +$$ + +where $\theta^{\prime}$ represents parameters of a target network that is periodically updated to match $\theta$ (in deep Q-learning algorithms such as DQN [26], or $\theta^{\prime} = \theta$ in simpler settings like traditional Q-learning). The term $r + \gamma \max_{a'} Q(s', a'; \theta')$ is often called the "target" value, while $Q(s, a; \theta)$ is the "prediction". The parameters are updated according to: + +$$ +\theta_ {t + 1} = \theta_ {t} - \alpha \nabla_ {\theta} \mathcal {L} (\theta_ {t}), \tag {18} +$$ + +where $\alpha$ is the learning rate, and $\nabla_{\theta}\mathcal{L}(\theta_t)$ is the gradient of the loss function with respect to $\theta_t$ . In practice, these updates are performed using stochastic gradient descent or variants on mini-batches of experience to improve computational efficiency and stability. + +# A.2.3 Linear Function Approximation + +A straightforward approach is linear function approximation, where state-action pairs are mapped into a feature space via a feature function $\phi : S \times \mathcal{A} \to \mathbb{R}^d$ , and Q-values are computed as: + +$$ +Q (s, a; \theta) = \theta^ {\top} \phi (s, a). \tag {19} +$$ + +This formulation is computationally efficient and, under specific conditions, enjoys theoretical convergence guarantees in on-policy settings [35]. However, linear approximation's representational + +capacity is fundamentally constrained by the chosen feature representation $\phi(s, a)$ . For discrete action spaces, an alternative formulation represents a separate linear function for each action: + +$$ +Q (s, a; \theta) = \theta_ {a} ^ {\top} \phi (s), \tag {20} +$$ + +where $\phi(s) \in \mathbb{R}^d$ is a state feature vector and $\theta_a \in \mathbb{R}^d$ is a parameter vector specific to action $a$ . + +The transition from linear to neural approximation typically retains this foundational structure while enhancing the feature extraction process. Instead of relying on handcrafted features, neural networks learn feature representations automatically through their hidden layers. + +# A.2.4 Neural Function Approximators + +To capture richer, nonlinear dependencies, neural networks can be used to parameterize Q-functions. In the field of DRL, for any state-action pair $(s,a)$ , the Q-value $Q(s,a)$ can be expressed as a function of the input state through a function approximation algorithm $\Phi_{\theta}(s)$ . This notation evolves from the linear case but generalizes to allow for arbitrary nonlinear transformations of the input. Formally: + +$$ +Q (s, a) = \Phi_ {\theta} (s) = \theta^ {\top} x (s), \tag {21} +$$ + +where $x(s)$ corresponds to the (learned) feature representation of the state $s$ (e.g., the outputs of the penultimate layer), and $\theta$ represents the weights of the final layer. + +With this design, neural networks learn both the feature extraction $(x(s))$ and the final mapping $(\theta)$ end-to-end through backpropagation, enabling highly expressive function approximators that can generalize across large or complex state-action domains. + +# A.3 Policy Gradient Methods with Ring Attractors + +While previous sections focused on value-based methods, ring attractors can also enhance policy gradient methods, such as Proximal Policy Optimization (PPO) [29]. The integration follows similar principles, but adapts to the unique characteristics of policy-based approaches [36]. + +# A.3.1 Proximal Policy Optimization with Ring Attractors + +In standard PPO, the policy is directly parameterized by a neural network that outputs action probabilities. When integrating ring attractors, we modify this structure to incorporate the spatial relationships between actions through the ring topology. + +Ring Attractor Policy Network. For PPO integration, we adapt the DL ring attractor structure from Section 3.2. While the value-based methods use the network to process Q-values, in PPO we process policy logits. For a policy network with parameters $\theta$ , the initial policy logits are computed as: + +$$ +Z (s) = \Phi_ {\theta} (s) \tag {22} +$$ + +This parallels Section 3.2, where $Q(s,a) = \Phi_{\theta}(s)$ represents the output of the network for value-based methods. The difference is that $Z(s)$ represents policy logits rather than action-values. + +These logits are then processed through the ring attractor using the same weighted connection structure. The input signal $V(s)$ uses $Z(s)$ as input and follows the same form as in Eq. (12): + +$$ +V (s) _ {m, n} = \frac {1}{\tau} Z (s) ^ {T} w _ {m, n} ^ {I \rightarrow H} = \frac {1}{\tau} Z (s) ^ {T} e ^ {- \frac {d (m , n)}{\lambda}} \tag {23} +$$ + +Similarly, the hidden state $U(v)$ maintains the same recurrent dynamics as in Eq. (12). The key difference occurs in the final output layer, where instead of scaling the output to represent Q-values, we apply a softmax function to directly obtain action probabilities: + +$$ +\pi (a | s) = \operatorname {s o f t m a x} (\tanh ((V (s) + U (v)))) \tag {24} +$$ + +# A.3.2 Policy Gradient Experimental Results + +As shown in Figure 4, PPO [29] with ring attractors (PPO-RA) demonstrates significant performance improvements over standard PPO in the OpenAI Gym Super Mario Bros environment [15]. + +![](images/b3029f627be3cc9c558771b2451f137022d24d77c15b7e6e5e4fb9471e6d23ed.jpg) +Figure 4: Performance comparison: PPO-RA vs standard PPO in OpenAI Gym Super Mario Bros environment [15] with discrete action space, demonstrating adaptability to complex, game-like scenarios. The plot shows PPO-RA (red) consistently outperforming standard PPO (blue), with shaded regions representing standard deviation across 10 averaged seeds. + +Our experimental results reveal an average performance increase of $+71.4\%$ over standard PPO, with a final performance improvement of $+67.5\%$ and a maximum performance gain of $+161.4\%$ during training. These results demonstrate that the spatial awareness provided by ring attractors is beneficial across different RL paradigms, not just value-based methods. The performance gains in PPO-RA can be attributed to more efficient policy updates due to the preservation of spatial relationships between actions, smoother policy gradients resulting from the structured representation of the action space, and enhanced exploration through the natural diffusion of policy preferences across similar actions in the ring. + +PPO-RA converges faster and achieves higher rewards than standard PPO. This suggests that ring attractors provide a general enhancement mechanism applicable to multiple RL approaches, showcasing that their utility may extend beyond the value-based methods explored in previous sections. + +# A.4 Validating Ring Attractor Contributions Through Ablation Studies + +# A.4.1 Exogenous Ring Attractor Model Ablation Study + +To isolate the impact of the ring attractor structure, we conducted an ablation study comparing our full BDQNRA model against versions with the action space overlay in an incorrect distribution in the ring, Fig. 5. This incorrect distribution involves randomly rearranging the placement of actions within the ring, disrupting the natural topology of the action space. + +For instance, this could mean placing opposing or unrelated actions side by side in the ring, such as pairing "move left" with "move down" instead of its natural opposite "move right". More generally, this incorrect distribution breaks the inherent relationships between actions that are typically preserved in the ring structure. + +![](images/9798edfae6d31bf4f45ada0268e3ea9271cd774c630682131e1dc15b25e08a2a.jpg) +Figure 5: Ablation study comparing BDQN variants in OpenAI Gym Super Mario Bros [15]. The plot shows cumulative reward over 1 million frames for three models: Standard BDQN [1]; BDQNRA-UA with RA and Uncertainty Awareness (UA) implementing both the ring attractor behavior policy from Section 3.1.2 and the uncertainty quantification model from Section 3.1.3; and BDQNRA-RM, applying the same concepts from BDQNRA-UA, but randomly distributing the action space across the ring in each experiment. Displaying mean episodic returns over 10 averaged seeds. + +# A.4.2 Deep Learning Ring Attractor Model Ablation Study + +This ablation study focused on isolating the impact of the ring-shaped connectivity in our RNN-based ring attractor model. The key aspect of our experiment was to remove the circular weight distribution in both the forward pass (input-to-hidden connections) and the recurrent connections (hidden-to-hidden), while maintaining all other aspects of the RNN architecture. This approach allows us to directly assess the contribution of the spatial ring structure to the model's performance. + +![](images/71d920882aee1d8e09bbec2ba3503d31183a010a11d228fb15cb6445a27092f9.jpg) +Figure 6: Ablation study results comparing the performance of the full RNN-based ring attractor model against a version with the circular weight distribution removed. The graph illustrates a significant performance drop for the Ms Pacman and Chopper Command environments in the Atari 100K benchmark [5]. This emphasizes the role of the circular topology in encoding spatial information and enhancing learning. Displaying mean episodic returns over 10 averaged seeds. + +![](images/1173fb1b6afd31fed3da26fda8fed731f288285ba8b05e398608fa18626cb7b6.jpg) + +In our original model, the weights between neurons were determined by a distance-dependent function that created a circular topology. This function assigned stronger connections between neurons that were close together in the ring and weaker connections between distant neurons. For the ablation, we replaced this distance-dependent weight function with standard weight matrices for both the input-to-hidden and hidden-to-hidden connections. This modification effectively transforms our ring + +attractor RNN into a standard RNN, where the weights are not constrained by the circular topology. We retained other key elements of the model, such as the learnable time constant and the non-linear transformation, to isolate the effect of the ring structure specifically. + +# A.4.3 Deep Learning Ring Attractor Model Evolution + +In this appendix section, we analyze model dynamics with both forward pass $(V(s))$ and hidden-to-hidden $(U(v))$ weights made trainable, rather than the standard approach of fixed forward pass connections, as presented in Section 3.2. As shown in Fig. 7, the forward-pass connections preserve the ring structure over training time, with strong distance-dependent decay patterns maintained throughout the learning process. This may indicate that the network naturally favors maintaining spatial topology for transmitting sensory information on a per-frame basis. + +![](images/f871bf721437342ea261fe46f853a73631a9b13e36ff9617d17c6a8ca337456d.jpg) +Figure 7: Evolution of forward pass weights showing preserved distance-dependent decay over training time, maintaining ring structure for spatial information transmission. + +The hidden-to-hidden connections, depicted in Fig. 8, exhibit markedly different behavior. These connections evolve beyond their initial ring structure, developing specialized patterns that enable the encoding of environment-specific relationships between neurons in the hidden space. This flexibility in hidden-layer connectivity supports the learning of complex action relationships while building upon the structured spatial representation from the forward pass. + +These findings validate our standard implementation approach described in Section 3.2, where forward pass connections are fixed and only hidden-to-hidden weights remain trainable. The natural preservation of ring structure in trainable forward weights suggests that this topology is inherently beneficial for processing spatial information, while adaptable hidden weights enable the task-specific learning demonstrated in our experimental results, Section 4.2. + +![](images/64e09800929c2359402600446b5755387c114e86acb4460d37a0b824df89d37f.jpg) +Neuron Weights in RNN Hidden Space (Hidden to Hidden) +Step:0 + +![](images/a282b9c6b793211c92b682781b9fc21420b23771b22d0b2970509cf606425621.jpg) +Ring Architecture + +![](images/2922132a3382b02be2e914664816d48409e519764d15ad14e150aee7724b3233.jpg) +Weight vs Distance Profile + +![](images/0e691b1df6ff723aa31a8a5e7c31d50468d9053560563f8a3d983d1a62961c3b.jpg) +Step: 300000 + +![](images/54c137e43e9a1c06c97412702ff675d898f6c99aa9d0c69765a6e598cd399a05.jpg) +Ring Architecture + +![](images/bbd8753c7b2b559b4d5c39530375b90d0feed7500dd80daed941db510b0b5212.jpg) +Weight vs Distance Profile + +![](images/c3442d2db7f83c7d5de0e735774c223aadbf07fb06e976998d3e873734ecfe14.jpg) +Step: 600000 + +![](images/173002c66c71cf811c81691ca2c4ceb09b1141d5e3ec33b183763fa565329849.jpg) +Ring Architecture + +![](images/d71b4543cf0219102f5818974e6f10bb40efead5ca41d1bd62ed41f6102c3a7e.jpg) +Weight vs Distance Profile + +![](images/f270710e1407575783e187abbf5a708bfb7c120ae40a48d0111ddc525e76ff13.jpg) +Step: 1000000 +Figure 8: Development of hidden-to-hidden connections over time, demonstrating emergence of learned relationships between neurons beyond initial ring topology. + +![](images/9c0b8bca7bc632c67af058516d80ecb6a435c05ef087b2614528b2917d2e6bb2.jpg) +Ring Architecture + +![](images/dbb7b8c286085b7762116ce489c23a4098bf578cf00abdcaf56edca1498683c4.jpg) +Weight vs Distance Profile + +# A.5 Attractor Dynamics Validation + +To validate the preservation of CANN dynamics in our DL implementation, we conducted controlled experiments examining the temporal evolution of neural states under dynamic input conditions. Ring attractors, as demonstrated in biological systems [18, 38], maintain stable activity patterns through recurrent connectivity that enables persistent neural states [16]. + +We present four experimental tables tracking hidden states and outputs across all eight neurons (DLN1-DLN8) in the ring attractor. Tables 2 and 3 examine pre-training dynamics under a two-stimulus protocol with asymmetric $(K_{2} = 0.1\cdot K_{1})$ and matched $(K_{1} = K_{2})$ input amplitudes, respectively, demonstrating the network's ability to maintain stable attractor states and transition smoothly between them. Tables 4 and 5 present post-training dynamics after 315,000 and 750,000 training steps in the OpenAI highway [21], revealing how task-specific learning modifies the ring attractor structure while preserving attractor properties. + +Table 2: Temporal dynamics of DL-based ring attractor with asymmetric input cues ( $K_{2} = 0.1 \cdot K_{1}$ ). The network maintains stable bumps and executes smooth transitions despite order-of-magnitude differences in input amplitude between the first cue (centered at neuron 2, steps 1-6) and second cue (centered at neuron 5, steps 7-14). Bold values indicate peak activation at each timestep. + +
StepCueCTRNNDLDLN1DLN2DLN3DLN4DLN5DLN6DLN7DLN8
CtrArgMaxArgMaxInHidOutInHidOutInHidOutInHidOutInHidOutInHidOutInHidOutInHidOut
12221.630.7552.281.730.7855.211.630.7552.281.380.6744.981.090.5836.860.930.5532.471.090.5836.861.380.6744.98
22221.630.8253.831.730.8656.971.630.8253.831.380.7246.021.090.6137.380.930.5632.751.090.6137.381.380.7246.02
32221.630.7151.501.730.7554.701.630.7151.501.380.6043.541.090.4934.740.930.4430.051.090.4934.741.380.6043.54
42221.630.6550.121.730.6953.341.630.6550.121.380.5442.111.090.4233.270.930.3728.561.090.4233.271.380.5442.11
52221.630.6449.941.730.6853.171.630.6449.941.380.5341.921.090.4133.060.930.3628.351.090.4133.061.380.5341.92
62221.630.6550.141.730.6953.371.630.6550.141.380.5442.121.090.4233.260.930.3728.551.090.4233.261.380.5442.12
75220.140.6517.510.150.7018.600.150.6517.750.160.5415.420.160.4312.890.160.3711.680.150.4312.740.150.5415.17
85220.14-0.022.720.15-0.003.130.15-0.023.020.16-0.052.430.16-0.081.750.16-0.101.370.15-0.081.550.15-0.052.11
95330.14-0.091.240.15-0.081.460.15-0.081.560.16-0.091.500.16-0.101.340.16-0.101.190.15-0.101.130.15-0.091.15
105340.140.013.280.150.013.430.150.013.600.160.013.700.160.013.700.160.013.610.150.013.480.150.013.34
115450.140.064.500.150.074.640.150.074.820.160.074.970.160.075.010.160.074.940.150.074.790.150.064.61
125550.140.074.660.150.074.790.150.074.980.160.085.140.160.085.190.160.085.130.150.074.970.150.074.78
135550.140.064.480.150.064.610.150.074.800.160.074.960.160.075.020.160.074.960.150.074.800.150.064.60
145550.140.064.360.150.064.480.150.064.680.160.064.840.160.064.900.160.064.840.150.064.680.150.064.48
+ +Table 3: Temporal ring attractor dynamics with uniform input cues ( $K_{1} = K_{2}$ ). Bold values indicate peak activation at each timestep. + +
StepCueCTRNNDLDLN1DLN2DLN3DLN4DLN5DLN6DLN7DLN8
CtrArgMaxArgMaxInHidOutInHidOutInHidOutInHidOutInHidOutInHid
12221.570.7651.291.590.7751.841.570.7651.291.520.7449.851.460.7248.041.420.7248.04
22221.570.8252.551.590.8253.151.570.8252.551.520.8051.001.460.7749.081.420.7647.84
32221.570.6949.821.590.7050.431.570.6949.821.520.6748.231.460.6446.281.420.6345.02
42221.570.6248.231.590.6348.851.570.6248.231.520.6046.641.460.5744.681.420.5643.42
52221.570.6148.031.590.6248.641.570.6148.031.520.5946.431.460.5644.461.420.5543.20
62221.570.6248.251.590.6348.861.570.6248.251.520.6046.651.460.5744.681.420.5643.42
75231.420.6344.931.460.6446.111.520.6347.311.570.6047.091.590.5846.741.570.5746.05
85241.420.5943.991.460.6045.241.520.6146.991.570.6248.281.590.6047.821.570.6148.04
95351.420.5743.701.460.5844.961.520.6146.861.570.6348.371.590.6348.721.570.6248.30
105351.420.5743.611.460.5844.861.520.6146.811.570.6348.381.590.6448.981.570.6348.37
115451.420.5743.581.460.5844.841.520.6046.801.570.6348.391.590.6449.001.570.6348.39
125551.420.5743.571.460.5844.831.520.6046.801.570.6348.391.590.6449.011.570.6348.39
135551.420.5743.571.460.5844.831.520.6046.801.570.6348.401.590.6449.011.570.6348.40
145551.420.5743.571.460.5844.831.520.6046.801.570.6348.401.590.6449.011.570.6348,40
+ +Table 4: Post-training dynamics after 315,000 steps in OpenAI highway [21]. Temporal dynamics of DL-based ring attractor with uniform input cues ( $K_{1} = K_{2}$ ). Bold values indicate peak activation at each timestep. + +
StepCueDLDLN1DLN2DLN3DLN4DLN5DLN6DLN7DLN8
CtrArgMaxInHidOutInHidOutInHidOutInHidOutInHidOutInHidOutInHidOutInHidOut
1281.57-0.2628.821.590.3140.351.570.0635.991.520.2538.971.460.2036.481.42-0.3922.461.46-0.3324.791.520.3641.32
2221.57-0.3925.921.590.0936.941.57-0.2828.481.52-0.0732.041.46-0.1029.821.42-0.6616.621.46-0.6417.971.52-0.2627.89
3221.57-0.3925.951.590.0435.931.57-0.3027.991.52-0.0831.841.46-0.1528.891.42-0.6417.071.46-0.5220.731.52-0.2029.20
4221.57-0.3526.991.590.0937.021.57-0.2429.421.52-0.0233.051.46-0.0830.461.42-0.5718.581.46-0.4921.421.52-0.1230.91
5221.57-0.3526.861.590.1037.181.57-0.2329.441.52-0.0233.151.46-0.0730.531.42-0.5818.291.46-0.5021.111.52-0.1230.87
6221.57-0.3626.691.590.0936.971.57-0.2529.161.52-0.0332.891.46-0.0830.261.42-0.6018.041.46-0.5120.911.52-0.1430.48
7521.42-0.3623.241.460.0934.031.52-0.2528.071.57-0.0333.981.59-0.0833.161.57-0.5921.561.52-0.5122.401.46-0.1429.10
8541.42-0.3323.821.460.0533.281.52-0.2527.921.57-0.0533.671.59-0.0733.321.57-0.5921.531.52-0.5321.921.46-0.1528.87
9551.42-0.3423.701.460.0433.091.52-0.2527.981.57-0.0633.311.59-0.0633.611.57-0.6021.431.52-0.5222.191.46-0.1429.08
10551.42-0.3423.751.460.0433.091.52-0.2528.041.57-0.0633.311.59-0.0633.721.57-0.6021.461.52-0.5222.151.46-0.1429.07
11551.42-0.3423.741.460.0433.081.52-0.2528.041.57-0.0633.321.59-0.0633.731.57-0.6021.431.52-0.5222.161.46-0.1429.07
12551.42-0.3423.741.460.0433.071.52-0.2528.041.57-0.0633.311.59-0.0633.731.57-0.6021.421.52-0.5222.151.46-0.1429.06
13551.42-0.3423.741.460.0433.071.52-0.2528.041.57-0.0633.311.59-0.0633.731.57-0.601.52-0.5222.161.46-0.1429.06
14551.42-0.3423.741.460.0433.071.52-0.2528.041.57-0.0633.311.59-0.0633.731.57-0.60 21.421.52-0.5222.161.46-0.1429.07
+ +Table 5: Post-training dynamics after 750,000 steps in OpenAI highway [21]. Uniform input cues ( $K_{1} = K_{2}$ ). Bold values indicate peak activation. + +
StepCueDLDLN1DLN2DLN3DLN4DLN5DLN6DLN7DLN8
CtrArgMaxInHidOutInHidOutInHidOutInHidOutInHidOutInHidOutInHidOutInHidOut
1221.57-0.1931.241.590.4243.781.570.1337.461.520.3140.521.460.2738.911.42-0.3224.831.46-0.2826.351.520.2939.17
2221.57-0.3128.471.590.1539.261.57-0.2131.021.52-0.0234.811.46-0.0532.371.42-0.5819.141.46-0.5620.521.52-0.1930.46
3221.57-0.3228.511.590.1138.191.57-0.2330.541.52-0.0334.581.46-0.0931.421.42-0.5719.611.46-0.4623.181.52-0.1431.73
4221.57-0.2829.541.590.1639.381.57-0.1831.961.520.0335.621.46-0.0332.981.42-0.4921.131.46-0.4223.941.52-0.0733.42
5221.57-0.2829.391.590.1739.561.57-0.1731.981.520.0435.731.46-0.0233.061.42-0.5120.821.46-0.4323.611.52-0.0633.38
6221.57-0.2929.211.590.1639.341.57-0.1931.681.520.0235.451.46-0.0332.781.42-0.5320.561.46-0.4423.411.52-0.0832.97
7521.42-0.2925.761.460.1636.411.52-0.1930.591.570.0236.521.59-0.0335.681.57-0.5224.081.52-0.4424.921.46-0.0831.62
8541.42-0.2626.341.460.1235.641.52-0.1930.431.57-0.0136.191.59-0.0235.841.57-0.5224.051.52-0.4624.431.46-0.0931.38
9551.42-0.2726.211.460.1135.431.52-0.1930.491.57-0.0235.821.59-0.0136.141.57-0.5323.941.52-0.4524.711.46-0.0831.59
10551.42-0.2726.271.460.1135.431.52-0.1930.561.57-0.0235.821.59-0.0136.261.57-0.5323.971.52-0.4524.671.46-0.0831.58
11551.42-0.2726.261.460.1135.421.52-0.1930.561.57-0.0235.831.59-0.0136.271.57-0.5323.941.52-0.4524.681.46-0.0831.58
12551.42-0.2726.261.460.1135.411.52-0.1930.561.57-0.0235.821.59-0.0136.271.57-0.5323.931.52-0.4524.671.46-0.0831.57
13551.42-0.2726.261.460.1135.411.52-0.1930.561.57-0.0235.821.59-0.0136.271.57-0.531.52-0.4524.681.46-0.0831.57
14551.42-0.2726.261.460.1135.411.52-0.1930.561.57-0.0235.821.59-0.0136.271.57-0.53\1.52-0.4524.681.46-0.0831.58
+ +# A.5.1 Experimental Protocol + +We present four experimental tables that track hidden states and outputs across all eight neurons (DLN1-DLN8) in the ring attractor. The experimental setup consisted of an 8-neuron ring configuration with parameters tuned for single stable bump formation: $\tau = 120.0$ (temporal integration constant) and $\beta = 22.0$ (output scaling). The protocol involved an initial Gaussian input centered on neuron 2 for $t \in [1,6]$ , followed by an instantaneous transition to a Gaussian input centered on neuron 5 for $t \geq 7$ . This experiment tests the network's ability to maintain stable attractor states and transition smoothly between them, as predicted by theoretical models of ring attractor dynamics [17]. Tables 2 and 3 assess pre-training dynamics under this two-stimulus protocol with asymmetric $(K_{2} = 0.1 \cdot K_{1})$ and matched $(K_{1} = K_{2})$ input amplitudes, respectively. Tables 4 and 5 present post-training dynamics after 315,000 and 750,000 training steps in the OpenAI Highway environment [21], revealing how task-specific learning modifies the ring attractor structure while preserving fundamental attractor properties. + +# A.5.2 Preservation of Attractor Dynamics + +Table 2 presents the temporal evolution of the DL-based ring attractor under the two-stimulus protocol with asymmetric input amplitudes where $K_{2} = 0.1 \cdot K_{1}$ , following the input signal formulation in Eq. 5. Despite the order-of-magnitude difference in input strength, the network transitions between attractor states. During the initial phase (steps 1-6), neuron 2 maintains the highest output values (Out ≈ 55.21), forming a stable activity bump. Following the input shift at step 7, the argmax tracks positions through intermediate neurons ( $2 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5$ over steps 7-11), exhibiting the characteristic temporal resistance of ring attractors [42]. By step 12, neuron 5 stabilizes at Output ≈ 5.19 despite the weaker input signal. Table 3 presents the same protocol with matched input amplitudes ( $K_{1} = K_{2}$ ). The dynamics exhibit three characteristic phases consistent with CANN behavior [4]. During the initial attractor formation phase ( $t \in [1,6]$ ), the network maintains a stable activity bump centered at neuron 2, with the argmax operation consistently identifying the correct peak location. The hidden state magnitudes remain comparable between the two stable phases (DLN2 Hidden ≈ 0.77 at step 1 vs. DLN5 Hidden ≈ 0.64 at step 12), confirming that the asymmetries observed in Table 2 arise from input differences rather than structural biases. Following the input shift at $t = 7$ , the network demonstrates smooth state transition dynamics ( $t \in [7,11]$ ), where the activity bump migrates from neuron 2 to neuron 5. The network subsequently converges to a new stable attractor state ( $t \in [12,14]$ ), settling into a configuration centered at neuron 5. This temporal lag reflects the $\tau = 120.0$ integration constant and should demonstrate the network's ability to maintain stable representations while integrating new information. + +Tables 4 and 5 present post-training dynamics after 315,000 and 750,000 training steps in the Highway environment, respectively. In Table 4, following the input shift at step 7, the peak transitions from neuron 2 (step 2, Output $\approx$ 36.94) through neuron 4 (step 8, Output $\approx$ 33.67) to neuron 5 (steps 9-14, Output $\approx$ 33.73). The hidden states include negative values (e.g., DLN5 Hidden $= -0.06$ at convergence), indicating that task-specific learning has modified internal dynamics while maintaining the output behavior necessary for action selection. The attractor state of a neuron is determined by the aggregate influence of all incoming connections through the hidden-to-hidden weights. As the hidden state undergoes transformation through $y_{t} = f(W_{hh}\cdot h_{t} + b_{y})$ , negative or non-maximal hidden values can produce the appropriate output behavior after passing through the tanh activation and scaling parameter $\beta$ . Table 5 shows stabilization of these learned dynamics at 750k steps, with similar structural features confirming that the adaptations represent stable task-specific encoding rather than transient learning dynamics. Neuron 6, corresponding to the "move backwards" action in Highway, consistently shows lower activation values across both checkpoints (Output $\approx$ 22.46 at step 1 in Table 4), reflecting the penalty associated with this action. + +# A.5.3 Analysis + +The preservation of attractor dynamics post-training aligns with findings in biological systems, where synaptic plasticity modifies connection strengths while maintaining functional circuit properties [37, 24]. We may infer from these experimental results that our DL implementation preserves essential CANN dynamics while enabling task-specific adaptation through learning. The maintained bump migration and state persistence properties, combined with the evidence of spatial representations provided in the ablation studies in Section A.4, indicate that performance improvements stem from the ring attractor's inherent spatial encoding capabilities and attractor states rather than merely adding recurrent connections. The preservation of attractor-like dynamics post-training, despite substantial weight evolution away from the initial Gaussian profile, reveals several computational mechanisms at work. The architecture maintains spatial awareness through the explicit ring topology: the circular arrangement of neurons preserves spatial relationships between actions throughout training, as evidenced by the natural preservation of distance-dependent weights in the input-to + +hidden connections and the smooth bump migration patterns observed across all experimental conditions in Section A.4. The ablation studies demonstrate that removing the ring structure causes performance to drop below baseline levels, confirming that spatial encoding provides functionally significant advantages over standard RNNs and feed-forward networks. Simultaneously, the learnable $\tau$ parameter introduces temporal filtering dynamics that serve distinct functions across training phases. During early training, the results suggest that the architecture also functions as a low-pass filter, where $\tau$ creates momentum that reduces jitter in action exploration and enables the agent to reach distant state-space regions more efficiently. + +# A.6 Deep Learning Ring Attractor Recurrent Neural Network Modeling + +As seen before, excitatory neurons are organized in a circular pattern, with connection weights between neurons determined by a distance-weighted function mimicking the synaptic connection of biological neurons. Fig. 9 illustrates the structured connectivity of the RNN, which mimics the circular topology of biological ring attractors. + +![](images/45b6dad825a49e4f01ceea4bdce0b676e7220b4290175faab36935eb84b1ad9f.jpg) + +![](images/f50954a64040507a8adf8f8e2c602e3c4a34548f7e172b551e31ed87a3b4197e.jpg) +Figure 9: RNN modeling ring attractor synaptic connections: The top panel shows the forward pass (input-to-hidden) as the agent output layer. The bottom panel depicts the hidden-to-hidden recurrent connection between inference time steps. Weighted connections are illustrated for the sample neuron $n_3$ . + +# A.7 Deep Learning Ring Attractor Model Implementation Details + +As presented in Section 3.2, both the input-to-hidden connections $V(s)$ and hidden-to-hidden connections $U(v)$ are constructed using the distance-dependent weight functions defined in Eq. (12). + +These equations establish the circular topology of the ring attractor and determine how information flows through the network. However, a special case arises when dealing with neutral actions in certain Atari games, requiring a modification to the standard distance function. + +# A.7.1 Neutral Inhibitory Action Implementation + +For games in the Atari benchmark with neutral actions (like 'no-op'), the ring attractor maintains its circular structure with a neutral action positioned centrally. This central position creates equal connections of strength 1 to all other actions in the ring, as if it were a direct neighbor to each action simultaneously. The distance between the neutral action $n$ and any other action $m$ is fixed at 1: + +$$ +d (m, n) = \left\{ \begin{array}{l l} 1, & \text {i f} m \text {o r} n \text {i s t h e n e u t r a l a c t i o n} \\ d (m, n), & \text {o t h e r w i s e}, \end{array} \right. \tag {25} +$$ + +where $d(m,n)$ remains as defined in Eq. (12) for all other action pairs. The weight matrices $w_{I\rightarrow H}$ and $w_{H\rightarrow H}$ maintain the same exponential decay based on this distance function. + +For example, in games like Seaquest or Asterix, this central positioning means the "no-op" action has consistent, strong connections to all directional actions. This arrangement preserves the spatial relationships between directional actions while ensuring the neutral action remains equally accessible from any game state. The constant distance of 1 to all other actions makes transitioning to or from the neutral action as natural as moving between adjacent directional actions in the ring. + +# A.7.2 Deep Learning Double Ring Attractor Equations + +For a double ring configuration in our DL implementation as presented in the experiments, Section 4.2, the weighted connections are defined as follows: + +Let $\mathbf{V}^{double} \in \mathbb{R}^{2N \times 2M}$ be the complete input-to-hidden weight matrix for both rings, where $N$ is the number of output neurons per ring and $M$ is the number of input features per ring. The matrix is structured as: + +$$ +\mathbf {V} ^ {\text {d o u b l e}} = \left[ \begin{array}{l l} \mathbf {V} _ {1 1} & \kappa \mathbf {V} _ {1 2} \\ \kappa \mathbf {V} _ {2 1} & \mathbf {V} _ {2 2} \end{array} \right], \tag {26} +$$ + +where $\mathbf{V}11 = \mathbf{V}22 = \mathbf{V}12 = \mathbf{V}21$ , $\kappa = 0.1$ is the cross-coupling learnable parameter initialised to 0.1. This allows the network to learn the optimal strength of interaction between the two rings during training. + +Developing from Eq. (12), each ring maintains identical connectivity patterns, preserving the spatial relationships of their respective action dimensions, each submatrix $\mathbf{V}_{ij}$ represents: + +$$ +\left[ \mathbf {V} _ {i i} \right] _ {m, n} = \frac {1}{\tau} \Phi_ {\theta} (s) ^ {T} e ^ {- d (m, n) / \lambda}, \tag {27} +$$ + +where $d(m,n)$ is defined as per the forward pass weighted connections in Eq. (12). + +Similarly, let $\mathbf{U}^{double} \in \mathbb{R}^{2N \times 2N}$ be the complete hidden-to-hidden weight matrix: + +$$ +\mathbf {U} ^ {\text {d o u b l e}} = \left[ \begin{array}{l l} \mathbf {U} _ {1 1} & \kappa \mathbf {U} _ {1 2} \\ \kappa \mathbf {U} _ {2 1} & \mathbf {U} _ {2 2} \end{array} \right] \tag {28} +$$ + +For primary connections $(\mathbf{U}_{11}$ and $\mathbf{U}_{22})$ + +$$ +\left[ \mathbf {U} _ {i i} \right] _ {m, n} = h (v) ^ {T} e ^ {- d (m, n) / \lambda}, \tag {29} +$$ + +where $d(m,n)$ is defined as per the hidden state weighted connections in Eq. (12). + +The complete forward pass for both rings is given by: + +$$ +Q (s, a) = \beta \tanh \left(\left[ \begin{array}{l l} \frac {1}{\tau} \Phi_ {\theta} \left(s _ {t}\right) ^ {T} \mathbf {V} _ {1 1} + h _ {t - 1} (v) ^ {T} \mathbf {U} _ {1 1} & \frac {\kappa}{\tau} \Phi_ {\theta} \left(s _ {t}\right) ^ {T} \mathbf {V} _ {1 2} + \kappa h _ {t - 1} (v) ^ {T} \mathbf {U} _ {1 2} \\ \frac {\kappa}{\tau} \Phi_ {\theta} \left(s _ {t}\right) ^ {T} \mathbf {V} _ {2 1} + \kappa h _ {t - 1} (v) ^ {T} \mathbf {U} _ {2 1} & \frac {1}{\tau} \Phi_ {\theta} \left(s _ {t}\right) ^ {T} \mathbf {V} _ {2 2} + h _ {t - 1} (v) ^ {T} \mathbf {U} _ {2 2} \end{array} \right]\right), \tag {30} +$$ + +where $\tau$ is the learnable time constant; $\beta$ is the learnable scaling factor; $\Phi_{\theta}(s_t)$ is the feature representation of state $s_t$ ; $h_{t - 1}(v)$ is the previous hidden state; and $\kappa = 0.1$ is the coupling strength between rings. + +The cross-coupling matrices $(\kappa \mathbf{V}12, \kappa \mathbf{V}21, \kappa \mathbf{U}12, \text{and}\kappa \mathbf{U}21)$ maintain a circular topology similar to the individual rings. A neuron at a particular position in the first ring connects most strongly to the neuron at the corresponding position in the second ring, with connection strength decreasing based on circular distance. This structured cross-coupling preserves spatial alignment between the two action dimensions while allowing semi-independent operation through the learnable coupling factor $\kappa$ . The final output provides action-values for both action dimensions simultaneously, preserving spatial relationships within each ring while allowing weak coupling between them. + +# A.7.3 Extension to N Ring Configurations + +The double ring implementation extends to $R$ rings through a block matrix structure, where each ring encodes a distinct action dimension. For $R$ rings, the architecture uses block matrices $V_{\mathrm{multi}} \in \mathbb{R}^{RN \times RM}$ and $U_{\mathrm{multi}} \in \mathbb{R}^{RN \times RN}$ , with diagonal blocks preserving individual ring dynamics and off-diagonal blocks handling cross-ring interactions via coupling parameter $\kappa$ , as seen in Section A.7.2. While computational complexity scales as $O(R^2)$ , selective coupling between only related dimensions creates a sparse structure with effective $O(R)$ complexity. This makes the approach viable for complex action spaces where actions decompose into multiple semi-independent planes, for example, in games that combine movement, combat, and resource-management dimensions. + +# A.8 Models and Environments Implementation + +Table 6: Implementation details for ring attractor architectures across environments. The table shows the environment (Env); ring configuration (Ring); number of actions or continuous 1D action space (Actions); inhibitory neuron placed equidistant to other neurons for "no action" term (Inhib); whether uncertainty estimation is used (Uncert); the implemented model (Model); and type of Neural Network used (Type). + +
Game +EnvironmentRingConfigurationImplementation
ActionsInhib.Uncert.ModelType
HighwaySingleContinuousNoYesBDQNRA-UACTRNN
Mario BrosSingle8NoYesBDQNRA-UACTRNN
HighwaySingle8NoNoDDQNRADL-RNN
Mario BrosSingle8NoNoDDQNRADL-RNN
AlienDouble18YesNoEffZeroRADL-RNN
AsterixSingle9YesNoEffZeroRADL-RNN
Bank HeistDouble18YesNoEffZeroRADL-RNN
BattleZoneDouble18YesNoEffZeroRADL-RNN
BoxingDouble18YesNoEffZeroRADL-RNN
Chopper C.Double18YesNoEffZeroRADL-RNN
Crazy ClimberSingle9YesNoEffZeroRADL-RNN
FreewayDouble18YesNoEffZeroRADL-RNN
FrostbiteDouble18YesNoEffZeroRADL-RNN
GopherDouble18YesNoEffZeroRADL-RNN
HeroDouble18YesNoEffZeroRADL-RNN
JamesbondDouble18YesNoEffZeroRADL-RNN
KangarooDouble18YesNoEffZeroRADL-RNN
KrullDouble18YesNoEffZeroRADL-RNN
Kung Fu M.Double18YesNoEffZeroRADL-RNN
Ms PacmanSingle9YesNoEffZeroRADL-RNN
Private EyeDouble18YesNoEffZeroRADL-RNN
Road RunnerDouble18YesNoEffZeroRADL-RNN
SeaquestDouble18YesNoEffZeroRADL-RNN
+ +We provide implementation details for both our models and the tested environments. For model implementations, EffZeroRA was applied across the Atari benchmark suite, while BDQNRA-UA and DDQNRA were specifically implemented for Highway and Mario Bros. Table 6 details the configuration of action spaces and ring architectures for each environment. The environments required different ring configurations based on their control schemes, ranging from single-ring implementations for basic movement to double-ring setups for more complex action spaces that combine movement and specialized actions. Each ring topology was designed to preserve the natural + +relationships between actions, with central inhibitory actions included where appropriate as "no action". + +# A.8.1 Computational Resources + +For the Atari 100K benchmark [5] experiments, we utilized a high-performance computing cluster equipped with 10 NVIDIA A100 GPUs (each with 80 GB memory), 512 GB of RAM, and 128 Intel Xeon CPU cores running at 2.4 GHz. The cluster environment enabled parallel execution of experiments across multiple seeds and games simultaneously, with training times ranging from 8 to 10 hours per game, depending on complexity. + +For the other environments, Highway [21] and Super Mario Bros [15], we employed a local workstation with an Intel Xeon processor (28 cores, 2.1 GHz), 125 GB RAM, and dual NVIDIA RTX A5000 GPUs (total 48 GB combined memory). Training on the local machine typically required 6-12 hours per environment, with the CTRNN implementation incurring additional computational overhead as noted in Section 4.1. All experiments were conducted using PyTorch 1.12 with CUDA 11.6. The total computational cost is estimated at approximately 8,000 GPU-hours across all experiments and ablation studies. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: See Abstract and contributions in Section 1. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer:[Yes] + +Justification: See Sections 4, 5. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. 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Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [NA] + +Justification: The paper does not include theoretical results. + +Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. 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propose Spatial-MLLM, a method that significantly enhances the visual-based spatial intelligence of existing video MLLMs. As shown, Spatial-MLLM is capable of understanding and reasoning about the underlying scene from video input, achieving state-of-the-art performance across a wide range of tasks. + +![](images/db1dc01b8439a3bfb3733a4154cea1fe7cebbca19bd1eb6c52cb1a950177f7f6.jpg) + +Example: Route Plan + +You are a robot beginning at the stairs and facing into the room. You want to navigate to the nightstand closer to the window. You will perform the following actions (Note: for each [please fill in], choose either 'turn back', 'turn left', or 'turn right').1: Go forward until the foot of the bed. 2. [please fill in] 3. Go forward until the wall. 4. [please fill in] 5. Go forward until the nightstand. You have reached the final destination. + +A. Turn Left, Turn Left C. Turn Back, Turn Left + +B. Turn Left, Turn Right D. Turn Right, Turn Left + +![](images/2dba6ad8446ebda8f8ea82147155fafc35ec07d99ef3840316ab517e20679ccd.jpg) + +: The goal is to reach the nightstand closer to the window. From my current position to the left side of the bed, I need to find this specific nightstand. I've already gone past a lamp on the left, so I should be looking around the wall area before the bed. I'm not sure how far away from the right side of the bed, it seems logical that the nightstand is further along than that side. + +![](images/3c4f89045b167b11efdc48f2e354ff8ab5dc51812fe34c18d00c06e547cf5c76.jpg) + +![](images/63acb4f052c4ec528498d34d8c608376d64dfd70101149721a4ea41a444392da.jpg) + +![](images/7dd8dae2ece56adb792fe180d0e798cdbfe59cfa9de72a6bb9670600f09080d1.jpg) + +![](images/a30a3a7d005322a2041c9fdcb6a1bd10a87067156cacc2e3bfe87d44c459db77.jpg) + +![](images/c97b46838ed656d181b10bf22e293cd0b1f70fc3e40a62efbe89f8459bd34adc.jpg) + +![](images/350c6063b90760f0527a44122699b6805d87c8f2de052b082abaf7c1e9493d33.jpg) + +![](images/5df28f3727fcfa00d25dfa632d2e524fe158fd3ac296468741212cff4872e612.jpg) + +![](images/5c85a8d2f26dd7b571c6fd4c69f1a26526d419e2c08a2181eaf64f1a2c69d67e.jpg) + +Current Position? + +Get to where? window...How? + +Walk along wall +->window + +from window to rightstand + +![](images/785d012a8175f589a267cecbceca578f36d83a4f23be13345212af13211049f5.jpg) + +# Spatial MLLM + +...To get there, I need to turn right. The wall will be on my right side. I once I'm along the wall, I need to turn left when I reached the window, which is where the nightstand is located. + + D + +![](images/23978af632c8ffe2a17602ee981e225450d3289024c7103a39920d8385c4f646.jpg) + +Appearance Order + +Absolute Distance + +Relative Distance + +Relative Direction + +![](images/7f4b4528c4771d96525327f20645a1d2b0849817246fed5c75d2b525c1264d65.jpg) + +Thinking + +Obj + +(2) + +Obj.Count + +# + +Room Size + +C. toilet, basket, bed, ceiling light + +![](images/4d4f1cfa395221503d871054e17cbe9bc744a006d31f1ca3476182cf5a0b9f00.jpg) + +![](images/b96c911ea8a31983bddc7111d1cdc031f14cbbe94cdb91ae8658d6b6d2ac1a26.jpg) + +# Abstract + +Recent advancements in Multimodal Large Language Models (MLLMs) have significantly enhanced performance on 2D visual tasks. However, improving their spatial intelligence remains a challenge. Existing 3D MLLMs always rely on additional 3D or 2.5D data to incorporate spatial awareness, restricting their utility in scenarios with only 2D inputs, such as images or videos. In this paper, we present Spatial-MLLM, a novel framework for visual-based spatial reasoning from purely 2D observations. Unlike conventional video MLLMs which rely on CLIP-based visual encoders optimized for semantic understanding, our key insight is to unleash the strong structure prior from the feed-forward visual geometry foundation model. Specifically, we propose a dual-encoder architecture: a pretrained 2D visual encoder to extract semantic features, and a 3D spatial encoder—initialized from the backbone of the visual geometry model—to extract 3D structure features. A connector then integrates both features into unified visual tokens for enhanced spatial understanding. Furthermore, we propose a space-aware frame sampling strategy at inference time, which selects the spatially informative frames of a + +video sequence, ensuring that even under limited token length, the model focuses on frames critical for spatial reasoning. Beyond architecture improvements, we construct a training dataset from multiple sources and train the model on it using supervised fine-tuning and GRPO. Extensive experiments on various real-world datasets demonstrate that Spatial-MLLM achieves state-of-the-art performance in a wide range of visual-based spatial understanding and reasoning tasks. Project page: https://diankun-wu.github.io/Spatial-MLLM/. + +# 1 Introduction + +Multimodal Large Language Models (MLLMs) [1, 2, 3] have achieved significant progress in processing multimodal inputs to generate contextually aware and semantically coherent responses. While proprietary models such as Gemini [4] and GPT-4o [5] exhibit state-of-the-art performance, the open-source community continues to advance the field by improving these models' ability to interpret diverse content modalities, including images [6, 7, 8], videos [9, 10, 11, 12, 13, 14], and audio [15, 16, 17]. Although these models excel at a wide range of 2D tasks, their capacity to perceive, understand, and reason about 3D scenes, i.e., 3D spatial intelligence, remains limited [18, 19]. + +The requirement of spatial understanding and reasoning typically arises in two scenarios. In the first scenario, the model has access to additional 3D or 2.5D data (e.g., point clouds, camera parameters, or depth maps) alongside 2D visual inputs (e.g., images or videos). These supplementary modalities enhance the model's spatial awareness, enabling more accurate spatial reasoning. However, this setup limits the model's applicability in many real-world scenarios where only monocular video of the scene is available, which is the second scenario. The model's ability to perform spatial understanding and reasoning under such conditions is referred to as visual-based 3D spatial intelligence [18, 20]. A major challenge in this setting is that each frame provides only a partial observation of the scene, and no global representation (e.g., the point clouds [21, 22, 23] or posed depth maps [24, 25]) is available as input. This requires the model to infer the global spatial layout from incomplete cues and internally integrate these partial observations into a coherent and implicit global representation, which demands strong spatial awareness. However, most existing video MLLMs pretrain their visual encoders on image-text pairs—primarily image-caption data [13, 14, 26]—following the CLIP [27] paradigm. This makes the visual encoder excel at capturing high-level semantic content but lack structure and spatial information when only 2D video inputs are available [28, 29, 30]. Consequently, current video MLLMs generally perform worse on spatial reasoning tasks than on other tasks, such as temporal understanding. Moreover, their performance still significantly lags behind human capabilities [18]. + +In this paper, we introduce Spatial-MLLM, a method that significantly improves the visual-based 3D spatial intelligence of existing video MLLMs. To address the limitations of visual encoders in general-purpose video MLLMs, our key insight is to unleash the strong structure prior provided by the feed-forward visual geometry foundation model [31, 32, 33, 34]. These models, typically trained on pixel-point pairs, complement the general-purpose video MLLM visual encoders that are trained primarily on image-text data [14]. Based on this insight, we design a dual-encoder architecture consisting of a 2D encoder—initialized from the visual encoder of a general-purpose video MLLM—to extract 2D semantic information, and a spatial encoder—leveraging the VGGT feature extractor [32]—to recover implicit 3D structural information from 2D video inputs. We then use a connector to integrate features from both branches into unified visual tokens. The resulting representation enables the Large Language Model (LLM) backbone to perform effective spatial reasoning without requiring explicit 3D data as input. + +Furthermore, we fully exploit the additional information provided by the introduced feed-forward visual geometry model [32], and propose a space-aware frame sampling strategy at inference time, which selects the most spatially informative frames from the video sequence when the total number of input frames is limited (e.g., due to the VRAM limitation). Specifically, we first feed a relatively large number of frames into the spatial encoder and decode the resulting 3D features into voxels. The frame selection task is then reformulated as a maximum coverage problem over these voxels, which we solve using a greedy algorithm. To train Spatial-MLLM, we construct a visual-based 3D spatial question-answering dataset and perform supervised fine-tuning on it. We further apply a simple cold-start [35] to help the model adapt to the correct reasoning format, and then train it using Group Relative Policy Optimization (GRPO) [36, 35] to enhance its long-chain-of-thought (long-CoT) spatial reasoning capability [37]. We conduct extensive evaluations on the VSIBench [18], + +ScanQA [38], and SQA3D [39] benchmarks and demonstrate that the proposed spatial-MLLM achieves state-of-the-art performance in a wide range of visual-based spatial understanding and reasoning tasks. + +In summary, our main contributions are: + +- We introduce Spatial-MLLM, a method that significantly enhances the visual-based 3D spatial intelligence of existing video MLLMs, demonstrating strong spatial understanding and reasoning capabilities without requiring any 3D or 2.5D data input. +- We design a dual-encoder and connector that effectively integrates semantic information from a standard 2D visual encoder with structural information extracted by a spatial encoder, which is initialized using a feed-forward visual geometry foundation model. +- We fully exploit the additional information provided by the feed-forward visual geometry model and design a space-aware frame sampling strategy that selects spatially informative frames, thereby improving model performance under input length constraints. +- We train our model on the constructed dataset with a two-stage pipeline. Extensive experiments demonstrate that our method achieves state-of-the-art performance on a wide range of visual-based spatial understanding and reasoning tasks. + +# 2 Related Work + +# 2.1 MLLMs for Video Understanding + +Multimodal Large Language Models have made significant progress in integrating vision and language. Early works such as BLIP-2 [2] and Flamingo [1] introduce token-level fusion (e.g., Q-Former) and feature-level fusion (e.g., cross-attention layers) to bridge modalities. Other approaches, including the LLaVA series [3, 40], MiniGPT-4 [41], and subsequent models [13, 42, 43], leverage MLPs to project visual features into the language space. Recent advancements in MLLMs have extended their capabilities from static images to videos, typically by introducing video-language alignment through large-scale pretraining [9, 44]. Later models, such as Qwen2.5-VL [14], enhance temporal reasoning via dynamic resolution and absolute time encoding. Although existing video MLLMs excel at capturing high-level semantics and temporal patterns, they struggle to interpret the underlying 3D scene from video input, which inspires our work to enhance their spatial understanding capabilities. + +# 2.2 3D MLLMs for Scene Understanding + +Recent advances in MLLMs have sparked interest in extending their capabilities from 2D to 3D scene understanding [23, 24, 25, 45, 46, 47, 22, 48, 49, 50, 51, 52]. LL3DA [23] extracts scene-level features from 3D point clouds using a Q-Former, while Grounded 3D-LLM [45] integrates 3D detectors to generate object proposals. Methods like Chat3D [46], LEO [47], and Chat-Scene [22] first segment 3D objects and encode object-centric features for LLM fusion. Alternatively, 3D-LLM [48] and Scene-LLM [49] aggregate CLIP features from pre-segmented multi-view object patches into 3D point representations, leveraging multi-view images and camera parameters. LLaVA-3D [24] projects 2D multi-view patch features into voxel space for 3D-aware aggregation, and GPT4Scene [50] enhances 3D reasoning by first reconstructing scenes and then using BEV images as input. While these methods advance 3D scene understanding, most of them require additional 3D or 2.5D input data that is difficult to acquire in real-world scenarios. In contrast, our approach only requires 2D videos as input. + +# 2.3 Visual-based 3D Spatial Intelligence + +Visual-based 3D spatial intelligence aims to enable video multimodal large language models (MLLMs) to perceive, infer, and reason about 3D structures and spatial relationships purely from 2D visual inputs. Despite recent advances, most existing video MLLMs are still primarily designed for 2D understanding tasks, and their extension to visual-based 3D reasoning, e.g., 3D question answering [6, 53] and robotic manipulation [54], remains relatively underexplored. To address this limitation, a new wave of specialized benchmarks has emerged to systematically evaluate the spatial reasoning capabilities of these models [18, 20, 55, 56, 57, 58]. Among them, VSI-Bench [18] serves as a + +![](images/09aca31a6e24f6d791fc9bfe20f759c440c8bf8cb2043f1ef06fb373503ffad6.jpg) +Figure 2: Overview of Spatial-MLLM. Our model is composed of a 2D visual encoder $\mathcal{E}_{\mathrm{2D}}$ , a 3D spatial encoder $\mathcal{E}_{\mathrm{Spatial}}$ , which is initialized from a feed-forward visual geometry foundation model, a connector, and a large language model backbone. At inference time, we incorporate a space-aware frame sampling strategy to select spatially informative frames when the number of input frames is limited due to GPU memory constraints. + +pioneering benchmark that comprehensively assesses visual-spatial intelligence across multiple dimensions. STI-Bench [20] introduces physics-aware questions, such as velocity estimation, to quantify a model's spatial and kinematic reasoning abilities. Ego-ST Bench [55] evaluates the model's spatial intelligence from an egocentric perspective, while VLM4D [56] emphasizes motion dynamics, such as trajectory prediction, to probe 4D spatiotemporal interactions. Collectively, these benchmarks signify a shift toward a more holistic evaluation of visual-based spatial intelligence in MLLMs. + +# 3 Method + +In this section, we introduce Spatial-MLLM. Given a video of $N$ frames depicting a scene, denoted as $\mathcal{V} = \{\mathbf{f}_i\}_{i=1}^N$ , where $\mathbf{f}_i \in \mathbb{R}^{H \times W \times 3}$ , Spatial-MLLM is designed to understand spatial relationships, perform spatial reasoning, and generate appropriate responses. We begin by describing the model architecture in Section 3.1, which comprises a 2D visual encoder, a 3D spatial encoder, a connector, and a large language model backbone. Then we present the space-aware frame sampling strategy in Section 3.2, which selects $N_k$ spatially informative frames $\{\mathbf{f}_i^k\}_{i=1}^{N_k}$ , where $N_k \ll N$ . Finally, we introduce the training dataset construction process and two-stage training pipeline in Section 3.3. + +# 3.1 Spatial-MLLM Architecture + +In this section, we present the architecture of Spatial-MLLM, which is shown in Figure 2. We adopt Qwen2.5-VL-3B [14] as our base model and explore strategies to enhance its spatial understanding and reasoning capability. Before diving into the details, we first briefly introduce the key insights that motivate our design. + +What hinders visual-based spatial intelligence in existing video MLLMs? Existing video MLLMs [14, 13, 12] typically employ a pre-trained 2D visual encoder $\mathcal{E}_{2D}$ to extract 2D patch features $\mathbf{e}_{2D}$ . These features are then projected into visual tokens through a lightweight connection module. A large language model backbone $f_{\theta}$ subsequently generates the final response by conditioning on both visual and textual tokens. A critical bottleneck in this process lies in the nature of the visual features extracted. The required type of information varies by task: high-level semantic representations are essential for 2D recognition and understanding, whereas fine-grained structural cues are crucial for spatial reasoning. However, the visual encoders used in current video MLLMs are primarily pre-trained on image-text datasets (mainly image-caption pairs) [14, 26] following the CLIP [27] paradigm. As a result, these models predominantly capture semantic content and often lack spatial awareness when no additional 3D or 2.5D data are available [28, 29, 30]. To address this, our key insight is to unleash feed-forward visual geometry foundation models [32], which are trained on pixel-point pairs and can recover rich 3D structural information from 2D inputs, which complements the semantic features extracted by the 2D visual encoder. We design a dual-encoder architecture that + +exploits the strengths of both models and a connector to fuse semantic and structural information into unified visual tokens. Below, we introduce the core components of our design. + +Dual-Encoder. The proposed dual-encoder consists of a 2D encoder $\mathcal{E}_{2\mathrm{D}}$ and a 3D spatial encoder $\mathcal{E}_{\mathrm{Spatial}}$ . For the 2D encoder branch, we adopt the same design as the visual encoder of Qwen2.5-VL [14] to encode input frames into semantically rich features: + +$$ +\mathbf {e} _ {\mathrm {2 D}} = \mathcal {E} _ {\mathrm {2 D}} \left(\left\{\mathbf {f} _ {i} \right\} _ {i = 1} ^ {N _ {k}}\right), \quad \mathbf {e} _ {\mathrm {2 D}} \in \mathbb {R} ^ {N _ {k} ^ {\prime} \times \left\lfloor \frac {H}{p _ {\mathrm {2 D}}} \right\rfloor \times \left\lfloor \frac {W}{p _ {\mathrm {2 D}}} \right\rfloor \times d _ {\mathrm {2 D}}}, \tag {1} +$$ + +where $p_{2\mathrm{D}}$ and $d_{2\mathrm{D}}$ denote the patch size and feature dimension of the 2D visual encoder, respectively. The two consecutive frames are grouped for video input, thus $N_k' = \lceil N_k / 2 \rceil$ . + +For the spatial encoder branch, we utilize the feature backbone of VGGT [32]. Specifically, given $N_{k}$ frames of the scene video, we first patchify the input and then extract 3D features with alternating frame-wise self-attention and global self-attention [59]. This process allows $\mathcal{E}_{\mathrm{spatial}}$ to aggregate spatial information across different frames to get the final 3D features: + +$$ +\mathbf {e} _ {\mathrm {3 D}}, \mathbf {e} _ {c}, \mathbf {e} _ {\text {r e g i s t e r}} = \mathcal {E} _ {\text {s p a t i a l}} \left(\left\{\mathbf {f} _ {i} \right\} _ {i = 1} ^ {N _ {k}}\right), \quad \mathbf {e} _ {\mathrm {3 D}} \in \mathbb {R} ^ {N _ {k} \times \left\lfloor \frac {H}{p _ {\mathrm {3 D}}} \right\rfloor \times \left\lfloor \frac {W}{p _ {\mathrm {3 D}}} \right\rfloor \times d _ {\mathrm {3 D}}}, \tag {2} +$$ + +where $\mathbf{e}_{3\mathrm{D}}$ , $\mathbf{e}_c$ , and $\mathbf{e}_{\mathrm{register}}$ represent the dense 3D feature, the camera feature for each frame, and the register tokens [60], respectively. We only use $\mathbf{e}_{3\mathrm{D}}$ in the feature fusion stage as it captures the dense structure information of the input frames. + +Connector. After obtaining the 2D and 3D features, we use a connector to integrate the semantic and structural information from both branches. Specifically, we first align $\mathbf{e}_{\mathrm{3D}}$ with $\mathbf{e}_{\mathrm{2D}}$ in both spatial and temporal dimensions: + +$$ +\mathbf {e} _ {\mathrm {3 D}} ^ {\prime} = \text {R e a r r a n g e} \left(\mathbf {e} _ {\mathrm {3 D}}\right), \quad \mathbf {e} _ {\mathrm {3 D}} ^ {\prime} \in \mathbb {R} ^ {N _ {k} ^ {\prime} \times \left\lfloor \frac {H}{p _ {\mathrm {2 D}}} \right\rfloor \times \left\lfloor \frac {W}{p _ {\mathrm {2 D}}} \right\rfloor \times d _ {\mathrm {3 D}} ^ {\prime}}. \tag {3} +$$ + +Here, the spatially and temporally adjacent information in $\mathbf{e}_{3\mathrm{D}}$ is aggregated into the feature channel dimension, enabling alignment with $\mathbf{e}_{2\mathrm{D}}$ . Next, we employ a lightweight connector to fuse the information to obtain the unified visual tokens: + +$$ +\mathbf {e} = \operatorname {C o n n e c t o r} \left(\mathbf {e} _ {\mathrm {2 D}}, \mathbf {e} _ {\mathrm {3 D}} ^ {\prime}\right), \tag {4} +$$ + +where $\mathbf{e} \in \mathbb{R}^{S \times d_{llm}}$ denotes the final visual tokens and $S = N_k' \times \left\lfloor \frac{H}{p_{2D}} \right\rfloor \times \left\lfloor \frac{W}{p_{2D}} \right\rfloor$ is the sequence length. In practice, we adopt a MLP-based design (detailed in Section B.2). Although more complex feature fusion methods, e.g., cross-attention [59, 61, 51], could be applied, we find that this approach is effective to enhance the model's spatial understanding and reasoning capabilities. We leave the exploration of more advanced fusion strategies for future work. + +# 3.2 Space-Aware Frame Sampling + +Due to GPU memory constraints, video MLLMs can process only a limited subset of frames from a scene video sequence. For example, in the VSI-Bench setup [18], only 8 to 32 frames are sampled as input to the video MLLM, while a typical scene video in VSI-Bench contains over 2,000 frames. A widely adopted solution is uniform frame sampling [13, 14, 18], which is effective for general-purpose video understanding. However, as spatial videos represent 3D scenes, the sampling strategy for spatial understanding tasks should focus on capturing most information of the underlying scene, which uniform sampling fails to achieve. + +Benefiting from the feed-forward visual geometry foundation model, we design a straightforward space-aware frame sampling strategy at inference time. Specifically, given a scene video $\mathcal{V} = \{\mathbf{f}_i\}_{i=1}^N$ , our objective is to select $N_k$ frames, $\{\mathbf{f}_i^k\}_{i=1}^{N_k}$ that have most coverage of the underlying scene. To achieve this, we first uniformly subsample $N_m$ frames, $\{\mathbf{f}_i^m\}_{i=1}^{N_m}$ , where $N_m$ satisfies $N_k < N_m < N$ , and is determined by the available GPU memory. In practice, we choose $N_m = 128$ and $N_k = 16$ . We then leverage $\mathcal{E}_{3D}$ to extract their corresponding 3D features $\mathbf{e}_{3D}^m$ and camera features $\mathbf{e}_c^m$ . Subsequently, we use the pretrained camera head $f_c$ and depth head $f_d$ of the VGGT model [32] to decode a set of camera parameters and depth maps: + +$$ +\left\{\mathbf {E} _ {i} ^ {m}, \mathbf {K} _ {i} ^ {m} \right\} _ {i = 1} ^ {N _ {m}} = f _ {c} \left(\mathbf {e} _ {c}\right), \text {a n d} \left\{\mathbf {D} _ {i} ^ {m} \right\} _ {i = 1} ^ {N _ {m}} = f _ {d} \left(\mathbf {e} _ {\mathrm {3 D}}\right). \tag {5} +$$ + +This allows us to calculate the voxels $V(f_{i}^{m})$ covered by each frame $f_{i}^{m}$ , and formulate frame selection as a maximum coverage problem [62], i.e., select $N_{k}$ frames $\{\mathbf{f}_{i}^{k}\}_{i = 1}^{N_{k}} \subseteq \{\mathbf{f}_{i}^{m}\}_{i = 1}^{N_{m}}$ such that the total number of unique covered voxels $\left|\bigcup_{i = 1}^{N_k}V(\mathbf{f}_i^k)\right|$ is maximized. In practice, we apply a greedy algorithm to accelerate computation [63, 25]. Once the $N_{k}$ frames are selected, it is not necessary to recompute their 3D features $\mathbf{e}_{\mathrm{3D}}^{k}$ and the corresponding features from the precomputed set $\mathbf{e}_{\mathrm{3D}}^{m}$ can be directly reused. We provide the complete algorithm and detailed explanation in Section B.1. + +# 3.3 Training + +Training Data Construction. We first construct a visual-based 3D spatial question-answering dataset. The dataset has approximately $120\mathrm{k}$ QA pairs and is constructed from three sources: the training set of ScanQA [38], SQA3D [39], as well as additional self-created spatial QA data. All items in our training dataset are derived from scenes in the training set of ScanNet [64] and are each represented as a quadruple $\mathcal{I}_i = \langle \mathcal{Q}_i, \mathcal{A}_i, \mathcal{V}_i, \mathcal{M}_i \rangle$ , denoting the question, answer, video (scene) ID, and meta-information (e.g., task type), respectively. For the self-created QA data, we follow the data processing pipeline proposed in VSI-Bench [18]. Specifically, we first convert ScanNet scenes into continuous video clips at 24 FPS and $640 \times 480$ resolution. Then we generate spatial reasoning QA pairs leveraging the meta-annotations of Scannet. The generated QA pairs cover various spatial understanding and reasoning tasks, including object counting, object size, room size, absolute distance, appearance order, relative distance, and relative direction. Since the QA pair construction process is similar to that of VSI-Bench [18], we exclude the QA pair $\mathcal{I}_i$ if its scene video $\mathcal{V}_i$ is used in VSI-Bench (these videos are sourced from the validation set of Scannet) to prevent data leakage. Finally, the self-created data contains approximately $70\mathrm{k}$ QA pairs in total. We provide additional details on training data construction in the section B.3. Figure 3 shows a brief summary of key statistics of the training dataset. + +Supervised Fine-tuning. Leveraging the constructed training dataset, we first perform supervised fine-tuning (SFT) on our model. Since $\mathcal{E}_{2\mathrm{D}}$ and $\mathcal{E}_{\mathrm{spatial}}$ are pre-trained on large-scale image-text and pixel-point pairs, respectively, we freeze them to preserve their ability to extract rich semantic and structural information. We jointly train the connection module and the LLM backbone to enable the model to adaptively fuse 2D and 3D features and enhance its spatial understanding and reasoning capability. During this stage, we employ the standard cross-entropy loss $\mathcal{L}_{\mathrm{ce}}$ between the model-generated answers and the ground-truth annotations: + +$$ +\mathcal {L} _ {\mathrm {c e}} (\theta) = - \sum_ {i} \log P \left(o ^ {(i)} \mid o ^ {(1: i - 1)}, q, \{f _ {j} \} _ {j = 1} ^ {N _ {k}}\right) \tag {6} +$$ + +where $\{f_j\}_{j=1}^{N_k}$ denotes input video frames, $q$ denotes the system prompt and question, $o^{(i)}$ represents the $i$ -th token in the ground-truth answer, and $o^{(1:i-1)}$ denotes the preceding answer tokens. + +![](images/e175eedba500964bcf3ce878b9732427a159576bf2781c10c2076c9a5ad81a09.jpg) +Figure 3: Basic statistic of training dataset. + +RL Training. Following the SFT stage, we first perform a simple cold start [35] to help the model adapt to the correct reasoning format. Then we train the model using Group Relative Policy Optimization (GRPO) [36] to enhance its long-CoT [37] spatial reasoning capability. During training, we first sample a set of output $\{o_1,o_2,\dots,o_G\}$ for each question $q$ from the policy model $\pi_{\theta_{\mathrm{old}}}$ . Then we optimize the policy model by maximizing the following objective: + +$$ +\mathcal {J} _ {\mathrm {G R P O}} (\theta) = \mathbb {E} _ {q, o _ {i}} \left[ \frac {1}{G} \sum_ {i = 1} ^ {G} \min \left(\frac {\pi_ {\theta} (o _ {i} \mid q)}{\pi_ {\theta_ {\mathrm {o l d}}} (o _ {i} \mid q)} A _ {i}, \operatorname {c l i p} \left(\frac {\pi_ {\theta} (o _ {i} \mid q)}{\pi_ {\theta_ {\mathrm {o l d}}} (o _ {i} \mid q)}, 1 \pm \epsilon\right) A _ {i}\right) - \beta \mathrm {K L} [ \pi_ {\theta} \| \pi_ {\mathrm {r e f}} ] \right] \tag {7} +$$ + +where $A_{i} = \frac{r_{1} - \mathrm{mean}(r_{1},r_{2},\ldots,r_{G})}{\mathrm{std}(r_{1},r_{2},\ldots,r_{G})}$ is the advantage function computed using the group rewards. + +In GRPO, the design of the reward function is critical. In addition to a formatting reward applied to all task types, we introduce task-dependent reward modeling to ensure that it accurately reflects + +Table 1: Evaluation Results on VSI-Bench [18]. For Spatial-MLLM and Qwen2.5-VL series [14], we use 16 frames as input and report micro average scores. For other open-source methods and GPT-4o [5], the number of frames is the same as VSI-Bench setting (ranging from 8 to 32 frames). For Gemini-1.5 Pro [4], it samples video frames at 1 FPS. Bold and underline denote the best-performing and second-best-performing open-source models, respectively. + +
MethodsNumerical QuestionMultiple-Choice QuestionAvg. Rank
Obj. Cnt.Abs. Dist.Obj. SizeRoom SizeRel. Dist.Rel. Dir.Route PlanAppr. Order
Proprietary Models
GPT-4o [5]46.25.343.838.237.041.331.528.534.0 7
Gemini-1.5 Pro [4]56.230.964.143.651.346.336.034.645.4 2
Open-source Models
InternVL2-40B [7]34.926.946.531.842.132.234.039.636.0 6
LongVILA-8B [66]29.19.116.70.029.630.732.525.521.6 12
VILA-1.5-40B [67]22.424.848.722.740.525.731.532.931.2 9
LongVA-7B [68]38.016.638.922.233.143.325.415.729.2 11
LLaVA-OneVision-72B [6]43.523.957.637.542.539.932.544.640.2 4
LLaVA-Video-72B [12]48.922.857.435.342.436.735.048.640.9 3
Spatial-MLLM and Qwen2.5-VL Series
Qwen2.5-VL-3B [14]24.324.731.722.638.341.626.321.230.6 10
Qwen2.5-VL-7B [14]40.914.843.410.738.638.533.029.833.0 8
Qwen2.5-VL-72B [14]25.129.354.538.838.237.034.028.937.0 5
Spatial-MLLM-4B65.334.863.145.141.346.233.546.348.4 1
+ +the proximity between the predicted and ground-truth answers. Specifically, we categorize the data into three types based on answer format: numeric answer questions, multiple-choice questions, and verbal answer questions. For numeric questions, we compute the mean relative accuracy [18]. For multiple-choice questions, we employ an exact match reward. For verbal answer questions, we use fuzzy matching based on Levenshtein distance. Further details on reward calculation are provided in Section B.5. + +# 4 Experiments + +# 4.1 Implementation Details + +Training details. Spatial-MLLM is built on Qwen2.5-VL [14] and VGGT [32] and has approximately 4.9B parameters in total. We use the visual encoder of Qwen2.5-VL [14] to initialize $\mathcal{E}_{\mathrm{2D}}$ , and the LLM backbone of it to initialize $f_{\theta}$ . We then use the feature backbone of VGGT [32] to initialize $\mathcal{E}_{\mathrm{spatial}}$ . During training, we use $640 \times 480$ resolution and limit video frames to 16. In the SFT stage, we train the model using Adam optimizer [65] for one epoch. We set the global batch size to 16 and use a linear learning-rate schedule, with a peak value of $10^{-5}$ . In the cold start stage, we first construct a small CoT dataset. Specifically, we prompt Qwen2.5-VL-72B [14] to generate multiple thinking processes and answers according to the scene video and question. Then we use the GT answer to filter a correct thinking-answer pair (more details are provided in Section B.4). We use a similar setting as in the SFT stage to train the model for 200 steps. In the RL stage, we perform 8 rollouts per question and set the default sampling temperature to 1. The KL divergence coefficient, $\beta$ , is set to 0.04. Due to computational resource limitations, we train the model for 1,000 steps with a learning rate of 1e-6. We show the training curve of SFT Stage and RL Stage in Figure 4. + +Inference Details. During inference, we set $N_{m} = 128$ and $N_{k} = 16$ for space-aware frame sampling. Since spatial reasoning requires a certain level of determinism, we set the temperature to 0.1 and the top- $p$ to 0.001. The default input resolution from the scene video is $640 \times 480$ . + +# 4.2 Comparisons on VSI-Bench + +Setup. VSI-Bench [18] contains more than 5,000 question-answer pairs derived from egocentric videos sourced from ScanNet [64], ScanNet++[69], and ARKitScenes[70]. The task types are divided into Multiple-Choice Answer (MCA) and Numerical Answer (NA). For the MCA tasks, we compute mean accuracy, and for the NA tasks, we calculate relative accuracy across confidence thresholds $\mathcal{C} = \{0.5, 0.55, \dots, 0.95\}$ . We report the final average score and individual metrics on eight task types of VSI-Bench, including: (1) configurational reasoning tasks (object counting, relative direction, absolute direction, and route planning), (2) measurement estimation tasks (object size, room size, and absolute distance), and (3) spatiotemporal reasoning tasks (appearance order). For Spatial-MLLM + +Table 2: Evaluation Results on ScanQA [38] and SQA3D [39]. We use the val set of ScanQA and test set of SQA3D for evaluation following common practice [22, 47, 25]. Bold and underline denote the best-performing and second-best-performing models in each model category, respectively. + +
MethodsScanQA (val)SQA3D (test)Video-Input Only
BLEU-1BLEU-4METEORROUGE-LCIDErEM-1EM-R1
Task-Specific Models
ScanQA [38]30.210.113.133.364.947.2-X
SQA3D [39]30.511.213.534.5-46.6-X
3D-Vista [71]--13.935.7-48.5-X
3D/2.5D-Input Models
3D-LLM [48]39.312.014.535.769.4--X
LL3DA [23]-13.515.937.376.8--X
Chat-Scene [22]43.214.318.041.687.754.657.5X
3D-LLaVA [21]-17.118.443.192.654.556.6X
Video-3D LLM [25]47.116.219.849.0102.158.6-X
Video-Input Models
Qwen2.5-VL-3B [14]22.53.89.725.447.443.445.9✓
Qwen2.5-VL-7B [14]27.83.011.429.353.946.549.8✓
Qwen2.5-VL-72B [14]26.812.013.035.266.947.050.9✓
LLaVA-Video-7B [12]39.73.117.744.688.748.5-✓
Oryx-34B [53]38.0-15.037.372.3--✓
Spatial-MLLM-4B44.414.818.445.091.855.958.7✓
+ +and Qwen2.5-VL series, we report micro average scores in Table 1 and macro average scores in Table 5. + +Baseline Models. We compare our model with a broad range of video-input MLLMs. For proprietary model, we include GPT-4o [5] and Gemini-1.5 Pro [4] for comparison. For open-source video-input MLLMs, we compare our model with InternVL2 [7], LongVILA [66], VILA [67], LongVA [68], LLaVA-NeXT-Video [12], LLaVA-OneVision [6], and the Qwen2.5-VL [14] series. The parameter count of the baseline models is reported in Table 1. + +Results. We present the quantitative results on VSI-Bench [18] in Table 1 and Table 5. Despite having 4.9B parameters, Spatial-MLLM significantly outperforms all proprietary and open-source MLLMs, including those with substantially larger parameter counts (e.g., 32B or 72B). Among the remaining models, the best-performing one is the proprietary Gemini-1.5 Pro [4]. Notably, Spatial-MLLM is provided with only 16 input frames per video, while Gemini-1.5 Pro [4] samples videos at 1 FPS (i.e., an average of 85 frames per video on VSI-Bench) according to its API instructions [18]. Despite the significantly lower number of input frames, Spatial-MLLM still achieves higher average accuracy than Gemini-1.5 Pro [4]. + +# 4.3 Comparison on ScanQA and SQA3D + +Setup. ScanQA [38] and SQA3D [39] are two 3D question-answering benchmarks built upon ScanNet [64]. Since the authors did not provide a test set for ScanQA, we evaluate it using the validation set, which consists of 4,675 QA pairs focused on understanding spatial relationships such as object alignment and orientation, as well as the ability to accurately identify objects in 3D scenes based on textual questions. We follow standard practice [25, 50] by evaluating answer quality using the following metrics: CiDEr, BLEU-1, BLEU-4, METEOR, and ROUGE-L. For SQA3D, we evaluate the model on its test set, which contains 3,519 QA pairs. The task requires the model to first understand its position and orientation within the 3D scene, as described by text, then reason about its environment and answer a question under those conditions. Since SQA3D contains definitive answers, we use exact match accuracy (EM) and its refined version (EM-R) as evaluation metrics. We provide the evaluation results using additional metrics for both benchmarks in Section C.2. + +Baselines. Since both the ScanQA [38] and SQA3D [39] benchmarks provide additional 3D annotations (e.g., point clouds and depth maps of the scene), we compare Spatial-MLLM with several other model types in addition to video-input MLLM. These include task-specific models designed for 3D question-answering tasks, such as ScanQA [38], SQA3D [39], 3D-VisTA [71], and LLMs that require point clouds or depth maps as input, such as Chat-Scene [22], Video-3D LLM [25], and 3D-LLaVA [21]. + +![](images/6ec14f19d2f0d78a792de9ee35a686cccab868ae263ffdd156391169b1643fb4.jpg) +Figure 4: Visualization of Training Curves in the SFT and RL Stages. For the SFT stage, we present the mean token accuracy and loss curves. For the RL stage, we show the dynamics of completion length and reward. + +![](images/89ff9093e8098a1de369fe0eabfe9a5543c5a3e47186cf0d5e774e17135bb1dd.jpg) + +![](images/7182a5064f928bb4a17d97345460794752c835b9cbe9499967ebd1f1ac7d37e3.jpg) + +![](images/33568558a98d1b7f3a64f3d7023798b1bbe2206b4bda335a1cfc18f807bc3370.jpg) + +Results. We present the quantitative results on the ScanQA [38] and SQA3D [39] benchmarks in Table 2. As shown, Spatial-MLLM significantly outperforms all video-input models across all metrics on both ScanQA and SQA3D. Our model also surpasses all task-specific models. Among models utilizing 3D or 2.5D input, only 3D-LLaVA [21] (on ScanQA) and Video-3D-LLM [25] (on ScanQA and SQA3D) achieve better performance than Spatial-MLLM. However, 3D-LLaVA requires additional point cloud input, and Video-3D-LLM depends on depth maps. Despite not relying on any additional 3D or 2.5D input, our model still outperforms other 3D-dependent models such as 3D-LLM [48], LL3DA [23], and Chat-Scene [22]. + +# 4.4 Ablation Study and Analysis + +Ablation on Input Frame Number. We evaluate the effect of the number of input frames on VSI-Bench across different models, including Spatial-MLLM, Gemini-1.5 Pro [4], and Qwen2.5-VL3B [14]. The result is shown in Table. 4. For the 1 fps setting of Gemini-1.5 Pro, we upload the entire video to the model following the VSI-Bench [18], where the video is sampled at 1 fps according to the API instructions. For the 0.1 fps and 0.25 fps settings, we first manually sample the video frames and then upload these sampled frames to the model. As shown, all models exhibit improved performance as the number of input frames increases, particularly when the number of frames is small. + +Effectiveness of Space-aware Frame Sampling. We evaluate different frame sampling configurations in Table 4, including 8, 16, and 32 frames using uniform sampling and space-aware frame sampling. As shown, increasing the number of sampled frames improves performance for both space-aware frame sampling and uniform sampling. Compared with uniform sampling, space-aware frame sampling consistently outperforms it when the number of input frames is the same. + +We further provide a visualization of our space-aware frame sampling in Fig. 5, which shows the point maps (predicted by the VGGT [32]) corresponding to the frames selected by different sampling strategies. As shown, the proposed space-aware frame sampling strategy consistently yields more spatial coverage than uniform sampling, which often overlooks transient regions that appear briefly in the video and tends to produce redundant viewpoints when the camera remains static. + +Effectiveness of RL Training. We evaluate SpatialMLLM's performance before and after GRPO training on VSI-Bench. The results are presented in the second (SFT + GRPO) and third (SFT) rows of Table 3. As shown, even though we conduct only small scale RL training (i.e., 1,000 steps), the GRPO-trained model + +Table 3: Ablation Study. We report micro average results for numerical questions and multiple-choice questions on VSI-Bench [18] in different settings. + +
MethodsNumericalMultiple-ChoiceAvg.
Spatial-MLLM52.743.848.4
Spatial-MLLM (w/o sa sampling)51.642.347.1
Spatial-MLLM (w/o sa sampling & GRPO)51.540.446.1
Qwen2.5-VL-3B (SFT)‡49.240.344.9
Qwen2.5-VL-3B (SFT)†47.132.640.0
Qwen2.5-VL-3B34.426.930.6
+ +still achieves performance gains, suggesting that long chain-of-thought reasoning enhances the spatial reasoning capabilities required by VSI-Bench [18]. + +Effectiveness of the Spatial-MLLM Architecture and Training Dataset. We compare the supervised fine-tuned version of Spatial-MLLM, two supervised fine-tuned versions of Qwen2.5-VL-3B [14] (the base model of Spatial-MLLM) and original Qwen2.5-VL-3B model in Table 3. $\dagger$ denotes results obtained with the R1-V [72] training framework. $\ddagger$ denotes results which we further apply a question token mask during the loss computation process within R1-V [72], which aligns better with Spatial-MLLM training process. As shown 3, both SFT versions of Qwen2.5-VL-3B + +![](images/77c96e7a57ed94ea8b3f432881799ffeae5eb9c7617ed2533af21fd92b3125ef.jpg) +Figure 5: Visualization of different frame sampling strategies. For clarity of visualization, we set $N_{m} = 128$ and $N_{k} = 8$ in the visualization example. + +![](images/858cd35a165920c4b7a836ecccc06d77631163f958d82f9c55af2eec01de7dc2.jpg) + +![](images/9320fb1352c3a9a5f185e897a7f806f2fe56b391933d34e77210e7aa764adbbf.jpg) + +![](images/7d9d0ae95ced2069345d8acf91fe53e1fafffba9e0f1b81df619807eeebe710b.jpg) + +Table 4: Ablation on Input Frame Number. We report micro average results on VSI-Bench [18] using different input frame numbers and frame rates (FPS). For Gemini-1.5 Pro, the input frame number is averaged over questions. + +
MethodsFramesFPSNumericalMultiple-ChoiceAvg.
Spatial-MLLM8N/A50.841.246.1
16N/A52.743.848.4
32N/A53.145.349.3
Spatial-MLLM (w/o sa sampling)8N/A48.239.243.8
16N/A51.642.347.1
32N/A52.444.248.4
Gemini-1.5 Pro [4]12.2 (avg.)0.143.135.739.5
29.6 (avg.)0.2548.837.843.5
117.1 (avg.)149.744.046.9
Qwen2.5-VL-3B [14]8N/A33.120.226.5
16N/A34.426.930.6
32N/A35.728.331.9
+ +show improvements, indicating the effectiveness of our proposed dataset to enhance the model's spatial reasoning capabilities. Furthermore, both models underperform compared to the supervised fine-tuned version of Spatial-MLLM, which validates the effectiveness of the proposed architecture. + +# 5 Conclusion + +We introduce Spatial-MLLM, a method that enables effective spatial understanding and reasoning from purely 2D visual inputs. By combining a semantic 2D encoder with a structure-aware spatial encoder initialized from a visual geometry foundation model, our dual-encoder design captures both semantic and spatial cues. Additionally, our proposed space-aware frame sampling strategy further enhances performance under limited input constraints. Trained on the collected dataset, our model achieves state-of-the-art results across multiple benchmarks. + +Limitations and Future Work. Although Spatial-MLLM demonstrates significant improvements over previous video MLLMs across a wide range of visual-based spatial understanding and reasoning tasks, there remains room to scale Spatial-MLLM further in terms of model size and training data. Moreover, as this work primarily addresses visual-based spatial intelligence, we have trained and evaluated our model specifically on relevant datasets and benchmarks. An interesting direction for future work would be to explore how integrating spatial structural information might further benefit general video understanding and reasoning tasks. + +# Acknowledgments + +This work was supported in part by the Beijing Natural Science Foundation under Grant L252011, and by the National Natural Science Foundation of China under Grant 62206147. + +# References + +[1] J.-B. Alayrac, J. Donahue, P. Luc, A. Miech, I. Barr, Y. Hasson, K. Lenc, A. Mensch, K. Millican, M. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Technical Appendices and Supplementary Material + +# A Broader Impacts + +This work advances spatial reasoning in multimodal large language models by enabling 3D understanding purely from 2D visual inputs. Such capability may broaden the accessibility of spatially aware AI in domains such as robotics, autonomous systems, and visual content understanding, without requiring costly 3D data. As with other vision-language models, considerations of data privacy and ethical deployment remain important to ensure positive social outcomes. + +# B Additional Method Details + +# B.1 Details of Space-Aware Frame Sampling + +Our space-aware frame sampling algorithm consists of three stages: (1) Scene geometry preprocessing, (2) Voxelization and coverage calculation, and (3) Greedy maximum coverage selection. Beginning with the original video sequence $\mathcal{V} = \{\mathbf{f}_i\}_{i=1}^N$ , we first perform uniform subsampling to obtain $N_m = 128$ candidate frames $\{\mathbf{f}_i^m\}_{i=1}^{N_m}$ . For each subsampled frame, we leverage the backbone and head of VGGT [32] to compute $\{\mathbf{E}_i^m, \mathbf{K}_i^m\}_{i=1}^{N_m}$ and $\{\mathbf{D}_i^m\}_{i=1}^{N_m}$ as illustrated in the main paper. Then we reconstruct 3D point maps $\mathcal{P}_i^m$ through depth reprojection: + +$$ +\mathcal {P} _ {i} ^ {m} = \mathbf {D} _ {i} ^ {m} \cdot \mathbf {K} _ {i} ^ {- 1} [ \mathbf {u} | \mathbf {v} | 1 ] ^ {\top} \cdot \mathbf {E} _ {i} ^ {- 1}, \tag {8} +$$ + +where $(\mathbf{u},\mathbf{v})$ denote pixel coordinates. In practice, we also obtain a confidence value $c(p)\in [0,1]$ for each point $p\in \mathcal{P}_i^m$ from the depth head. Although VGGT [32] can also directly decode point maps from 3D dense features, we find that using depth and camera produces more accurate results. + +The voxelization and coverage calculation process first establishes a 3D bounding box encompassing all valid scene points: + +$$ +\mathcal {P} _ {\text {valid}} = \bigcup_ {i = 1} ^ {N _ {m}} \left\{p \in \mathcal {P} _ {i} ^ {m} \mid c (p) > 0.1 \wedge c (p) \geq \text {Percentile} (\{c (p) \}, 50 \%) \right\}. \tag{9} +$$ + +We then discretize the bounding box into voxels. To handle relative scales in VGGT [32] outputs, we use an adaptive way to set the voxel size $\Delta$ to $\frac{1}{\lambda}$ of the minimum dimension of the scene's bounding box: + +$$ +\Delta = \frac {1}{\lambda} \cdot \min \left(\max \left(\mathcal {P} _ {\text {v a l i d}}\right) - \min \left(\mathcal {P} _ {\text {v a l i d}}\right)\right), \tag {10} +$$ + +where $\lambda$ is a hyperparameter and we set it to 20. Each frame's voxel coverage $V(\mathbf{f}_i^m)$ is then calculated by discretizing its valid points: + +$$ +V \left(\mathbf {f} _ {i} ^ {m}\right) = \left\{\left\lfloor \frac {p - \operatorname* {m i n} \left(\mathcal {P} _ {\text {v a l i d}}\right)}{\Delta} \right\rfloor \mid p \in \mathcal {P} _ {i} ^ {m} \cap \mathcal {P} _ {\text {v a l i d}} \right\}. \tag {11} +$$ + +Finally, we can formulate frame selection as the typical maximum coverage problem [62]: + +$$ +\max _ {\mathcal {S} \subseteq \{1, \dots , N _ {m} \}} \left| \bigcup_ {i \in \mathcal {S}} V \left(\mathbf {f} _ {i} ^ {m}\right) \right| \quad \text {s . t .} \quad | \mathcal {S} | = N _ {k}, \tag {12} +$$ + +In practice, we set $N_{k} = 16$ and use a greedy approach [63, 25] to iteratively select the frame that provides the maximum new coverage, which is illustrated in Algorithm 1. + +# B.2 Details of Feature Fusion + +Both the 2D and 3D encoders use a spatial patch size of 14. The 2D encoder further reduces the token sequence length by merging tokens spatially ( $2 \times 2$ adjacent tokens) and temporally (every 2 consecutive frames). As a result, the 2D encoder outputs exactly one-eighth the number of tokens compared to the 3D encoder (we exclude register and camera tokens). To align these tokens, we first apply the same spatial-temporal merging strategy as used in the 2D encoder. After merging, we rearrange the tokens into sequence, ensuring the two sets of tokens are precisely aligned in both position and number. Then we project both tokens into language model's hidden dimension with a two-layer MLP and fuse them by element-wise addition. + +Algorithm 1 Greedy Maximum Coverage Sampling +Input Frame voxel sets $\{V(\mathbf{f}_i^m)\}_{i = 1}^{N_m}$ target selection size $N_{k}$ +Output Selected frame indices $\mathcal{S}\subseteq \{1,\dots ,N_m\}$ +1: $\mathcal{S}\gets \emptyset$ $\triangleright$ Selected frames +2: $\mathcal{C}\gets \emptyset$ $\triangleright$ Covered voxels +3: $\mathcal{R}\gets \{1,\ldots ,N_m\}$ $\triangleright$ Remaining candidates +4: for $t\gets 1$ to $N_{k}$ do +5: if $\mathcal{R} = \emptyset$ then +6: break $\triangleright$ No remaining candidates +7: end if +8: $i^{*}\gets \mathrm{argmax}|V(\mathbf{f}_{i}^{m})\backslash \mathcal{C}|$ Max coverage gain +9: if $|V(\mathbf{f}_{i^{*}}^{m})\backslash \mathcal{C}| = 0$ then +10: break $\triangleright$ No additional coverage +11: end if +12: $\mathcal{S}\gets \mathcal{S}\cup \{i^{*}\}$ Update selection +13: $\mathcal{C}\gets \mathcal{C}\cup V(\mathbf{f}_{i^{*}}^{m})$ Update covered voxels +14: $\mathcal{R}\gets \mathcal{R}\backslash \{i^{*}\}$ Remove from candidates +15: end for +16: return S + +# B.3 Details of Dataset Construction + +We follow a similar approach to that used in [18] to construct the self-created part of training dataset. Specifically, the construction involves three main processes: video preprocessing, metadata computation, and QA pair generation. + +Video Preprocessing. In this stage, we extract frames from the raw ScanNet [64] scans and convert them into videos at 24 FPS with a resolution of $640 \times 480$ . + +Metadata Computation. In this stage, we extract spatial and semantic metadata from raw ScanNet scans and their associated semantic annotations. First, we align each raw scene mesh using the provided axis alignment matrices and convert it to the Open3D [73] point cloud. At the room level, we compute the room size using the alpha-shape algorithm and determine the center coordinates. At the object level, we generate oriented bounding boxes (OBBs) for each valid object instance and assign semantic labels from the annotations, excluding structural elements (e.g., walls, floors) and ambiguous categories (e.g., otherstructure). To ensure consistency across categories, we remap the original ScanNet semantic labels to a new label set based on the NYU40 classes [74, 75] (which we manually add and remove some categories to align with VSIBench [18]). In addition, we collect the projected 2D semantic annotation of each scene video for the appearance order task. The final metadata for each scene includes: (1) room size and center coordinates; (2) the projected 2D semantic annotation of the scene video; (3) object instances and their OBB parameters, including rotation matrices, extents, and centers; and (4) semantic labels for each object. + +QA Pair Generation. Finally, we generate QA pairs of different tasks, including object counting, object size, room size, absolute distance, appearance order, relative distance, and relative direction. + +- Object counting (numerical): We first count how many times each object category appears in the scene, then randomly sample a category that appears at least twice. Question template: "How many (s) are in this room?" +- Object size (numerical): We randomly sample a unique object in the scene and take the longest side of its oriented bounding box (OBB) as the ground-truth length (in cm). Question template: "What is the length of the longest dimension (length, width, or height) of the , measured in centimeters?" +- Room size (numerical): We use the pre-computed room size (in $\mathrm{m}^2$ ) as the ground-truth value. +Question template: "What is the size of this room (in square meters)?" +- Absolute distance (numerical): For a pair of objects, we uniformly sample points inside each OBB and take the minimum Euclidean distance between the two point clouds as the ground-truth + +(in m). Question template: "Measuring from the closest point of each object, what is the direct distance between the and the (in meters)?" + +- Appearance Order (multiple choice): We calculate the first appearance timestamp of each category, which is the timestamp when its visible pixel count exceeds a predefined threshold. Using these timestamps, we generate the correct order of appearance among the categories, along with other options. Question template: What will be the first-time appearance order of the following categories in the video: , , , +- Relative distance (multiple choice): We use an "anchor" object that is unique in the scene and then select four additional objects while enforcing 15-30cm separation thresholds between options. Question template: "Which of these objects (, , , ) is closest to the ?" +- Relative direction (multiple choice): For triple $\{position, facing, query\}$ of unique categories, we compute the horizontal angle between the vectors position $\rightarrow$ facing and position $\rightarrow$ query. The angle is then discretized into directional classes (easy: left/right, medium: left/right/back, hard: front-left/front-right/back-left/back-right). Question template (easy example): "If I am standing by the and facing the , is the to the left or the right?" + +# B.4 Details of Cold Start + +To align the model with the desired reasoning format, we perform a simple cold start for 200 steps before GRPO training. The key to this stage is the construction of a spatial reasoning dataset with chain-of-thought (CoT) annotations. The construction process is as follows: + +Subset Sampling. We begin by sampling a subset $\mathcal{D}_0 = \{\mathcal{I}_i\}_{i=1}^{N_s} = \{\langle \mathcal{Q}_i, \mathcal{A}_i, \mathcal{V}_i, \mathcal{M}_i \rangle\}_{i=1}^{N_s}$ from our training dataset. + +Multi-path CoT Generation. For each item $\mathcal{I}_i\in \mathcal{D}_0$ , we utilize Qwen2.5-VL-72B [14] to generate $K$ independent reasoning processes $\hat{\mathcal{T}}_i^{(k)}$ and corresponding answers $\hat{\mathcal{A}}_i^{(k)}$ . We then compute a reward $r_i^{(k)} = \text{Reward}(\hat{\mathcal{A}}_i^{(k)},\mathcal{A}_i)$ for each reasoning-answer pair, where $\text{Reward}(\cdot ,\cdot)$ is the reward function described in Sec B.5. Consequently, we obtain a set of outputs $\mathcal{O}_i = \{(\hat{\mathcal{T}}_i^{(k)},\hat{\mathcal{A}}_i^{(k)},r_i^{(k)})\}_{k = 1}^K$ for each $\mathcal{I}_i\in \mathcal{D}_0$ . + +Adaptive Filtering. Since Qwen2.5-VL-72B [14] may generate incorrect reasoning processes and answers, we apply a filtering process based on the computed rewards. While using a global reward threshold is straightforward, it often results in an imbalance across question types in the selected subset. To mitigate this, we adopt an adaptive filtering strategy. Specifically, for each item $\mathcal{I}_i\in \mathcal{D}_0$ , we first keep the output with the highest reward to get $\hat{\mathcal{O}}_i = \{(\hat{T}_i^{(k^*)},\hat{\mathcal{A}}_i^{(k^*)},r_i^{(k^*)})\}$ where $k^{*} = \arg \max_{k}r_{i}^{(k)}$ . Let $\hat{r}_i = r_i^{(k^*)}$ denote the maximum reward. We then categorize all items based on their question type and compute a question type-dependent threshold $\tau_{t(i)}$ , where $t(i)$ denotes the type of problem $i$ . The item is added into the cold start set if and only if: + +$$ +\hat {r} _ {i} \geq \tau_ {t (i)} \quad \text {a n d} \quad \hat {r} _ {i} > 0, +$$ + +where the type-dependent threshold satisfies $\tau_{t(i)}\coloneqq \mathrm{Quantile}\bigl (\{\hat{r}_j\mid t(j) = t(i)\} ,0.5\bigr)$ . This rule preserves approximately the top $50\%$ of generations per question type while discarding degenerate (zero-reward) outputs. In practice, we set $N_{s} = 5000$ and $K = 3$ , and finally we get 2459 items in the cold start set. We provide a pseudocode for this process in Algorithm 2: + +# B.5 Details of SFT and GRPO Training + +Reward Calculation. Given predicted answer $\mathcal{A}_{\mathrm{pred}}$ and ground truth answer $\mathcal{A}_{\mathrm{gt}}$ , the reward function $\text{Reward}(\mathcal{A}_{\mathrm{pred}}, \mathcal{A}_{\mathrm{gt}})$ consists of a format reward $\mathcal{R}_{\mathrm{fmt}}$ and a task-specific reward: + +$$ +\operatorname {R e w a r d} \left(\mathcal {A} _ {\text {p r e d}}, \mathcal {A} _ {\mathrm {g t}}\right) = \lambda_ {1} R _ {\text {f o r m a t}} + \lambda_ {2} \left\{ \begin{array}{l l} \mathrm {R} _ {\mathrm {M C}}, & \text {m u l t i p l e - c h o i c e} \\ \mathrm {R} _ {\mathrm {M R A}}, & \text {n u m e r i c a l} \\ \mathrm {R} _ {\text {V e r b a l}}, & \text {v e r b a l} \end{array} \right. \tag {13} +$$ + +Algorithm 2 Cold Start Dataset Construction +Input Original dataset $\mathcal{D}$ +1: Qwen2.5-VL model $M$ +2: Reward function Reward(.,.) +3: Sample size $N_{s}$ , Paths per item $K$ +Output Filtered dataset $\mathcal{D}_{\mathrm{cold}}$ +4: Initialize $\mathcal{D}_0\gets$ Sampling $(\mathcal{D},N_s)$ +5: $\mathcal{D}_{\mathrm{cold}}\gets \emptyset$ +6: for each item $\mathcal{I}_i = \langle \mathcal{Q}_i,\mathcal{A}_i,\mathcal{V}_i,\mathcal{M}_i\rangle \in \mathcal{D}_0$ do +7: Generate $K$ reasoning paths: $\{\hat{T}_i^{(k)}\}_{k = 1}^K\gets M(\mathcal{Q}_i,\mathcal{V}_i)$ +8: Compute rewards: $r_i^{(k)}\gets$ Reward $(\hat{\mathcal{A}}_i^{(k)},\mathcal{A}_i),\forall k$ +9: Select best path: $k^{*}\gets$ arg max $k r_{i}^{(k)}$ +10: Record $\hat{r}_i\gets r_i^{(k^*)},\hat{\mathcal{O}}_i\gets (\hat{T}_i^{(k^*)},\hat{\mathcal{A}}_i^{(k^*)})$ +11: end for +12: Group items by type: $\{\mathcal{G}_t\} \gets$ GroupByType( $\{\hat{r}_i\}$ ) +13: for each question type $t$ do +14: Compute threshold: $\tau_t\gets$ Quantile( $\{\hat{r}_j|j\in \mathcal{G}_t\}$ ,0.5) +15: end for +16: for each item $\mathcal{I}_i\in \mathcal{D}_0$ do +17: if $\hat{r}_i\geq \tau_{t(i)}$ and $\hat{r}_i > 0$ then +18: $\mathcal{D}_{\mathrm{cold}}\gets \mathcal{D}_{\mathrm{cold}}\cup \{\hat{\mathcal{O}}_i\}$ +19: end if +20: end for +21: return $\mathcal{D}_{\mathrm{cold}}$ + +where $\lambda_{1}$ and $\lambda_{2}$ are hyperparameters, both of which are set to 1 in our implementation. For multiple-choice questions, we implement exact match criterion: + +$$ +\mathrm {R} _ {\mathrm {M C}} \left(\mathcal {A} _ {\text {p r e d}}, \mathcal {A} _ {\mathrm {g t}}\right) = \mathbb {I} \left(\psi \left(\mathcal {A} _ {\text {p r e d}}\right) = \psi \left(\mathcal {A} _ {\mathrm {g t}}\right)\right) \tag {14} +$$ + +where $\psi (\cdot)$ performs answer normalization through whitespace stripping and $\mathbb{I}(\cdot)$ denotes the indicator function. For numerical tasks, we compute mean relative accuracy (MRA) [18]: + +$$ +\mathrm {R} _ {\mathrm {M R A}} \left(\mathcal {A} _ {\text {p r e d}}, \mathcal {A} _ {\mathrm {g t}}\right) = \frac {1}{| \mathcal {T} |} \sum_ {\tau \in \mathcal {T}} \mathbb {I} \left(\frac {\left| \alpha \left(\mathcal {A} _ {\text {p r e d}}\right) - \alpha \left(\mathcal {A} _ {\mathrm {g t}}\right) \right|}{\left| \alpha \left(\mathcal {A} _ {\mathrm {g t}}\right) \right| + \epsilon} < \tau\right) \tag {15} +$$ + +where $\alpha (\cdot)$ normalizes numeric values, $\epsilon = 10^{-8}$ prevents division by zero, and $\mathcal{T} = \{0.50,0.55,\dots,0.95\}$ defines accuracy thresholds. For verbal answer questions, we compute a normalized similarity score using the Levenshtein ratio: + +$$ +\mathrm {R} _ {\text {V e r b a l}} \left(\mathcal {A} _ {\text {p r e d}}, \mathcal {A} _ {\mathrm {g t}}\right) = 1 - \frac {D _ {\text {L e v}} \left(\phi \left(\mathcal {A} _ {\text {p r e d}}\right) , \phi \left(\mathcal {A} _ {\mathrm {g t}}\right)\right)}{\left| \phi \left(\mathcal {A} _ {\text {p r e d}}\right) \right| + \left| \phi \left(\mathcal {A} _ {\mathrm {g t}}\right) \right|} \tag {16} +$$ + +where $D_{\mathrm{Lev}}$ denotes the Levenshtein edit distance, and $\phi (\cdot)$ represents the text normalization function. In practice, we use the implementation provided by the Levenshtein library. In addition to the format and task-specific rewards, we also incorporate a reasoning length reward following Video-R1 [12], which encourages the model to perform more thinking before generating the final answer. + +Other Details. Figure 6 presents the prompts used in the SFT and GRPO stages. For both stages, we adopt the default system prompt of Qwen2.5-VL [14], namely, "You are a helpful assistant." In the SFT stage, the user prompt consists of a question and a type template. In the GRPO stage, the user prompt comprises a question, a question post string, and a type template. We conduct all experiments on Intel(R) Xeon(R) Gold 6430 platform with 80G NVIDIA A800 GPUs. + +# Question + +# Example 1: + +Question : " How many chair(s) are in this room?" + +# Example 2: + +Question : " If I am standing by the sofa and facing the stove, is the tv to my left, right, or back? InAn object is to my back if I would have to turn at least 135 degrees in order to face it. Options: A. back B. right C. left" + +# SFT Stage + +# System Prompt + +"You are a helpful assistant." + +# User Prompt + +{Question} + Type Template["problem type"] + +# Type Template + +- Multiple choice: "Please answer with the option's letter from the given choices (e.g., A, B, etc.) within the tags." +Numerical:"Please answer with the only numerical value (e.g., 42,3.14,etc.) within the tags." +- Verbal: "Please answer the question simply within the tags." + +# GRPO Stage + +# System Prompt + +"You are a helpful assistant." + +# User Prompt + +{Question} + + +"Please think about this question as if you were a human pondering deeply. Engage in an internal dialogue using expressions such as 'let me think', 'wait', 'Hmm', 'oh, I see', 'let's break it down', etc, or other natural language thought expressions. It's encouraged to include self-reflection or verification in the reasoning process." + Type Template["problem type"] + +# Type Template + +- Multiple choice: "Please provide your detailed reasoning between the tags, and then answer the question with the option's letter from the given choices (e.g., A, B, etc.) within the tags." +- Numerical: "Please provide your detailed reasoning between the tags, and then answer the question with the only numerical value (e.g., 42, 3.14, etc.) within the tags." +- Verbal: "Please provide your detailed reasoning between the tags, and then answer the question simply within the tags" + +Figure 6: Illustration of the prompts used in the SFT and GRPO stages. We use the default system prompt of Qwen2.5-VL [14] (i.e., "You are a helpful assistant") for both stages. In the SFT stage, the user prompt consists of a question and a type template. In the GRPO stage, the user prompt includes a question, a question post string, and a type template. + +Table 5: Macro average scores on VSI-Bench [18] for Qwen2.5-VL [14] series and Spatial-MLLM. + +
MethodsNumerical QuestionMultiple-Choice QuestionAvg.
Obj. Cnt.Abs. Dist.Obj. SizeRoom SizeRel. Dist.Rel. Dir.Route PlanAppr. Order
Qwen2.5-VL-3B [14]24.324.731.722.638.342.626.321.229.0
Qwen2.5-VL-7B [14]40.914.843.410.738.640.133.029.831.4
Qwen2.5-VL-72B [14]25.129.354.538.838.239.334.028.934.3
Spatial-MLLM-4B65.334.863.145.141.346.933.546.347.3
+ +# C Additional Experiments + +# C.1 Additional Results on VSI-Bench + +We present qualitative examples of Spatial-MLLM on the VSI-Bench [18] dataset in Figures 7 to 10. As illustrated, Spatial-MLLM is capable of reasoning with visual information across different task types and producing final answers accordingly. Furthermore, it demonstrates strong abilities in self-verification and task decomposition during the reasoning process. + +# C.2 Additional Results on ScanQA and SQA3D + +We present additional evaluation results on the ScanQA [38] and SQA3D [39] benchmarks in Table 6 and Table 7. As shown, our proposed method consistently outperforms all video-input models, including LLaVA-Video-7B [12] and Oryx-34B [53], both of which incorporate spatial reasoning datasets such as ScanQA [38] during training. + +Despite having only 4.2 billion parameters, Spatial-MLLM significantly surpasses Qwen2.5-VL72B [14] on the ScanQA benchmark, achieving substantial gains across multiple metrics—for instance, +2.3 EM-1, +17.6 BLEU-1, and +24.9CIDEr. Similarly, on the SQA3D benchmark, Spatial-MLLM consistently outperforms Qwen2.5-VL-72B across all question types and overall performance, including improvements of +4.2 EM-1 and +7.8 EM-R1, with notable gains in the Is (+15.3) and Which (+13.9) categories. + +Table 6: Additional evaluation results on ScanQA [38] for task-specific models, 3D/2.5D input models, and video-input models. Reported metrics include EM-1, BLEU-1 to BLEU-4,ROUGE-L, METEOR, and CIDEr. + +
MethodsScanQA (val)
EM-1BLEU-1BLEU-2BLEU-3BLEU-4ROUGE-LMETEORCIDEr
Task-Specific Models
ScanQA [38]21.130.220.415.110.133.313.164.9
3D-Vista [71]22.4---10.435.713.969.6
3D/2.5D-Input Models
3D-LLM [48]20.539.325.218.412.035.714.569.4
LL3DA [23]----13.537.315.976.8
Chat-Scene [22]21.643.229.120.614.341.618.087.7
3D-LLaVA [21]----17.143.118.492.6
Video-3D LLM [25]30.147.131.722.816.249.019.8102.1
Video-Input Models
Qwen2.5-VL-3B [14]15.422.513.18.13.825.49.747.4
Qwen2.5-VL-7B [14]19.027.813.66.33.029.311.453.9
Qwen2.5-VL-72B [14]24.026.817.814.612.035.213.066.9
LLaVA-Video-7B [12]-39.726.69.33.144.617.788.7
Oryx-34B [53]-38.024.6--37.315.072.3
Spatial-MLLM-4B26.344.428.821.914.845.018.491.8
+ +Table 7: Additional evaluation results on SQA3D [39] for task-specific models, 3D/2.5D input models, and video-input models. In addition to the average EM-1 and EM-R1 across all questions, we also report the average EM-1 for different question types, including What, Is, How, Can, Which, and Others. + +
MethodsSQA3D (test)
WhatIsHowCanWhichOthersAvg. (EM-1)Avg. (EM-R1)
Task-Specific Models
SQA3D [39]31.663.846.069.543.945.346.6-
3D-Vista [71]34.863.345.469.847.248.148.5-
3D/2.5D-Input Models
Scene-LLM [49]40.969.145.070.847.252.354.2-
Chat-Scene [22]45.467.052.069.549.955.054.657.5
Video-3D LLM [25]51.172.455.569.851.356.058.6-
Video-Input Models
Qwen2.5-VL-3B [14]34.852.139.852.745.647.043.445.9
Qwen2.5-VL-7B [14]39.756.641.155.947.647.246.549.8
Qwen2.5-VL-72B [14]41.756.341.555.644.548.047.050.9
LLaVA-Video-7B [12]42.756.347.555.350.147.248.5-
Spatial-MLLM-4B45.971.655.169.552.053.055.958.7
+ +![](images/d7daacac67d87fec30b2051cf8225c346dce7a96c61513497aeae79740f2be58.jpg) +Video + +# Question + +"How many chair(s) are in this room?" + +# Spatial-MLLM + +# "think> + +Let me think about this carefully. In the images provided, I need to identify any chairs present in the room. First, I'll scan the entire room to ensure I'm not missing anything. The room appears to be a living space with a kitchen area, a couch, a dining area, and other furniture. Now, let's break it down: 1. In the kitchen area, there is a small black stool or cushioned seat near the cabinets. This doesn't seem like a chair. + +2. There's a couch in the living room area, but that's a sofa, not a chair. +3. In the dining area, there is a small table with two chairs visible. They are wooden and have simple designs. So, after a thorough review of the images, I can confirm that there are two chairs in the room. Oh, I see. The question was asking for the number of chairs, and now I'm confident that there are two. + +# + +2" + +![](images/9e607f452ffbc4767b7888f54af3f5982a431490dbd594d90c8d215156714cbc.jpg) +Figure 7: Qualitative example on VSI-Bench [18]. +Video +Figure 8: Qualitative example on VSI-Bench [18]. + +# Question + +"How many sofa(s) are in this room?" + +# Spatial-MLLM + +# "think> + +Let me think about this for a moment. The images show a living room with various pieces of furniture. As I scan through them, I notice a few clearly identifiable sofas. + +First, in the initial image, there's a partially visible sofa near the window. It appears to be a traditional design, likely made of fabric and leather. Next, in the following image, there's another sofa positioned near the fireplace. This one seems to have a more modern design, possibly made of leather or a similar material. Then, in the third image, there's a third sofa visible in the background, near the bookshelf. It looks like it's a different style from the first two, possibly a sectional sofa. In the fourth image, there's a fourth sofa visible on the left side of the room. It appears to be a single-seater sofa, different from the others. Let's break it down: + +- Traditional sofa: 1 - Modern sofa: 1 - Sectional sofa: 1 - Single-seater sofa: 1 + +Oh, I see. There are four distinct sofas visible in these images. I should double-check to make sure I'm not missing any, but after reviewing again, I confirm there are no additional sofas hidden or partially obscured. Final count: 4 + +4" + +# Video + +![](images/16b3380e11c9254c42fe5af012819db2d15c1980d39be46243c2480b4b379feb.jpg) + +![](images/6d56c9f0eee4b23951fa483fc69a1b2527659a01944df3f8f09fb4735a9e7013.jpg) + +![](images/eb72813a6b9d36785ea35694329b79a088f88079b0ec4fa4a36ce7fea7b2a0c1.jpg) + +# + +![](images/9cc970d03725636b161a8ed258da6430384025c666c4cb607e4a0ef4455db36a.jpg) + +![](images/e285ab4acda2aec2559af4263cf2519f0dfa1371ab4363afc5168c3f62072a9d.jpg) + +![](images/6d7a5a14302054bd3837f321fc95b90cb1155ba3531216a78fb2a64ca98e8bd0.jpg) +Figure 10: Qualitative example on VSI-Bench [18]. + +![](images/24484cb6ef3fe491935ec7455ec14328f8dcc2fb95a0dc3861c8dce656e967ca.jpg) + +# Question + +"If I am standing by the sofa and facing the stove, is the tv to my left, right, or back? \nAn object is to my back if I would have to turn at least 135 degrees in order to face it. Options: A. back B. right C. left" + +# Spatial-MLLM + +# + +Let me think about this for a moment. If I'm standing by the sofa and facing the stove, I need to consider the layout of the room. The stove is in the kitchen area, which appears to be separated from the living room area by a counter and some cabinets. Now, where is the TV? The TV is positioned on the wall near the kitchen area, which means it's not directly in front of me but rather to my side. Wait, the question asks if the TV is to my left, right, or back. Let's break it down: if I'm facing the stove, and the TV is positioned in such a way that it's not in front of me, it must be either to the left or the right. Given the layout of room, the TV is on the opposite side of the kitchen area, which would be to the right of the stove if I'm facing it. Hmm, let's verify this. If I turn around from the sofa and face the kitchen area, the TV is indeed to the right of the kitchen appliances. This means that if I'm facing the stove from the sofa, the TV would be to my right. Oh, I see now. The TV is to my right when I'm facing the stove from the sofa. B " + +# Video + +![](images/ad78a2db16cb8a8e6f6d74ea526facc424111edce908e00bbe14eb5afa6165a7.jpg) + +![](images/ac39a48173d09551c1dc5744d6e0d8e5555288b4ab9c3bcbe8b0c29ce6ef4d2a.jpg) + +![](images/68036d89d025a1c9928351c142791ce7bbd2051bf987470a924b71489f83f197.jpg) + +# + +# + +![](images/681d26c8d4b43aaae5157c31b8442d8883b084a9d960516501c0f09cb23c1832.jpg) +Figure 9: Qualitative example on VSI-Bench [18]. + +![](images/5e02895c81e18d560e906c1f9e2d60c54b6a86d24e68ef63b8d0df18044a68cd.jpg) + +![](images/131176bd0ff93f02a3becb82192cb5a0f4863cb935fcb7e074db2e992f929a48.jpg) + +# Question + +"What will be the first-time appearance order of the following categories in the video: basket, pillow, toilet, tv? Options:\nA. basket, pillow, toilet, tv B. pillow, tv, toilet, basket C. toilet, pillow, tv, basket D. toilet, basket, pillow, tv" + +# Spatial-MLLM + +"Let me think about this carefully. The question asks for the first appearance order of certain objects in a video, but I need to focus on the video that shows the bathroom. In the video, I see various objects appear in different sequences. Let's break it down: + +1. Bathroom sequence: The video starts with the bathroom, showing a toilet, sink, shower curtain, and some shelves. There's also a basket visible in one part of the bathroom. 2. Transition to living room: The video then transitions to a living room with a bed, which is a different space entirely. The living room shows a bed with a pillow and some decorative items. The third part shows the living room with a couch, TV, and other decorations. + +Now, let's consider the appearance order. The toilet is in the bathroom, the basket appears in the bathroom, the pillow is in the bedroom, and the TV is in the living room. + +Let's verify this one more time: Bathroom -> toilet -> basket -> Bedroom -> Yes pillows, other bedroom items -> Living Room -> TV." + +Therefore, the correct answer is D. 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These outputs include architectural elements like walls, doors, windows, and oriented object boxes with their semantic categories. Unlike previous methods which exploit task-specific network designs, our model adheres to the standard multimodal LLM architecture and is fine-tuned directly from open-source LLMs. + +To train SPATIALLM, we collect a large-scale, high-quality synthetic dataset consisting of the point clouds of 12,328 indoor scenes (54,778 rooms) with ground-truth 3D annotations, and conduct a careful study on various modeling and training decisions. On public benchmarks, our model gives state-of-the-art performance in layout estimation and competitive results in 3D object detection. With that, we show a feasible path for enhancing the spatial understanding capabilities of modern LLMs for applications in augmented reality, embodied robotics, and more. + +# 1 Introduction + +3D indoor environments are ubiquitous in our daily lives. We spend a lot of time and perform various activities in such environments every day. Therefore, a long-standing goal of artificial intelligence is to teach machines to perceive, reason about, and interact with 3D indoor scenes as humans do. In this work, we focus on the task of structured indoor modeling, which aims to extract structured indoor scene descriptions from raw sensorial inputs (i.e., RGBD scans). Specifically, the scene description includes both the architectural layout (i.e., walls, doors, and windows) and 3D object bounding boxes in the indoor environment. Such 3D structure information has been shown to benefit a wide range of real-world applications such as scene editing [40], augmented reality [4, 2], and robot navigation [17]. + +Compared to 3D formats such as meshes, voxels, or implicit functions, the structured scene description provides a highly compact yet flexible scene representation. In this paper, our first contribution is to regard the structured descriptions as scripts of a general-purpose language (i.e., Python) and propose to predict the language in text form, as illustrated in Figure 1. This design choice has several advantages: (i) it is human interpretable and editable; (ii) it can be easily extended to incorporate any new classes without affecting existing content in the scripts; and (iii) it allows us to leverage the strong built-in coding capability of pre-trained large language models. + +Therefore, our goal is to implement our method by directly fine-tuning open-source LLMs. While modern (multimodal) LLMs have revolutionized fields like natural language processing, 2D image understanding and generation, few work has attempted to use LLMs for 3D structured scene modeling. This is partly due to the lack of large-scale, high-quality datasets for the task. In this paper, as our second contribution, we collect a new synthetic dataset, which consists of the point clouds of 12,328 + +![](images/ab3afc69c659890b134b15d42d08e6454d5552f1fe464c2443c1752577e2fb8c.jpg) +Figure 1: The overall pipeline of SPATIALLM. Given the point cloud input, it employs a standard "Encoder-MLP-LLM" architecture for multimodal feature alignment (left), and generates structured scene descriptions in pure text form as output (middle). The reconstructed 3D structure is further overlaid on the point cloud for visualization (right). + +distinct scenes (54,778 rooms) paired with 3D structure information, and conduct the first empirical study on the best strategy of aligning the point cloud input with LLMs for structured scene modeling. + +Building upon our findings, we introduce SPATIALLM, a large language model which is capable of processing point cloud input and generating structured scene descriptions. As our third contribution, we show that, by first training on our large-scale dataset and then on smaller downstream task data, our model gives competitive performance on public benchmarks for layout estimation and 3D object detection, respectively. Additionally, we provide proof-of-concept zero-shot experiments which show that our model can handle point clouds from diverse sources such as monocular video sequences. These results suggest that SPATIALLM may serve as a base for developing future solutions for scenarios where enhanced spatial understanding and reasoning is required (e.g., embodied AI). + +# 2 Related Work + +# 2.1 Structured Indoor Modeling + +Structured indoor modeling aims to recover structural elements (i.e., walls, doors, and windows) as well as 3D objects from unstructured point clouds. In the literature, some studies focus on room layout estimation from point clouds. Early methods [37, 38] rely on traditional geometric analysis algorithms (i.e., RANSAC) to detect wall planes. SceneCAD [1] jointly optimizes the layout estimation and CAD model alignment of the scanned scene. RoomFormer [60] proposes to train a Transformer-based network to predict corners and rooms. Meanwhile, a number of recent methods [35, 46, 51, 47, 25] have advanced the state-of-the-art on 3D object detection from point clouds. + +The closest work to ours is SceneScript [2], which also casts structured indoor reconstruction as a sequence modeling problem. It demonstrates great flexibility as the language can be tailored to different scenarios without changing the method. But unlike our method, SceneScript uses domain-specific tokens for language commands, thus requiring a specialized decoder. Instead, SPATIALLM strictly adheres to the "Encoder-MLP-LLM" architecture and is directly fine-tuned based on modern foundation models. + +# 2.2 Large Language Models for 3D Scene Understanding + +Recently, there has been a growing interest in empowering LLMs to understand and reason about 3D spaces, as defined in public benchmarks such as ScanRefer [9] and ScanQA [3]. One line of work [53, 20, 19, 29, 22] first uses 3D detectors to locate objects in the point cloud, then extracts features of individual objects using a pre-trained object-level encoder. In this paper, we study how such 3D detectors can be learned with modern LLMs. + +Other methods directly extract scene-level features from the point cloud and align the features with LLMs [18, 66, 10, 16]. 3D-LLM [18] uses pre-trained image encoder to obtain 2D features from multi-view images and back-projects onto the point cloud [21]. It then leverages a Q-former [28] to map them to a fixed number of tokens as input to the LLM. LL3DA [10] adopts a 3D visual encoder + +![](images/5377b0b39525e345ebb772c7782c6236e816a080091b66329a4eeb301d1e7cfa.jpg) +Figure 2: Definition of our structured representation for layouts and objects. + +![](images/e0a7d161528481de4b3947b3faf6ff7558671450d77b9723f1150140560c007a.jpg) + +![](images/23ea93eaeb248d193a624c63a64b31421ed7577a9b8b9b2fbc31dcc6b6ff11f0.jpg) + +![](images/9aab7b14dceb589512dcc89c5975d9ef7cfff38b6d942536619c66826bbfe6ef.jpg) + +pre-trained on ScanNet detection as the scene encoder. ScanReason [66] further employs a 3D point cloud encoder to obtain fine-grained geometric information for accurate object grounding. SceneLLM [16] tokenizes the 3D point cloud features by simply dividing the space into a fixed-resolution voxel grid and aggregating features in the same voxel. + +Besides point cloud-based representations, recent studies have explored other ways to integrate 3D information in LLMs. SpatialVLM [8] primarily focuses on synthesizing large-scale 3D VQA data to enhance the spatial reasoning capabilities of the VLM. LLaVA-3D [67] injects 3D position embeddings into 2D CLIP features to establish a unified architecture for 2D image understanding and 3D scene understanding. SpatialRGPT [11] extracts a 3D scene graph from a single image to enhance grounded spatial reasoning in LLMs. Video-3D LLM [63] learns representations from video frames and 3D coordinates to leverage the strong priors of pre-trained 2D Video LLMs. LSceneLLM [65] uses an adaptive self-attention module to capture fine-grained details in local areas. Our work differs from these studies in that we focus on analyzing the structural information of the scene. + +# 3 SPATIALLM + +Given the point cloud $\mathbf{P}$ (typically derived from RGBD scans), our goal is to recover the 3D layout $\mathbf{L}$ , which consists of all walls, doors and windows, and a set of objects $\mathbf{O}$ in the scene. As discussed before, we represent $\mathbf{L}$ and $\mathbf{O}$ as scripts of a general-purpose language, and fine-tune LLMs to predict the language in text form. Figure 2 shows all the parameters we use for the layout and objects. + +To the best of our knowledge, we are the first to employ LLMs for structured scene reconstruction from point clouds. Thus, we first propose a new dataset suitable for training the LLMs on (Section 3.1). Then, with the new dataset, we set out to investigate the following two key questions: (i) What point cloud features to use for the structured indoor modeling task (Section 3.2)? (ii) How to integrate the features into LLMs (Section 3.3)? + +# 3.1 Dataset and Metrics + +Table 1 provides a list of popular indoor RGBD scan datasets. We can make a few observations: + +First, the number of scenes in real datasets is relatively small, ranging from about one hundred to a few thousands. To collect such a dataset, one needs to take RGBD scans in real scenes and manually annotate 3D information, which is a highly laborious process. Further, most real datasets provide annotations on the objects (i.e., semantic labels and 3D boxes) only. One exception is SceneCAD [1], which manu + +
Table 1: Quantitative statistics of indoor RGBD scan datasets.
Dataset (year)source#scenesannotations
Layoutsobjects
ScanNet (2017) [13]real1,513●
Matterport3D (2017) [7]real90●
3RScan (2019) [50]real1,482●
SceneCAD (2020) [1]real1,151●●
ARKitScenes (2021) [4]real5,048●
MultiScan (2022) [34]real273●
ScanNet++ (2023) [59]real1,006●
HM3DSem (2023) [58]real181●
Structured3D (2020) [64]syn.3,500●
Hypersim (2021) [45]syn.461●
ProcTHOR (2022) [14]syn.12,000●●
HSSD (2024) [14]syn.211●●
ASE (2024) [2]syn.100,000●
SPATIALLM dataset (ours)syn.12,328●●
+ +ally adds wall annotations to ScanNet dataset. But it only contains 1,151 rooms. + +![](images/b7e517aea60afd86ca8fb7881ac3d1fd5e5a0de649c5b0881f4ca797dd4dcd50.jpg) + +![](images/b02cbb7f6ffe658ea904c930fa3d476d38add25a1f13206b679162786185926e.jpg) + +![](images/418fba1b8a9f3a1b9c0c08580f4dfd8bc90887de3b10509f1e405adb9708f8e3.jpg) + +![](images/7423248e4ce1848514d830674897a02ba2d1926482d6a30d6d0f2c253ab2500c.jpg) + +![](images/cd417eb6daf5050e8cf26b066670ea9506118e7b8c27e3e9fabe6e64ef825989.jpg) +ProcTHOR [14] + +![](images/a228e337003b6836d09790bc4b5aad057985236ed65e5394d64d506de5077024.jpg) +HSSD [24] + +![](images/fb8724d0e60815437106b2b95211039b63a8026d8120aac30534417b0a94b485.jpg) +ASE [2] + +![](images/fb27191c645820890f24a17fa0bf369a421557bcef6133d1f848651259cef349.jpg) +SPATIALLM dataset +Figure 3: Dataset visual quality comparison. The layout and object placements in ProcTHOR [14] and ASE [2] are program-generated, which exhibit noticeable differences from real-world statistics. The scenes in HSSD [24] and our dataset are fully human-authored. But HSSD only has 211 scenes. + +Second, for synthetic datasets, it is much cheaper to obtain ground-truth 3D annotations. But to ensure the realism of the RGBD scans, the 3D scenes must be professionally constructed. As a result, the diversity of 3D scenes in some datasets is still limited. For example, Hypersim [45] and HSSD [24] only have 461 and 211 scenes, respectively. On the other hand, one may create a large number of scenes by employing a rule-based procedural modeling pipeline, as did in ProcTHOR [14] and ASE [2]. But the quality of the scenes is not as high. + +SPATIALLM dataset. In this paper, we make an effort to build a dataset suitable for training LLMs for structured indoor scene reconstruction. To this end, we take advantage of access to a large repository of interior designs from an online platform in the interior design industry. Most designs are created by professional designers and used for real-world production. We parse each 3D house into individual rooms and use a few rules to filter the rooms. This results in 12,328 distinct scenes with 54,778 rooms. + +For the objects, we keep annotations on a selected set of 59 common categories (excluding wall, door, and window), and filter out small objects with side lengths all $< 15\mathrm{cm}$ . In the end, our dataset contains 412,932 annotated object instances of 35,426 unique CAD models. + +For each scene, we generate photo-realistic RGBD images utilizing an industry-leading rendering engine. At the time of rendering, a camera trajectory traversing each room is simulated, based on which the images are taken at an interval of $0.5\mathrm{m}$ . Figure 3 compares the visual quality of our dataset to several recent synthetic datasets. Finally, the dataset is divided into 11,328/500/500 scenes for training/validation/testing. + +Evaluation metrics. We evaluate our method on both layout estimation and 3D object detection. For layout estimation, let $\hat{\mathbf{L}} = \{\hat{l}_i\}_{i=1}^m$ and $\mathbf{L} = \{l_i\}_{i=1}^m$ denote the set of predicted structural elements and ground truth, respectively. Since each element is a plane segment in 3D space, we compute the 2D intersection-over-union (IoU2D) by projecting each predicted element to the plane of the ground-truth element to find the optimal assignment with the Hungarian algorithm, and report the F1-scores at 0.25 and 0.5 thresholds. For 3D object detection, we match the predictions $\hat{\mathbf{O}} = \{\hat{o}_i\}_{i=1}^n$ with the ground truth $\mathbf{O} = \{o_i\}_{i=1}^n$ via Hungarian matching. Then, we compute the 3D intersection-over-union (IoU3D) between each matched pair and report the F1-scores at 0.25 and 0.5 thresholds. + +# 3.2 Point Cloud Encoders + +Extracting meaningful representations for a point cloud is challenging because of its irregular format. Learning-based approaches to process 3D point clouds can be roughly classified into three groups: Mapping-based methods are inspired by the success of 2D image backbones such as CLIP [44] and the DINO series [6, 39], and map pixel-aligned image features to 3D. For example, Lexicon3D [33] + +Table 2: Experiment on the encoder design. + +
EncoderParams (M)F1 - layoutF1 - object
IoU2D@0.25IoU2D@0.5IoU3D@0.25IoU3D@0.5
Voxelize (P+DINOv2)0.010.85.50.10.0
Rand. sampling (P+DINOv2)0.010.64.90.20.0
3DCNN enc. (P)10.579.476.359.846.1
3DCNN enc. (P+DINOv2)11.681.579.457.145.0
3DCNN enc. (P) + Voxelize (DINOv2)11.983.881.662.946.7
Sonata/PTv3 enc. (P)108.884.682.665.149.4
+ +projects 3D points onto the image plane to obtain their corresponding pixel features; ConceptFusion [21] builds multimodal 3D maps via traditional SLAM and multi-view fusion approaches. Voxel-based methods first transform irregular point clouds to regular representations via 3D voxelization. Then, sparse convolution algorithms [12] can be readily used for efficient computation. Point-based methods, by contrast, process point clouds directly as sets embedded in continuous space. Recent approaches in this category [62, 54-56] have successfully explored the self-attention mechanism in Transformers and demonstrated superior performance on downstream applications. + +Note that all methods discussed above produce one feature per point by default. Given that a scene-level point cloud consists of hundreds of thousands of points or more, these features are too much in terms of quantity for LLMs to consume directly. Since most modern open-source LLMs take an "Encoder-MLP-LLM" architecture for multimodal feature alignment (e.g., the Llama series [32] and Qwen series [42, 43]), we can define a point cloud encoder for LLMs as a mapping which takes a set of $N$ points as input and generates $K$ feature embeddings, where $K \ll N$ : + +$$ +\mathbf {F} = \mathcal {E} (\mathbf {P}), \quad \mathbf {P} \in \mathbb {R} ^ {N \times 6}, \mathbf {F} \in \mathbb {R} ^ {K \times D}. \tag {1} +$$ + +Here, each input point is a 6-dimensional vector (XYZ and RGB), and $D$ is the feature dimension. + +Recently, a few attempts have been made to convert point representations into feature embeddings [2, 16]. However, this step has not been carefully investigated. In this section, we conduct experiments to study (i) the effectiveness of different encoders, and (ii) the impact of $K$ , i.e., the number of output features. Unless otherwise stated, Qwen2.5-0.5B [42] is employed as our base model. We use Sonata [56] as the point cloud encoder, and a two-layer MLP as the projector. + +Experiment on the encoder design. We consider six possible point cloud encoders. The first group of encoders adopts a mapping-based method to fuse 2D image features to 3D points. Following Lexicon3D [33], we pool pixel-level DINOv2 features to form 3D point representations. Then, one of the two routes may be taken to reduce the points into a manageable length of tokens: + +- Voxelize $(\mathbf{P} + \mathbf{DINOv2})$ : divides the space into a fixed-resolution voxel grid and aggregates all the points in the same voxel - similar to the approach used in Scene-LLM [16]; +- Rand. sampling (P+DINOv2): uses Farthest Point Sampling [41] to randomly sample a fixed number of points with their features. + +From Table 2 we can see, these two options result in extremely low F1-scores. We hypothesize that this is because too much spatial information is lost during the naive down-sampling step. While this may be acceptable for certain applications, such as visual Q&A, it is problematic for our task, which relies heavily on the spatial information to reconstruct the geometric coordinates of objects. + +The second group of encoders adopts a standard 3DCNN encoder to generate pooled features. We follow SceneScript [2] to build our encoder with a sparse 3D convolution library [48]. Here we consider three variants: + +- 3DCNN enc. (P): takes raw points (6-dim XYZRGB values) as input; +- 3DCNN enc. $(\mathbf{P} + \mathbf{DINOv2})$ : takes the points with the associated DINOv2 features as input; +- 3DCNN enc. (P) + Voxelize (DINOv2): concatenates the output of 3DCNN with aggregated DINOv2 features. + +As shown in Table 2, a learned encoder is effective in keeping the necessary semantic and geometric information for reconstruction. We also find that incorporating DINOv2 features yields slightly better results, possibly due to the enhanced contextual information. + +The last encoder we consider is Sonata [56], a variant of Point Transformer V3 (PTv3) [55] which removes the decoder and focuses on self-supervised learning on the encoder. Compared to other backbones, such an "encoder-only" design makes it quite convenient for adapting to LLMs. Indeed, as shown in Table 2, with raw points as input, it already achieves the best results among all encoders. + +# Experiment on the spatial resolution. Recall in Eq. (1) that the parameter $K$ denotes the number of feature embeddings or visual tokens, which is critical to the performance of LLMs. In this experiment, we study the effect of $K$ . + +However, fixing $K$ is difficult because different point clouds generally produce varying $K$ values. Instead, since the encoders employ a coarse-to-fine structure for point cloud pro + +Table 3: Experiment on the spatial resolution. Avg. $(K)$ indicates the average value of $K$ under the corresponding resolution. + +
Res.Avg.(K)F1 - layoutF1 - object
IoU2D@0.25IoU2D@0.5IoU3D@0.25IoU3D@0.5
1×12384.682.665.149.4
1.25×19784.882.665.352.2
1.5×28686.985.168.758.6
2×51087.485.371.661.1
2.5×78886.183.972.261.7
4×2,00085.883.769.958.9
+ +cessing, we set the resolution at the finest level at $5\mathrm{cm}$ and use a 5-level hierarchical structure by default. A detailed illustration of the point cloud encoder hierarchy is presented in the appendix (see Figure 11). Then, we gradually increase the resolution up to $4\times$ , and report the results in Table 3. As can be seen, model performance steadily improves from $1\times$ to $2\times$ resolutions, but further increasing the resolution hurts the performance. This is likely due to the excessively long token sequences. + +# 3.3 Training Schedules + +As most recent MLLMs start from pre-trained unimodal backbones, how to align the multimodal content with the language model plays a critical role in the performance of the resulting MLLMs. For example, a common practice adopted by many MLLMs [30, 49] is the inclusion of a two-stage training schedule: (i) an alignment stage that trains the randomly initialized MLP projector only, followed by (ii) a fine-tuning stage that trains both the projector and language model + +Table 4: Experiment on training schedule. For each training stage, we use three circles to represent the three components in the "Encoder-MLP-LLM" architecture. "○" and "●" indicate whether a component is frozen or trainable, respectively. + +
Training stagesF1 - layoutF1 - object
1st2nd3rdIoU2D@0.25IoU2D@0.5IoU3D@0.25IoU3D@0.5
○●●--73.868.440.222.7
●●●--84.682.665.149.4
○●○○●●-70.962.727.413.7
●●○●●●●-85.383.460.745.6
○●○●●○○●●76.970.131.116.0
○●○●●○○●●●83.280.357.141.2
+ +while keeping the vision encoder frozen. Other studies (e.g., [23]) advocate a single-stage training protocol, suggesting that including the projector training stage is unnecessary. Furthermore, when adapting the LLMs to rich visual content such as long videos, some research [68] has shown that a three-stage pipeline, which progressively unfreezes different components in different stages, improves the performance. + +In this experiment, we test six possible training configurations, ranging from one-stage to three-stage, to evaluate the influence of different training strategies. As shown in Table 4, we found that a single-stage fine-tuning yields the best results. Adding more stages generally degrades the model performance. We also found that making all parameters trainable in a single stage is important, especially for 3D object detection. This may indicate that, compared to modern image encoders such + +as SigLIP [61] and DINOv2 [39], current pre-trained point cloud encoders are not as versatile in supporting downstream tasks yet. + +# 4 Experiments + +In this section, we compare SPATIALLM to the current state-of-the-art on scene layout estimation (Section 4.1) and 3D object detection (Section 4.2), respectively. Following our empirical study, we use Qwen2.5-0.5B [42] as the base model and Sonata [56] as the point cloud encoder. We set of resolution at the finest level at $2.5\mathrm{cm}$ and employ a single-stage training on our dataset. + +Besides testing on established RGBD benchmarks, we also investigate whether our model can handle inputs from other sources, such as point clouds reconstructed from RGB videos (Section 4.3). Other extensions and more technical details can be found in the appendix. + +# 4.1 Layout Estimation + +Layout estimation focuses on predicting architectural elements, i.e., walls, doors, and windows, within an indoor scene. In this experiment, we include results from two representative baselines, namely RoomFormer [60] and SceneScript [2]. As the state-of-the-art specialist model for layout estimation, RoomFormer employs a highly specialized network architecture, using two-level queries to predict room polygons and their corners, respectively. SceneScript, on the other hand, casts 3D structure estimation as an auto-regressive "next-token-prediction" problem. However, unlike our method, it uses specialized structured language commands to describe the scene and trains a Transformer decoder from scratch. + +Following the setting of RoomFormer, we perform evaluation on the Structured3D benchmark [64], which contains 3,500 residential houses with diverse floorplans. We use the original data split of 3000/250/250 for training/validation/testing, respectively. For RoomFormer, we directly download the model checkpoint from its website3 for inference. Note that RoomFormer takes in 2D density maps projected from point clouds and outputs 2D layouts, which are converted to 3D by extruding based on the lowest/highest points in the point cloud. For SceneScript, we download the pre-trained model and code4, and fine-tune it on Structured3D with the default hyperparameter settings. + +For our model, we include three variants: (i) trained on Structured3D only, (ii) trained on our dataset only, and (iii) first trained on our dataset, then fine-tuned on Structured3D. As can be seen in Table 5, directly training LLMs on the smaller Structured3D dataset leads to poor + +Table 5: Experiment on layout estimation. + +
MethodParams (M)F1 - Structured3D
IoU2D@0.25IoU2D@0.5
RoomFormer [60]42.283.481.4
SceneScript [2]29.990.489.2
SPATIALLM (ft. Structured3D)603.532.618.1
SPATIALLM (ft. Ours)603.559.944.7
SPATIALLM (ft. Ours → Structured3D)603.594.393.5
+ +performance. In contrast, by first training on our new dataset and then on Structured3D, our model outperforms both baselines for this task. This demonstrates the benefits of our new dataset. + +Figure 4 provides some qualitative comparisons. Since RoomFormer detects room elements independently, we can see that some doors and windows are not attached to any wall. Auto-regressive methods like SceneScript and our method can preserve the relationships between structures by design. But we observe that SceneScript tends to miss some elements in its output. + +# 4.2 3D Object Detection + +Next, we compare SPATIALLM to existing methods on 3D object detection. We include results from two representative baselines, namely V-DETR [47] and SceneScript. As the state-of-the-art + +![](images/eec3bf4f971692b79dd9e0e12380cb4ded87490e5df455eb9ade0800e1d2654e.jpg) + +![](images/614c18cd6a8eada88e23d3a3085c9241bd54f8f4050619e79e9bfcf08d3c269b.jpg) + +![](images/aea8bc5e039ec0839b9e8cdfc0dc653f7ee652294d784f25a858f14412254c72.jpg) + +![](images/ce5e7d4c37506988d4f59f8ea31e9645817f5bfd286b934c10be5799db46a019.jpg) + +![](images/9ad96631962ccabd6c9594a917f8f5b802ef40cd47ab618ab03f3daf62d13c28.jpg) +Ground truth + +![](images/74955199d92a92204d835f1ad8cc6f57ca7eb9a7513c199c994300dd230a709f.jpg) +RoomFormer [60] + +![](images/1b1807e6d63b851f3fb0595060f85f4dfef20e7c6d77184d7752c92e507c9d0f.jpg) +SceneScript [2] + +![](images/e518f2d0067f36deaeeee442c173ce18058e652f0c2201dd7c2cb954d20cb72b.jpg) +SPATIALLM +Figure 4: Qualitative results on Structured3D dataset. + +specialist model for the task, V-DETR builds upon the DETR framework [5] and makes several task-specific improvements, including a 3D vertex relative position encoding (3DV-RPE), object-based normalization, among others. Following prior practice, we perform evaluation on ScanNet [13], which contains 1,513 3D indoor scans with annotations of 18 object categories. The training set is composed of 1,201 scenes, while 312 scenes are used for validation/testing. Since auto-regressive models (i.e., SPATIALLM and SceneScript) do not produce confidence scores, we report F1 scores instead of mean Average Precision (mAP), as suggested in [2]. + +For V-DETR, we directly use the model checkpoint downloaded from its website. A confidence threshold of 0.5 is applied for both objectness and semantic predictions, followed by non-maximum suppression (NMS). For SceneScript, we fine-tune the model on ScanNet with the default hyperparameter settings. During fine-tuning, we apply the same data augmentations as used in PTv3 [55] and V-DETR [47]. + +Table 6: Experiment on 3D object detection. “*”: numbers reported on the overlapping subset of object classes between our dataset and ScanNet. + +
MethodParams (M)F1 - ScanNet
IoU3D@0.25IoU3D@0.5
V-DETR [47]79.465.156.8
SceneScript [2]29.949.136.8
SPATIALLM (ft. ScanNet)603.52.90.7
SPATIALLM (ft. Ours)603.533.8*22.6*
SPATIALLM (ft. Ours → ScanNet)603.565.652.6
+ +For our model, we again include three variants: (i) trained on ScanNet only, (ii) trained on our dataset only, and (iii) first trained on our dataset, then fine-tuned on ScanNet. As can be seen in Table 6, the first variant has very low F1 scores, suggesting that ScanNet itself is too small to train LLMs on. By first training on our new dataset and then on ScanNet, our model outperforms SceneScript and achieves competitive results against the state-of-the-art specialist V-DETR. + +Figure 5 shows some qualitative results. Compared to SceneScript, SPATIALLM demonstrates improved detection accuracies across categories. In comparison to V-DETR, we observe that our method has some difficulty in locating objects in "picture", "sink" and "showercurtain" classes. Notably, "picture" and "sink" are the smallest objects in ScanNet by volume $(< 0.1\mathrm{m}^3)$ , whereas "showercurtain" category is not included in our dataset and has the fewest occurrences in the ScanNet. + +![](images/ae119c125bf994204003ee16c32f8b66b4c8ddf456d4e83b55c6c11ede2cc780.jpg) + +![](images/5714ed0fb91da0b9669e4ed6b6368592e4787218e21382d9d5340985627edb54.jpg) + +![](images/445374f7719676ef9ba57d43897672b5b1fba8586c346a725041b9df21686522.jpg) + +![](images/7e9f4a81e443a4a259618b571ddc353124bc4eaf7ab51348db83715abe662375.jpg) + +![](images/32c325f11cef51307ae8b9c880af615b31aba74569ea19869ff2c098c7da66e2.jpg) +Ground truth + +![](images/43ccdeba36dc797671ce74065caff15d9a08b75a0f47a229bc1bc0e4bbb176c4.jpg) +V-DETR [47] + +![](images/bf30a46fb698f7695b41daa514c7341502a4c51f936721daa5d7e7b49d59ce18.jpg) +SceneScript [2] +Figure 5: Qualitative results on ScanNet dataset. + +![](images/ff0d9aaef95a4bac9c713fc096df5170530271164ba94ea4da7bc7bbc80a57a5.jpg) +SPATIALLM + +![](images/f5082a4572dc3a15756231492dad52b8c5aa47f914bfc4fbc5fdaf435291cf6f.jpg) + +![](images/145a88e851afa6e3bfc561c3e1d0f80d1b2bf15474532062d709849da6375cdc.jpg) +Figure 6: Qualitative results of zero-shot detection on videos. + +![](images/e192d56ea9b539dde7647b2f6b4a84336777da4f60ad0cf241efdc1ca74f5c6b.jpg) + +![](images/870e665cef153db36406e4a268a91de261796f8a111fed0d0c5de312e5b833d4.jpg) + +![](images/9ae23f440924c06538c191dfcb369d296d9870a1ea37bc50dad811c387b741de.jpg) + +![](images/ee4f6a83558de3f658277dc4dca1dd9f282ccd892eb09d9db477d57aa15768bb.jpg) + +![](images/0b0e23bd5f768800c2182a399120721d2cc7662f0476a83534e3c230e5efb804.jpg) + +![](images/a11cfb595ab64ea725fa6a51c75f9034360c237ce7b988b6d5d49878106fcd7c.jpg) + +# 4.3 Zero-shot Detection on Videos + +Although significant progress has been made in SFM and SLAM techniques over the past decades, reliably inferring 3D structure from videos remains a highly challenging problem. Recently, DUSt3R [52] and its extensions [27, 15, 36, 31] provide a novel solution that employs deep networks for simultaneous point matching and camera motion estimation. These techniques are shown to be remarkably robust in dealing with real-world videos or unconstrained image collections. + +In this experiment, we collect a set of 107 videos of virtual indoor scene tours and use MASt3R-SLAM [36] to reconstruct a point cloud from each video. Then, we test SPATIALLM in a zero-shot setting on the reconstructed point clouds. Unlike RGBD scans, these reconstructions have significant noises, occlusions, and geometric artifacts, posing a greater challenge for structured 3D understanding. + +Figure 6 shows some qualitative results. Without fine-tuning, SPATIALLM demonstrates strong resilience to imperfect data and preserves consistent layout and object-level predictions. Trained on synthetic datasets with accurate bounding boxes, the model infers full object sizes and orientations from sparse or occluded inputs. In the first example, it predicts beds and nightstands extending to the floor, even when those regions are not visible. Leveraging the auto-regressive nature of LLMs, SPATIALLM is also capable of generating plausible room layouts from partial observations. In the second example, the model reasonably reconstructs areas such as the balcony and dining space, filling in missing elements based on contextual understanding. + +# 5 Conclusion + +In this study, we demonstrate the feasibility of training LLMs for structured indoor modeling tasks. We regard it as a meaningful step towards building future foundation models that can not only understand, but also reason about, interact with, and even create structured 3D scenes. + +Limitations. SPATIALLM has several limitations. First, it falls short of delivering a universal, state-of-the-art model for extracting 3D structure from arbitrary point clouds. There are significant differences between point clouds from different sources, such as monocular videos, RGBD scans, and LiDAR sensors. As shown in Section 4, although our model exhibits reasonable generalizability across datasets, fine-tuning on a specific dataset is still needed to achieve the best performance. Further scaling up the data and model size could be a promising future direction. Second, in this paper, we focus on training LLM for structured indoor modeling only, without considering how it affects the model's natural skills in natural language processing and reasoning. To address this, a more thorough benchmarking may be performed. Third, our current approach models indoor layouts using a set of predefined object categories, limiting the LLMs in leveraging their open-ended language capabilities. Future directions include extending our work to support open-vocabulary object detection and 3D visual question answering (VQA), enabling more flexible and generalizable scene understanding. + +Impacts. We envision that SPATIALLM will be used in real-world applications such as layout estimation and 3D object detection. For example, people may take our code and fine-tune the model on their own datasets. On the methodology side, we hope that our work will inspire future studies on MLLM models for 3D scene understanding, reasoning, and creation. + +# Acknowledgments and Disclosure of Funding + +This work was partially supported by the Key R&D Program of Zhejiang Province (2025C01001) and the HKUST project 24251090T019. + +# References + +[1] Armen Avetisyan, Tatiana Khanova, Christopher B. Choy, Denver Dash, Angela Dai, and Matthias Nießner. SceneCAD: Predicting object alignments and layouts in RGB-D scans. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The method development in this research does not involve LLMs. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A SPATIALLM Dataset + +SPATIALLM dataset comprises 12,328 distinct scenes with 54,778 rooms, featuring a total of 403,291 walls, 123,301 doors, and 48,887 windows, with an overall floor area of approximately $863,986\mathrm{m}^2$ + +Object categories. We have curated a set of 59 commonly occurring object categories, organized by functional and semantic similarity, as shown in Table 7. Statistics of these categories can be found in Figure 7. + +Table 7: Categories in the SPATIALLM dataset. + +
Super CategoriesCategories
Seatingssofa, chair, dining_chair, bar_chair, stool
Beddingsbed, pillow
Cabinetrywardrobe, nightstand, tv_cabinet, wine_cabinet, bathroom_cabinet, shoe_cabinet, entrance_cabinet, decorative_cabinet, washing_cabinet, wall_cabinet, sideboard, cupboard
Tablescoffee_table, dining_table, side_table, dressing_table, desk
Kitchen appliancesintegrated_stove, gas_stove, range_hood, microwave_oven, sink, stove, refrigerator
Bathroom fixtureshand_sink, shower, shower_room, toilet, tub
Lightingillumination, chandelier, floor Standing_lamp
Decorationwall_decoration, painting, curtain, carpet, plants, potted_bonsai
Electronicstv, computer, air_conditioner, washing-machine
Othersclothes_rack, mirror, bookcase, cushion, bar, screen, combination_sofa, dining_table Combination, leisure_table_and_chair Combination, multifunctional Combination.bed
+ +Figure 8 plots the object-to-room correlation heatmap, which reveals strong correlations between object distribution and specific room types. Additionally, each room type contains a variety of object categories, highlighting the diversity of our dataset. + +Dataset visualizations. We provide visualizations of the ground-truth point cloud and layout annotations in Figure 9 to showcase the quality of the photo-realistic scans and the realism of the layout. Additionally, we show rendered images of objects in Figure 10 to illustrate the diversity and quality of the objects in our dataset. + +# B Implementation Details of SPATIALLM + +In Figure 12, we show the full prompt and an example response of SPATIALLM. Note that we shift the point cloud and the corresponding layout and object box coordinates to ensure that they are all non-negative. Next, we quantize the coordinates into 1,280 bins, each with a resolution of $2.5\mathrm{cm}$ . LLM predicts integer values, which are mapped back to continuous values using the inverse transformation and normalization, restoring them to the original coordinate system. + +Data augmentations. We apply a number of augmentations to the point cloud, which are summarized in Table 8 and explained below. + +We begin with a cuboid crop following V-DETR [47]. This process involves randomly selecting a point as the center of the cuboid and then determining random dimensions for the cuboid relative to the input point cloud size. We maintain a minimum aspect ratio of 0.8 and ensure that at least 50,000 points remain after the cropping process. + +Then, we apply four types of jitter noises to the point cloud: (i) jitter that simulates the minor uncertainties inherent in sensor accuracy; (2) jitter that replicates noise in areas with greater uncertainty; (3) jitter that reflects noise along the edges, typically from floating points between foreground objects and the background; and (4) jitter that represents distant floating points outside the room, often attributed to glossy windows, reflective mirrors, and points along the path to the outdoors. + +![](images/a62562852cff5790a912ea4e888718ad6308b5ff587830d4811bfdc543ad6e57.jpg) +Figure 7: Statistics of the SPATIALLM dataset. Top: Number of objects in each category. Bottom: Distribution of physical sizes in each category. We compute the object size as the diagonal length of the object's bounding box (in meters). All objects in our dataset have realistic physical sizes. + +![](images/037ce517cb08dbbda2c9b7ea2e6c92a424f1e58b4c7bfaeaa63ac08257dbf975.jpg) +Figure 8: Room object correlation of SPATIALLM dataset. + +Compute resources. We train SPATIALLM for 4 epochs with a total batch size of 64. The learning rate is set at $10^{-4}$ , using a cosine scheduler with a warm-up ratio of 0.03. The parameters for AdamW optimizer are as follows: adam_beta1 is 0.9, adam_beta2 is 0.99, and adam_epsilon is $1 \times 10^{-8}$ . We utilized 32 NVIDIA H20 GPUs, and training on SPATIALLM dataset takes approximately one day. + +For experiments on Structured3D [64] and ScanNet [13], we use a batch size of 8. Fine-tuning on Structured3D involves 50 epochs, while keeping the other settings the same. For fine-tuning on ScanNet, we introduce an additional pre-training stage to mitigate the difference in label spaces across datasets. Specifically, we map the classes in our dataset to those of ScanNet, and further train the model on our dataset for 3 epochs, followed by fine-tuning on ScanNet for 30 epochs. + +# C Supplementary Results for SPATIALLM Experiments + +In this section, we present per-category quantitative results for (i) layout estimation, (ii) 3D object detection, and (iii) zero-shot detection on videos. + +Table 8: Point cloud data augmentations. + +
MethodParameters
cuboid cropmin_points=50,000, aspect=0.8, min_crop=0.75, max_crop=1.0, p=0.1
random jitterσ=0.025, clip=0.05, ratio=0.8, p=0.9
random jitterσ=0.2, clip=0.2, ratio=0.05, p=0.8
random jitterσ=0.4, clip=1.0, ratio=0.001, p=0.75
random jitterσ=0.5, clip=4.0, ratio=0.0005, p=0.65
elastic distortparams=[[0.2, 0.4], [0.8, 1.6]], p=[0.8, 0.5]
random rotateaxis=z, angle=[0, 90, 180, 270]
random scalescale=[0.75, 1.25]
auto contrastp=0.2
chromatic translationp=0.75, ratio=0.1
chromatic jitterstd=0.05; p=0.8
color dropp=0.1
+ +# C.1 Layout Estimation + +Per-category results. In Table 9, we report the F1 scores in percentages at 0.25 and 0.5 thresholds of $\mathrm{IoU}_{\mathrm{2D}}$ for architectural elements i.e., walls, doors and windows, and the averaged score across the categories on Structured3D test set. + +Table 9: Experiment results on layout estimation. + +
MethodIoU2D@0.25IoU2D@0.5
avg.walldoorwindowavg.walldoorwindow
RoomFormer [60]83.486.085.178.981.484.683.176.6
SceneScript [2]90.492.992.885.589.291.992.583.3
SPATIALLM (ft. Structured3D)32.654.119.424.318.133.59.411.5
SPATIALLM (ft. Ours)59.971.542.765.644.764.839.729.6
SPATIALLM (ft. Ours → Structured3D)94.395.395.991.793.594.595.790.2
+ +# C.2 3D Object Detection + +Per-category results. We report the F1 scores in percentages at 0.25 and 0.5 thresholds of $\mathrm{IoU}_{3\mathrm{D}}$ for 18 ScanNet categories and the averaged score across the categories in Table 10 and Table 11, respectively. + +For $\mathrm{IoU}_{3\mathrm{D}}@0.25$ , we have $+16.5\%$ overall gain compared to SceneScript, and $+0.5\%$ difference compared to V-DETR. For $\mathrm{IoU}_{3\mathrm{D}}@0.5$ , we have $+15.8\%$ overall gain compared to SceneScript, and $-4.2\%$ difference compared to V-DETR. The largest gaps between our model and V-DETR correspond to "picture", "sink", and "shower_curtain". Note that these are either the smallest or the least occurring objects in ScanNet. + +Table 10: 3D object detection results with IoU3DMethodaveragebathroombedbookshelfcabinetchaircountercurtaindeskdoorgarbin.picturerefriger.s.curtainsinksofatabletoiletwindowV-DETR [47]65.177.480.148.844.482.555.666.168.062.354.047.450.874.079.572.755.398.054.9SceneScript [2]49.164.571.139.730.381.143.430.953.441.742.611.828.258.648.968.355.877.536.5\( SPATIALLM^† \)2.96.57.02.51.24.01.42.34.82.50.30.01.42.30.91.74.75.03.7\( SPATIALLM^* \)33.857.049.224.412.250.9n/a15.348.1n/an/a9.723.4n/a3.638.331.475.6n/a\( SPATIALLM \)65.680.679.952.040.086.651.066.862.867.155.629.753.681.670.778.963.995.464.3 + +Table 11: 3D object detection results with IoU3D@0.5.†: trained on ScanNet only. *: trained on our dataset only. “n/a” indicates the category is not included in our dataset. + +
Methodaveragebathroombedbookshelfcabinetchaircountercurtaindeskdoorgarbin.picturerefriger.s.curtainsinksofatabletoiletwindow
V-DETR [47]56.868.875.550.439.176.243.654.152.150.947.842.749.261.561.465.450.688.643.9
SceneScript [2]36.854.865.937.321.273.725.316.944.523.530.53.023.319.524.962.050.164.221.7
\( SPATIALLM^† \)0.72.22.00.00.10.61.40.81.70.40.00.01.40.00.00.02.00.0
\( SPATIALLM^* \)22.637.641.013.56.830.7n/a1.034.3n/an/a2.617.7n/a1.128.419.559.0n/a
\( SPATIALLM \)52.674.271.747.729.579.340.153.352.646.641.710.242.950.640.271.160.390.545.0
+ +# C.3 Zero-shot Detection on Videos + +Per-category results. We collect 107 virtual room-tour videos and use MASt3R-SLAM [36] to reconstruct a point cloud from each video. Note that these videos are rendered from virtual scenes created by professional designers, thus ground-truth 3D annotations are available. To calibrate the reconstructed point cloud with ground truth 3D annotations, we align the estimated camera trajectory from MASt3R-SLAM to the ground-truth camera trajectory. Then, we perform zero-shot detection with SPATIALLM on the calibrated point clouds. + +In Table 12, we report the quantitative results on the top-20 occurring categories in the video test set. Note that this is a challenging task, because the reconstructed point clouds are often noisy and highly incomplete. Furthermore, because of the differences in scale and position due to imperfect alignment of camera trajectories, the objects in the point cloud may not match well with the ground truth 3D annotations, leading to lower F1 scores. + +Table 12: Zero-shot results on layout estimation and 3D object detection with IoU@0.25. +(a) Layout estimation + +
avg.walldoorwindow
55.768.247.451.4
+ +(b) 3D object detection + +
averagecurtainnightstandchandelierwardrobebedsofachaircabinetdiningtab.plantstvcab.coffeetab.sidetab.aircond.dresserstoolrefrigerate.paintingcarpettv
36.937.067.036.839.695.269.132.311.224.226.327.364.99.724.046.730.816.738.224.118.0
+ +# D Extensions of SPATIALLM + +In this section, we present more proof-of-concept experiment results on future extensions of SPATIALLM, including (i) handling point clouds from diverse sources, and (ii) adapting to various downstream tasks via language-based prompts. + +# D.1 Supporting Point Clouds from Diverse Sources + +Point clouds offer a lightweight and flexible 3D representation that can be readily generated from a variety of sources. Meanwhile, our dataset is large and diverse, enabling strong zero-shot performance over point cloud inputs in both synthetic and real-world data. In this experiment, we examine the following four types of input sources: + +- Text-to-3D: We first generate an image via Flux-dev [26] with prompt "a low poly 3D living room in cartoon style", then convert the image into 3D using TRELLIS [57]. + +- Hand-held video camera: We take a real-world video using a hand-held camera and perform 3D reconstruction using MASt3R-SLAM [36]. +- LiDAR-based reconstruction: We take point clouds reconstructed from captures using iPhone ARKit from MultiScan [34]. +- Synthetic mesh: We create point clouds by randomly sampling mesh surfaces from a synthetic dataset HSSD [24]. + +As can be seen in Figure 13, our model is robust to variations in point cloud origin, structure, and appearance. + +# D.2 Task Adaptation via Language-based Prompts + +A key idea of our work is to represent the 3D scene structures in pure text form. This enables SPATIALLM to be conveniently adapted to different downstream tasks via natural language instruction, without changing the underlying model architecture. Below, we demonstrate two such extensions. + +Detection with user-specified categories. We enable the model to perform object detection conditioned on user-specified categories by leveraging the flexibility of LLMs. During training, the input prompt is customized to indicate which object categories to detect. The model then learns to selectively predict oriented bounding boxes (OBBs) for instances matching the specified categories, all within a unified training procedure. This experiment highlights the model's adaptability through prompt-based control. Some qualitative results are provided in Figure 14. + +Semantic label completion. Besides natural language prompts, SPATIALLM also supports structured language scripts as input, enabling novel tasks such as semantic label completion. In this task, the model receives object bounding boxes and their poses written in a general-purpose language (i.e., Python), while the object categories are left unspecified. The model then predicts the semantic label of each object based on the spatial layout and the corresponding point cloud data. Note that this problem often arises in real-world design workflows that involve 3D assets with known positions but without semantic labels. + +Compared to 3D object detection, this task is relatively easy. Nevertheless, it demonstrates the versatility and potential use of large language models for structured 3D understanding. We show some qualitative results in Figure 15. Our model achieves an overall classification accuracy of $96.8\%$ on the test set of our dataset. + +![](images/64ea15eee83d78fa003400e34c47a5d1183890a3a76d170a72bd4c1f6243eee4.jpg) +Figure 9: Point clouds with overlaid 3D layout and object annotations in SPATIALLM dataset. + +![](images/4d517cc8542efbb7f974aa7faf7542ad78bd6a37e6544b6ae4ed6cb62d5a247c.jpg) +Figure 10: Example objects in SPATIALLM dataset. Note that each category has a diverse set of objects with high quality in both geometry and appearance. + +![](images/4ddcbdca44c1b6e926e56886c89aee877fcaff6bd1d0f8264479f89ad20b43ee.jpg) +Figure 11: Five-level hierarchical structure of the point cloud encoder [56]. $D$ indicates the feature dimension at each level, and grid pool downscales the point cloud resolution by 2 times along each dimension. + +![](images/e4e789cfc18f928810f8e99b64c3224dee8962552d165fe2bb9f45fde992d16b.jpg) +Prompt: +Figure 12: An example conversation of SPATIALLM. + + Detect walls, doors, windows, boxes. The reference code + +# is as follows: + +@dataclass + +class Wall: + +a_x: int + +ay: int + +z-y: int + +1 + +b_x: int + +b_y: int + +b_z: int + +height: int + +@dataclass + +class Door: + +wall_id: str + +position x: int + +position: int + +position $z$ int + +0031616n-5 width: int + +width: 10 + +height: int + +@dataclass + +class Window: + +wall id: str + +position x: int + +position v: int + +position_y: int position_x: int + +positionn_2 + +width:int + +height:int + +@dataclasss + +class Bbox: + +class: str + +position x: int + +position: int + +position $z$ int + +1081010n=2 + +angle_z: int + +scale_x: int + +scale y: int + +scale z: int + +# Response: + +wall_0 = Wall(144, 15, 30, 139, 15, 30, 101) + +wall_1=Wall(144,15,30,144,26,30,101) + +wall_2=Wall(139,15,30,128,16,30,101) + +wall_3=Wall(128,16,30,118,18,30,101) + +wall_4 = Wall(47, 18, 30, 118, 18, 30, 101) + +wall_5=Wall(47,18,30,47,42,30,101) + +wall_6 = Wall(144, 26, 30, 152, 26, 30, 101) + +wall_7=Wall(152,26,30,152,42,30,101) + +wall_8=Wall(47,42,30,67,42,30,101) + +wall_9=Wall(67,42,30,67,53,30,101) + +wall_10 = Wall(138, 42, 30, 152, 42, 30, 101) + +wall_11 = Wall(138, 42, 30, 138, 53, 30, 101) + +wall_12=Wall(49,53,30,67,53,30,101) + +wall_13 = Wall(49, 53, 30, 49, 151, 30, 101) + +wall_14=Wall(138,53,30,152,53,30,101) + +wall_15 = Wall(152, 53, 30, 152, 170, 30, 101) + +wall_16 = Wall(19, 151, 30, 49, 151, 30, 101) + +wall_17=Wall(19,151,30,19,269,30,101) + +wall_18=Wall(118,170,30,152,170,30,101) + +wall_19=Wall(118,170,30,118,269,30,101) + +wall_20 = Wall(19, 269, 30, 118, 269, 30, 101) + +door $0 =$ Door(wall 16.32.151.60.30.75) + +door 1=Door(wall 17,19,194,60,27,75) + +door 2=Door(wall 19.118.213.60.37.75) + +door 3=Door(wall 20.98.269.60.30.75) + +window $0 =$ Window(wall 4.94.18.74.73.75) + +window 1=Window(wall 17.19.162.60.17.75) + +bbox 0=Bbox(curtain,93,20,74,320,165,10,96) + +bbox_1=Bbox(plants,144,46,53,320,27,27,74) + +bbox_2=Bbox(air_conditioner,147,58,55,412,15,15,81) + +bbox_3=Bbox(plants,58,61,52,480,24,31,70) + +bbox_4=Bbox(tv_cabinet,55,103,41,480,111,19,34) + +bbox 5=Bbox (ty, 50, 103, 69, 480, 77, 6, 44) + +bbox_6=Bbox(combination_sofa,119,110,45,480,166,105,48) + +bbox_7=Bbox(painting,152,110,80,480,102,6,41) + +bbox_8=Bbox(floor-standing_lamp,145,163,55,320,21,21,80) + +bbox_9=Bbox(dining_chair,50,200,43,320,26,28,41) + +bbox_10=Bbox(diningchair,67,214,43,480,26,28,41) + +bbox_11=Bbox(dining_table,49,228,50,480,77,40,43) + +bbox_12=Bbox(diningchair,67,233,43,480,26,28,41) + +bbox_13=Bbox(painting,21,236,77,480,58,6,39) + +bbox_14=Bbox(shoe_cabinet,113,246,68,480,52,16,120) + +bbox_15=Bbox(dining_chair,41,252,43,480,26,28,41) + +bbox_16=Bbox(diningchair,67,252,43,480,26,28,41) + +bbox_17=Bbox(sideboard,54,264,68,320,98,16,120) + +![](images/3cb41611e74d4ec299083d9ab22d239480291654926e5e16b8b3774a62313690.jpg) +Figure 13: Qualitative results on various input sources. First row: Text-to-3D. Second row: Hand-held video camera. Third row: LiDAR-based reconstruction. Fourth row: Synthetic mesh. + +![](images/c30c19bb85613a54092edd578ac0183f10cc4d5c369f27cd8fc67ec9a743b422.jpg) +all objects + +![](images/a5919b4ee45e30794f625853351b9bfca5a8b3ddc45b4fbfcdd4bea2a378cc2f.jpg) +bed + +![](images/e488d321b652ac64246b55174f18a3fc2a47bb6f6b492f5d9293f9d61f1022e0.jpg) +chair + +![](images/47e0b2345c73c0f00751affcf88206b32e86ad7e66931fc81181609d71c96512.jpg) +desk + +![](images/38cac314e6408c2a16066fdb0b2c116596048fc0dbd96af8c8f7fb1ba9858fc2.jpg) +layout + +![](images/835657e7acc5cbcdb0c82ad96516781800eb135e581e26814b133c734985e0fa.jpg) +dining_chair + +![](images/6ece1e4076471f2a16a96e418b76a02ee933b29755d21c5013f0a384abfa7555.jpg) +tv, tv_cabinet + +![](images/42f64088b9171cf63004533a3f142977e82cb7a9446752557b8b1c6136eb350f.jpg) +sofa,coffee_table +Figure 14: Qualitative results of detection with user-specified categories. Labels below each figure indicate the category specified for detection. + +![](images/4f2c1429a829b9874124df6fb3d53bd5ed5f187c1107477c44ce8b0fe2df6074.jpg) + +![](images/c67cf62f435dbcced19922507bf25370d3b1c4c315de083da444fb6aae096f41.jpg) + +![](images/8ecaf53b253048ad97f3ccc35f4e8318b1306661d77af71755893c364ba8c46b.jpg) + +![](images/437fe84c39b534cbb935621b10d0fc33d57689bf2b75ef9da1c17a35b75773fa.jpg) + +![](images/d005ad6d838fcd5db01723b2d487f7cd623e673a8c4eaba80bb50b98613c85cd.jpg) + +![](images/edf1fea9c0e4b69047cc67f82b8d98e1adf81bdb413729da899a1da66be240f9.jpg) + +![](images/3d3a225c4fc567f20be7d64ba634313e6f2d65a98d96fd721e326bf5a6c02d8b.jpg) +Input +Figure 15: Qualitative results of semantic label completion. 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Recent studies explore data-driven approaches and achieve enhanced spatial reasoning performance by fine-tuning models on 3D-related visual question-answering data. However, these methods typically perform spatial reasoning in an implicit manner and often fail on questions that are trivial to humans, even with long chain-of-thought reasoning. In this work, we introduce SpatialReasoner, a novel large vision-language model (LVLM) that addresses 3D spatial reasoning with explicit 3D representations shared between multiple stages-3D perception, computation, and reasoning. Explicit 3D representations provide a coherent interface that supports advanced 3D spatial reasoning and improves the generalization ability to novel question types. Furthermore, by analyzing the explicit 3D representations in multi-step reasoning traces of SpatialReasoner, we study the factual errors and identify key shortcomings of current LVLMs. Results show that our SpatialReasoner achieves improved performance on a variety of spatial reasoning benchmarks, outperforming Gemini 2.0 by $9.2\%$ on 3DSRBench, and generalizes better when evaluating on novel 3D spatial reasoning questions. Our study bridges the 3D parsing capabilities of prior visual foundation models with the powerful reasoning abilities of large language models, opening new directions for 3D spatial reasoning. Our project page is available here. + +# 1 Introduction + +3D spatial reasoning studies how models perceive, understand, and reason about 3D object properties and spatial relationships. It is not only a fundamental task for vision-language models to achieve human-level intelligence, but also crucial to a range of downstream applications in robotics [14, 18] and embodied AI [10]. Despite the recent advancements of large multi-modal models, such as GPT-4o and Qwen2.5-VL, their 3D spatial reasoning capabilities remain limited and fall far behind human-level performance [9, 31, 51]. Recent 3D-aware large vision-language models (LVLMs) injected 3D knowledge by fine-tuning the model on synthetic 3D-related question-answer pairs [9, 33] and achieved improved performance on spatial reasoning benchmarks [31, 45, 51]. On the other hand, large proprietary models such as Gemini 2.0 [15] have advanced 3D parsing by directly predicting 3D object bounding boxes, enabling the development of powerful generalist robotics models [43]. + +We identify two key challenges in 3D spatial reasoning: (1) $3D$ thinking—the ability to decompose a complex 3D spatial reasoning question into small, manageable steps, and (2) $3D$ computation—the ability to solve these thinking steps in a consistent and accurate manner. As shown in Figure 1, prior reasoning methods adopt long chain-of-thought reasoning to tackle the problem but do not + +![](images/e8cb47874263637c9921ac73d2abe26274ebd6df2886dce44714875a78241e97.jpg) +Figure 1: Comparing 3D spatial reasoning of our SpatialReasoner with previous state-of-the-art models. Our SpatialReasoner builds on explicit 3D representations, performs 3D computation, and reasons about the final answer. Although Gemini 2.0 can also break down complex 3D spatial reasoning questions into small and tractable steps, it lacks reliable 3D computation and the long chain-of-thought reasoning ultimately leads to a wrong answer. + +have explicit 3D computation. ChatGPT o3 [36] fails to adopt a systematic approach to solving the problem and instead relies on other visual cues to assist the reasoning (e.g., location of the foil package). In contrast, the Gemini 2.0 thinking model [15] employs an organized strategy to tackle the problem but ultimately fails to arrive at the correct answer due to a lack of reliable 3D computation. + +In this work, we present SpatialReasoner, a novel large vision-language model (LVLM) built with (1) explicit 3D representations and (2) enhanced and generalizable 3D thinking. Specifically, our SpatialReasoner adopts explicit 3D representations, such as 3D locations and orientations, as an interface that enables coherent and reliable reasoning across multiple stages, i.e., 3D perception, computation, and reasoning. On the other hand, we would like to learn enhanced 3D thinking capabilities that generalize to novel question types not seen during training. Hence we explore a two-stage training strategy following prior works [17]. In Stage I, we apply supervised fine-tuning (SFT) to equip the LVLM with explicit 3D representations, enhancing 3D perception and computation capabilities of the model. Then in Stage II, we leverage reinforcement learning (RL) to develop robust and generalizable 3D thinking built on explicit 3D representations. + +Inspired by previous 3D pseudo-annotation pipelines in [9, 11, 33], we synthesize basic 3D perception, i.e., detection and pose estimation, and 3D computation question-answering data interleaved with explicit 3D representations. We further generate standard spatial reasoning question-answer pairs with chain-of-thought reasoning that breaks down complex 3D spatial reasoning questions into multiple steps-3D perception, computation, and reasoning. Experimental results demonstrate that our SpatialReasoner with explicit 3D representations can significantly enhance 3D spatial reasoning abilities of LVLMs and generalize to novel question types. + +Besides enhancing 3D spatial reasoning capabilities of LVLMs, reasoning with explicit 3D representations allows us to interpret the reasoning process and to study the failure modes of LVLMs. We find that the accuracy of 3D perception lags significantly behind that of 3D computation, suggesting that most errors in downstream VQA tasks still stem from failures in 3D perception. Moreover, by predicting key 3D information, such as 3D object locations and orientations, as intermediate results, our SpatialReasoner enables compositional reasoning for 3D spatial tasks [46]. This not only allows our method to generalize better to novel spatial reasoning questions, but also makes it easily extensible to other tasks that build on our explicit 3D representations. + +Besides improving spatial reasoning performance on a variety of benchmarks, we experiment on various LVLMs fine-tuned with different combinations of data and training methods to study the key + +factors toward improved 3D spatial reasoning. Our empirical results lead to the following insights: (1) For 3D-aware VLMs, SFT offers a more scalable approach than RL that requires high-quality 3D-aware data, which is often difficult to obtain. Our recipe of RL followed by SFT achieves the best overall performance; (2) 3D-aware LVLMs fine-tuned with RL generalize better than SFT when tested on novel 3D spatial reasoning questions, echoing prior findings [12]; (3) Standard LVLMs often exploit 2D reasoning as a shortcut to tackle 3D spatial reasoning problems, whereas our SpatialReasoner avoids the spurious correlations and always reasons with explicit 3D representations, achieving improved and robust performance on challenging real-world 3D spatial reasoning datasets. + +In summary, our contributions are as follows: (1) We introduce SpatialReasoner, a novel LVLM that solves 3D spatial reasoning problems with a compositional approach based on explicit 3D representations. (2) Reasoning with 3D representations allows us to interpret the reasoning process and to study the failure modes of LVLMs. (3) Our SpatialReasoner effectively improves the 3D spatial reasoning performance on a range of benchmarks and extensive experimental results provide valuable findings on data and training strategy designs for future development of 3D-aware LVLMs. + +# 2 Related Works + +3D Spatial reasoning. 3D spatial reasoning explores how models perceive and reason about 3D object properties and relationships. Early works [46, 47, 48] studied this in simulated environments or datasets with object-level annotations [20, 25, 7, 11, 45, 33]. More recent efforts construct benchmarks based on real-world imagery [31, 51] and improve model performance via synthetic QA data [9, 11, 33, 39]. However, these models often rely on implicit reasoning, offering little interpretability or intermediate 3D computation. Moreover, it remains unclear whether the 3D spatial reasoning capabilities acquired from data-driven fine-tuning can generalize to novel question types that require complex 3D computation over different combinations of 3D perception outputs. + +Explicit 3D representations. Explicit 3D representations simplify reasoning and expose model failures. Simulation-based works [35] used neural-symbolic methods to parse object-level structures, while PO3D [46] and its extensions [47, 48] showed that structured visual modules enable interpretable reasoning [32]. These results in simulation systems reveal the limitations of current LVLMs and highlight the importance of the visual module for a successful reasoning model. + +Post-training. Post-training aligns pre-trained models with downstream objectives via SFT [49, 54] and RL [5, 6, 37, 41], improving formatting and reward-guided alignment [21, 44]. Recent systems like GPT-4 [1], Claude-3.5 [2], and DeepSeek-R1 [17] showcase these techniques. Despite recent advances, post-trained models still struggle with generalization under distribution shifts and novel tasks. While SFT stabilizes outputs but often overfits, RL improves adaptability [12], and combining both shows promise [17], making post-training vital for robust, aligned LLMs. + +Test-time scaling. Scaling inference-time compute improves performance without retraining. Approaches include beam search [30, 16], best-of-N sampling [42], and MCTS [13], alongside prompting methods like CoT [50] and ToT [53]. Strategically allocated test-time compute has proven effective for enhancing reasoning in both unimodal and multimodal models. Building on this, we apply CoT reasoning to 3D spatial reasoning by fine-tuning models to generate step-by-step rationales, providing both accurate and interpretable results. + +# 3 SpatialReasoner + +# 3.1 Overview + +In this section, we introduce our SpatialReasoner for explicit and generalizable 3D spatial reasoning. Our SpatialReasoner features two key designs: (1) explicit 3D representations that serves as interface to support multi-stage spatial reasoning, i.e., 3D parsing, computation, and reasoning (see Figure 2), and (2) generalizable spatial reasoning from multi-stage training (see Figure 4). + +In Section 3.2, we present the explicit 3D representations and describe how our model is trained to predict and to interpret the 3D representations for spatial reasoning. Then in Section 3.3 we discuss our training strategies, exploring standard supervised fine-tuning, reinforcement learning, as well as + +![](images/d980e96c1bd47f756a5b756177e75ff8ebb81c4fdea9c3b30f5736dcbeac7034.jpg) +Figure 2: Overview of our SpatialReasoner design. Our SpatialReasoner adopts explicit 3D representations as an interface to enable coherent and reliable multi-stage spatial reasoning, i.e., 3D parsing, computation, and reasoning. + +3D-aware process rewards. Lastly we introduce our 3D-aware data generation pipeline and different variants of training data used at different stages in Section 3.4. + +# 3.2 Learning Explicit 3D Representations + +Despite improved spatial reasoning abilities achieved by 3D-aware VLMs, such as SpatialRGPT [11] and SpatialLLM [33], and advanced proprietary models like Gemini 2.0 [15], these methods lack explicit 3D representations and rely on natural language to perform 3D spatial reasoning. For example, Gemini 2.0 describes object poses as "facing towards the viewer and slightly to its right" and estimates 3D distances with phrases such as "far behind some other object". Such natural language descriptions are inefficient and often not accurate enough for complex 3D spatial reasoning. + +Therefore, we propose integrating LVLMs with explicit 3D representations, such as 3D locations and poses, to serve as an accurate and reliable interface shared across stages of 3D spatial reasoning (see Figure 2). Our SpatialReasoner can predict explicit 3D representations as intermediate results or take them as inputs to perform basic 3D computations or complex spatial reasoning tasks. + +Explicit 3D representations. We define explicit 3D representations in a calibrated camera 3D space, which is a standard camera 3D space calibrated with the extrinsics of the camera. As illustrated in Figure 3, the calibrated camera 3D space has its $z$ -axis aligned with the $z$ -axis of the 3D world space, and the origin on the $z$ -axis is positioned close to the ground plane. Although estimating 3D object locations is easier in original camera 3D space, adopting explicit 3D representations in calibrated camera 3D space offers many advantages for subsequent spatial reasoning: (1) the $z$ -coordinates directly correspond to object heights, (2) estimating 3D spatial relationships such as "above" and "below" is largely simplified, and (3) objects often are on a plane parallel to the ground plane, which reduces many 3D spatial relationships to simpler 2D problems. + +![](images/35df31586c7c43c7d43716a5333c5581e105f2c2c2f917a249d16842e502c402.jpg) +Figure 3: Comparison between original and calibrated camera 3D space. Our explicit 3D representations are defined within calibrated camera 3D space that simplifies subsequent 3D computation and reasoning. + +A unified interface for explicit 3D spatial reasoning. Explicit 3D representations serve as an interface enabling coherent and accurate 3D spatial reasoning across stages (see Figure 2). For 3D perception, the model predicts object locations and orientations as 3D vectors, then estimates explicit distances or angles based on these predictions. Finally, the model aggregates explicit 3D information from earlier stages to reason about and answer 3D spatial questions. + +# 3.3 Training Strategies + +Supervised fine-tuning (SFT). We adopt a two-stage post-training strategy to equip the model with explicit, generalizable 3D spatial reasoning capabilities. In the first stage, SFT serves as a critical initialization step, aligning the pre-trained LVLM with curated 3D-annotated datasets. By + +![](images/dc033555914f40a1f0257f64fc3d0c195a720f6541244ba25681b5edfa9cd561.jpg) +Figure 4: Overview of our SpatialReasoner training. We adopt a multi-stage training strategy: In Stage I, we apply supervised fine-tuning (SFT) to equip the LVLM with explicit 3D representations; then in Stage II, we leverage reinforcement learning (RL) to develop robust and generalizable 3D spatial reasoning built on explicit 3D representations. + +optimizing a maximum likelihood objective over paired input-output sequences, the model learns to identify objects in 3D scenes and predict explicit 3D representations such as object locations and orientations. This structured supervision enables the model to expose interpretable intermediate reasoning traces during spatial computations—e.g., calculating relative distances and angles. However, due to its reliance on static demonstrations, SFT shows limited capacity to generalize beyond observed reasoning patterns. As shown in Section 4.3, SFT-trained models tend to memorize spatial templates in the training set and struggle with novel compositions or combinatorial variations in 3D queries. + +Reinforcement learning (RL). To overcome this limitation, we further post-train the SFT-initialized model using reinforcement learning (RL) with rule-based rewards. Treating the spatial reasoning task as a sequential decision process, each reasoning step is framed as an action within a Markov Decision Process (MDP), and policy gradients are optimized using GRPO [41]. Importantly, we integrate a reward scheme that provides structured reward signals reflecting both the correctness of final answers and the coherence of intermediate 3D computation steps. This enables multi-turn optimization where the model learns to revise inaccurate inferences and explore alternative reasoning paths. Empirically, this RL stage substantially improves generalization to out-of-distribution 3D spatial questions (Table 3), particularly in settings involving multi-object arrangements. The combined SFT+RL strategy therefore balances response formatting and perceptual grounding with adaptive reasoning and robustness, making it especially suited for complex 3D spatial understanding. + +Reward and policy optimization design. To train SpatialReasoner via reinforcement learning, we design a composite reward scheme capturing both answer correctness and reasoning quality. For outcome rewards, we use an accuracy reward—aligned with the multiple-choice evaluation metric from MMBench [28]—which gives positive feedback only when the model selects the correct answer, and a format reward [17] that encourages structured, readable outputs. We also explore process rewards to assess whether models can learn structured reasoning without SFT. These include a reasoning steps reward, which promotes use of structured indicators (e.g., "First", "Next"), and a 3D-aware process reward that checks for necessary spatial terms like distance or orientation. The reward signal is then calculated through the accuracy of the presence of each term. Though not used in the final model, these rewards serve as diagnostics for emergent reasoning behavior. Finally, we ablate the KL divergence term in GRPO and find that it harms training accuracy despite stabilizing output lengths (Section 4.3). We thus remove it in our final setting. + +# 3.4 Training Data + +To enable LVLMs to predict and reason with explicit 3D representations and to post-train LVLMs to solve various challenging 3D spatial reasoning questions, we generate a series of 3D-aware training data. We extend the data generation pipeline in [9, 11, 33]. Our process begins with generating 3D pseudo-annotations, followed by optional human verification, and ends with constructing various VQAs and chain-of-thought reasoning based on the 3D pseudo-annotations. + +Pseudo 3D ground-truths. We extend the 3D pseudo-annotation pipeline proposed in [9, 33] and generate various 3D annotations, such as object category, 3D location, and 3D pose, on unlabeled images from the OpenImages dataset [22]. Based on the object-level 3D annotations, we then apply rule-based methods to derive ground-truth labels for a range of 3D spatial relationships. Despite significant progress in visual foundation models for segmentation [38], metric depth estimation [52], and object pose estimation [34], we notice many factual errors in the generated 3D ground-truth, + +
MethodMeanHeightLocationOrientationMulti-Object
Open-Sourced Generalist
LLaVA-v1.5-7B [26]38.139.146.928.734.7
LLaVA-Next-8B [24]48.450.659.936.143.4
Cambrian-1-8B [45]42.223.253.935.941.9
Qwen2.5-VL-3B-Instruct [4]43.945.256.835.735.7
Qwen2.5-VL-7B-Instruct [4]48.444.162.740.640.5
Qwen2.5-VL-72B-Instruct [4]54.953.371.043.146.6
Open-Sourced Specialist
SpaceLLaVA [40]42.049.354.427.635.4
SpatialBot [8]41.040.454.431.933.5
SpatialLLM [33]44.845.861.630.036.7
SpatialRGPT [11]32.755.939.027.820.0
SpatialRGPT w/ depth [11]48.455.960.034.242.3
Proprietary
GPT-4o-mini39.744.352.421.036.5
GPT-4o44.253.259.621.639.0
Claude 3.5 V Sonnet48.253.563.131.441.3
Gemini 2.0 Flash49.849.768.932.241.5
Gemini 2.0 Flash (thinking)51.153.067.135.843.6
QwenVLMax52.045.170.737.744.8
Ours
SpatialReasoner-Zero54.046.467.348.447.2
SpatialReasoner-SFT58.351.973.550.750.3
SpatialReasoner60.352.575.255.251.8
+ +Table 1: Comparison with previous state-of-the-art methods on 3DSRBench [31]. Our SpatialReasoner outperforms previous open-source and proprietary methods on challenging 3D spatial reasoning problems in 3DSRBench. + +particularly when mistakes (e.g., missing objects or inaccurate object poses) propagate through the data pipeline. Therefore, we adopt a series of aggressive filtering steps to ensure the quality of our training data, including: (1) removing images with densely cluttered scenes, (2) excluding object categories that are difficult to segment or estimate pose for, and (3) discarding boundary cases that could lead to ambiguity. + +Human verification. Despite leveraging state-of-the-art visual foundation models in our data generation pipeline and applying multiple filtering strategies, the 3D pseudo-annotations remain susceptible to factual errors. Specifically, small issues such as missing objects or inaccurate 3D pose predictions propagate through later stages of the pipeline, leading to factual errors in the final spatial relationship pseudo-annotations. To assess the impact of data quality, we create a smaller but higher-quality dataset by manually verifying the correctness of the 3D pseudo-annotations. + +Training data variants. Based on the obtained 3D ground-truth, we can generate different variants of data for fine-tuning. Specifically, we consider the following: (1) Basic3D-QA consists of basic 3D perception and 3D computation question-answering data. This can be used to learn explicit 3D representations without training on various 3D spatial relationships considered in downstream tasks. (2) SR-QA contains visual question-answering pairs about various 3D spatial relationships, following previous 3D-aware datasets [11, 33]. (3) SR-CoT extends SR-QA and comprises chain-of-thought reasoning with explicit 3D representations. Questions are answered in a step-by-step manner. + +# 4 Results + +# 4.1 Experimental Setup + +Baselines. We compare our SpatialReasoner with the following three types of baseline models. (1) Open-sourced generalists: such as LLaVA [26], Cambrian-1 [45], and Qwen2.5 [4] that are trained on + +
MethodCV-Bench-3DGQA
DepthDistanceMeanChooseCompareLogicalQueryVerify
Qwen2.5-VL-7B-Instruct82.583.258.882.771.578.940.682.5
SpatialReasoner-Zero77.581.860.281.667.979.243.882.1
SpatialReasoner-SFT85.271.562.082.872.281.445.583.2
SpatialReasoner87.373.361.883.281.171.845.282.9
+ +Table 2: Performance on CV-Bench-3D and GQA. Our SpatialReasoner also improves the spatial reasoning performance on GQA [19] and depth-related questions in CVBench-3D [45]. For distance-related questions, unlike Qwen2.5 that exhibits excessive dependence on 2D shortcuts, our SpatialReasoner employs rigid 3D spatial reasoning and achieves compelling performance. Regarding the performance on distance-related questions, see Section 4.2 and Section C for detailed discussions. + +
MethodMeanHeightIn-Distribution LocationOrientationNovel Multi-Object
Qwen2.5-VL-7B-Instruct48.444.162.740.640.5
SpatialReasoner-Zero53.740.668.450.246.6
SpatialReasoner-SFT52.244.969.548.940.0
SpatialReasoner56.452.572.654.143.4
+ +Table 3: Evaluation of generalization ability by finetuning on simpler (in-distribution) 3D spatial reasoning questions and evaluating on complex (novel) questions types unseen during training. Our SpatialReasoner-Zero and SpatialReasoner demonstrates superior zero-shot generalization, indicating RL fosters more robust and flexible reasoning than SFT. + +general vision-language data. (2) Open-sourced specialists: We evaluate SpaceLLaVA [40] (a public re-implementation of SpatialVLM [9]), SpatialBot [8] that enhances fine-grained spatial reasoning and robot control with RGB and depth images, and SpatialLLM [33] that fine-tunes a LLaVA model with multi-stage 3D-informed training. Note for fair comparison, we evaluate SpatialBot with RGB inputs only. (3) Proprietary models such as GPT-4o [1] and Claude 3.5 [3] that were trained on abundant web-scale data and for Gemini 2.0 [15], additional 3D-aware post-training. + +Evaluation benchmarks. We evaluate spatial reasoning abilities of various models on three spatial reasoning benchmarks. 3DSRBench [31] is a comprehensive 3D spatial reasoning benchmark with 2,100 questions and studies various 3D awareness and reasoning abilities with a robust evaluation setup. CVBench [45] is a vision-centric benchmark that assesses models at classic vision tasks with a range of 2D and 3D understanding VQAs. In this work, we focus exclusively on 3D-related questions, i.e., CVBench-3D, as 2D left-right relationships can lead to ambiguity with 3D left-right questions considered in 3DSRBench. GQA [19] is a widely adopted benchmark that studies visual reasoning and compositional question answering on a range of spatial relationships between objects. + +# 4.2 Advancing 3D Spatial Reasoning + +We evaluate SpatialReasoner on 3DSRBench [31], CVBench-3D [45], and GQA [19] to assess 3D perception, computation, and reasoning. As shown in Table 1, SpatialReasoner achieves a new state-of-the-art $60.3\%$ mean accuracy on 3DSRBench, outperforming Gemini 2.0 Flash $(49.8\%)$ and Claude 3.5 Sonnet $(48.2\%)$ . It shows notable improvements on Location $(+6.3\%)$ and Orientation $(+14.6\%)$ questions, and achieves $51.8\%$ $(+8.2\%)$ on Multi-Object reasoning relative to the second-best model, indicating stronger 3D perception and complex spatial understanding. + +On CVBench-3D and GQA (Table 2), SpatialReasoner achieves $87.3\%$ on depth-related questions in CVBench-3D, and improved performance on GQA's Compare category, further supporting its multi-object reasoning capability. These gains validate our multi-stage training: SFT equips the model with explicit 3D representations, while RL enhances adaptive reasoning and generalization. + +Despite this, we observe a performance drop on CVBench-3D distance questions, in contrast to substantial gains on similar 3DSRBench cases ("multi-object-closer-to", $34.3\% \rightarrow 70.9\%$ ). We attribute + +
MethodMean
Qwen2.5-VL-7B-Instruct48.4
SR-SFT58.3
SR-SFT (w/o exp. 3D rep.)51.9
+ +Table 4: Comparisons between SpatialRea- soner with and without explicit 3D repre-ssentations and CoT reasoning. Results highlight the benefits of explicit 3D representations as an interface to support enhanced 3D spatial reasoning. + +
MethodMean
Qwen2.5-VL-7B-Instruct48.4
SpatialReasoner-SFT58.3
SpatialReasoner-SFT (+HQ SFT)54.7
SpatialReasoner-Zero54.0
SpatialReasoner-Zero (w/ KL)52.4
SpatialReasoner-Zero (w/ 3D Rwd)54.6
+ +Table 5: Ablation study on various design choices in RL and SFT. Notably with 3D-aware rewards, SpatialReasoner-Zero produces coherent chain-of-thought reasoning on explicit 3D representations and improves benchmark performance. + +this gap to shortcut-driven 2D biases in CVBench-3D. Unlike existing LVLMs, SpatialReasoner relies on explicit 3D reasoning, offering greater robustness to such spurious cues (Section C). + +# 4.3 Analyses and Findings + +Generalization abilities. While SFT stabilizes outputs, it often overfits training distributions, limiting generalization to out-of-distribution (OOD) variations [12]. In contrast, outcome-based RL promotes transferable reasoning strategies and perceptual adaptability. To examine this, we compare SpatialReasoner-SFT (SFT-only), SpatialReasoner-Zero (RL-only), and SpatialReasoner (SFT+RL) on multi-object reasoning, with all multi-object data removed during training. + +As shown in Table 3, SpatialReasoner-Zero achieves $46.6\%$ , outperforming SpatialReasoner-SFT $(40.0\%)$ and SpatialReasoner $(43.4\%)$ , confirming RL's superior zero-shot generalization ability. Although SFT+RL improves over SFT, it does not fully match RL-only performance, indicating that RL fosters more flexible reasoning than memorization-prone SFT. + +When multi-object training data are available (Table 1), SFT attains $50.3\%$ via pattern matching, but drops to $40.0\%$ when data are withheld, whereas RL-based models retain stable performance $(46.6\%)$ vs. $47.2\%$ . Mean accuracy further supports this: SpatialReasoner-Zero maintains performance $(53.7\%)$ vs. $54.0\%$ , while SpatialReasoner-SFT degrades significantly $(58.3\%)$ to $52.2\%$ , reinforcing that RL enables more robust generalization. + +Compositional reasoning. We assess whether structured 3D reasoning can emerge from process rewards alone, without curated CoT. As shown in Figure 8, SpatialReasoner-Zero trained with 3D-aware rewards generates coherent multi-step reasoning grounded in 3D cues, improving interpretability over the baseline relying on shallow shortcuts. This demonstrates the model's ability to synthesize novel compositional reasoning strategies from basic 3D perception and computation skills. Quantitatively, it achieves a performance gain from $54.0\%$ to $54.6\%$ (Table 5), indicating that outcome-driven reward shaping can induce coherent reasoning without human-crafted CoT. Nonetheless, for the final SpatialReasoner, we use SFT to guide reasoning traces, as it outperforms process rewards alone. + +Scaling of training computation. We ablate on the scaling of training computation for SpatialReasoner-SFT and SpatialReasoner in Figure 5. Results show that with more training steps, SpatialReasoner-SFT starts to overfit and exhibit decreased performance, while SpatialReasoner trained with RL retains a stable and competitive performance. + +Scaling of training data. 3D-related tasks often lack high-quality data with 3D (pseudo-)annotations due to the cost and expertise required. While synthetic or pseudo-labeled data is more scalable, they introduce noise and domain gaps. To study data scaling, we mix 1.2K verified and 24K unverified samples and compare SpatialReasoner with and without RL. As shown in Figure 6, RL performs best with only the 1.2K verified data, whereas SpatialReasoner-SFT benefits from more unverified data. This suggests that SFT is more tolerant to noisy pseudo-annotations, while RL favors fewer but higher-quality samples. + +![](images/b901e69d91eff66bc8bb728df7e4461279ad254a5fb9a65f19d07c50d7069667.jpg) +Figure 5: Scaling of training computation. SpatialReasoner-SFT may overfit to training data while SpatialReasoner trained with RL retains a stable and competitive performance. + +![](images/80f27bb29537003929a58abdd46a16d0fb736c94a0209ad47c5919f77f9487d7.jpg) +Figure 6: Scaling of training data by mixing 1.2K human-verified with 24K unverified data. Results show SFT remains a compelling choice when scaling with potentially noisy training data. + +Ablation study on explicit 3D representations. To study the importance of explicit 3D representations and step-by-step 3D computations, we train a variant of SpatialReasoner on the same data but without explicit 3D representations. Results in Table 4 show that explicit 3D representations can enhance 3D spatial reasoning by a wide margin. + +Ablation study on KL divergence. We evaluate KL divergence regularization in GRPO and find it harms learning in our setting. As shown in Figure 9, adding KL causes training to collapse, while removing it yields stable improvements. The shorter completions without KL may reflect mitigated length bias [29], improving token efficiency. As shown in Table 5, SpatialReasoner-Zero reaches $54.0\%$ accuracy without KL vs. $52.4\%$ with it. We thus omit KL for better spatial reasoning. + +Ablation Study on SFT+RL vs. SFT+SFT To verify the performance gains are not merely from additional data exposure, we compare sequential SFT+RL against SFT+SFT. From Table 5, while SpatialReasoner-SFT achieves $58.3\%$ accuracy, applying a second round of SFT (SFT+SFT) degrades performance to $54.7\%$ . In contrast, RL after SFT (SFT+RL) improves accuracy to $60.3\%$ . These results indicate repeated SFT exacerbates overfitting to training patterns, whereas RL effectively builds on the structured outputs from SFT to promote adaptive and generalizable 3D spatial reasoning. + +# 4.4 Interpreting Failure Modes + +Explicit 3D representations enhance both spatial reasoning and interpretability [46]. We categorize 3D spatial reasoning into two stages: 3D perception (Parsing key 3D information from the image input) and 3D reasoning (computing 3D metrics and deriving final answer). To quantitatively analyze model behavior, we manually verify 300 questions with human-verified 3D answers. For 3D perception, we evaluate questions on object location and orientation; for 3D reasoning, models estimate distances, depths, and angles from 3D inputs, with accuracy (within $\pi /6$ ) for angles and mean error for positions. As shown in Table 7, models reason more accurately than they perceive-indicating most VQA errors stem from flawed 3D perception, consistent with our qualitative visualizations. Additionally, RL slightly degrades perception and reasoning accuracy, likely due to rewards focusing only on final answers and reasoning format, not intermediate 3D outputs. + +# 5 Conclusions + +In this paper, we presented SpatialReasoner that performs explicit and generalizable 3D spatial reasoning by predicting intermediate 3D representations across perception, computation, and reasoning stages. Our two-stage post-training pipeline with SFT followed by RL achieves state-of-the-art results on multiple benchmarks while generalizing well to novel tasks. By analyzing model behavior, we show that explicit 3D reasoning improves both accuracy and interpretability, and reveals key limitations in 3D perception. 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Pulsecheck457: A diagnostic benchmark for comprehensive spatial reasoning of large multi-modal models. arXiv preprint arXiv:2502.08636, 2025. +[49] Jason Wei, Maarten Bosma, Vincent Y Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, Andrew M Dai, and Quoc V Le. Finetuned language models are zero-shot learners. arXiv preprint arXiv:2109.01652, 2021. +[50] Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al. Chain-of-thought prompting elicits reasoning in large language models. Advances in neural information processing systems, 35:24824-24837, 2022. +[51] Jihan Yang, Shusheng Yang, Anjali W Gupta, Rilyn Han, Li Fei-Fei, and Saining Xie. Thinking in space: How multimodal large language models see, remember, and recall spaces. arXiv preprint arXiv:2412.14171, 2024. +[52] Lihe Yang, Bingyi Kang, Zilong Huang, Zhen Zhao, Xiaogang Xu, Jiashi Feng, and Hengshuang Zhao. Depth anything v2. Advances in Neural Information Processing Systems, 37:21875-21911, 2024. +[53] Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Tom Griffiths, Yuan Cao, and Karthik Narasimhan. Tree of thoughts: Deliberate problem solving with large language models. Advances in neural information processing systems, 36:11809-11822, 2023. +[54] Chunting Zhou, Pengfei Liu, Puxin Xu, Srinivasan Iyer, Jiao Sun, Yuning Mao, Xuezhe Ma, Avia Efrat, Ping Yu, Lili Yu, et al. Lima: Less is more for alignment. Advances in Neural Information Processing Systems, 36:55006-55021, 2023. + +# A Limitations + +Our system has a few limitations. First, despite the use of reinforcement learning (RL), supervised finetuning data with chain-of-thought reasoning and explicit 3D representations remains crucial as a warm-up. Training with RL alone-i.e., SpatialReasoner-Zero-fails to produce high-quality reasoning processes. Second, SpatialReasoner consistently performs explicit multi-step 3D reasoning, regardless of the difficulty of the problems. Ideally the model should adopt a hybrid strategy-leveraging common sense and visual cues for simpler questions while reserving explicit 3D computation and reasoning for more challenging problems. This would retain many strong and generalizable knowledge from the base Qwen2.5-VL model while improving inference efficiency. + +# B Implementation Details + +We train SpatialReasoner using different combinations of curated datasets and training objectives. Starting from the Qwen2.5-VL-7B [4] base model, we first apply SFT with 24k curated SR-CoT data alongside 24k randomly sampled LLaVA [27] data, resulting in SpatialReasoner-SFT. Next, we further train SpatialReasoner-SFT with RL using 1.2k SR-QA examples, leading to our final SpatialReasoner. For comparison, if we instead fine-tune SpatialReasoner-SFT with SFT on the same 1.2k SR-QA set, we obtain SpatialReasoner-SFT (+HQ SFT), serving as an ablation study baseline. In parallel, we directly fine-tune the base model with RL using the 1.2k SR-QA data without prior SFT, resulting in SpatialReasoner-Zero. Additionally, to investigate if explicit 3D perception ability can enable the model to self-organize reasoning trajectories under process rewards, we train the base model with SFT using 12k Basic3D-QA data alongside 12k randomly sampled LLaVA data before applying RL training on the 1.2k SR-QA data, yielding SpatialReasoner-Zero (w/ 3D Rwd). + +We conduct all training experiments using 4×NVIDIA H100 80GB HBM3 GPUs. For SFT, we train the model for 10 epochs (approximately 20K steps with a batch size of 6) on the combined 24k SR-CoT and 24k LLaVA datasets. For RL training, we train for 100 epochs (approximately 13K steps with a batch size of 12) on the 1.2k SR-QA dataset, using 1 GPU with vLLM [23] for efficient inference acceleration. For the experiments withholding multi-object training examples, we double the number of training epochs to compensate for the reduced training set size. We set the learning rate to 5e-6 for SFT and 5e-7 for RL, both following a cosine learning rate scheduler with a warm-up ratio of 0.1. In the KL divergence ablation study, we set the KL penalty weight to 0.04. We monitored the model every 1K training steps and reported results based on the best-performing checkpoint. + +# C 2D Reasoning as a Shortcut + +As shown in Table 1 and Table 2, our SpatialReasoner outperforms previous open-source and proprietary models on 3DSRBench [31], and achieves notable improvements on GQA [19] (from $58.8\%$ to $61.8\%$ ) and depth-related questions in CVBench-3D [45] (from $82.5\%$ to $87.3\%$ ). However, if we focus on multi-object 3D distance-related questions in CVBench-3D [45] and 3DSRBench [31], we observe contradictory results: SpatialReasoner achieves a substantial improvement of $21.5\%$ on 3DSRBench, but exhibits a notable performance drop of $9.9\%$ on CVBench-3D (see Table 6). + +We attribute this discrepancy to the abundant shortcuts in distance-related questions in CVBench-3D. From the qualitative examples in Figure 7, the provided 2D bounding boxes in CVBench-3D can be exploited as shortcuts to answer the 3D spatial reasoning question. Rather than reasoning about 3D distances between objects, we can easily derive the correct answer by comparing the 2D distances between the red and blue boxes and between the red and green boxes. Meanwhile, 3DSRBench is a human-collected VQA dataset and manually avoid such spurious correlation, e.g., "objects closer in 3D space are also closer in 2D image plane". For the 3DSRBench example in Figure 7, the bounding box of the dog is actually closer to the bounding box of the man in black that farther away in 3D space. + +Given the 2D bounding box annotations in CVBench-3D and 3DSRBench, we derive a simple heuristic to answer distance-related spatial reasoning questions by simply comparing L2 distances between 2D centers of object bounding boxes. We achieve an $80.2\%$ accuracy on distance questions in CVBench-3D and $34.3\%$ in 3DSRBench. This demonstrates that baseline models such as Qwen2.5-VL are largely exploiting 2D spatial reasoning as a shortcut to answer complex 3D spatial + +CVBench-3D (Distance) + +3DSRBench (Multi-Object Closer to) bounding box added for visualization) + +![](images/7baa26d0ead55b1d8fe4ef1085ef4e26d11ad9c248be5daeb5a4ff4bc99a49e3.jpg) +Figure 7: Comparison between (multi-object) distance-related questions in CVBench-3D [45] and 3DSRBench [31]. + +Question: Estimate the real-world distances between objects in this image. Which object is closer to the lamp (highlighted by a red box), the sofa (highlighted by a blue box) or the television (highlighted by a green box)? + +Answer: Sofa. + +![](images/abbb87c0be6d5fc3bacd8df1e3b6c7dfda37e316be1d1d7bced3e82739074c44.jpg) + +Question: Consider the real-world 3D locations of the objects. Which is closer to the dog, the person in red or the person in black? + +Answer: Person in red. + +
MethodCVBench3D Distance3DSRBench multi-object-closer-to
2D Heuristic80.234.3
Qwen2.5-VL-7B-Instruct [4]83.249.4
SpatialReasoner73.370.9
+ +Table 6: Comparison between Qwen2.5-VL [4] and SpatialReasoner on (multi-object) 3D distance-related questions in CVBench-3D [45] and 3DSRBench [31]. + +reasoning questions. Meanwhile, with our 3D-aware post-training, our SpatialReasoner adopts explicit 3D representation for 3D spatial reasoning questions. The trade-off between exploiting 2D shortcuts and adopting explicit 3D representations results in slightly lower performance of SpatialReasoner on test data with abundant spurious correlations, but more importantly, leads to a robust and largely improved performance on a challenging real-world dataset. + +Visual cues or explicit 3D reasoning? LVLMs that exploit 2D reasoning as shortcuts may achieve improved performance on test data with abundant spurious correlations (e.g., distance questions in CVBench-3D). However, they cannot genuinely solve 3D spatial reasoning problems and fall far behind SpatialReasoner that builds on explicit 3D spatial reasoning when tested on challenging real-world problems in 3DSRBench. + +# D Additional Results + +We present the following additional results: + +1. Table 7. We study the failures modes of SpatialReasoner. Results show that 3D computation is more accurate than 3D perception, indicating that most VQA errors stem from flawed 3D perception. +2. Figure 8. Qualitative comparisons between different variants of SpatialReasoner. +3. Figure 9. Ablation study results on KL divergence. + +![](images/32e3a6206e4018854732cf0a38c271bce6959b8c5074653ad6c3bb23d0c21434.jpg) + +# User + +Consider the real-world 3D locations and orientations of the objects. Which side of the bus is facing the green board? A. right, B. front, C. left, D. back. + +SpatialReasoner-SFT w/o explicit 3D representation + +D. back. + +SpatialReasoner-Zero + +The green board is on the front of the bus.B. + +![](images/18fb1c0660bb0dc17379f1d701f7b3ea7dbf2dc9b423095c9960eda30e73a796.jpg) + +# SpatialReaons-Zero w/ 3D-aware process reward + +First, I will determine the front direction of the bus. The front direction of the bus is (-0.17, 0.25, -0.96). Next, I will calculate the 3D location of the bus and the green board. The 3D location of the bus is (3.97, 2.35, 17.15), and the 3D location of the green board is (1.75, 1.65, 13.95). Then, I will calculate the vector from the bus to the green board. The vector is (-1.70, 0.00, 0.99). After that, I will determine the left direction of the bus. The left direction of the bus is (-0.92, 0.37, -0.29). Finally, I will check the angle between the front direction and the green board and compare it with the angle between the left direction and the green board. The angle between the front direction and the green board is 0.29 and the angle between the left direction and the green board is 0.33. Therefore, the correct answer is the front side of the bus facing the green board.B.front + +![](images/a58c83eff12a3f28d3fa90b290e04879217f8824251e1b1d93fa418569d7fb58.jpg) + +# SpatialReaonser-SFT + +The 3D location of the bus is $(-4.3, 1.9, 17.7)$ . The 3D location of the green board is $(4.9, 2.9, 22.1)$ . The vector from the bus to the green board is hence $(9.2, 0.9, 4.4)$ . The left direction of the bus is $(-0.7, 0.0, -0.7)$ . The cosine similarity between the vector pointing to the green board and the left direction is $-0.94$ , corresponding to an angle of $160.87$ degrees. Thus the angle between the vector pointing to the green board and the right direction is $19.13$ degrees. The front direction of the bus is $(0.7, 0.0, -0.7)$ . The cosine similarity between the vector pointing to the green board and the front direction is $0.34$ , corresponding to an angle of $70.30$ degrees. Thus the angle between the vector pointing to the green board and the back direction is $109.70$ degrees. Among the four directions, the smallest angle is right direction, with an angle of $19.13$ degrees. Thus right side of the bus is facing the green board. Therefore, the final answer is A. right. + +![](images/ca2a4f661d22b9d32db458020a457f3d9083300ff92166296a5bf760d96cf7dd.jpg) +Figure 8: Qualitative comparisons. Our explicit 3D spatial reasoning improves interpretability over the baseline that relies on shallow shortcuts. + +# SpatialReaonser + +The 3D location of the bus is (-3.7, 2.1, 14.7). The 3D location of the green board is (2.2, 3.1, 19.8). The vector from the bus to the green board is hence (5.9, 1.0, 5.1). The left direction of the bus is (-0.9, 0.0, -0.4). The cosine of the angle between the vector pointing to the green board and the left direction is -0.99, corresponding to an angle of 172.09 degrees. Thus the angle between the vector pointing to the green board and the right direction is 7.91 degrees. The front direction of the bus is (0.1, 0.0, -1.0). The cosine of the angle between the vector pointing to the green board and the front direction is -0.15, corresponding to an angle of 98.47 degrees. Thus the angle between the vector pointing to the green board and the back direction is 81.53 degrees. Among the four angles, the smallest is the right direction (7.91 degrees), which is the answer to this question.right + +# E Open Access + +All code, data, and models will be available on our project page to support reproducibility and benefit the research community. + +1. Codebase for our full 3D-aware data generation pipeline. +2. Codebase for our SFT and RL finetuning. +3. Synthesized 3D-aware training data. +4. Weights of our SpatialReasoner and SpatialReasoner-SFT. + +
Method3D Perception3D Reasoning
Orientation (↑)Location (↓)Angle (↑)Distance (↓)Depth (↓)
SpatialReasoner-SFT35.50.9155.00.170.13
SpatialReasoner31.01.0552.50.190.25
+ +Table 7: Studying failure modes of SpatialReasoner. We observe that 3D reasoning can estimate angles, distances, and depths a lot more accurately than 3D perception that predicts orientations and locations from visual features. + +![](images/802372ac73c49a7710d76777b85da783ac0d6d5a103e97227d58e723083f3c0b.jpg) +(a) Accuracy of SpatialReasoner-Zero $w/$ and $w/o$ KL divergence. + +![](images/7c6538369cef3e9f6c3ee4fde77ee250d398eb3deb43defd48b844663449ffba.jpg) +(b) Completion length of SpatialReasoner-Zero $w/$ and $w/o$ KL divergence. +Figure 9: Training curve between SpatialReasoner-Zero and SpatialReasoner-Zero w/ KL. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Claims made in the abstract and introduction are supported by our quantitative results in Section 4 and qualitative results in Figure 1. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. 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In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. 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For example +(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. +(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. +(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). +(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +# Answer: [Yes] + +Justification: All our code and data will be made public as detailed in Section E. + +# Guidelines: + +- The answer NA means that paper does not include experiments requiring code. +- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). +- The instructions should contain the exact command and environment needed to run to reproduce the results. 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Recent works propose understanding NeRFs with natural language using Multimodal Large Language Models (MLLMs) that directly process the weights of a NeRF's MLP. However, these approaches rely on a global representation of the input object, making them unsuitable for spatial reasoning and fine-grained understanding. In contrast, we propose weights2space, a self-supervised framework featuring a novel meta-encoder that can compute a sequence of spatial tokens directly from the weights of a NeRF. Leveraging this representation, we build Spatial LLaNA, a novel MLLM for NeRFs, capable of understanding details and spatial relationships in objects represented as NeRFs. We evaluate Spatial LLaNA on NeRF captioning and NeRF Q&A tasks, using both existing benchmarks and our novel Spatial ObjNeRF dataset consisting of 100 manually-curated language annotations for NeRFs. This dataset features 3D models and descriptions that challenge the spatial reasoning capability of MLLMs. Spatial LLaNA outperforms existing approaches across all tasks. + +# 1 Introduction + +Neural Radiance Fields (NeRFs) [45] can encode the 3D geometry and photorealistic appearance of an object compactly within the weights of a neural network, typically implemented as a Multi-Layer Perceptron (MLP). The ability of NeRFs to capture intricate visual details has led to widespread adoption in vision, graphics, and robotics [21, 59]. + +With NeRFs emerging as an effective and popular modality to represent 3D objects, what if one could effortlessly interact through words with an AI assistant capable of providing detailed information about a complex object stored in a computer as a NeRF? For instance, one may ask the assistant to describe specific parts of the object or to explain intricate spatial relationships precisely. Such intuitive, language-based interaction with NeRF representations has recently been investigated by LLaNA [6, 7]. LLaNA employs a meta-encoder [53] to extract a global embedding of an object + +![](images/d59f82ee8070eaccf8ce835e8498c4f6e861ef2c6aff67b942d86d842bdfbe64.jpg) +Figure 1: (left) LLaNA relies on a single global representation of the input NeRF. (right) S-LLaNA processes multiple spatially-aware tokens extracted by our new meta-encoder and, thus, can provide detailed descriptions that capture the relationships between objects. In particular, the answer of Spatial LLaNA contains more fine-grained information (e.g., it hints at the "wooden shelf with a red ceramic pot" and enumerates the "four wheels" and the "three wooden benches") while also reasoning spatially (e.g., it discriminates between objects "on one side" and objects "inside the carriage"). + +stored as a NeRF by directly processing the weights of the MLP. Compared to rendering views from the given NeRF so as to feed them to a Vision-Language Model, the strategy introduced by LLaNA is remarkably faster and avoids decisions that impact object understanding, such as the number of views to render and their resolution. The global representation computed by the meta-encoder is then projected into the embedding space of a Large Language Model (LLM) and prepended to a language instruction to realize NeRF-Language reasoning. However, we reckon that reliance on a single token to represent a whole NeRF limits the ability of LLaNA to capture fine-grained details and understand precise spatial relationships among objects, as illustrated in Figure 1, left. + +Multimodal Large Language Models (MLLMs) that operate on explicit spatial representations, such as images [4, 8, 15, 32, 33, 39, 63, 70] and point clouds [49, 50, 62], typically achieve fine-grained spatial reasoning by dividing inputs into localized regions – images into patches and point clouds into sets – in order to extract localized, spatially-aware tokens that capture geometric and visual context. However, the weights of a NeRF inherently lack such explicit spatial structure, as the spatial information is distributed across all of them, without any weight being directly associated with any specific object part. Consequently, traditional spatially-aware tokenization approaches are not directly applicable to NeRF representations. + +This observation raises an intriguing research question: Is it possible to elicit spatially-aware representations from the information distributed across the weights of a NeRF? + +To answer this question, we propose a novel self-supervised framework, weights2space, trained to reorganize the information encoded within the weights of a NeRF into spatially-aware tokens. Specifically, weights2space employs a novel Transformer-based meta-encoder that ingests the MLP weights and outputs a sequence of tokens that are reorganized into a tri-plane [13] structure during training, so as to associate each token with a specific, localized spatial region. The meta-encoder is trained alongside a decoder to render images from the tri-plane representations. This approach allows the meta-encoder to extract a localized, spatially-grounded representation of the input NeRF. By leveraging this novel representation, we introduce Spatial LLaNA (S-LLaNA), a novel Multimodal Large Language Model for NeRFs, capable of reasoning about fine-grained object details and spatial relationships, as shown in Figure 1, right. + +To validate our approach, we thoroughly evaluate Spatial LLaNA on the existing NeRF-Language datasets: ShapeNeRF-Text [53], HST [5], and ObjNeRF-Text [7]. Moreover, we propose a novel benchmark, Spatial ObjNeRF, consisting of 100 NeRF models from Objaverse [18] paired with detailed language descriptions focusing on object parts and their spatial relations. Experimental results show a substantial improvement in NeRF-Language reasoning with our approach. + +In summary, we make the following contributions: + +- weights2space: a self-supervised framework featuring a novel Transformer-based meta-encoder capable of extracting spatially-aware token representations directly from the weights of a NeRF. +- Spatial LLaNA: a novel MLLM for NeRFs that leverages the localized representations from the weights2space meta-encoder to achieve detailed spatial reasoning. +- Spatial ObjaNeRF: a new manually annotated dataset of 100 NeRFs explicitly designed to evaluate detailed spatial reasoning tasks involving NeRFs. +- Spatial LLaNA achieves state-of-the-art performance on the new spatial reasoning tasks of our dataset as well as on existing benchmarks for NeRF captioning and NeRF Q&A, demonstrating the effectiveness of our approach. + +# 2 Related work + +Deep learning on neural networks. Processing the weights of a neural network poses distinct challenges compared to standard input formats, primarily due to the high dimensionality of the weight space, inherent symmetries [23], and the influence of randomness on the convergence point of training [3, 20]. Early approaches to meta-networks use group theory to design architectures equivariant to permutation symmetries in network weights [46, 69], but were limited to specific architectures like MLPs or CNNs. To generalize across architectures, Graph Meta-Networks (GMNs) were introduced [31, 37], reframing the problem as one of converting neural networks into graphs, which GMNs can process due to their inherent permutation equivariance. nf2vec [53] is the first method to perform tasks on NeRFs by directly processing its weights. This model computes a global embedding from the weights of the NeRF's MLP, leveraging an encoder-decoder architecture supervised with a rendering loss. LLaNA [6] uses the feature vector computed by this encoder to perform NeRF-Language tasks, such as NeRF captioning and NeRF Q&A. While nf2vec is designed to ingest MLPs, [11] proposes a method to process a NeRF architecture consisting of a tri-plane structure, performing the tasks of classification and segmentation on NeRF data. All these methods compute a single feature vector from the weights of the input NeRF, encapsulating global information about the object. In contrast, the weights2space meta-encoder proposed in this work extracts spatially-aware tokens from the weights of the input NeRF, providing a richer set of information. + +Multimodal Large Language Models. Multimodal Large Language Models (MLLMs) expand the capabilities of existing LLMs [1, 29, 52, 56, 68] by combining language with other modalities like images, audio, or 3D data. Vision-language models [4, 8, 15, 32, 33, 39, 63, 70] encode images using pre-trained Vision Transformers (ViT) [51] that convert image patches into sequences of visual tokens. These local representations capture fine-grained spatial information across the image. 3D MLLMs [25, 49, 50, 62] divide the input point cloud into local patches. Transformer-based encoders like Point-BERT [64] and Recon++ [49] compute discrete token sequences that preserve spatial relationships. MLLMs for audio [19, 26, 65] split acoustic inputs into temporal patches and encode them using audio-specific encoders [14], producing sequences of tokens that retain temporal structure. Similarly, MLLMs for video-language understanding [34, 41, 43, 44, 66] use visual encoders to compute local features on each video frame, preserving both spatial and temporal information. In contrast to these local representation approaches, the only existing MLLM for Neural Radiance Fields (NeRFs), LLaNA [6], computes a single global embedding from the weights of a NeRF's MLP. Although this approach enables basic language tasks on NeRFs, it critically overlooks spatial information since it does not adopt the local token-based representation strategy successfully used in other modalities. In this work, we propose S-LLaNA, a novel MLLM framework for NeRF which relies on our new weights2space meta-encoder to compute a set of spatial tokens from the weights of the input NeRF's MLP, preserving crucial spatial information while also allowing to extract richer and more detailed information from the input NeRF. + +Language and Neural Radiance Fields. Given a textual description of an object, generative models produce realistic objects represented as NeRFs [28, 48, 60, 67]. Similarly, other approaches focus on editing an input scene represented as a NeRF, through text control [17, 55, 57, 58]. The interaction between NeRF and language is also leveraged for 3D semantic understanding. On this task, LeRF [30] predicts language features for each spatial location alongside density and color. While the primary focus of LeRF is on object localization, 3D-OVS [40] distills the reasoning capability of CLIP [51] + +![](images/e89608a685eb2d36abe482ff12bba3e463597388c9cee3cbb1bb0beb3993328f.jpg) +Figure 2: weights2space overview. + +![](images/9809149b145b5e3ce0902dd01cfb3a34619b0ab96ee244a9519c9288ed8c96c2.jpg) +Figure 3: Spatial LLaNA overview. + +and DINO [12] into a NeRF, to perform 3D segmentation. However, these methods train a semantic neural field, relying on the noisy and view-inconsistent semantics from CLIP. In contrast to this methods, OV-NeRF [36] addresses the task of open-vocabulary 3D segmentation by leveraging a regularization technique to improve the single-view semantics and a self-enhancement strategy for cross-view semantics. The neural field adopted by these methods is parametrized by a neural network. The first method to perform captioning and Q&A tasks on NeRF is LLaNA [6] while [7] expands the capabilities of LLaNA to a larger dataset, ObjaNeRF-Text, by scaling its underlying LLM. + +# 3 weights2space: spatially-aware NeRF weights tokenization + +Our goal is to compute a spatially-aware representation of an object directly from the weights of its NeRF model. The only existing MLLM for NeRF, LLaNA [6], utilizes a global representation of the input NeRF, in the form of a single embedding. In contrast, state-of-the-art MLLMs typically operate on spatially-aware, token-based representations computed by Transformer encoders [49, 51, 64]. These architectures allow the model to capture localized features crucial for detailed understanding and downstream language tasks. + +Inspired by these approaches, we introduce a novel self-supervised framework, weights2space, that learns how to transform the weights of a NeRF into a sequence of spatially-aware embeddings, as shown in Figure 2 - more implementation details are in the Appendix. + +NeRF encoding. Similarly to LLaNA [6], we assume NeRFs to be MLPs consisting of $L$ hidden layers with $H$ neurons, an input layer, and an output layer. The input layer has a weight matrix $\mathbf{W}_i\in \mathbb{R}^{I\times H}$ and biases $\mathbf{b}_i\in \mathbb{R}^H$ , where $I$ depends on the frequency encoding [45] used to encode the input coordinates. Hidden layers have the same number of input and output neurons, $H$ , thus their squared weight matrix has dimension $\mathbf{W}_l\in \mathbb{R}^{H\times H}$ and biases $\mathbf{b}_l\in R^H$ . The output layer, which predicts the RGB colour and density $\sigma$ at a given 3D coordinate, has a weight matrix of $\mathbf{W}_o\in \mathbb{R}^{H\times 4}$ and biases $\mathbf{b}_{out}\in \mathbb{R}^4$ . The parameters of this MLP are packed as in nf2vec [53] to provide the input to the meta-encoder. In particular, the weight matrices and biases of the input MLP are stacked along the row dimension to form a matrix $\mathbf{W}\in \mathbb{R}^{S\times H}$ , where $S = I + L*(H + 1) + H + 2$ . The weights and biases of the output layer, $\mathbf{W}_o$ and $\mathbf{b}_o$ are padded with zeros to obtain $H$ columns. + +Meta-encoder. It processes the parameters of the input MLP packed into matrix $\mathbf{W}$ through two components: a weights2seq module and a seq2space module. + +The weights2seq module, consisting of a series of linear layers - 4 in our experiments -, batch normalizations [27] and ReLU activations [2], processes each row of $\mathbf{W}$ independently for efficient computation. The module yields a set of tokens $\mathbf{V} \in \mathbb{R}^{S \times G}$ , each representing a row of parameters in the input MLP, with $S = 336$ and $G = 1024$ in our setup. We note that this per-row processing module is the same as nf2vec [53], yet with one key modification: our architecture does not perform a final max-pooling to aggregate all row-level features into a single global representation. We conjecture that the pooled vector, which is the sole input utilized by LLaNA [6] to represent a NeRF, + +contains compact and coarse information on the underlying object. Instead, by skipping this pooling step, we aim at retaining more comprehensive and fine-grained information. + +In standard MLLM pipelines for visual or 3D modalities, the encoder [51, 64] computes a sequence of spatially localized tokens, each representing a specific image patch or set of 3D points. However, the tokens computed by the weights2seq module have no direct correspondence to any specific spatial location or object part. + +To elicit spatial information from the weights of the NeRF, we deploy the seq2space module, a Transformer decoder with 12 decoder layers, where the keys and values of the cross-attention layers are obtained from the sequence of tokens $\mathbf{V}$ . Input to seq2space is a sequence of learnable queries $\mathbf{X} = (\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N)\in \mathbb{R}^{N\times C}$ , where $N$ is the number of queries and $C$ is the feature dimension. The Transformer decoder, hence, outputs a new sequence of tokens $\mathbf{Y} = (\mathbf{y}_1,\mathbf{y}_2,\dots,\mathbf{y}_N)\in \mathbb{R}^{N\times C}$ . In our experiments, we employ $N = 3072$ and $C = 516$ . + +Tri-plane representation and decoder. To force each token of $\mathbf{Y}$ to contain localized information, we reshape the $N$ tokens of $\mathbf{Y}$ into a tri-plane feature map [13], defined as $M = (F_{xy}, F_{zx}, F_{yz})$ . Each plane $F \in \mathbb{R}^{H \times W \times C}$ , where $H, W$ are the spatial dimensions, is obtained by arranging a subset of the tokens into a 2D grid. In particular, we reshape the 3072 tokens into three $32 \times 32$ spatial planes. In this way, each token of $\mathbf{Y}$ corresponds to a specific location in one of the three orthogonal planes. This design introduces spatial structure and locality into the otherwise unstructured token sequence originally obtained by the weights2seq module. + +Given a 3D point $\mathbf{p} \in \mathbb{R}^3$ , we can obtain its 3D feature by projecting it onto the three orthogonal planes to its corresponding 2D coordinates: $\mathbf{p}_{\mathbf{xy}}, \mathbf{p}_{\mathbf{xz}}, \mathbf{p}_{\mathbf{yz}}$ . Then, for each projection, bilinear interpolation is performed on the four nearest neighbors' features within the corresponding plane to extract three vectors of dimension $C$ : $\mathbf{f}_{\mathbf{xy}}, \mathbf{f}_{\mathbf{xz}}, \mathbf{f}_{\mathbf{yz}}$ . These are summed element-wise to obtain a unified point-wise feature vector $\mathbf{f} \in \mathbb{R}^C$ . The detailed explanation of this step is reported in the Appendix. + +This feature vector is then concatenated with a positional encoding [45] of $\mathbf{p}$ , $\mathbf{PE}(\mathbf{p})$ , and fed into an MLP decoder $\Phi(\cdot)$ that predicts the radiance field value $\hat{\mathbf{q}} = \Phi(\mathbf{p}, \theta)$ , consisting of color and density components $(R, G, B, \sigma)$ . $\Phi$ consists of 4 linear layers as the decoder used in [53]. + +Training. All modules of weights2space are optimized end-to-end using differentiable volumetric rendering [35]. At each training step, a set of camera poses is sampled. For each pose, multiple rays are cast – together forming a training batch $\mathcal{R}$ – and 3D points are sampled along these rays. Each point is processed through the tri-plane module and the decoder as described above. The predicted radiance and density values are then aggregated along each ray using volumetric rendering to produce final RGB values. The framework is supervised using a smooth L1 loss between rendered and ground truth colors: + +$$ +\mathcal {L} _ {\text {r e n d e r}} = \sum_ {\mathbf {r} \in \mathcal {R}} \operatorname {S m o o t h L 1} \left(\hat {C} (\mathbf {r}) - C ^ {*} (\mathbf {r})\right) \tag {1} +$$ + +where $C^*(\mathbf{r})$ is the ground truth RGB color for ray $\mathbf{r}$ and $\hat{C}(\mathbf{r})$ is the predicted color by the decoder. Since NeRF can be a standalone modality with ground truth images unavailable, we adopt a self-supervised approach. Specifically, we render images directly from the input NeRFs using volumetric rendering. These pseudo ground-truth images serve as reference images, making the entire training protocol fully self-supervised in the standalone NeRF scenario. The model is trained on 300K NeRFs from ShapeNeRF-Text and ObjNeRF-Text for 500 epochs with a learning rate of 0.00001, as done in LLaNA [7]. The training needs $\sim 2$ days on four 64GB A100 GPUs to reach convergence. + +# 4 Spatial LLaNA: spatially-aware MLLM for NeRF + +Architecture. S-LLaNA is the MLLM that performs NeRF-Language tasks by taking as input the NeRF tokens $\mathbf{Y} \in \mathbb{R}^{N \times C}$ computed by the weights2space meta-encoder. Inspired by [6], we process these tokens with a shared projector network $\Psi$ , composed of a stack of 2 trainable linear layers, interleaved with $GELU$ activation functions [24]. This network maps each NeRF token into the internal embedding space of the language model, LLaMA-2 [56]. The input to the LLM consists of the projected NeRF tokens, encapsulated between two learnable special tokens $\langle t\_start\rangle$ and $\langle t\_end\rangle$ , and concatenated with a sequence of $k$ language tokens corresponding to the user's prompt. + +The language model LLaMA processes the combined sequence to autoregressively generate relevant textual output. + +Training. We adopt the training protocol introduced in [7], using the ShapeNeRF-Text and ObjNeRF-Text datasets. In the first stage, $S$ -LLaNA is trained using brief textual descriptions, optimizing only the projection layers $\Psi$ and the trainable special tokens $$ and $$ . In the second stage, the full model, including the LLM, is fine-tuned using both brief and detailed descriptions, as well as Q&A annotations. Importantly, the encoder responsible for computing the NeRF tokens $\mathbf{Y}$ is kept frozen during both training stages. The total training set consists of approximately 300K NeRFs with textual annotations. We supervise $S$ -LLaNA by minimizing the negative log-likelihood of the text token at each position: + +$$ +\mathcal {L} _ {\mathrm {C E}} = - \sum_ {t = 1} ^ {T} \log P \left(w _ {t} \mid \hat {w} _ {0 \dots}, \hat {w} _ {t - 2}, \hat {w} _ {t - 1}, \Psi (\mathbf {Y})\right) \tag {2} +$$ + +where $P(w_{t} \mid \hat{w}_{0}, \dots, \hat{w}_{t-2}, \hat{w}_{t-1}, \Psi(\mathbf{Y}))$ is the probability assigned by the LLM to the correct token $w_{t}$ at step $t$ , conditioned on the previously predicted tokens and the projected NeRF token sequence $\Psi(\mathbf{Y})$ . + +Implementation details. S-LLaNA shares the same LLM backbone as LLaNA [7], i.e. the 7B Vicuna and 13B Vicuna [16] of LLaMA-2 [56]. For experimental fairness, S-LLaNA follows the same training protocol as [7] and is trained on 300K samples from ShapeNeRF-Text and ObjNeRF-Text for 3 epochs for each training stage. The projection network $\Psi$ has 2 trainable linear layers which bring each token from dimension $C = 516$ to the hidden dimension of the LLM, i.e., 4096 for LLaMA-7B and 5192 for LLaMa-13B. S-LLaNA is implemented in PyTorch and trained on NVIDIA A100 GPUs with 64GB of VRAM each. The 7B model requires 8 GPUs for training, while the 13B model requires 16 GPUs. Training either version of the model takes approximately one day. + +Ablation experiments on the tri-plane resolution and tokens dimensionality can be found in the Appendix. + +# 5 Experimental setup + +Baselines. The only direct competitor of S-LLaNA is LLaNA [7], as this is the only MLLM performing NeRF-Language tasks without rendering any 2D image or extracting an explicit 3D structure from input NeRFs. We evaluate also the same reference baselines as LLaNA [7], i.e., state-of-the-art MLLMs ingesting images or 3D explicit data like point clouds and meshes, with these models processing explicit data rendered or extracted from the input NeRF. As for 3D MLLMs, included baselines are GPT4Point [50], PointLLM [62], 3D-LLM [25] and ShapeLLM [49]. As for 2D MLLMs, we consider LLaVA-1.6 [39] and BLIP-2 [33]. Since multiple views can be rendered from an input NeRF, these 2D baselines are evaluated using different viewpoints, i.e. front-view (FV), back-view (BV), random-view (RV), and in a multi-view (MV) scenario where we concatenate image tokens from $N = 3$ images rendered from random viewpoints. For each method, we report standard metrics employed in MLLM literature [7, 50, 62]: Sentence-BERT [54] and SimCSE [22] based on pre-trained encoders; BLEU-1 [47], ROUGE-L [38], and METEOR [9] based on n-gram statistics. + +NeRF-Language datasets. As in [7], we evaluate the baselines on the test sets of the existing NeRF-Language datasets ShapeNeRF-Text and ObjNeRF-Text. Two different test sets for ObjNeRF-Text are considered to ensure fair comparison with PointLLM and GPT4Point models. Results on these datasets are divided into three tasks: brief captioning, detailed captioning, and single-round Q&A conversations. In addition, within the brief captioning task, we evaluate the performance the methods on HST [5], a subset of ShapeNeRF-Text objects with manually annotated short captions. + +Spatial ObjNeRF. To rigorously evaluate the ability of NeRF-Language models to move beyond simple object recognition and capture complex spatial relationships, we introduce Spatial ObjNeRF, a manually curated benchmark. This dataset consists of 100 human-annotated 3D models from ObjNeRF-Text. Crucially, to fully assess the spatial reasoning capability of Multi-modal Large Language Models (MLLMs) within a realistic context, every data sample represents a complex scene featuring an arrangement of multiple interacting objects. For each scene, we created a detailed textual + +Table 1: NeRF brief captioning on ShapeNeRF-Text and HST datasets. Best results are in bold, runner-up is underlined. (FV: front-view, BV: back-view, MV: multi-view) + +
ShapeNeRF-TextHST
ModelModalityS-BERTSimCSEBLEU-1ROUGE-LMETEORS-BERTSimCSEBLEU-1ROUGE-LMETEOR
LLaVA-vicuna-7bImage (FV)59.8562.3522.6723.2423.3554.3156.2810.0814.7114.53
LLaVA-vicuna-7bImage (BV)55.6858.4621.9722.4622.5051.7552.298.1313.9614.18
LLaVA-vicuna-7bImage (MV)59.7761.4223.1622.6823.0154.1556.8710.3212.7615.13
BLIP-2 FlanT5-xxlImage (FV)56.1358.215.4618.699.6757.1159.438.2118.0212.14
BLIP-2 FlanT5-xxlImage (BV)52.4854.055.6718.209.5054.1156.379.0917.3811.79
LLaVA-vicuna-13bImage (FV)61.0061.1614.3020.0023.3155.6255.566.5611.8114.52
LLaVA-vicuna-13bImage (BV)54.3556.0921.9421.6722.0950.0050.799.3912.7614.46
LLaVA-vicuna-13bImage (MV)59.6461.0122.8422.1723.0854.2555.569.7814.1314.99
3D-LLM FlanT5-x1Mesh + MV59.4656.4212.6921.4914.3256.0752.1315.9420.7115.22
GPT4Point-Opt-2.7bPoint cloud41.8540.2211.7616.5411.6343.1542.2212.0218.7313.69
PointLLM-7bPoint cloud49.5948.8416.7417.9214.5643.4044.508.5311.649.97
ShapeLLM-7bPoint cloud39.1638.3420.7121.3517.5032.7235.669.3911.339.76
PointLLM-13bPoint cloud51.5450.3517.2018.5114.9245.4146.399.5712.3811.92
ShapeLLM-13bPoint cloud42.2940.1821.5921.8717.9625.0938.2410.0313.269.94
LLaNA-7bNeRF74.9476.4142.7343.6441.9564.7865.0120.6523.2424.10
S-LLaNA-7bNeRF78.9179.9147.8149.0847.5867.7668.6320.3123.6124.47
LLaNA-13bNeRF75.0976.4542.8343.7042.2165.6666.0920.8024.2825.90
S-LLaNA-13bNeRF78.9879.9847.9749.2247.8767.4368.2620.9123.8625.03
+ +Table 2: NeRF brief captioning on ObjNeRF-Text. Best results are in bold, runner-up is underlined. (RV: random view, MV: multi-view) + +
ObjaNeRF-Text (PointLLM test set)ObjaNeRF-Text (GPT4Point test set)
ModelModalityS-BERTSimCSEBLEU-1ROUGE-LMETEORS-BERTSimCSEBLEU-1ROUGE-LMETEOR
LLaVA-vicuna-7bImage (RV)39.2940.935.228.3512.4941.5942.387.2411.7416.01
LLaVA-vicuna-7bImage (MV)40.1541.095.629.0713.2242.3443.218.5911.9418.52
BLIP-2 FlanT5-xxlImage (RV)35.4835.197.6913.7511.6837.2437.3210.6316.8714.69
LLaVA-vicuna-13bImage (RV)38.5739.054.627.6311.8542.0841.046.5310.5715.67
LLaVA-vicuna-13bImage (MV)41.0141.104.878.0312.3544.1543.196.7510.8616.10
3D-LLM FlanT5-x1Mesh + MV38.2239.605.477.2910.7541.2637.7411.0812.2615.58
GPT4Point-Opt-2.7bPoint cloud-----34.4231.898.7613.6213.55
PointLLM-7bPoint cloud38.8140.115.558.3911.24-----
ShapeLLM-7bPoint cloud30.1332.425.807.7211.3825.9225.578.049.6013.23
PointLLM-13bPoint cloud39.6440.635.948.4311.63-----
ShapeLLM-13bPoint cloud31.4432.696.017.5911.6626.8727.038.319.6514.10
LLaNA-7bNeRF41.3642.2813.2116.9315.6343.7343.0920.2225.1522.21
S-LLaNA-7bNeRF45.2345.8015.7219.9817.7050.1249.5524.4930.7226.15
LLaNA-13bNeRF42.0842.4013.8617.5116.1844.2643.7520.6125.6222.36
S-LLaNA-13bNeRF45.4446.1815.7820.1717.6150.6850.3024.5330.5226.41
+ +description that emphasizes spatial structure, highlighting the size, shape, and relative positioning of specific objects. + +Examples of such annotations and their comparison with those present in ObjNeRF-Text are provided in the Appendix. + +# 6 Experiments + +Results on ShapeNeRF-Text, HST, and ObjNeRF-Text. We evaluate S-LLaNA on the tasks of NeRF brief captioning, detailed captioning, and Q&A, using the datasets ShapeNeRF-Text, HST, and ObjNeRF-Text. Results are presented in Tables 1 to 4. Across all benchmarks and tasks, S-LLaNA consistently outperforms LLaNA and all other 2D and 3D MLLM baselines, often by substantial margins. The 13B variant of S-LLaNA achieves the highest performance in nearly every setting, while the 7B variant is typically the second-best model, outperforming all the other baselines. These results demonstrate the effectiveness of the NeRF spatially-aware tokenized representation learned by our weights2space meta-encoder in existing public NeRF-Language datasets. On the brief captioning task, reported in Tables 1 and 2, S-LLaNA outperforms LLaNA by approximately $\sim 4$ points on Sentence-BERT and SimCSE metrics, consistently across all datasets and model sizes. This indicates that even in tasks requiring high-level object descriptions, the spatially-aware NeRF tokens produced by the weights2space meta-encoder provide more discriminative and semantically rich input than the global representation of LLaNA. Table 4 shows that, on the Q&A task, S-LLaNA outperforms LLaNA with performance improvements comparable to those observed in brief captioning. This suggests that the ability to reason over object structure and part-level features, supported by our spatial encoding, is also beneficial to this task. We observe larger gains on the detailed captioning task, in Table 3, where S-LLaNA surpasses LLaNA by up to 6 points on Sentence-BERT and SimCSE and over 10 points on reference-based metrics. These results confirm that the rich, fine-grained geometric + +Table 3: NeRF detailed captioning on ShapeNeRF-Text. Best results are in bold, runner-up is underlined. (FV: front-view, BV: back-view, MV: multi-view) + +
ModelModalityS-BERTSimCSEBLEU-1ROUGE-LMETEOR
LLaVA-vicuna-7bImage (FV)57.5557.6814.9922.8214.36
LLaVA-vicuna-7bImage (BV)53.1154.4614.7322.4714.05
LLaVA-vicuna-7bImage (MV)55.2658.4615.0724.0514.85
BLIP-2 FlanT5-xxlImage (FV)41.2740.690.187.832.60
BLIP-2 FlanT5-xxlImage (BV)38.4937.890.197.722.58
LLaVA-vicuna-13bImage (FV)59.0858.8723.6323.5522.55
LLaVA-vicuna-13bImage (BV)60.0950.3313.7721.3613.18
LLaVA-vicuna-13bImage (MV)61.2159.5113.6732.4614.44
3D-LLM FlanT5-xlNeRF+MV58.0053.911.5814.405.28
GPT4Point-Opt-2.7bPoint cloud42.4438.333.729.215.13
PointLLM-7bPoint cloud59.0258.3010.2819.2610.55
ShapeLLM-7bPoint cloud43.4540.928.1818.4710.15
PointLLM-13bPoint cloud59.6458.5510.5219.4410.83
ShapeLLM-13bPoint cloud43.2640.9810.4718.4210.20
LLaNA-7bNeRF75.2577.4219.5732.9620.45
3-LLNa-7bNeRF81.0383.0044.7440.0836.03
LLaNA-13bNeRF75.5177.6319.8732.9320.46
3-LLNa-13bNeRF80.9882.9044.7840.0936.24
+ +Table 4: NeRF single-round Q&A on ShapeNeRF-Text. Best results are in bold, runner-up is underlined. (FV: front-view, BV: back-view, MV: multi-view) + +
ModelModalityS-BERTSimCSEBLEU-1ROUGE-LMETEOR
LLAVA-vicuna-7bImage (FV)71.7971.9625.7934.0434.86
LLAVA-vicuna-7bImage (BV)70.8870.9325.1733.3034.22
LLAVA-vicuna-7bImage (MV)72.2670.6724.0934.6735.64
BLIP-2 FlanTS-xxxlImage (FV)45.2047.9211.5020.1613.49
BLIP-2 FlanTS-xxxlImage (BV)45.0647.6611.5019.9813.44
LLAVA-vicuna-13bImage (FV)71.6170.9820.1930.4232.53
LLAVA-vicuna-13bImage (BV)68.2569.0620.0329.8432.27
LLAVA-vicuna-13bImage (MV)71.8471.1620.0430.2033.46
3D-LLM FlanTS-xiMesh + MV69.6267.5532.1940.9535.83
GPT4point-Opt-2.7bPoint cloud27.6231.416.269.385.41
PointLLM-7bPoint cloud74.7074.4036.8144.4139.76
ShapeLLM-7bPoint cloud46.6046.3219.4717.1312.83
PointLLM-13bPoint cloud74.6574.1837.1644.8640.13
ShapeLLM-13bPoint cloud47.2246.5920.0317.4513.22
LLANa-7bNeRF81.0381.6145.9853.2749.97
S-LLaNA-7bNeRF83.4984.0650.8958.3954.52
LLANa-13bNeRF81.0581.6046.0853.4449.98
S-LLaNA-13bNeRF83.6084.1750.8658.2554.58
+ +Table 5: Detailed spatial captioning on Spatial ObjNeRF. Best results are in bold, runner-up is underlined. (RV: random view, MV: multi-view) + +
ModelModalityS-BERTSimCSEBLEU-IROUGE-LMETEOR
LLaVA-VCuica-7bImage (RV)70.1173.2521.9524.2620.09
LLaVA-VCuica-7bImage (MV)72.1374.9822.4426.8721.33
BLIP-2 FlanT5-xxlImage (RV)54.1260.238.4710.236.59
LLaVA-VCuica-13bImage (RV)70.5575.8221.3129.4519.74
LLaVA-VCuica-13bImage (MV)73.4277.0624.2030.8520.47
3D-LLM FlanT5-xlMesh + MV53.2454.6912.0514.5716.88
GPT4Point-Opt-2.7bPoint cloud40.0840.724.6511.715.88
PointLLM-7bPoint cloud72.6773.9325.9024.7519.05
ShapeLLM-7bPoint cloud62.4864.649.2621.5517.69
PointLLM-13bPoint cloud73.0875.0726.3925.0219.18
ShapeLLM-13bPoint cloud64.2267.989.5521.0716.84
LLaNA-7bNeRF70.6171.7725.6125.6619.21
5-LLaNA-7bNeRF75.0177.3029.4328.8521.70
LLaNA-13bNeRF73.2074.7626.7829.1520.37
5-LLaNA-13bNeRF78.2580.5832.3431.0324.56
+ +![](images/80b047398596af31a6b160ed2059490fccb675e9045890fc9dbd693c0205dc29.jpg) +Figure 4: Qualitative examples on Spatial ObjNeRF + +information captured by the weights2space meta-encoder helps guide the language model toward more complete and structured descriptions. All the MLLMs operating on explicit data structures, i.e., Vision Language Models and 3D MLLMs, show significantly worse performance than LLaNA and S-LLaNA, highlighting the effectiveness of building NeRF-Language models operating directly on NERF weights. Additional qualitative examples are in the Appendix. + +Results on Spatial ObjNeRF. Unlike previous captioning tasks, Spatial ObjNeRF targets explicitly the understanding of relative positions, part-whole hierarchies, and size comparisons: skills that require a structured and localized representation of geometry. This benchmark, described in Section 5, is proposed as a key tool to assess whether NeRF-Language models can achieve fine-grained 3D understanding. Quantitative results on Spatial ObjNeRF are reported in Table 5. S-LLaNA-13b achieves the best performance across all metrics, followed by S-LLaNA-7b. The performance gap between S-LLaNA and LLaNA highlights the importance of explicitly modeling spatial features within the representation provided as input to the LLM. While LLaNA relies on a global embedding computed by processing the weights of the NeRF's MLP with linear layers, S-LLaNA benefits from spatially-aware tokens extracted by the weights2space meta-encoder. Qualitative examples in Figure 4 illustrate this advantage. In the first example, only S-LLaNA accurately recognizes the objects on the table and correctly describes their spatial arrangement. Indeed, it notes that the book is "next to the bowl". LLaNA mistakenly reports the presence of "two bowls" and "two books" and describes their relative position by stating that the books are "under the bowl of fruit". In the second example, S-LLaNA demonstrates precise part-level reasoning, describing the hat of the character as a "regal crown", while LLaNA incorrectly describes it as a "sombrero". These examples further demonstrate the superior ability of S-LLaNA to interpret complex scenes with fine-grained details, particularly when multiple objects interact spatially. More qualitative examples are in the Appendix, together with experiments on the generalization capabilities of S-LLaNA. + +Ablation on LLM finetuning. As described in Section 5, both LLaNA and S-LLaNA are trained in two stages: first, the projection network $\Psi$ is trained independently; then, it is jointly fine + +Table 6: Experiments on LLM finetuning. Best results are in bold, runner-up is underlined. (SN-T: ShapeNeRF-Text, ON-T: ObjNeRF-Text PointLLM test set) + +
SN-T brief captioningON-T brief captioningSN-T detailed captioningSN-T Q&ASpatial test
ModelLLM Finetuned# Trainable paramsS-BERTSimCSES-BERTSimCSES-BERTSimCSES-BERTSimCSES-BERTSimCSE
LLaNA-7b✓6.75B74.9476.4141.3642.2875.2577.4281.0381.6170.6171.77
LLaNA-7b143M67.7568.8833.2633.7371.2571.7471.2571.7462.4861.44
S-LLaNA-7b✓6.75B78.9179.9145.2345.8081.0383.0083.4984.0673.2074.76
S-LLaNA-7b140M79.2980.3645.1745.8181.0382.9082.4383.0173.2770.99
LLaNA-13b✓13.03B75.0976.4542.0842.4075.5177.6381.0581.6075.0177.30
LLaNA-13b178M70.4868.8535.2136.6872.6670.4172.2472.6262.3461.50
S-LLaNA-13b✓13.03B78.9879.9845.4446.1880.9882.9083.6084.1778.2580.58
S-LLaNA-13b175M79.0580.1744.8745.4180.6382.5082.1082.6777.7280.47
+ +![](images/22c1faa5bd7b86bfe89236f7844b3f077ebc10997741d4e3bf4af5af252d0f89.jpg) +Figure 5: Visualization of the self-attention maps of the LLM (LLaMA 2) + +tuned with the LLM. This protocol, common among state-of-the-art MLLMs [39, 62], leads to high computational costs, particularly prohibitive when using large LLMs with tens of billions of parameters. This section investigates whether comparable performance can be achieved without fine-tuning the LLM, by optimizing only the projection network $\Psi$ on the full dataset. Results are shown in Table 6, where rows marked "LLM Finetuned" correspond to the two-stage protocol, while the remaining rows represent models trained solely with the first stage on the full dataset. For the brief captioning task on ObjNeRF-Text, we evaluate the models on the PointLLM test set, which is larger than the GPT4Point test set. Interestingly, S-LLaNA maintains excellent performance even without finetuning LLaMA. In fact, the non-finetuned S-LLaNA-7b and S-LLaNA-13b closely match their finetuned counterparts with performance gaps consistently under 1 point across all metrics. In some cases, the non-finetuned variants even outperform the fine-tuned ones. For instance, on the brief captioning task of ShapeNeRF-Text, the Sentence-BERT score improves from 78.91 to 79.29 for S-LLaNA-7B, and from 78.98 to 79.05 for S-LLaNA-13B. In contrast, LLaNA struggles without LLM fine-tuning. Its non-finetuned versions exhibit substantial drops across all benchmarks, with gaps exceeding 10 points on challenging tasks like Spatial ObjNeRF. This trend persists even for the larger 13B model. We hypothesize that the failure of LLaNA in the non-finetuned regime stems from the global NeRF representation used as input to the LLM. This representation cannot be directly mapped to a specific language token, making fine-tuning of the LLM — and thus adaptation of its input token space — a crucial step. In contrast, to represent objects, S-LLaNA uses $N = 3072$ spatial tokens that encode more local and lower-level concepts. Thus, it is more likely that the projector can learn a mapping from these tokens to the LLM's input space without requiring LLM fine-tuning. This property of S-LLaNA allows for significantly faster and more cost-effective training, while still achieving comparable performance. In the Appendix, a discussion on the general language understanding of our model is proposed. + +Visualization of attention maps in LLaMA. In Figure 5, we visualize the self-attention maps within the LLM of S-LLaNA-13b when prompted with questions designed to force the model to focus on specific object parts of a NeRF from the ObjNeRF test set. Specifically, we extract the self-attention maps computed on the input sequence fed into LLaMa-13b, which consists of $N = 3072$ NeRF tokens, concatenated with the text tokens of the question, for a total of $B$ tokens. For visualization, we focus on the final decoder layer of LLaMA-13b and average the attention scores across all 40 attention heads, obtaining a matrix $A \in \mathbb{R}^{B \times B}$ . We visualize the part of a column in $A$ corresponding to the attention between a word in the question and the 3072 spatial tokens. To better visualize these scores, we reshape them into three images corresponding to each tri-plane of dimension $H \times W$ , where $H = W = 32$ . Several patterns emerge from these attention maps. First, in each plane of the tri-plane, the outline of the object is distinctly visible. This indicates that the supervision of the weights2space meta-encoder successfully injects spatial object information into the NeRF token representation. Moreover, attention is clearly localized: tokens with the highest attention scores to a given word correspond to spatial regions where the referenced element is located. For instance, in the right example, the word "flower" activates tokens in the upper region of the + +object, and "vase" focuses attention on the lower portion. These visualizations highlight the ability of S-LLaNA to ground linguistic concepts into spatially localized NeRF representations. + +# 7 Conclusions + +In this work, we demonstrated that extracting spatially-aware tokens directly from the weights of a NeRF's MLP is possible, enabling fine-grained language interaction on NeRF data. To achieve this goal, we introduced weights2space, a self-supervised framework featuring a novel meta-encoder that computes a sequence of spatial tokens, and S-LLaNA, a novel Multimodal Large Language Model for NeRF. Our model significantly outperforms existing methods on NeRF-Language tasks, whether they operate directly on NeRFs or use explicit representations. Notably, S-LLaNA achieves strong performance even without finetuning its underlying LLM, making it extremely efficient to train. Moreover, we propose Spatial ObjaNeRF, a benchmark designed to evaluate fine-grained spatial understanding in NeRF-Language tasks. Finally, we show qualitatively that our approach leads to interpretable and meaningful attention patterns. This work contributes to the nascent field of Multimodal Large Language Models for Radiance Fields. + +Limitations and Future work. Our spatially-aware representation may have the potential to address additional tasks, such as spatial Q&A on 3D scenes, 3D grounding, and 3D object detection. At present, the scarcity of large-scale NeRF-text datasets limits progress in these directions. Expanding this representation to such tasks remains a promising direction for future research. + +# Acknowledgements + +We acknowledge ISCRA for awarding this project access to the LEONARDO supercomputer, owned by the EuroHPC Joint Undertaking, hosted by CINECA (Italy). + +This work was partially funded by "FSE+ 2021-2027 ai sensi dell'art. 24, comma 3,lett. a), della Legge 240/2010 e s.m.i. e del D.G.R. 693/2023 (RIF. PA: 2023-20090/ER - CUP: J19J23000730002)". + +# References + +[1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report. arXiv preprint arXiv:2303.08774, 2023. +[2] Abien Fred Agarap. Deep learning using rectified linear units (relu). arXiv preprint arXiv:1803.08375, 2018. +[3] Samuel K Ainsworth, Jonathan Hayase, and Siddhartha Srinivasa. 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Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +Answer: [Yes] + +Justification: In Section 3 and Section 4, we provide all the details about the computational resources of our experiments. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The paper should indicate the type of compute workers CPU or GPU, internal cluster or cloud provider, including relevant memory and storage. +- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. +- The paper should disclose whether the full research project required more computer than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). + +# 9. 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Safeguards + +Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? + +# Answer: [Yes] + +Justification: the large language models employed in our approach, i.e., LLaMA2-7b and LLaMA2-13b, are already equipped with safeguards mechanisms, and our finetuning does not affect such features. Indeed, the datasets we used for training are not scraped from the web. They contain only textual annotations on safe contexts curated by humans. + +# Guidelines: + +- The answer NA means that the paper poses no such risks. +- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. +- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. +- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. + +# 12. 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Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +Justification: the paper does not involve crowdsourcing nor research with human subjects. + +# Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: the development of our core method did not involve LLMs. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# A Bilinear Interpolation of tri-plane Features + +In this Section, we describe the algorithm used to obtain feature vectors from our tri-plane representation, introduced in Section 3. + +# Inputs: + +- Tri-plane feature maps: three orthogonal 2D feature planes $(F_{xy}, F_{xz}, F_{yz})$ of dimension $\mathbb{R}^{H \times W \times C}$ . +- A 3D point $p = (x, y, z) \in \mathbb{R}^3$ . +- Axis-aligned bounding box for coordinate normalization: $aabb_{\min}, aabb_{\max} \in \mathbb{R}^3$ . + +Output: Interpolated feature vector at point $p$ , $f \in \mathbb{R}^C$ . + +# A.1 Coordinate Normalization + +The 3D point $p$ is normalized to the unit cube $[0,1]^3$ : + +$$ +\operatorname {c o o r d s} _ {0 1} = \frac {p - a a b b _ {\min}}{a a b b _ {\max} - a a b b _ {\min}} \in [ 0, 1 ] \tag {3} +$$ + +# A.2 Projection to 2D Triplane Coordinates + +The normalized 3D point is projected onto the three feature planes: + +$$ +p _ {x y} = \left(u _ {x y}, v _ {x y}\right) = \left(\text {c o o r d s} _ {0 1}. x, \text {c o o r d s} _ {0 1}. y\right), \tag {4} +$$ + +$$ +p _ {x z} = \left(u _ {x z}, v _ {x z}\right) = \left(\text {c o o r d s} _ {0 1}. x, \text {c o o r d s} _ {0 1}. z\right), \tag {5} +$$ + +$$ +p _ {y z} = \left(u _ {y z}, v _ {y z}\right) = \left(\operatorname {c o o r d s} _ {0 1}. y, \operatorname {c o o r d s} _ {0 1}. z\right). \tag {6} +$$ + +# A.3 Bilinear Interpolation on Each Plane + +Given a feature plane $F_{xy}$ and the projected coordinate $p_{xy}$ : + +Step 1: Map to image grid coordinates + +$$ +u _ {\mathrm {i m g}} = u _ {x y} (W - 1), \quad v _ {\mathrm {i m g}} = v _ {x y} (H - 1) \tag {7} +$$ + +Step 2: Compute integer and fractional parts + +$$ +i = \left\lfloor u _ {\mathrm {i m g}} \right\rfloor , \quad j = \left\lfloor v _ {\mathrm {i m g}} \right\rfloor \tag {8} +$$ + +$$ +\delta_ {u} = u _ {\mathrm {i m g}} - i, \quad \delta_ {v} = v _ {\mathrm {i m g}} - j \tag {9} +$$ + +Step 3: Retrieve four neighboring features + +$$ +f _ {0 0} = F _ {x y} [ j, i,: ] \quad (\text {t o p - l e f t}) +$$ + +$$ +f _ {1 0} = F _ {x y} [ j, i + 1,: ] \quad (\text {t o p - r i g h t}) +$$ + +$$ +f _ {0 1} = F _ {x y} [ j + 1, i,: ] \quad (\text {b o t t o m - l e f t}) +$$ + +$$ +f _ {1 1} = F _ {x y} [ j + 1, i + 1,: ] \quad (\text {b o t t o m - r i g h t}) +$$ + +Step 4: Compute bilinear interpolation + +$$ +f = \left(1 - \delta_ {u}\right) \left(1 - \delta_ {v}\right) f _ {0 0} + \delta_ {u} \left(1 - \delta_ {v}\right) f _ {1 0} + \left(1 - \delta_ {u}\right) \delta_ {v} f _ {0 1} + \delta_ {u} \delta_ {v} f _ {1 1} \tag {10} +$$ + +Steps 1-4 are repeated for the other planes to obtain $f_{xz}$ and $f_{yz}$ . + +# A.4 Feature Combination + +The final feature vector is obtained by summing the interpolated features from the three planes: + +$$ +f = f _ {x y} + f _ {x z} + f _ {y z} \in \mathbb {R} ^ {C} \tag {11} +$$ + +# B Additional implementation details + +# B.1 weights2space: spatially-aware NeRF weights tokenization + +NeRF encoding. Each NeRF is implemented as an MLP composed of an input layer, $L = 3$ hidden layers with $H = 64$ neurons each, and an output layer. The input to the network is a $I$ -dimensional vector, with $I = 75$ , obtained by applying frequency encoding [45] to the input coordinates. The input layer has a weight matrix $\mathbf{W}_i \in \mathbb{R}^{75 \times 64}$ and bias vector $\mathbf{b}_i \in \mathbb{R}^{64}$ . Each hidden layer uses weights $\mathbf{W}_l \in \mathbb{R}^{64 \times 64}$ and biases $\mathbf{b}_l \in \mathbb{R}^{64}$ . The output layer, which predicts RGB values and density $\sigma$ , has weight matrix $\mathbf{W}_o \in \mathbb{R}^{64 \times 4}$ and bias vector $\mathbf{b}_o \in \mathbb{R}^4$ . The weights2seq module takes as input all weight matrices and bias vectors, stacked row-wise into a matrix $\mathbf{W} \in \mathbb{R}^{S \times H}$ , where $H = 64$ and $S = I + L(H + 1) + H + 2 = 336$ . + +Tri-plane representation. The sequence of tokens $\mathbf{Y} = (\mathbf{y}_1, \mathbf{y}_2, \dots, \mathbf{y}_N) \in \mathbb{R}^{N \times C}$ , with $N = 3072$ and $C = 516$ , produced by seq2space, is reshaped into a tri-plane feature representation $M = (F_{xy}, F_{zx}, F_{yz})$ . Each plane $F$ is a tensor of shape $\mathbb{R}^{H \times W \times C}$ , where $H = W = 32$ and $C = 516$ . + +Decoder. The decoder $\Phi$ consists of 4 linear layers. A skip connection is applied after the second layer by adding a projection of the input. Each linear layer is followed by a ReLU activation, except for the final output layer, which is followed by a sigmoid for $RGB$ and a shifted exponential for $\sigma$ . + +Training. At each training step, for every camera pose, 55000 rays are cast, forming a training batch $\mathcal{R}$ . + +# B.2 Spatial LLaNA: spatially-aware MLLM for NeRF + +Training. S-LLaNA-7B and S-LLaNA-13B are trained on NVIDIA A100 GPUs following a two stages protocol as described in the main manuscript. The hyperparameters of each training stage are the following: + +- **Stage 1:** Training on brief textual descriptions from ShapeNeRF-Text and ObjNeRF-Text for 3 epochs, using a learning rate of 0.002. We leverage AdamW [42] as optimizer, and a cosine learning rate scheduler. +- **Stage 2:** Training on brief and detailed textual descriptions, along with Q&A conversations from the same datasets, for 3 epochs, with a learning rate of 0.00002. We leverage AdamW [42] as optimizer, and a cosine learning rate scheduler. + +Datasets. The full training set of Spatial LLaNA includes approximately 300K NeRFs with textual annotations. While the ObjNeRF-Text training set [7] contains annotations across all categories—brief, detailed, and Q&A—its test sets (PointLLM test set and GPT4Point test set) include only manually annotated brief descriptions. Therefore, in the main paper, the brief captioning task is evaluated on both ShapeNeRF-Text and ObjNeRF-Text, while the other tasks are evaluated solely on ShapeNeRF-Text. Spatial ObjNeRF provides detailed spatial annotations for a subset of 100 NeRFs from the GPT4Point test set of ObjNeRF-Text. The question paired to each ground-truth annotation of this dataset is: "Provide a very detailed description of the object. Focus on the relative position of the parts of the object. Include as many details as possible." This question does not belong to the training set of S-LLaNA. + +# C Ablation experiment on tri-plane resolution and token dimensionality + +We conducted several experiments to analyze the sensitivity of Spatial LLaNA to the number of queries N (and consequently tri-plane resolution) and token dimensionality C. We report results in Table 7, where the first row refers to the official version of S-LLaNA presented in the main paper. The second row shows an experiment to analyze the impact of the number of queries, and thus of the + +Table 7: Ablation on tri-plane and token configurations. Best results are in bold, runner-up is underlined. (SN-T: ShapeNeRF-Text, ON-T: ObjNeRF-Text PointLLM test set) + +
SN-T brief captioningON-T brief captioningSN-T detailed captioningSN-T Q&ASpatial ObjNeRF
ModelTri-plane Res.Token Dim.S-BERTSimCSES-BERTSimCSES-BERTSimCSES-BERTSimCSES-BERTSimCSE
S-LLaNA-13b3251678.9879.9845.4446.1880.9882.9083.6084.1778.2580.58
S-LLaNA-13b1651676.6577.6841.7742.1176.7478.6983.3683.8375.4877.26
S-LLaNA-13b3225276.7776.3342.2542.3676.4778.0783.2783.4976.6278.43
+ +tri-plane resolution. weights2seq and S-LLaNA were trained using N=768 queries. As each plane is obtained by reshaping the queries into 3 planes of spatial dimension $H \times W$ , with $H = W$ in our experiments, the corresponding tri-plane resolution is $\sqrt{\frac{768}{3}} = 16$ . Finally, the third row analyzes the sensitivity of our model to the token dimensionality $C$ . S-LLaNA was trained with $N = 3072$ tokens (i.e., tri-plane resolution 32) as in the main paper, yet with a smaller token dimensionality $C = 252$ . + +Each column reports the S-BERT and SimCSE results on various tasks and datasets: ShapeNeRF-Text (SN-T) brief and detailed captioning and QA, ObjNeRF-Text (ON-T) brief captioning, and Spatial ObjNeRF. + +We observe that reducing the tri-plane resolution or token dimensionality leads to a marginal performance drop (around $2\%$ across all metrics). This behaviour suggests that models with higher capacity (i.e., higher resolution and dimensionality) can encode richer and more fine-grained features from the input NeRF. However, the relatively small drop also indicates that S-LLaNA remains robust and effective even with more compact and computationally efficient representations. + +# D Ablation experiment on multi-token NeRF-Language Models + +In this section, we report a set of experiments designed to isolate the impact of the spatial information encoded within multi-token NeRF representations for NeRF-Language Models. Originally, LLaNA [7] leverages a single-token representation computed from the nf2vec encoder [53]. To investigate on the impact of our proposed spatially-aware NeRF encoding, we train LLaNA after removing the max-pooling layer at the end of this encoder, resulting in a multi-token representation. However, these tokens lack explicit spatial information, unlike those in Spatial LLaNA, where the spatial information is encoded within the NeRF tokens thanks to the specific training methodology of the weights2space meta-encoder. The purpose of this experiment is to determine whether the superior performance of Spatial LLaNA compared to LLaNA stems primarily from the multi-token representation or from the spatial information that is explicitly encoded within the NeRF tokens. The results of this experiment are reported in Table 8. Once again, S-LLaNA emerges as the best performing model. For LLaNA, the transition to multiple tokens yields minimal performance improvements and, in most tasks, actually degrades performance. The only exception is the detailed captioning task on ShapeNeRF-Text, where multiple tokens produce limited gains of approximately 2 points in Sentence-BERT and 0.20 in SimCSE metrics. A marginal 1-point improvement in Sentence-BERT is also observed for the Q&A task on ShapeNeRF-Text. Overall, these results suggest that employing multi-token representations for the multimodal input instead of single-token representations contributes negligibly to performance improvements in NeRF-Language tasks. In contrast, the spatially-aware tokens utilized by S-LLaNA prove crucial for achieving superior performance across the considered tasks. These findings strongly suggest that the ability to extract spatially-aware representations from the weights of the input NeRF is significantly more important than increasing the number of tokens in the representation. + +Table 8: Ablation experiment on multi-token representation Best results are in bold, runner-up is underlined. (SN-T: ShapeNeRF-Text, ON-T: ObjNeRF-Text PointLLM test set) + +
SN-T brief captioningON-T brief captioningSN-T detailed captioningSN-T Q&ASpatial ObjaNeRF
ModelS-BERTSimCSES-BERTSimCSES-BERTSimCSES-BERTSimCSES-BERTSimCSE
S-LLaNA-13b78.9879.9845.4446.1880.9882.9083.6084.1778.2580.58
LLaNA-13b75.0976.4542.0842.4075.5177.6381.0581.6075.0177.30
LLaNA-13b multi-token74.0674.8436.3138.6277.0277.8382.0481.5174.6976.88
+ +Table 9: Zero-shot NeRF classification task on ShapeNeRF-Text. Best results are in bold. + +
ModelInput modalityAccuracy (%)
LLaNA-7bNeRF67.56
S-LLaNA-7bNeRF68.67
LLaNA-13bNeRF69.27
S-LLaNA-13bNeRF71.85
+ +# E Experiments on the general language understanding of Spatial LLaNA + +While our primary focus is spatial reasoning, we have included downstream tasks that go beyond spatial understanding to assess whether S-LLaNA retains general reasoning capabilities. First, the brief captioning task on ShapeNeRF-Text, HST, and ObjNeRF-Text involves generating concise textual descriptions focused on identifying the main subject of a scene. This task primarily evaluates global scene understanding, rather than spatial reasoning. As shown in the main paper, in this setting, S-LLaNA consistently outperforms LLaNA, demonstrating its ability to preserve a holistic understanding of the input NeRF. + +Furthermore, we evaluated S-LLaNA on a zero-shot NeRF classification task on ShapeNeRF-Text following the protocol in [6], where the model must infer the object class based on its multimodal input. This task does not require fine-grained spatial reasoning. Table 9 provides results on this task. + +These results confirm that S-LLaNA does not sacrifice holistic visual or linguistic understanding. Finally, the ablation study on LLM finetuning of Section 6 demonstrates that S-LLaNA performs robustly even without this step, achieving results comparable to the finetuned variant. Consequently, in this version of our framework, the pre-trained LLM remains untouched, and thus certainly preserves its original reasoning capabilities and world knowledge. + +# F Examples from Spatial ObjNeRF + +Figure 6 presents a comparison between the textual annotations from the test set of ObjaNRF-Text [7] and those from Spatial ObjaNRF. Specifically, since Spatial ObjaNRF includes 100 annotated NeRFs from the GPT4Point test split of ObjaNRF-Text, we provide examples from this subset. The annotations of Spatial ObjaNRF clearly offer rich descriptions with spatial relationships between the objects in the scene. In contrast, the original ObjaNRF-Text test set provides brief descriptions that focus primarily on the main subject of the scene. For completeness, we include a JSON file alongside this document that contains all textual annotations from Spatial ObjaNRF, each paired with the corresponding object_id of the 3D model from the Objaverse dataset [18]. + +![](images/32453c4d6021fc16c2ec12e93f9440a9a931ee9e797b578bdbe54b392e37096f.jpg) +Figure 6: Comparison between ObjaNRF-Text (GPT4Point test set) and Spatial ObjaNRF annotations. In green, the fine-grained and spatial details. + +# G Real-world generalization of Spatial LLaNA + +Although ObjaNeRF-Text already includes 3D models of real scenes, to further test the real-world generalization of S-LLaNA, we perform qualitative experiments on OmniObject3D [61] and mip-NeRF360 [10] datasets. The former provides meshes of real-scanned objects, captured in a controlled environment with neutral lighting and background, while the latter includes complex real-world unbounded scenes. mip-NeRF360 is one of the most commonly used datasets for novel view synthesis tasks. Figure 7 shows qualitative examples on OmniObject3D. Such predictions highlight the generalization ability of S-LLaNA, which is able to provide meaningful and spatially-grounded descriptions even for scenes that do not belong to the training distribution of the model. In Figure 8, we can notice how S-LLaNA has the potential to generalize to unbounded in-the-wild NeRF scenes. Such data are extremely different from the training set of the model, from different points of view. First of all, our training data are purely object-centric with a single large object in the center, while in this case the vase of flowers on the table is quite far from the camera view. Secondly, the 3D models of our training set have a neutral back background, while in-the-wild scenes have very complex backgrounds with rich details. Such features make the task of generalizing to in-the-wild scenes very challenging. Despite this, S-LLaNA is able to recognize the main subjects of the scene ("a round wooden table with a vase of flowers on top") and their spatial relations ("A vase is placed on top of the table"). Interestingly, S-LLaNA can also provide some information about the background of the scene ("The background is green, covered with plants.", which has never seen in any training example. + +![](images/9ef3ff369ce11cda48bd8fc867efed608b6a5f48776978600ed1a919ec8ae331.jpg) +Figure 7: Qualitative examples of S-LLaNA on OmniObject3D + +![](images/6b464df2098079c5427b56a1d5e3d8eb00ae53fd41c2a5255f03ea0710b74754.jpg) +Figure 8: Qualitative examples of S-LLaNA on Mip-NeRF 360 + +![](images/75dc707db5bc24fe9331955442b461d305a6e63ab5a36a0a781c014bd1fe378c.jpg) +Figure 9: Qualitative evaluation on Spatial ObjNeRF. In red, the hallucinations generated by the MLLMs. In green, the correct fine-grained and spatial details. + +# H Qualitative results on ShapeNeRF-Text, ObjaNeRF-Text and Spatial ObjaNeRF + +Figure 9 shows qualitative results from the evaluation of the baselines on Spatial ObjNeRF. This dataset contains a large set of scenes involving multiple objects, as shown in the Figure (first, second, and fourth row). It can be noticed how Spatial LLaNA (S-LLaNA in short) is capable of generating detailed and accurate descriptions of the input scenes, including fine-grained spatial information, without exhibiting signs of hallucination. In contrast, the baseline methods either omit such details or produce incorrect descriptions. The ground-truth annotation column further confirms that Spatial ObjNeRF provides rich and comprehensive textual descriptions of the input 3D models. Figures 10 to 12 report qualitatives examples on the language tasks evaluated on the test sets of ShapeNeRF-Text and ObjNeRF-Text [7]. In particular, since the test set of ObjNeRF-Text contains only brief descriptions, this is the only task evaluated on this dataset, as in the experiments reported in the main paper. Results show that also on these tasks Spatial LLaNA is able to recognize and describe very specific details about the input scene, while the competing baselines are not able to do so. + +![](images/77d2903afc02df6a5baca19039194585a427497ae45f0107aac5925827f53fc1.jpg) +Figure 10: Qualitative evaluation of the single-round Q&A task on ShapeNeRF-Text. In red, the hallucinations generated by the MLLMs. In green, the correct fine-grained and spatial details. + +![](images/d964a4187251d0a49c4dfbd129fef364841311ee8663ca4204f2d91473900f8e.jpg) +Figure 11: Qualitative evaluation of the brief captioning task on ShapeNeRF-Text and ObjNeRF-Text. In red, the hallucinations generated by the MLLMs. In green, the correct fine-grained and spatial details. + +# I Failure cases of Spatial LLaNA + +Figure 13 presents some failure cases of Spatial LLaNA on Spatial ObjNeRF. These failures can be broadly categorized into two main categories: (i) the model fails to recognize the main object in the scene (first and second row), or (ii) the model generates short descriptions lacking specific details (third row). Such failures may happen in cases where out-of-distribution objects are provided as input to the model. For example, in the first row, the model is fooled by the unusual shape of the teapot, leading to the weird description "a square device for making tea", while the "pink teapot" had already been correctly included in the answer. In the second row, since most of the snowmen of ObjNeRF-Text are simple white snowballs with outstretched arms made of sticks, this particular snowman is incorrectly described as a "cartoon character". Another interesting case appears in the last row, where Spatial LLaNA correctly detects a "blue toy truck" but omits any distinctive features in the generated description. This may be attributed to the complexity or uniqueness of the object, which challenges the ability of Spatial LLaNA to generate a detailed description. + +# J Additional visualization examples of the attention maps in LLaMA + +In this section, we report additional visualization examples of the self-attention maps of LLaMA, the underlying LLM of Spatial LLaNA. While in the main manuscript we present the attention patterns from a single question word to the 3072 spatial NeRF tokens, here we show the opposite. More precisely, for a given NeRF token at a specific object location, we visualize the normalized attention scores to each text token in the question (Figure 14). These maps are derived from the final decoder layer of LLaMA-13B, with attention scores averaged across all 40 heads. These attention maps confirm the ability of the weights2space meta-encoder to generate spatially-aware tokens from the weights of the input NeRF. As shown in the first row of the Figure, tokens located at the base of + +![](images/93bb8112f9924d2121fa0553d60069509ef17db9cd61c1dd64be48f02151fd29.jpg) +Figure 12: Qualitative evaluation of the detailed captioning task on ShapeNeRF-Text. In red, the hallucinations generated by the MLLMs. In green, the correct fine-grained and spatial details. + +![](images/ff8cfc8955334b529c3e7a628bdc20eb6e502971d7fbebc831cfde00ba979ecb.jpg) +Figure 13: Examples of failure cases of Spatial LLaNA on Spatial ObjNeRF. In red, the hallucinations generated by the model. In green, the correct fine-grained and spatial details. + +the cup and on the vase of the flower exhibit the strongest attention to the words "base" and "vase", respectively, while showing much lower scores for the remaining words. Similarly, in the second row, tokens covering the top of the objects correlate most strongly with the words "cup" and "flower". These visualizations further highlight the ability of S-LLaNA to connect the words of the question with the NeRF tokens representing specific parts of the object. + +![](images/d3ab872a2b25a8d250260d7886e940e03f7f6e51f63661429a5fd03672a3fc06.jpg) +Figure 14: Visualization of the normalized self-attention scores of single NeRF tokens towards the text tokens of the question. In yellow, surrounded by a red circle, the reference NeRF token. + +![](images/46edb6a9a3239d0f04b17e65a2c248bb468bdb6d71f9a8ad4227a01f8da48131.jpg) + +# K Details about datasets, models and source code licenses + +This section provides details about the datasets, models, and source code licenses used in the work, ensuring proper credit to the creators or original owners, and adherence to license terms. + +Datasets: the datasets employed in our work and the relative licenses are listed below: + +- ShapeNeRF-Text: licensed under MIT License. +- ObjNeRF-Text: licensed under MIT License. +- ShapeNet: licensed under GNU Affero General Public License v3.0. +- Objaverse: licensed under Open Data Commons Attribution License v1.0 (ODC-By v1.0). +- GPT2Shape HST: licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. + +Models: the models used in our experiments and their relative licenses are detailed below: + +- LLaNA: licensed under MIT License. +- nf2vec: licensed under MIT License. +- PointLLM: licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. +- GPT4Point: licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. +- ShapeLLM: licensed under Apache License 2.0. +- 3D-LLM: licensed under MIT License. +- BLIP-2: licensed under BSD 3-Clause License. +- LLAVA: licensed under Apache License 2.0. +- LLAMA-2: licensed under META LLAMA 2 COMMUNITY LICENSE AGREEMENT1. \ No newline at end of file diff --git a/NeurIPS/2025/Spatially-aware Weights Tokenization for NeRF-Language Models/images.zip b/NeurIPS/2025/Spatially-aware Weights Tokenization for NeRF-Language Models/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..5a020caf3dad86316d62e2f37177e628d36950ec --- /dev/null +++ b/NeurIPS/2025/Spatially-aware Weights Tokenization for NeRF-Language Models/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Consensus with Scene Prior for Unsupervised Domain Adaptive Person Search/full.md b/NeurIPS/2025/Spatiotemporal Consensus with Scene Prior for Unsupervised Domain Adaptive Person Search/full.md new file mode 100644 index 0000000000000000000000000000000000000000..8ba5f20102302ca2a07e2b3a478b09479e767e03 --- /dev/null +++ b/NeurIPS/2025/Spatiotemporal Consensus with Scene Prior for Unsupervised Domain Adaptive Person Search/full.md @@ -0,0 +1,559 @@ +# Spatiotemporal Consensus with Scene Prior for Unsupervised Domain Adaptive Person Search + +Yimin Jiang† + +Dalian Maritime University yimin_jiang@dlmu.edu.cn + +Huibing Wang + +Dalian Maritime University huibing.wang@dlmu.edu.cn + +Jinjia Peng† + +Hebei University pengjinjia@hbu.edu.cn + +# Abstract + +Person Search aims to locate query persons in gallery scene images, but faces severe performance degradation under domain shifts. Unsupervised domain adaptation transfers knowledge from the labeled source domain to the unlabeled target domain and iteratively rectifies the pseudo-labels. However, the pseudo-labels are inevitably contaminated by the source-biased model, which misleads the training process. This, in turn, reduces the quality of the pseudo-labels themselves and ultimately affects the search performance. In this paper, we propose a Spatiotemporal Consensus with Scene Prior (STCSP) framework that effectively eliminates the interference of noise on pseudo-labels, establishes positive feedback, and thus gradually bridging the domain gap. Firstly, STCSP uses a Spatiotemporal Consensus pipeline to suppress the noise from being mixed into the pseudo-labels. Secondly, leveraging the scene prior, STCSP employs our designed Iterative Bilateral Extremum Matching method to prevent the occurrence of some incorrect pseudo-labels. Thirdly, we propose a Scene Prior Contrastive Learning module, which encourages the model to directly acquire the scene prior knowledge from the target domain, thereby mitigating the generation of noise. By suppressing noise contamination, avoiding noise occurrence and mitigating noise generation, our framework achieves state-of-the-art performance on two benchmark datasets, PRW with $50.2\%$ mAP and CUHK-SYSU with $87.0\%$ mAP. + +# 1 Introduction + +Person search aims to localize and identify a query person from a gallery of scene images. It can be taken as a joint task of person detection and re-identification (re-id) in an end-to-end manner, where supervised learning has made significant advancements. However, notable performance degradation is observed in these methods when deployed to new application scenarios, primarily attributed to domain gaps induced by factors such as camera configuration. Additionally, the process of annotating an adequate amount of training data for a particular domain is both arduous and costly. Consequently, unsupervised domain adaptation (UDA) holds considerable promise in practical real-world scenarios for person search. + +UDA in person search aims to achieve model generalization from labeled source data to unlabeled target deployment scenarios. DAPS [11], a pioneering work in applying UDA to person search, employs implicit domain alignment in conjunction with a pseudo-labeling method to effectively bridge the gap between the source and target domains. Building upon DAPS, Almansoori et al. proposed DDAM [1], which generates a hybrid domain and learns within it to minimize the distance between the two domains. Although these methods achieve satisfactory performance, they failed to + +![](images/8ba6eb626f2387ec767722b5cb724c17449ff60db0599d4625f99cd1bbeac6f0.jpg) +(a) Correct labeling. + +![](images/c84e8365727b02650ead0c7f3460f53e0789cc6f7dd086f9c3a4ddf2edf7d37c.jpg) +Figure 1: Display of incorrect pseudo-labels. (a) A correct pseudo-label has an accurate BBox and a correct person identity. (b) An inaccurate pseudo-BBox manifests in the form of either encapsulating an erroneous region or exhibiting the overlap of multiple boxes. (c) An incorrect identity label comes from situations where an identity is assigned to different individuals or where different instances of a person are assigned different identities. (d) Assigning the same identity label to intra-scene persons is a clear error. + +![](images/25f7197d9ab359b7d87c9420329e2710baac48ae33c183cf50f53921d7cc00ad.jpg) +(b) Inaccurate BBoxes. + +![](images/7b971fc9d6bf439f195f02ad57134f7ed92a9e55cd426c3ad71b249db5009e8c.jpg) +(c) Incorrect Identities. +(d) Identity Conflict. + +effectively eliminate the interference of noise stemming from the source-biased models on pseudolabels. As illustrated in Fig. 1b, the inaccurate bounding boxes (BBoxes) inevitably lead to detection failures, which can induce significant degradation in person search performance. Additionally, as shown in Fig. 1c, the incorrect labels will directly cause confusion in identifying the person identities. Furthermore, the noise accumulates with the increase of epochs, forming a negative feedback mechanism that will cause performance degradation of the model during the later training phase. Therefore, eliminating the interference of the noise on pseudo-labels is a major challenge. In addition, previous studies failed to realize that a scene can provide relationships for its inner persons. Specifically, the identities of persons within a scene image are inherently distinct, and this is referred to as scene prior. The omission of considering and exploiting this scene prior results in a conspicuous error as illustrated in Fig. 1d. Consequently, the utilization of the scene prior is another crucial issue. + +To address these issues, we propose a novel Spatiotemporal Consensus with Scene Prior (STCSP) framework that eliminates the impact of the noise through suppressing noise contamination, preventing some erroneous pseudo-labels and mitigating noise generation. Our framework progressively bridges the domain gap via three core innovations. First, we devise a Spatiotemporal Consensus (STC) pipeline to suppress noise propagation in pseudo-labels. STC maintains a memory bank of the previous detection and re-id information. To filter out temporal jitters, STC leverages the temporal consensus between the information in the memory bank and the current state. In addition, for the detection and re-id subtasks respectively, STC exploits the spatial consensus in the scene space of the image and the latent space of clustering to eliminate spatial outliers. Second, leveraging the intrinsic scene prior, STCSP conducts bipartite matching for the instances in any two of all the images, thus averting the error depicted in Fig. 1d. However, deriving the optimal solutions for bipartite matching across thousands of images is computationally prohibitive. To tackle this challenge, we propose an Iterative Bilateral Extremum Matching (IBEM) method, leveraging GPU acceleration, to seek the approximate solutions. Third, we propose a Scene Prior Contrastive Learning (SPCL) module that encourages the model to learn discriminative person features within the target domain. Specifically, besides proxy learning, SPCL steers the model to learn identity heterogeneity within the same scene, enabling the model to directly absorb knowledge from the target domain. The existence of this knowledge effectively reduces the noise output of the model. + +Our major contributions are summarized below: + +- A Spatiotemporal Consensus pipeline is proposed to suppress the noise origination from the domain gap, thereby generating reliable pseudo-labels. +- An Iterative Bilateral Extremum Matching method is designed. Within a few seconds, it can match the instances in any two of all thousands of images. +- A Scene Prior Contrastive Learning module is proposed, which encourages the model to directly acquire knowledge from the target domain. + +With the spatiotemporal consensus and the scene prior, STCSP achieves state-of-the-art (SOTA) performance on two benchmark datasets. Specifically, it attains a mean average precision of $50.2\%$ on the PRW dataset and $87.0\%$ on the CUHK-SYSU dataset. + +![](images/28bced8d5b67ef9b0a1759607e26eab9173c45491fe7673fece1da5b57202b77.jpg) +Figure 2: Architecture of the proposed framework. This figure is structured into two rows. The first row presents the input images and the spatiotemporal consensus pipeline. The second row, arranged from left to right, encompasses the backbone network, detection module, re-id module, task cooperation module, and scene prior contrastive learning module. + +# 2 Method + +# 2.1 Framework Overview + +Our UDA person search model is an end-to-end architecture based on SeqNet [12]. Differently, we replaced the backbone network and person feature embedder with ConvNeXt-B [16] and our designed attention embedder, respectively. Following the approach of DAPS [11], we introduce gradient reversal layers (GRLs) and domain discriminators (DDs) for three level features. We adhere to the principle of "sample quality first" and use our STC pipeline to obtain reliable pseudo-labels. The SPCL and task cooperation (TC) modules are employed to steer our model to learn discriminative features. + +Prior to the training of each epoch, the STC pipeline preprocesses the unlabeled target domain. As in the first row of Fig. 2, the temporal consensus updates the memory bank with model-inferred BBox scores, BBoxes, and embeddings. The spatial consensus then screens out pseudo-BBoxes from the memory bank. Subsequently, their embeddings are calculated by Re-Ranking [30] to form a cross-distance matrix. This matrix undergoes bipartite matching via the IBEM method, and the result is used for matrix sparsity regularization. The regularized matrix generates spatial consensus clusters, which combine with those from the previous epoch to form spatiotemporal ones. The center of each cluster serves as a proxy sent to the SPCL. + +During the training phase, the model is fed source and target domain mini-batches in an alternating sequence, with parameters being updated every two mini-batches. As shown in the second row of Fig. 2, the first four ConvNeXt-B [16] stages are responsible for extracting scene features. The feature map is then fed into the Faster-RCNN-based [20] detection module for BBox generation, where RoI-Align pools the instance feature map for each BBox and the Attention Embedder encodes it into an embedding. Finally, the embedding is sent to the TC and SPCL modules for loss calculation. + +# 2.2 Spatiotemporal Consensus for Detection + +Temporal Consensus for Detection. For the estimated BBoxes $\mathcal{H}$ inferred by the model and the anchor BBoxes $\mathcal{G}$ stored in the memory bank, we use the intersection-over-union (IoU) scores as the basis for the matching between them. The unmatched BBoxes and their accompanying BBox scores and instance embeddings in the memory bank will be removed, and the remaining contents will be + +![](images/60c0fc82ae579f1ba8624c7e9b922e83c960b3501abfb6891212e602931e126e.jpg) +Figure 3: An example of IBEM. On the left and right are the cost matrix and the optimal solution of a minimum bipartite matching task, respectively. The first BEM extracts the primary pair, and the secondary BEM resolves the residuals. + +updated with momentum $\eta$ . The matching and updating operations are expressed as follows: + +$$ +\widetilde {\mathcal {G}} = \left\{\eta g + (1 - \eta) h \left| \begin{array}{l} g \in \mathcal {G}, h \in \mathcal {H}, \rho (g, h) > \theta , \\ \operatorname {a r g m a x} _ {a \in \mathcal {G}} \rho (h, a) = g, \operatorname {a r g m a x} _ {b \in \mathcal {H}} \rho (g, b) = h \end{array} \right. \right\}, \tag {1} +$$ + +where $\widetilde{\mathcal{G}}$ is a set of updated anchor BBoxes, $\rho$ is a box IoU function and $\theta$ is a threshold for the IoU score. In this equation, the anchor BBox $g$ and the estimated BBox $h$ are mutually IoU-optimal, and the score also meets the threshold. Matching the anchor with the maximum IoU for each estimate is to avoid updating multiple anchors for one estimate. Based on the assumption that the pseudo-labels are true, we consider each anchor to be true. Therefore, selecting the estimate with the maximum IoU is equivalent to maximizing the likelihood probability of generating that anchor. + +In addition, the estimated BBoxes in $\mathcal{H}$ that are independent of all anchor BBoxes are used to generate new anchors. Whose definition is as follows: + +$$ +\widehat {\mathcal {H}} = \left\{h \in \mathcal {H} \mid \max _ {g \in \mathcal {G}} \rho (g, h) \leq \theta \right\}. \tag {2} +$$ + +The boxes in $\widehat{\mathcal{H}}$ with a high IoU score between them will be merged to ensure the independence among the new anchors. First, a variant of non-maximum suppression (NMS) operation with threshold $\theta$ is applied to divide $\widehat{\mathcal{H}}$ into multiple subsets, as follows: + +$$ +\widehat {\mathcal {H}} \xrightarrow {\text {S p l i t b y N M S} _ {\theta}} \left\{\widehat {\mathcal {H}} _ {1}, \widehat {\mathcal {H}} _ {2}, \dots \right\}. \tag {3} +$$ + +Specifically, a box retained by the NMS and the boxes it suppresses are grouped together. Finally, the mean values of each subset from the new anchors $\widehat{\mathcal{G}}$ . Notably, the score and embedding corresponding to each BBox will undergo the same operations during this process. Each update of the memory bank is a deep integration of historical and current knowledge. Therefore, the result $\mathcal{G} \gets \widetilde{\mathcal{G}} \cup \widehat{\mathcal{G}}$ is a temporal consensus. + +Spatial Consensus for Detection. The comprehensive analysis of Eqs. (1) to (3) reveals that the maximum value of the IoU scores among the anchor BBoxes in the memory bank is approximately $\theta$ . This indicates that there may be multiple BBoxes stacked on one person instance, as shown in the right of Fig. 1b. Therefore, we apply a NMS with threshold $\varphi$ to screen out reliable BBoxes from the memory bank, thereby establishing spatial consensus on each image. + +# 2.3 Measurement with Scene Prior + +Following previous works, we adopt Re-Ranking [30] method to compute pairwise distances between instance embeddings. To optimize computational performance, we have developed a GPU-accelerated implementation using PyTorch. However, different from previous works, we find that scenes prior can sparsely regularize the distance matrix, thus avoiding the error as shown in Fig. 1d. Specifically, our approach constructs inter-scene cost matrices from the instance distances, performing minimum bipartite matching to set unmapped pair distances to infinity, thereby enforcing intra-scene identity uniqueness. However, the $O\left(n^{3}\right)$ complexity of exact solvers like Hungarian algorithm [10] becomes prohibitive at scale ( $\approx$ 2hours/10k images). + +Iterative Bilateral Extremum Matching. To relieve this dilemma, we propose an IBEM method to calculate the approximate solution. Specifically, as shown in algorithm 1, the IBEM iteratively identifies entries that are extrema (e.g., minima or maxima) in both their row and column, then removes the matched entries' rows and columns for further iterations until the conditions are met. + +# Algorithm 1 Iterative Bilateral Extremum Matching + +Require: Cost matrix $\mathcal{C} = \{c_{1,1},\dots \}$ and its row and column indices $I_{1}$ and $J_{1}$ . + +Ensure: Matching pairs: $\mathcal{M} = \mathcal{M}_1\cup \mathcal{M}_2\cup \dots \cup \mathcal{M}_{\delta}$ + +$$ +\begin{array}{l} \text {w h i l e} t = 1 \text {t o} \delta , | I _ {t} | \times | J _ {t} | / | I _ {1} | \times | J _ {1} | < \vartheta \text {d o} \\ \mathcal {M} _ {t} \leftarrow \left\{(i, j) \mid c _ {i, j} = \min _ {k \in I _ {t}} c _ {k, j}, c _ {i, j} = \min _ {k \in J _ {t}} c _ {i, k} \right\} \\ I _ {t + 1} \leftarrow \{i \in I _ {t} | i \neq p, (p, q) \in \mathcal {M} _ {t} \} \\ J _ {t + 1} \leftarrow \{j \in J _ {t} | j \neq q, (p, q) \in \mathcal {M} _ {t} \} \\ \end{array} +$$ + +end while + +Although some rows and columns may remain unmatched during each iteration of bilateral extremum matching (BEM), we rigorously prove by contradiction that a scenario with complete unmatched of all rows and columns is theoretically impossible. Consequently, full instance matching is guaranteed via the iterative process. To balance computational efficiency and convergence speed, we introduce two hyperparameters, an unmatched proportion threshold $\vartheta$ and a maximum number of iterations $\delta$ as limitations. Fig. 3 shows an example of this method. Notably, by leveraging GPU acceleration, IBEM completes cross-image matching across thousands of instances within seconds. + +# 2.4 Spatiotemporal Consensus for Re-ID + +Spatial Consensus for Re-ID. Conceptually, a discriminative cluster requires sufficient separation from other instances. However, the uneven density of the latent space renders a fixed-distance threshold inadequate for the specificity evaluation. To address this challenge, we propose a spatial clustering consensus strategy, which identifies reliable clusters through dual-granularity pattern analysis. Specifically, we implement a dual-density DBSCAN strategy: A conservative density with radius $\epsilon$ and min sample size $\kappa$ captures coarse-grained clusters, while an aggressive density with radius $\alpha \cdot \epsilon$ and min sample size $\alpha \cdot \kappa$ detects fine-grained clusters. Then, through the cluster consensus method, our strategy synthesizes spatial consensus clusters, as shown on the first row of Fig. 2. This cross-validation mechanism effectively distinguishes reliable clusters, ensuring robustness against the latent space with uneven density. + +Temporal Consensus for Re-ID. As the instances iteratively update their latent representations, the temporally stable clusters maintain structural cohesion despite positional shifts. Therefore, to screen out the clusters with temporal robustness, we propose a temporal clustering consensus strategy, which integrates the spatial consensus results across previous and current epochs. This cross-epoch fusion process identifies the persistent clusters by the cluster consensus method, thereby forming temporal consensus results. Specifically, as shown on the first row of Fig. 2, the temporal consensus window is constrained to two adjacent epochs, enforcing gradual cluster evolution while preserving temporal consensus across training iterations. + +Cluster Consensus. For a certain element $\xi$ in a set $\Xi = \{\xi_1, \xi_2, \dots, \xi_n\}$ containing $n$ instances, there are two different strategies, $\Phi$ and $\Psi$ , to obtain its cluster labels $\Phi(\xi, \Xi)$ and $\Psi(\xi, \Xi)$ respectively. So, in strategy $\Phi$ , the cluster which $\xi$ belongs to is as follows: + +$$ +F (\xi , \Xi , \Phi) = \{x \in \Xi | \Phi (x, \Xi) = \Phi (\xi , \Xi) \}. \tag {4} +$$ + +Based on $\mathrm{F}$ , an intersection-over-union score of the clusters where $\xi$ is located in $\Phi$ and $\Psi$ can be calculated as follows: + +$$ +f (\xi , \Xi , \Phi , \Psi) = \frac {\left| F (\xi , \Xi , \Phi) \cap F (\xi , \Xi , \Psi) \right|}{\left| F (\xi , \Xi , \Phi) \cup F (\xi , \Xi , \Psi) \right|}, \tag {5} +$$ + +where $\mathrm{F}(\xi, \Xi, \Phi)$ and $\mathrm{F}(\xi, \Xi, \Psi)$ represent the clusters where $\xi$ is placed in $\Phi$ and $\Phi$ , respectively. Then, the maximum IoU score among all instances in $\mathrm{F}(\xi, \Xi, \Phi)$ is as follows: + +$$ +\mathrm {g} (\xi , \Xi , \Phi , \Psi) = \max _ {x \in \mathrm {F} (\xi , \Xi , \Phi)} \mathrm {f} (x, \Xi , \Phi , \Psi). \tag {6} +$$ + +Finally, a cluster consensus label for $\xi$ is obtained through the following conditional mapping: + +$$ +\mathrm {h} (\xi , \Xi , \Phi , \Psi) = \left\{ \begin{array}{l l} \Phi (\xi , \Xi), & \mathrm {f} (\xi , \Xi , \Phi , \Psi) = \mathrm {g} (\xi , \Xi , \Phi , \Psi) > 0. 5 \\ - 1, & \text {e l s e} \end{array} \right., \tag {7} +$$ + +where $\mathrm{f}(\xi, \Xi, \Phi, \Psi) = \mathrm{g}(\xi, \Xi, \Phi, \Psi)$ means that $\xi$ lies in the intersection of $\mathrm{F}(\xi, \Xi, \Phi)$ and $\mathrm{F}(\xi, \Xi, \Psi)$ , which has the maximum IoU score within $\mathrm{F}(\xi, \Xi, \Phi)$ . Moreover, we enforce a 0.5 IoU threshold to validate high confidence in $\Phi$ and $\Psi$ consensus. Besides, an instance with cluster label of -1 denotes an outlier. + +# 2.5 Scene Prior Contrastive Learning + +Existing works have achieved satisfactory results through contrastive learning using person feature proxies. However, they failed to realize and utilize scene prior to guide the model learning. Therefore, we design a triplet loss function to encourage the model to learn this prior knowledge. Specifically, we introduce a person proxy table (PT) $V = \{v_{1}, v_{2}, \dots \}$ for the target domain. Before each training epoch, the PT will be re-formed by taking the mean value of embeddings in each cluster. And, during the training, it is updated online with a momentum $\lambda$ . Given a set $\mathcal{I}$ of the instance feature embeddings in a scene image, the margin $m$ of an embedding $x$ in $\mathcal{I}$ is calculated as follows: + +$$ +m = \left\langle v _ {\mathrm {L} (x)}, x \right\rangle - \max _ {a \in \mathcal {A}} \left\langle a, x \right\rangle , \mathcal {A} = \left\{a \in \mathcal {I} | \mathrm {L} (a) \neq \mathrm {L} (x) \right\}, \tag {8} +$$ + +where $\langle \cdot, \cdot \rangle$ denotes the inner product between two embeddings and $\mathrm{L}(\cdot)$ represents taking the label corresponding to the embedding. $v_{\mathrm{L}(x)}$ is the proxy in $V$ corresponding to the $x$ and $\mathcal{A}$ is all the embeddings in $\mathcal{I}$ that have different labels from $x$ . Besides, we use a hyper-parameter $M$ to evaluate the margin. And the triplet loss objective is to increase the margin $m$ to $M$ : + +$$ +\mathcal {L} _ {\text {T r i p l e t}} = \max (M - m, 0). \tag {9} +$$ + +In addition, we also employ a cross-entropy loss in this module. It calculates the probability that the label of $x$ is $L(x)$ , and uses cross-entropy to supervised the probability. + +$$ +\mathcal {L} _ {C E} = - \log p, p = \frac {\varepsilon (v _ {\mathrm {L} (x)} , x)}{\sum_ {a \in V} \varepsilon (a , x) + \sum_ {b \in U} \varepsilon (b , x) + \sum_ {c \in Q} \varepsilon (c , x)}, \varepsilon (a, b) = \exp (\langle a, b \rangle / \tau), \tag {10} +$$ + +where $\tau = 1 / 30$ is hyper-parameter which adjusts the softness of the probability distribution. $U$ and $Q$ are respectively the lookup table (LUT) and circular queue (CQ) proposed by OIM, which are updated by the embeddings generated from the source domain. Finally, the scene prior contrastive learning loss is obtained by adding two terms together: + +$$ +\mathcal {L} _ {S P C L} = \mathcal {L} _ {\text {T r i p l e t}} + \mathcal {L} _ {C E}. \tag {11} +$$ + +# 2.6 Task Cooperation + +BNR [9] loss utilizes the foreground and background labels from the detection head to supervise the embeddings generated by the re-id head. It can encourage the backbone to enhance the foreground features and suppress the background features, thereby improving the model's detection and re-id performance simultaneously. We improved the equation for calculating the probability that an embedding $x$ belongs to the foreground as follows: + +$$ +p = \sigma \left[ \mathrm {B N} \left(\| x \| _ {2} ^ {2} / 2\right) \right], \tag {12} +$$ + +where $\sigma (\cdot)$ is a sigmoid function, BN $(\cdot)$ is an batch normalization [8] layer, and $\| \xi \| _2$ is the $\mathrm{L}_2$ -norm value of the feature embedding $x$ . We employ the square $\mathrm{L}_2$ -norm as the regularization term to impose independent constraints on each dimension of $x$ and prevent outliers in some dimensions from influencing others through cross-term interactions. And dividing by 2 is to simplify the derivative operation. Finally, Focal Loss [14] is calculated for $p$ , which focus training on the hard negatives: + +$$ +\mathcal {L} _ {T C} = - \left(1 - p _ {t}\right) ^ {2} \log \left(p _ {t}\right), p _ {t} = \left\{ \begin{array}{l l} p, & y = 1 \\ 1 - p, & y = 0 \end{array} \right., \tag {13} +$$ + +where $y$ is a label (either 0 or 1) indicating whether its corresponding embedding is marked as a background or person. + +![](images/35cb648eee7d62754de349e9f9df11ed7b888def9985601b5581a4b758cdd608.jpg) +Figure 4: Details of the proposed attention embedder. "Pos. Encod.", "Concat." and "Batch Norm." respectively represent "Positional Encodings", "Concatenate" and "Batch Normalization". + +# 2.7 Attention Embedder + +Inspired by the attention pool module in CLIP [19], we propose a novel attention embedder as shown in Fig. 4. Distinct from the attention pool module, our design introduces two critical enhancements to the multi-head attention output: Batch Normalization [8] (BN) and Dropout [21]. BN standardizes the distribution of each embedding dimension by centering it near zero with unit variance, mitigating its congregation in positive or negative ranges. This normalization encourages a discriminative embedding and enhance its expressivity in the latent space. Subsequently, Dropout injects stochasticity during training, preventing over-reliance on specific embedding dimensions. Through the synergism of these components, a discriminative and robust embedding is obtained. + +# 3 Experiments + +# 3.1 Experimental Setup and Implementation Details + +CUHK-SYSU [25] is a large-scale person search dataset that contains 18,184 images collected from handheld cameras, movies, and TV shows, resulting in significant scene diversity. It encompasses 8,432 unique person identities and 96,143 annotated bounding boxes. The dataset is split into a training set with 5,532 identities and 11,206 images, and a test set with 2,900 queries and 6,978 gallery images. For each query, the dataset defines a gallery size ranging from 50 to 4,000, with a default gallery size of 100 images. + +PRW [29] is composed of video frames captured by six fixed cameras on a university campus. It contains 11,816 scene images with 932 distinct person identities and 43,110 annotated bounding boxes. The training set consists of 483 identities with 5,704 images, and the test set contains 2,057 queries and 6,112 scene images. For each query, the dataset uses all images in the test set except for the query as the gallery. + +Evaluation Metric. Following the established setting in previous work, the mean Average Precision (mAP) and top-1 accuracy (top-1) are employed to evaluate the performance for person search. + +Implementation Details. We conduct all experiments on a NVIDIA A800 GPU and implement our model with PyTorch [18]. ConvNeXt [16] pre-trained on ImageNet [5] is adopted as the backbone network. During training, we set the batch size to 4, and use Automatic Mixed Precision (AMP). Adam is used to optimize our model for 20 epochs with an initial learning rate of 0.0001 and a weight decay of 0.01, which is warmed up during the first epoch and reduced by a factor of 0.1 at epochs 8 and 14. When PRW is used as the target domain, our model undergoes pre-training on the source domain CUHK-SYSU for 5 epochs before commencing joint training. Conversely, it pre-train on PRW for 3 epochs. The thresholds $\theta$ and $\varphi$ in Sec. 2.2 are set to 0.7 and 0.4, respectively. For algorithm 1, the limitations $\delta$ and $\vartheta$ are set to 2 and 0.2, respectively. The hyper-parameter $\epsilon$ , $\kappa$ and $\alpha$ in Sec. 2.4 is set to 0.5, 4 and 0.8, respectively. For Eq. (9), we initialize the hyper-parameter $M$ as 0.35. For the attention embedding, the probability of the Dropout is set to 0.1. + +# 3.2 Comparison to the State-of-the-Arts + +Comparison on PRW. The 3rd column of Tab. 1 shows the performance of all methods on the PRW test set. As shown in the table, our framework outperforming all the other UDA and weakly unsupervised methods, whether based on ConvNeXt or ResNet. Especially for the current best-performing DDAM [1], STCSP outperforms it by $13.5\%$ and $6.1\%$ in mAP and top-1 accuracy, respectively. And when both employ ResNet-50 as the backbone, our method still outperforms it by $5.7\%$ and $0.9\%$ . Notably, our method outperforms most fully supervised methods, despite they using ground truth labels to train models. + +Table 1: Comparison of mAP(%) and top-1(%) accuracy with the fully-supervised, weakly-supervised and UDA methods on the PRW and CUHK-SYSU test sets. For UDA methods, the best and second best scores are shown in bold and underlined, respectively. + +
MethodBackbonePRWCUHK-SYSU
mAPtop-1mAPtop-1
Fully Supervised Learning
OIM [25]ResNet-5021.349.475.578.7
IAN [24]ResNet-5023.061.976.380.1
NPSM [15]ResNet-5024.253.177.981.2
RCAA [2]ResNet-50--79.381.3
CTXG [28]ResNet-5033.473.684.186.5
QEEPS [17]ResNet-5037.176.788.989.1
NAE+ [3]ResNet-5044.081.192.192.9
AlignPS+ [27]ResNet-5046.182.194.094.5
SeqNet [12]ResNet-5046.783.493.894.6
SEAS [9]ResNet-5052.085.796.297.1
SEAS [9]ConvNeXt-B60.589.597.197.8
Weakly Supervised Learning
R-SiamNet[6]ResNet-5021.475.286.087.1
CGPS [26]ResNet-5016.268.080.082.3
SSL [22]ResNet-5030.780.687.488.5
KCD [13]ResNet-5040.583.686.888.2
DICL [23]ResNet-5035.580.987.488.8
Unsupervised Domain Adaptation
DAPS [11]ResNet-5034.780.677.679.6
FOUS [4]ResNet-5035.480.878.780.5
DDAM [1]ResNet-5036.781.279.581.3
STCSP (ours)ResNet-5042.482.180.482.5
STCSP (ours)ConvNeXt-B50.287.387.087.9
+ +![](images/7d090b790bc958f5ee75018c6a2ee6fc83f7cd6e2481247f19bbf7fa99200fbd.jpg) +(a) Comparison of mAP. + +![](images/c57b4715325ed05bbe0a5160ba2d83199319aa3cb1b07dc8369dbcff4f59dea1.jpg) +(b) Comparison of top-1 accuracy. +Figure 5: Comparison of mAP and top-1 accuracy on CUHK-SYSU across various gallery sizes. + +Comparison on CUHK-SYSU. The performance of our method and the other methods on the CUHK-SYSU test set is shown in the 4th column of Tab. 1. STCSP based on ConvNeXt or ResNet outperforms all the other UDA methods. It achieves the best scores of $87.0\%$ in mAP and $87.9\%$ in top-1 accuracy, surpassing DDAM [1] by $7.5\%$ and $6.6\%$ in mAP and top-1 accuracy, respectively. The CUHK-SYSU training set boasts greater diversity and a larger quantity of scene images compared to the PRW training set. As a result, UDA from PRW to CUHK-SYSU presents a formidable challenge. Despite this, our method still outperforms several weakly-supervised and fully-supervised methods on the CUHK-SYSU test set. + +We further perform a comparison between our STCSP and other methods on the CUHK-SYSU test set with gallery sizes ranging from 50 to 4,000. Fig. 5 shows the performance curve of all the methods in terms of mAP and top-1 as the gallery size increases. Since it is a challenge for all compared methods to consider more distracting persons in the gallery set, the performance of them is reduced as the gallery size increases. However, our method consistently outperforms all the other methods in + +![](images/c9118467e37eede4f901dcfc56a2a4626a6923fbe7c4031b2b55027c093fcdf2.jpg) +Figure 6: Qualitative comparison of our method with DAPS on the PRW test set. Each column represents a comparison result. The blue bounding boxes denote the queries, while the green and red bounding boxes denote correct and incorrect top-1 matches, respectively. + +Table 2: Comparative results of the different strategies for our framework. + +
StrategyPRWCUHK-SYSU
mAPtop-1mAPtop-1
w/o Detection Spatial Consensus46.885.785.485.4
w/o Detection Temporal Consensus47.786.086.487.5
w/o Re-ID Spatial Consensus46.985.886.787.7
w/o Re-ID Temporal Consensus49.087.486.687.5
w/o IBEM48.285.985.986.6
w/o Triple Loss of SPCL47.987.086.287.4
50.287.387.087.9
+ +mAP and top-1, whether based on ConvNeXt or ResNet. This indicates that our method has excellent generalization ability. + +Qualitative Results. Some example person search results are illustrated in Fig. 6. The 5th column of the figure shows the result of using a low-resolution image as the query. When multiple highly similar characters appear in a scene, the search results are illustrated in the 1st and 4th columns. The 2nd column shows the result of the person's body being obstructed. The 3rd and 6th columns show the result when the scene and viewpoint change. It can be observed that our successfully handles low-resolution, similar-character, occlusion, cross-scene and viewpoint variation, while other state-of-the-art methods such as DAPS fail in these scenarios. This demonstrates the robustness of our method. + +# 3.3 Ablation Study + +Spatiotemporal Consensus for Detection. To verify the effectiveness of our STC strategy for detection, we evaluate separately the performance of our framework in two scenarios: detection without spatial consensus and without temporal consensus, and report the results in Tab. 2. We find that STCSP exhibits significant performance degradation when lacking the spatial or temporal consensus for detection. This results indicate that the STC strategy for detection significantly elevates the quality of pseudo-BBoxes. + +Spatiotemporal Consensus for Re-ID. Tab. 2 shows the performance of our framework being employed with different strategies on the two benchmarks. First, to validate the effectiveness of the spatial consensus, we replace it with a DBSCAN method. The results reveal that our strategy of employing a loose and compact clustering space to form spatial consensus can generate reliable clusters. Furthermore, The results after removal of the temporal consensus are also reported in Tab. 2. Through comparison, we can see that the temporal consensus improve the performance of our model + +Table 3: Comparative results of the different components for our framework. RN, CN, NAE and AE indicate ResNet-50, ConvNeXt-B, Norm-Aware Embedding and Attention Embedder, respectively. + +
BackboneEmbedderPRWCUHK-SYSU
RNCNNAEAEmAPtop-1mAPtop-1
✓✗✓✗41.182.280.382.3
✓✗✗✓42.482.280.482.5
✗✓✓✗45.885.685.685.5
✗✓✗✓50.287.387.087.9
+ +on both of PRW and CUHK-SYSU. We attribute these improvements to the stable clusters formed by the temporal consensus. + +Scene Prior. We evaluated the influence exerted by the lack of scene priors within the related components on the performance manifestations of our framework, and report the results in Tab. 2. As presented in the 5th row of the table, when the IBEM is absent from our framework, entailing the potential occurrence of the error depicted in Fig. 1d, the performance of our framework experiences a substantial decline across both datasets. Notably, this decline is more pronounced on the PRW dataset. Similarly, as indicated in the 6th row of the table, a comparable scenario transpires when the triplet loss within the SPCL is lacking. The significant performance degradation observed on PRW can be attributed to the existence of persons with a high similarity within some scene images in this dataset. This indicates that scene prior effectively steers the model to learn discriminative features. + +Model Architecture. As shown in Tab. 3, in order to assess the influence of diverse model architectures on our framework, we report the performance outcomes under all cross-combination modes of ResNet-50 (RN) [7] and ConvNeXt-B (CN) [16] with Norm-Aware Embedding (NAE) [3] and Attention Embedder (AE). When comparing backbone networks, ConvNeXt-B significantly outperforms ResNet-50 across all configurations. Furthermore, the AE leads to a further performance improvement compared to the NAE, especially when it is integrated with CN. Notably, the performance gains of AE are relatively less pronounced when combined with RN, suggesting that the compatibility between the backbone network and the embedder affects the performance of the framework. The results indicate that the combination of CN and AE is the optimal configuration. + +# 4 Conclusion + +In this paper, we focus on UDA Person Search, where existing methods suffer from noise accumulation caused by source-biased pseudo-labels. To bridge the domain gap, we propose the STCSP framework, which systematically suppresses noise contamination, avoids noise occurrence, and mitigates noise generation. With STC pipeline, IBEM method and SPCL module, our framework achieves SOTA performance. For future research, the IBEM algorithm can be extended to other tasks that demand the rapid completion of extensive bipartite matchings. + +# Acknowledgments and Disclosure of Funding + +This work was supported in part by the National Natural Science Foundation of China Grant 62576067 and 62501226, National Key Research and Development Program of China Grant 2024YFB4710800, Liaoning Provincial Natural Science Foundation Grant 2025-YQ-01 and 2024-MS-012, Dalian Science and Technology Talent Innovation Support Plan Grant 2024RY010, Natural Science Foundation of Hebei Province Grant F2025201037. + +# References + +[1] Mohammed Khaleed Almansoori, Mustansar Fiaz, and Hisham Cholakkal. Ddam-ps: Diligent domain adaptive mixer for person search. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 6688-6697, 2024. +[2] Xiaojun Chang, Po-Yao Huang, Yi-Dong Shen, Xiaodan Liang, Yi Yang, and Alexander G Hauptmann. 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In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1318–1327, 2017. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We introduced the research content and listed the contributions of this paper in Sec. 1. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: We derived the theoretical results in Sec. 2 and validated them in Sec. 3. + +# Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. 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$^{1}$ , Yue Yu $^{1*}$ , Ping Luo $^{1,2,3}$ + +Peng Cheng Laboratory + +$^{2}$ Key Lab of Intelligent Information Processing, Institute of + +Computing Technology, Chinese Academy of Sciences (ICT/CAS) + +3University of Chinese Academy of Sciences + +$^{4}$ Tianjin Laboratory Autonomous Intelligence Technology and Systems, School of + +Computer Science and Technology, Tiangong University + +$^{5}$ Institute for Advanced Algorithms Research, Shanghai + +lvbo19@mails.ucas.ac.cn + +# Abstract + +Ensembles of generative large language models (LLMs) are a promising way to compensate for individual model limitations, integrating the strengths of different LLMs. Existing LLM ensemble methods, however, face limitations such as first-token delay and challenges in long-range semantic collaboration between models. Moreover, they typically assume equal voting weights for all models during ensemble, ignoring task-specific performance differences among models. In this work, we propose SpecEM, a training-free, plug-and-play LLM ensemble framework that dynamically adjusts each model's model contribution in real time based on task performance. Inspired by speculative decoding, SpecEM iteratively performs drafting and verification, allowing models to collaborate semantically at the segment level for integrated output. Furthermore, we introduce an online feedback mechanism with multiplicative weight updates, where each model's voting weight is adjusted on-the-fly according to how often it outperforms others during verification stage, ensuring that stronger models exert greater influence during ensembling. Experimental results on five LLM families (ranging from 7B to 72B parameters) and six benchmark datasets, spanning open-domain instruction following, reasoning, commonsense, demonstrate consistent performance improvements compared to state-of-the-art LLM ensemble methods. Our code is available at https://github.com/lvbotenbest/SpecEM. + +# 1 Introduction + +Generative large language models (LLMs) [AI@Meta, 2024, Yang et al., 2024] have been widely adopted due to their impressive performance across a broad range of domains. Owing to differences in training data and model architectures, off-the-shelf generative LLMs often exhibit strengths in different areas. Consequently, ensembling multiple LLMs at inference time can help mitigate individual biases and errors, resulting in a more robust and reliable user experience. + +Existing LLM ensemble approaches [Chen et al., 2025, Jiang et al., 2023b, Huang et al., 2024, Lv et al., 2024a] can be broadly categorized into generate-then-ensemble [Jiang et al., 2023b, Lv et al., 2024b] and ensemble-while-generation [Huang et al., 2024, Yao et al., 2025] paradigms. The former typically generates full responses from all base models for a given query, then leverages an additional + +fusion model to summarize or select the best outputs. The latter adopts a more interleaved approach, greedily aggregating the output probabilities from different models at certain timesteps to decide next tokens, which is then broadcast to all models. + +Despite promising progress, generate-then-ensemble methods suffer from first-token delay, as users must wait until all models complete their responses before receiving integrated output. In contrast, ensemble-while-generation methods mitigate this latency but may fall short in enabling long-range semantic communications across models. Moreover, existing methods focus solely on aggregating model outputs, typically assuming equal contribution from all models, and overlooking the fact that different models may perform differently depending on the task. We argue that incorporating an online learning mechanism to dynamically assign higher weights to better-performing models while down-weighting weaker ones can lead to higher-quality ensemble outputs. + +Based on the above observations, we propose SpecEM, a training-free, plug-and-play LLM ensemble framework that performs segment-level fusion of model outputs and dynamically adjusts model weights on-the-fly based on task-specific performance. Inspired by speculative decoding [Xia et al., 2023, Leviathan et al., 2023a], SpecEM iteratively executes two key stages: drafting and verification. In the drafting stage, each base LLM generates a candidate text segment given the prior context, with a predefined maximum length per iteration. In the verification stage, all LLMs receive these candidate segments with prior context and mutually evaluate them in parallel based on their output logits. The top-ranked segment is then broadcast to all models, guiding them to generate higher-quality text in subsequent rounds. This iterative drafting-verification process eliminates the need for training fusion modules or selection aggregators, and allows for effortless integration of base LLMs without any fine-tuning. + +Furthermore, we introduce an online feedback mechanism in SpecEM to dynamically adjust each model's influence during the verification stage. It is based on the assumption that models capable of generating higher-quality segments in drafting are also better at evaluating others in verification. Specifically, we treat the number of times a model outperforms its peers in verification as a reward signal to update its voting weight using a multiplicative weight update algorithm. This ensures that stronger models progressively exert greater influence during generation. + +We evaluate SpecEM on five popular LLM families (ranging from 7B to 72B parameters) across six benchmark datasets, covering open-domain instruction following, reasoning, and commonsense. Experimental results demonstrate consistent performance improvements over state-of-the-art LLM ensemble methods. In summary, our contributions are as follows: + +- We propose SpecEM, a training-free and plug-and-play ensemble framework that integrates outputs by iteratively coordinating drafting and verification across multiple LLMs. +- We propose an online feedback mechanism that dynamically adjusts each model's contribution to inference and verification during generation, ensuring that stronger models exert greater influence in the ensemble. +- We conduct comprehensive evaluations on five LLM families and six benchmarks, showing that SpecEM consistently outperforms state-of-the-art ensemble methods. + +# 2 Related Work + +# 2.1 LLM Ensembling + +Recent efforts in LLM ensembling have shown that combining multiple models can enhance performance by leveraging their complementary strengths. These methods can be broadly categorized into generate-then-ensemble and ensemble-while-generation, depending on when and how the ensembling of model outputs occurs. + +Generate-then-ensemble methods first let each base LLM generate a complete response, and then aggregate the outputs through selection [Freitag et al., 2023] or fusion [Lv et al., 2024b]. Selection-based methods, such as MBR [Freitag et al., 2023] and PairRank [Jiang et al., 2023b], rank or compare candidates among all outputs to select the best one as the output. Fusion-based methods, such as GenFuse [Jiang et al., 2023b] and MOA [Wang et al., 2024], generate new outputs by using base model responses as input to a fusion model or aggregator. + +![](images/51f80acfcfa899f0a6aab46faf25da1a6d4844012574a997dca539b8a4cbbaf4.jpg) +Figure 1: Overview of SpecEM, a training-free plug-and-play LLM ensemble framework with three components: drafting, verification, and online feedback. The blue solid lines indicate a single iteration; dashed lines denote input refreshing for the next round. + +Ensemble-while-generation methods [Yu et al., 2024, Huang et al., 2024, Xu et al., 2024] aggregate model outputs during the generation process, typically by fusing output probability distributions to produce ensembling results incrementally. Due to vocabulary mismatches across different LLMs that hinder the combination of multiple probability distributions, Yu et al. [2024] construct a new union vocabulary by combining the vocabularies of multiple models to include all tokens from each model. They then project the distribution information from each model onto this merged vocabulary for averaging aggregation. Similarly, DeePEn [Huang et al., 2024] and EVA [Xu et al., 2024] project the output distributions of multiple models into a shared relative/pivot space, followed by averaging aggregation. While, these methods operate over all LLM's vocabulary at each timestep, which may leads to some computational overhead. More efficiently, UniTe [Yao et al., 2025] focuses only on the top-K portion of each model's output distribution and uses the union vocabulary strategy [Yao et al., 2025] to reduce alignment costs while maintaining good performance. + +While recent progress has been made, challenges remain in balancing efficiency with effective cross-model collaboration. In this work, we introduce SpecEM, a plug-and-play training-free framework that performs segment-level collaboration via iterative drafting and verification. Unlike previous methods that rely on static model contributions, SpecEM incorporates an online feedback mechanism to dynamically adjust each model's influence based on performance during generation, promoting more adaptive and effective ensembling. + +# 2.2 Speculative Decoding + +Speculative Decoding [Xia et al., 2023, Chen et al., 2023, Sun et al., 2024] aims to accelerate inference in LLMs by leveraging a lightweight draft model to propose multiple candidate tokens, which are then verified by a larger target model [Leviathan et al., 2023b]. Concretely, at each decoding step, the draft model efficiently generates a sequence of potential tokens, and the target model accepts only those that match its own predictions [Miao et al., 2024]. This significantly reduces the number of expensive forward passes through the larger model without compromising output quality. + +Inspired by this idea, we propose SpecEM, which reimagines speculative decoding for model assembling rather than acceleration. Instead of a small model drafting for a large one, multiple LLMs iteratively generate draft segments and verify each other's outputs in parallel. This collaborative refinement allows stronger models to guide weaker ones. In addition, SpecEM incorporates an online feedback mechanism that dynamically adjusts each model's influence during verification. + +# 3 Methodology + +Figure 1 presents an overview of SpecEM. SpecEM performs LLM ensembling through iterative drafting, verification, and online feedback, which are described in detail in the following subsections. + +![](images/7e51428c34a13a09216db8dfb043a804f115c867ca29405853f30f059a754d90.jpg) +Original Attention Mask Matrix + +![](images/e79a6143fa8124bd9ea482e988b3d20c8b4c766878a347bcdea6166bcefd099d.jpg) +Verify-In-line Attention Mask Matrix +Figure 2: An overview of verify-in-line attention mask and position id. + +# 3.1 Drafting Stage + +During the drafting stage, in each generation round, all models are simultaneously activated to perform parallel inference based on the task query and the best candidate segment broadcast from previous rounds. Formally, let the ensemble consist of models $\{M_i\}_N$ , where $M_{i}$ denotes the $i$ -th base LLM. In the $k$ -th iteration round, the draft candidate segment generated by $M_{i}$ is denoted as $C_i^{(k)} = \{t_1^{(k)},\dots,t_l^{(k)}\}$ , where the number of generated tokens $l$ is constrained by a predefined maximum segment length $L$ . + +$$ +C _ {i} ^ {(k)} = M _ {i} \left(I ^ {(0)}, \dots , I ^ {(k - 1)}\right) \tag {1} +$$ + +Here, $I^{(k-1)}$ denotes the best candidate segment broadcast in the $k$ -th round, and $I^{(0)}$ corresponds to the initial task query. + +# 3.2 Verification Stage + +During the verification stage, all models perform mutual evaluation on the candidate segments generated during the drafting stage. At iteration $k$ , each model receives the full set of candidate segments $\{C_i^{(k)}\}_N$ , along with the prior context $\{I^{(0)}, \dots, I^{(k-1)}\}$ . Each model scores the candidates, and the one with the highest aggregated score is selected as the output of the current round. This selected segment is then broadcast to all models as part of the context for the next generation round. As illustrated in the mutual evaluation matrix in Figure 1, let $s_{i,j}^{(k)}$ denote the score assigned by model $i$ to candidate $j$ in generation round $k$ . Concretely, for each candidate segment, we compute the average of the logits produced by the model over the tokens in the segment, which serves as the model's evaluation score [Lv et al., 2023, Varshney et al., 2023, Lv et al., 2025] for that candidate: + +$$ +s _ {i, j} ^ {(k)} = \frac {1}{l} \sum_ {u = 1} ^ {l} p \left(t _ {i, j, u} ^ {(k)}\right) \tag {2} +$$ + +where $p(t_{i,j,u}^{(k)})$ denotes the logit score by model $i$ in round $k$ for the $u$ -th token of candidate $j$ . To mitigate scoring bias caused by some models producing systematically higher or lower logits, we normalize the scores that the model assigns to all candidates before aggregation: + +$$ +s _ {i, j} ^ {(k)} \leftarrow \frac {s _ {i , j} ^ {(k)}}{\sum_ {j = 1} ^ {N} s _ {i , j} ^ {(k)}} \tag {3} +$$ + +Finally, the overall score $\{y_{j}\}_{N}$ of each candidate is computed as a weighted sum of the scores from all verifier models: + +$$ +y _ {j} ^ {(k)} = \sum_ {i = 1} ^ {N} \omega_ {i} ^ {(k)} s _ {i, j} ^ {(k)} \tag {4} +$$ + +with weights $\{\omega_{i}\}_{N}$ dynamically updated by the online feedback mechanism described in Section 3.3. + +Specially, we introduce a verify-in-line mechanism, avoiding redundant attention computations over the prior context and the increased time complexity caused by serial model-wise scoring. Concretely, at the $k$ -th generation iteration, we concatenate the prior context and all candidate segments along the sequence dimension to construct a unified input sequence: + +$$ +L I N E = \left[ \mathbb {I} ^ {(k - 1)}: C _ {1} ^ {(k)}: C _ {2} ^ {(k)}: C _ {N} ^ {(k)} \right] \tag {5} +$$ + +where $\mathbb{I}^{(k - 1)}$ denotes the prior context $[I^{(0)}:\dots :I^{(k - 1)}]$ for brevity, and $[\cdot ]$ denotes sequence-wise concatenation. We then modify the attention mask and position IDs in the Transformer such that each model can process LINE and output scores for all candidates in parallel, as shown in Figure 2: + +Verify-in-line attention mask. During verification, each candidate segment should only attend to the shared prior context $\mathbb{I}^{(k - 1)}$ , without accessing information from other candidate segments. To enforce this, we augment the standard triangular attention mask of the decoder-only Transformer with an additional masking scheme that blocks attention across candidate segments. This ensures that tokens in $C_i^{(k)}$ can only attend to $\mathbb{I}^{(k - 1)}$ , not to tokens in other candidates $C_j^{(k)}$ for $j \neq i$ . As a result, although all candidates are concatenated into LINE, model $M_i$ effectively "sees" only $[\mathbb{I}^{(k - 1)} : C_i^{(k)}]$ during scoring, enabling efficient and parallel evaluation. + +Verify-in-line position IDs. While the modified attention mask guarantees the correct visibility, the default position encoding would still reflect the segments' physical positions in the concatenated LINE, which may distort modeling. For instance, the actual input sequence for scoring $C_2^{(k)}$ is $[\mathbb{I}^{(k-1)} : [mask] : C_2^{(k)} : [mask] : \ldots]$ rather than $[\mathbb{I}^{(k-1)} : C_2^{(k)}]$ . To address this, we further adapt LLMs' relative positional encoding so that each candidate segment is positioned as if it were immediately following the prior context. That is, the position IDs for each $C_i^{(k)}$ are reset to be consecutive with $\mathbb{I}^{(k-1)}$ , ensuring the position modeling remains consistent with the actual evaluation context. + +# 3.3 Online Feedback Mechanism + +It is generally difficult to anticipate which model performs best on a given query. Due to differences in model architectures and training corpora, different models may exhibit varying strengths across domains, and may produce low-quality outputs when encountering unfamiliar or challenging topics. Since the verification stage relies on each model's scoring of candidate segments, models with weaker generation capabilities may also produce unreliable evaluations when acting as validationators. + +We propose a core assumption: models that perform better in generation tend to offer more reliable judgments during verification. This assumption is empirically supported in Appendix A. Building on this, we introduce an online feedback mechanism based on the multiplicative weights update algorithm. It dynamically adjusts each model's contribution in the verification stage by tracking and weighting its validation performance during the generation process. As a result, decisions from better-performing models are prioritized, enhancing the overall ensemble effectiveness. + +Formally, let there be $N$ models. Denote the verification weight of model $M_{i}$ at generation round $k$ as $\omega_{i}^{(k)}$ . All models are initially assigned uniform weights: $\omega_{i}^{(0)} = \frac{1}{N}$ . At round $k$ , model $M_{i}$ receives a feedback reward $\gamma_{i}^{(k)}$ , and its weight is updated according to: + +$$ +\omega_ {i} ^ {(k)} = \omega_ {i} ^ {(k - 1)} \cdot e ^ {\eta \gamma_ {i} ^ {(k)}} \tag {6} +$$ + +To address the fact that more models (i.e. $N$ ) lead to smaller initial weights, and to ensure that updates become more stable over time (i.e. $k$ ), we define the learning rate $\eta$ in Eq. 6 as: + +$$ +\eta = \alpha \cdot \frac {\sqrt {1 / k}}{N} \tag {7} +$$ + +where $\alpha$ is a hyperparameter. Then all weights are normalized as: + +$$ +\omega_ {i} ^ {(k)} \leftarrow \frac {\omega_ {i} ^ {(k)}}{\sum_ {j = 1} ^ {n} \omega_ {j} ^ {(k)}} \tag {8} +$$ + +This feedback-driven reweighting ensures that more credible placeholders exert greater influence on the selection process, leading to progressively improved collective decisions over time. + +Reward Definition. For each model $M_{i}$ , we define its reward $\gamma_{i}^{(k)}$ in Eq. 6 based on how often its generated segment is preferred over others in the evaluations conducted by the remaining models. Specifically, we count the number of times model $M_{i}$ 's candidate $C_{i}^{(k)}$ is scored higher than another + +candidate $C_r^{(k)}$ by a third model $M_j$ , where $j \neq i$ and $r \neq i$ , and normalize this count as reward $\gamma_i^{(k)}$ : + +$$ +\gamma_ {i} ^ {(k)} = \frac {\sum_ {j \neq i} \sum_ {r \neq i} b o o l \left(s _ {j , i} ^ {(k)} > s _ {j , r} ^ {(k)}\right)}{\sum_ {i = 1} ^ {N} \sum_ {j \neq i} \sum_ {r \neq i} b o o l \left(s _ {j , i} ^ {(k)} > s _ {j , r} ^ {(k)}\right)} \tag {9} +$$ + +Here, $bool(\cdot)$ returns 1 if the condition inside is true, and 0 otherwise. Intuitively, if a model's candidate frequently outperforms others in peer evaluations, it is considered better suited to the current task. In Section 4.3, we empirically compare this reward formulation against alternative segment selection strategies. Finally, the comprehensive score for each candidate $\{y_{j}^{(k)}\}_{N}$ is computed as the weighted sum of its scores assigned by all validationators, following Equation 4. The highest-scoring candidate is selected as the best output $I^{(k)}$ , and it is appended to the prior context for use in the next round of drafting and verification. + +# 4 Experiments + +# 4.1 Experimental Setup + +Datasets. We evaluate SpecEM on six datasets that reflect key capabilities of LLMs, including open-domain instruction following, commonsense, and reasoning. FuseEval: A multilingual instruction response benchmark we construct by combining Dolly-15k [Conover et al., 2023] and AlpacaGPT4 [Peng et al., 2023] for English, and Human-Value and Ruozb from COIG-CQIA [Bai et al., 2024] for Chinese. IFEval [Zhou et al., 2023]: Evaluates instruction adherence under four granular settings, prompt-strict/loose, instruction-strict/loose. AlpacaEval 2.0 [Dubois et al., 2024]: Measures alignment with human preferences via GPT-4 based pairwise comparisons against GPT-4 outputs. MMLU (5-shot) [Hendrycks et al., 2021] and ARC-C (5-shot) [Clark et al., 2018]: Multiple-choice benchmarks that test factual knowledge and general commonsense. GSM8K (3-shot) [Cobbe et al., 2021]: Focuses on arithmetic and multi-step reasoning through grade-school math problems. x-shot refers to providing x examples as in-context during inference. Please refer to Appendix B.1 for a detailed description of datasets. + +Base LLMs. We use top-performing open-source instruction-tuned models (7B-9B) as base LLMs in our ensemble, including Llama-3-8B-instruct [AI@Meta, 2024], Mistral-7B-v0.3-instruct [Jiang et al., 2023a], Qwen2-7B-instruct [Yang et al., 2024], Glm-4-9b-instruct [GLM et al., 2024], and Gemma-2-9b-instruct [Gemma et al., 2024]. Moreover, to assess scalability, we also evaluate SpecEM with larger base models, including Qwen2-72B-instruct, Llama3-70B-instruct, Qwen2.5-32B-instruct[Qwen et al., 2025], and Mistral-24B-instruct-2501 [Team., 2025]. + +Metrics. We follow standard evaluation protocols for each task in previous works. For FuseEval, we use BARTScore[Yuan et al., 2021], BERTScore[Zhang et al., 2019], BLEU[Papineni et al., 2002], ROUGE [Lin, 2004], and GPT4-Rank [OpenAI et al., 2024] to assess reference-based generation quality. MMLU and ARC-C report accuracy by selecting the option with the highest likelihood. For GSM8K, we compute exact match accuracy based on the predicted answer. For IFEval, we use the provided evaluation files to test under prompt/instruction-strict and -loose conditions. AlpacaEval 2.0 reports length-controlled (LC) win rates against GPT-4 outputs using gpt-4-1106-preview. + +Comparative methods. We compare our proposed SpecEM with several strong recent LLM ensemble methods, including PairRank [Jiang et al., 2023b], Minimum Bayes Risk (MBR) [Freitag et al., 2023], Generation Fusion (GF) [Jiang et al., 2023b], Mixture-of-Agents (MOA) [Wang et al., 2024], Majority Voting [Davani et al., 2022], and Unite [Yao et al., 2025]. Please refer to Appendix B.3 for a detailed description of these baseline methods. + +Implement details. SpecEM requires no training and operates purely during inference. All models are loaded using bfloat16 precision, with do_sample = True, temperature = 0.6, and top_p = 0.9 generation settings. For experiments with 7B-9B models, we use A100 GPUs, while larger models (24B-72B) are evaluated on H200 GPUs. The maximum number of candidate segments is set to $L = 10$ , and the online feedback hyperparameter is set to $\alpha = 1$ . All reported results are averaged over three independent runs to ensure stability. + +Table 1: Results on the English and Chinese subsets of the FuseEval benchmark. Pink highlights the best overall result, and Blue marks the best result among base LLMs. The upward arrow $\uparrow$ means higher is better, and the downward arrow $\downarrow$ means lower is better. + +
ModelROUGE-1↑ROUGE-2↑ROUGE-L↑BLEU↑BARTScore↑BERTScore↑GPT4-Rank↓
English Scenario
Base LLMs
Llama-3-8B-instruct [AI@Meta, 2024]25.169.7723.313.57-2.9869.999.52
Glim-4-9B-instruct [GLM et al., 2024]25.8510.2623.903.48-2.9670.519.24
Qwen2-7B-instruct [Yang et al., 2024]26.6210.8124.493.86-2.9471.448.48
Gemma-2-9B-instruct [Gemma et al., 2024]25.3110.0123.594.19-2.9371.529.01
Mistral-7B-instruct-v0.3 [Jiang et al., 2023a]27.7510.7525.574.82-2.9471.887.62
Larger LLMs
Llama-3-70B-instruct [AI@Meta, 2024]26.7710.8724.564.10-2.8470.985.22
Qwen2-72B-instruct [Yang et al., 2024]27.2611.2325.114.29-2.7671.734.21
Mixtral-8x7B-instruct [Jiang et al., 2023a]29.0412.2526.754.08-2.8172.194.16
Methods of Ensembling Base LLMs
GF (Qwen2) [Jiang et al., 2023b]23.088.9221.283.19-2.9569.7010.10
GF (Gemma-2) [Jiang et al., 2023b]21.817.6620.083.00-3.0268.2010.11
GF (Mistral) [Jiang et al., 2023b]24.929.5822.973.92-2.9370.388.56
MBR [Freitag et al., 2023]27.1210.4025.334.56-2.8971.637.66
PairRank [Jiang et al., 2023b]28.2110.8625.944.99-2.8672.096.84
MOA [Wang et al., 2024]27.6111.3025.475.12-2.8871.907.48
UniTE [Yao et al., 2025]27.8110.9125.734.53-2.9071.776.00
SpecEM (Ours)31.1914.4028.865.81-2.8873.343.98
Chinese Scenario
Base LLMs
Gemma-2-9B-instruct [Gemma et al., 2024]29.157.6518.353.36-4.2868.738.58
Mistral-7B-instruct-v0.3 [Jiang et al., 2023a]30.998.6520.664.42-4.4870.106.55
Qwen2-7B-instruct [Yang et al., 2024]29.938.0920.033.62-4.3369.996.42
Glim-4-9B-instruct [GLM et al., 2024]30.888.7120.424.47-4.3070.255.18
Larger LLMs
Llama-3-70B-instruct [AI@Meta, 2024]27.787.0520.224.14-4.5568.527.38
Qwen2-72B-instruct [Yang et al., 2024]31.448.9722.484.88-4.3470.653.63
Methods of Ensembling Base LLMs
GF (Mistral) [Jiang et al., 2023b]30.298.1220.333.88-4.5470.047.28
GF (Qwen2) [Jiang et al., 2023b]28.697.8718.933.32-4.4169.818.41
GF (Glim-4) [Jiang et al., 2023b]30.268.7020.514.27-4.3370.235.40
MBR [Freitag et al., 2023]30.938.7120.634.31-4.3170.235.33
UniTE [Yao et al., 2025]27.468.2220.542.83-4.6067.589.37
MOA [Wang et al., 2024]30.968.5020.604.36-4.3170.115.93
SpecEM (Ours)32.159.9424.004.75-4.2971.033.25
+ +# 4.2 Main Results + +Results on Diverse Evaluation Benchmarks. Table 1 presents results on the English and Chinese subsets of the FuseEval benchmark. SpecEM, built on 7B-9B base models, outperforms all individual LLMs and existing ensemble methods across all metrics. + +Notably, it achieves over 3-point average gains in ROUGE1/2/L and ranks highest on GPT4-Rank. Despite using only several 7B-9B LLMs, SpecEM performs comparably to 70B-scale single models while remaining more parameter efficient. The improvements are consistent in the English and Chinese scenarios, indicating the generalizability of SpecEM across languages. We further assess SpecEM on MMLU, ARC-C, GSM8K, and IFEval benchmarks using base + +Table 3: Performance on FuseEval and AlpacaEval 2.0 benchmarks. Win rates on FuseEval are measured relative to the outputs of Qwen2-72b-instruct. + +
ModelEnglish FuseEval (winrate)Chinese FuseEval (winrate)AlpacaEval 2.0 (LC-winrate)Avg
Base LLMs
Qwen2-72b-instruct--38.10-
Qwen2.5-32b-instruct49.3137.4843.8243.54
Llama3-70b-instruct43.1010.6634.4029.39
Mistral-24b-instruct-250152.5531.7948.4644.27
Methods of Ensembling Base LLMs
MOA53.63 (+1.08)53.12 (+15.64)46.98 (-1.48)51.24
GenFuse51.06 (-1.49)52.41 (+14.93)49.06 (+0.60)50.84
UniTE54.79 (+2.24)19.13 (-18.35)49.20 (+0.74)41.04
SpecEM55.46 (+2.91)56.77 (+19.29)51.32 (+2.86)54.52
+ +Table 2: Results on MMLU, ARC-C, GSM8K, and IFEval benchmarks. Values in parentheses indicate performance difference from the best-performing base model in ensemble in each column. + +
ModelMMLUARC-CGSM8KIFEval
prompt-avginstruct-avg
Base LLMs
Qwen2-7B-instruct68.2384.7374.2241.7053.88
GLM-4-9B-instruct67.1685.1571.8056.0167.14
Gemma-2-9B-instruct71.5188.1477.2661.6472.26
Methods of Ensembling Base LLMs
Majority-Voting71.78 (+0.27)88.38 (+0.24)77.29 (+0.03)--
MBR--76.98 (-0.28)54.96 (-6.68)66.21 (-6.05)
MOA70.43 (-1.08)88.28 (+0.14)77.30 (+0.04)60.80 (-0.84)68.81 (-3.45)
UniTE71.94 (+0.43)88.54 (+0.40)76.52 (-0.74)56.72 (-4.92)62.08 (-10.18)
SpecEM (Qwen2+GLM4)70.73 (+2.50)87.54 (+2.39)75.44 (+1.22)51.15 (-4.86)63.07 (-4.07)
SpecEM (Qwen2+Gemma2)72.18 (+0.67)88.40 (+0.26)78.70 (+1.44)56.01 (-5.63)67.39 (-4.87)
SpecEM (GLM4+Gemma2)71.82 (+0.31)88.74 (+0.60)75.82 (-1.44)66.89 (+5.25)75.52 (+3.26)
SpecEM (All)73.01 (+1.50)89.08 (+0.94)77.41 (+0.15)62.11 (+0.47)71.56 (-0.70)
+ +LLMs with varied strengths. + +As shown in Table 2, SpecEM consistently surpasses all baseline ensemble methods across these benchmarks. In particular, SpecEM (Qwen2 + GLM4) achieves +2.5 and +2.4 improvements on MMLU and ARC-C, respectively, leveraging complementary model capabilities. These results demonstrate the effectiveness of SpecEM across diverse task formats beyond open-ended generation. + +Scaling to Larger Models. To further evaluate the scalability of our framework, we conduct experiments by integrating four larger LLMs ranging from 24B to 72B parameters on FuseEval and AlpacaEval 2.0. As shown in Table 3, SpecEM consistently outperforms all base models and prior ensemble baselines, achieve the best performance. In particular, it surpasses the strongest single base model by an average win rate margin of 10.3 points. These results demonstrate that SpecEM generalizes robustly across model sizes scale. + +Table 4: Win-rate comparisons between SpecEM and ablations on English (EN) and Chinese (CN) FuseEval. Each cell shows the win/loss percentage judged by GPT-4o-2024-11-20. $\Delta$ denotes average win-rate improvement over ablations, and Avg reflects the mean win rate across EN and CN. + +
ComparisonEN (win/lose)CN (win/lose)Δ (EN / CN)Avg Δ
SpecEM vs w/o Online feedback53.66 / 46.3452.98 / 47.02+7.32 / +5.96+6.64
SpecEM vs w/ Feedback (Score-based reward)52.17 / 47.8351.63 /48.37+4.34 / +3.26+3.80
SpecEM vs w/o Feedback, Top-win selection53.40 / 46.6052.35 / 47.65+6.80 / +4.70+5.75
+ +# 4.3 Analysis + +Ablation analysis. We perform ablation studies to assess the effect of the online feedback mechanism in the verification stage. Using GPT-4 as the evaluator, we compare the full SpecEM with three variants: (a) w/o online feedback: Removes feedback; final output is selected solely based on verification scores. (b) w/ score-based reward: Replaces win count $\gamma_{i}$ with the normalized average verification score as the reward. (c) w/o feedback, top-win selection: Selects the candidate with the highest win count without reward accumulation. As shown in Figure 4, the full SpecEM outperforms all variants. Online feedback yields a 6.6-point average gain; using win count as the reward offers an additional 2.8-point improvement over score-based reward. Directly selecting the top-win segment leads to a 5.8-point drop, likely due to close or tied win counts, which are more reliable as soft rewards than as decisive selection criteria. + +Inference Latency Analysis. We evaluate the inference efficiency of SpecEM against other ensemble methods and base models, focusing on two key metrics shown in Figure 3: (1) First Token Latency, the time from user input to the generation of the first token, which is crucial for interactive user experience; (2) Total Generation Time, the time to generate a complete response across varying output lengths. SpecEM achieves the lowest total response time among all ensemble methods, with + +![](images/097d20da2d253d67a1b0306c09182f404237a53dc7efedffd972ccfcb0856f00.jpg) +Figure 3: Comparison of inference latency performance across methods. Left: total generation time (seconds); Right: first-token latency (seconds), both plotted against the number of generated tokens. + +![](images/b58fea0862328bc079e0271b6b12d7f5b5cc57eb70953bbac2158444b1e368ec.jpg) + +only a $20\%$ overhead compared to the slowest single model. This is because, under parallel inference settings, the ensemble latency is bottlenecked by the slowest model's output time. + +Crucially, SpecEM maintains consistently low first token latency (under 0.6s) across all lengths, enabling fast user feedback. In contrast, methods that wait for full outputs before fusion suffer from rapidly increasing first token latency, making them unsuitable for real-time applications. + +Base Model Number Analysis. We evaluate how SpecEM scales with the number of integrated base LLMs on the English FuseEval dataset. As shown in Figure 4, performance consistently improves as more base models are added. The improvements are more pronounced when stronger models are introduced, while even weaker models still contribute positively. These results highlight the flexibility and scalability of SpecEM, where new models can be seamlessly integrated without additional training or adaptation, making the system robust and extensible in real-world deployment. + +![](images/9a055b6c4056465a4ff6d393ea5d46e72e058468ecf4aa3b3485d249d687c6b2.jpg) +Figure 4: The variation in SpecEM's ROUGE-L score as the number of base LLMs increases. $+$ [model] indicates the incremental addition of a specific model to the ensemble. + +Candidate Segment Length Analysis. We study the impact of the maximum generation length $L$ of + +candidate segments on SpecEM's performance using the English FuseEval development set. As shown in Figure 5 (left), both BERTScore and ROUGE-L improve as $L$ increases, peaking at $L = 10$ , then gradually decrease. This trend arises because shorter segments lack sufficient information, which weakens the judgment of the verification component and limits the mutual inspiration between models. Conversely, overly long segments reduce the frequency of cross-model interactions, hindering effective knowledge fusion and ultimately degrading the final output quality. + +![](images/ac11e63d71bf9d093469163b5eb9d57346863440a5d8f6aeb17bae47c65c657f.jpg) +Figure 5: Performance trends of SpecEM. Left: Varying maximum candidate segment length. Right: Varying hyperparameter $\alpha$ in the online feedback. + +![](images/f0534aed3dc6ed82264932e0cc1ca2779d9dec5525feaf90b681c3785b356038.jpg) + +Online Feedback Hyperparameter Analysis. To investigate the impact of the hyperparameter $\alpha$ in learning rate in online feedback mechanism, we vary its value from 0.1 to 2 and evaluate the + +results on the English FuseEval development set. As shown in Figure 5 (right), both BERTScore and ROUGE-L scores initially increase with larger $\alpha$ , stabilize around the range of 0.8-1.3, and then begin to decline. A very small $\alpha$ leads to insufficient updates, limiting the effectiveness of online feedback, while an overly large $\alpha$ causes instability and degrades performance. + +# 5 Conclusion + +We propose SpecEM, a training-free and plug-and-play ensemble framework for generative LLMs. SpecEM integrates model outputs through an iterative drafting-verification process at the segment level, enabling semantic collaboration across models without additional training. Further, we introduce an online feedback mechanism that dynamically adjusts each model's influence during generation based on real-time performance. Experiments across six datasets and five LLM families ranging from 7B to 72B parameters show that SpecEM can effectively coordinate multiple LLMs, demonstrating strong generalization across model scales, task types, and languages. + +# 6 Limitations + +Although SpecEM enables flexible integration of new models without additional training, it still faces challenges in the ensemble process. Specifically, introducing a model that performs poorly on the current task can degrade the overall performance, sometimes even falling below that of an ensemble excluding the weaker model. For instance, as shown in Table 2, on the IFEval dataset, SpecEM integrating Qwen2, GLM4, and Gemma2 underperforms compared to the ensemble of only GLM4 and Gemma2. While we adopt an online feedback mechanism to dynamically adjust each model's contribution, low-quality outputs from weaker models can still negatively affect the initial generation phase. In future work, we plan to explore rejection sampling and resampling strategies to more effectively identify and amplify stronger models in the generation process, while suppressing the influence of weaker ones. + +# 7 Acknowledgments + +This work was supported by the National Key Research and Development Program of China (No. 2022ZD0115301), Major Key Project of PCL via grant No. PCL2025AS11, National Natural Science Foundation of China (NSFC) via grant 62206140,62406223. Thanks for the support provided by OpenI Community (https://openi.pcl.ac.cn). + +# References + +AI@Meta. Llama 3 model card, 2024. URL https://github.com/meta-llama/llama3/blob/main/MODEL_CARD.md. +Yuelin Bai, Xinrun Du, Yiming Liang, Yonggang Jin, Ziqiang Liu, Junting Zhou, Tianyu Zheng, Xincheng Zhang, Nuo Ma, Zekun Wang, et al. Coig-cqia: Quality is all you need for chinese instruction fine-tuning, 2024. +Charlie Chen, Sebastian Borgeaud, Geoffrey Irving, Jean-Baptiste Lespiau, Laurent Sifre, and John Jumper. 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Enabling ensemble learning for heterogeneous large language models with deep parallel collaboration, 2024. +Albert Q. Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, Lélio Renard Lavaud, Marie-Anne Lachaux, Pierre Stock, Teven Le Scao, Thibaut Lavril, Thomas Wang, Timothée Lacroix, and William El Sayed et al. Mistral 7b, 2023a. URL https://arxiv.org/abs/2310.06825. +Dongfu Jiang, Xiang Ren, and Bill Yuchen Lin. Llm-blender: Ensembling large language models with pairwise ranking and generative fusion, 2023b. URL https://arxiv.org/abs/2306.02561. +Yaniv Leviathan, Matan Kalman, and Yossi Matias. Fast inference from transformers via speculative decoding, 2023a. URL https://arxiv.org/abs/2211.17192. + +Yaniv Leviathan, Matan Kalman, and Yossi Matias. Fast inference from transformers via speculative decoding, 2023b. URL https://arxiv.org/abs/2211.17192. +Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Ves Stoyanov, and Luke Zettlemoyer. Bart: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension, 2019. URL https://arxiv.org/abs/1910.13461. +Chin-Yew Lin. Rouge: A package for automatic evaluation of summaries. In Text summarization branches out, pages 74-81, 2004. +Bo Lv, Xin Liu, Shaojie Dai, Nayu Liu, Fan Yang, Ping Luo, and Yue Yu. DSP: Discriminative soft prompts for zero-shot entity and relation extraction. In Anna Rogers, Jordan Boyd-Graber, and Naoaki Okazaki, editors, Findings of the Association for Computational Linguistics: ACL 2023, pages 5491-5505, Toronto, Canada, July 2023. Association for Computational Linguistics. doi: 10.18653/v1/2023-findings-acl.339. URL https://aclanthology.org/2023-findings-acl.339/. +Bo Lv, Xin Liu, Kaiwen Wei, Ping Luo, and Yue Yu. TAcEKD: Teacher assistant enhanced knowledge distillation for closed-source multilingual neural machine translation. In Nicoletta Calzolari, Min-Yen Kan, Veronique Hoste, Alessandro Lenci, Sakriani Sakti, and Nianwen Xue, editors, Proceedings of the 2024 Joint International Conference on Computational Linguistics, Language Resources and Evaluation (LREC-COLING 2024), pages 15530-15541, Torino, Italia, May 2024a. ELRA and ICCL. URL https://aclanthology.org/2024.lrec-main.1350/. +Bo Lv, Chen Tang, Yanan Zhang, Xin Liu, Ping Luo, and Yue Yu. URG: A unified ranking and generation method for ensembling language models. In Lun-Wei Ku, Andre Martins, and Vivek Srikumar, editors, Findings of the Association for Computational Linguistics ACL 2024, pages 4421-4434, Bangkok, Thailand and virtual meeting, August 2024b. Association for Computational Linguistics. doi: 10.18653/v1/2024-findings-acl.261. URL https://aclanthology.org/2024.findings-acl.261. +Bo Lv, Nayu Liu, Yang Shen, Xin Liu, Ping Luo, and Yue Yu. Whether LLMs know if they know: Identifying knowledge boundaries via debiased historical in-context learning. In Wanxiang Che, Joyce Nabende, Ekaterina Shutova, and Mohammad Taher Pilehvar, editors, Findings of the Association for Computational Linguistics: ACL 2025, pages 19516-19528, Vienna, Austria, July 2025. Association for Computational Linguistics. ISBN 979-8-89176-256-5. doi: 10.18653/v1/2025-findings-acl.999. URL https://aclanthology.org/2025-findings-acl.999/. +Xupeng Miao, Gabriele Oliaro, Zhihao Zhang, Xinhao Cheng, Zeyu Wang, Zhengxin Zhang, Rae Ying Yee Wong, Alan Zhu, Lijie Yang, Xiaoxiang Shi, Chunan Shi, Zhuoming Chen, Daiyaan Arfeen, Reyna Abhyankar, and Zhihao Jia. Specinfer: Accelerating large language model serving with tree-based speculative inference and verification. 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URL https://arxiv.org/abs/2304.03277. +Qwen,:, An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, Huan Lin, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang, Jiaxi Yang, Jingren Zhou, Junyang Lin, Kai Dang, Keming Lu, Keqin Bao, Kexin + +Yang, Le Yu, Mei Li, Mingfeng Xue, Pei Zhang, Qin Zhu, Rui Men, Runji Lin, Tianhao Li, Tianyi Tang, Tingyu Xia, Xingzhang Ren, Xuancheng Ren, Yang Fan, Yang Su, Yichang Zhang, Yu Wan, Yuqiong Liu, Zeyu Cui, Zhenru Zhang, and Zihan Qiu. Qwen2.5 technical report, 2025. URL https://arxiv.org/abs/2412.15115. +Ziteng Sun, Ananda Theertha Suresh, Jae Hun Ro, Ahmad Beirami, Himanshu Jain, and Felix Yu. Spectr: Fast speculative decoding via optimal transport, 2024. URL https://arxiv.org/abs/2310.15141. +Mistral AI Team. Mistral small 3: Apache 2.0, 81 URL https://mistral.ai/news/mistral-small-3. +Neeraj Varshney, Wenlin Yao, Hongming Zhang, Jianshu Chen, and Dong Yu. 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URL https://arxiv.org/abs/2407.10671. +Yuxuan Yao, Han Wu, Mingyang Liu, Sichun Luo, Xiongwei Han, Jie Liu, Zhijiang Guo, and Linqi Song. Determine-then-ensemble: Necessity of top-k union for large language model ensembling, 2025. URL https://arxiv.org/abs/2410.03777. +Yao-Ching Yu, Chun-Chih Kuo, Ziqi Ye, Yu-Cheng Chang, and Yueh-Se Li. Breaking the ceiling of the llm community by treating token generation as a classification for assembling, 2024. URL https://arxiv.org/abs/2406.12585. +Weizhe Yuan, Graham Neubig, and Pengfei Liu. Bartscore: Evaluating generated text as text generation. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 27263-27277. Curran Associates, Inc., 2021. URL https://proceedings.neurips.cc/paper/2021/file/e4d2b6e6fdeca3e60e0f1a62fee3d9dd-Paper.pdf. +Tianyi Zhang, Varsha Kishore, Felix Wu, Kilian Q. Weinberger, and Yoav Artzi. Bertscore: Evaluating text generation with BERT. CoRR, abs/1904.09675, 2019. URL http://arxiv.org/abs/1904.09675. +Jeffrey Zhou, Tianjian Lu, Swaroop Mishra, Siddhartha Brahma, Sujoy Basu, Yi Luan, Denny Zhou, and Le Hou. Instruction-following evaluation for large language models, 2023. URL https://arxiv.org/abs/2311.07911. + +# A Empirical Validation of the Correlation Between Generation and Verification Capabilities of LLMs. + +![](images/ba59f6ec2a8cb5360e7da42aacba40d8811fff94f4f5fd81ffc1b0e116140a68.jpg) +Figure 6: Win rates of four LLMs compared to Llama-3-8B-instruct, evaluated by GPT-4. Orange bars indicate generation performance (as response generators); Blue bars indicate verification performance (as candidate scorers for LLaMA). The consistent ordering supports the hypothesis that stronger generators are also stronger verifiers. + +In Section 3.3, we hypothesize that models with stronger generation capabilities also possess stronger verification abilities, to conduct online feedback mechanism in SpecEM. To verify the assumption, we design an empirical experiment. Specifically, we use GPT-4 as an external evaluator, and evaluate four models, Mistral-7B-v0.3-instruct, Qwen2-7B-instruct, Gemma-2-9b-instruct, and Glm-4-9b-instruct, by comparing their responses against those generated by Llama-3-8B-instruct on FuseEval. The orange bars in Figure 6 report the win rates of these models over LLaMA, as judged by GPT-4. The results indicate a consistent performance ranking: Mistral $>$ Qwen $>$ Genma $>$ GLM, with all models outperforming LLaMA. + +To assess verification ability, we treat each of the four models as a verifier. For every input example, we sample four candidate responses from LLaMA. Each verifier independently scores all candidates using the method described in Section 3.2, and selects the one with the highest score. The selected responses are then evaluated by GPT-4 against the original single-response baseline from LLaMA. The blue bars in Figure 6 show the win rates of the verifier-selected responses, again consistently ranked as Mistral $>$ Qwen $>$ Genma $>$ GLM. + +This alignment in performance ordering across generation and verification supports our central hypothesis: models with stronger generation capabilities are more competent as verifiers. For example, Mistral not only achieves the highest generation win rate, but also, when acting as a verifier, selects the strongest responses for LLaMA, demonstrating its superior verification ability. + +# B Experimental Setup + +# B.1 Dataset Details + +We evaluate all the models on six datasets that represent different core capabilities of LLMs, including open-domain instruction-following, commonsense, and reasoning. + +- FuseEval: We evaluate the model's instruction-response capability by constructing this category on both English and Chinese scenarios. For English, we choose the Dolly-15k [Conover et al., 2023] and Alpaca-gpt4 [Peng et al., 2023] datasets for evaluation, both of which have inputs that consist of human instructions. We select these two datasets because their response sources differ: the Dolly-15k dataset features human-provided responses, while the Alpaca-GPT4 contains responses generated by the state-of-the-art GPT-4 [OpenAI et al., 2024] model, which provides neutral reference answers to each question and can refuse to answer inappropriate or harmful ones. Using both types of responses for scoring allows us to more thoroughly compare the advantages of our ensemble system. Additionally, due to the large size of these datasets, we randomly sample portions from each to create a new test set and a development set. For Chinese, we utilize the Human-Value and Ruozb datasets from the COIG-CQIA [Bai et al., 2024] benchmark for testing. The instructions in these two datasets consist of human-posed questions, with answers provided either by humans or generated by GPT-4. The COIG-CQIA authors manually review and filter the responses, retaining only the correct answers generated by GPT-4. +- AlpacaEval 2.0 [Dubois et al., 2024]: This is an automated benchmark for evaluating large language models' instruction-following capabilities. It employs GPT-4 Preview (11/06) as an evaluator to compare model responses against a baseline (also GPT-4 Preview (11/06)), computing win rates with a length-controlled scoring mechanism to reduce morbidity bias. Given the strong performance of 24-72B foundation models, we further conducted direct comparisons between ensemble model outputs and GPT-4 generated responses. +- MMLU (5-shot) [Hendrycks et al., 2021]: A widely-used massive multitask language understanding benchmark for evaluating knowledge and commonsense reasoning across 57 subjects, including STEM, humanities, and social sciences. It assesses models' breadth of understanding across a diverse set of multiple-choice questions. x-shot refers to providing x examples as in-context during inference. +- ARC-C (5-shot) [Clark et al., 2018]: A subset of the AI2 Reasoning Challenge benchmark consisting of grade 3–9 science exam questions that require non-trivial logical reasoning. The task is formulated as a multiple-choice question answering problem. +- GSM8K (3-shot) [Cobbe et al., 2021]: A high-quality dataset of linguistically diverse grade school math word problems, curated to evaluate arithmetic reasoning capabilities. Each question requires multi-step reasoning to arrive at the correct solution. +- IFEval [Zhou et al., 2023]: A targeted benchmark for assessing instruction-following proficiency. It contains prompts with explicit directives and uses GPT-4 to evaluate how well model outputs comply with the given instructions. + +# B.2 Evaluation Methods + +We use a variety of metrics for different tasks, following the test scripts from the Openllm leaderboard. For FuseEval, we apply BARTScore (Bart-S) [Yuan et al., 2021], BERTScore (Bert-S) [Zhang et al., 2019], GPT4-Rank (GPT4-R) [OpenAI et al., 2024], BLEU [Papineni et al., 2002], and ROUGE (R-n) [Lin, 2004]. For multiple-choice tasks such as MMLU and ARC-C, we select the option with the highest likelihood to calculate accuracy (Acc). For the reasoning dataset GSM8K, we evaluate exact match (EM) accuracy. For IFEVAL, we rely on the evaluation files provided by the dataset creators [Zhou et al., 2023], testing under prompt-strict, instruction-strict, prompt-loose, and instruction-loose conditions. For AlpacaEval 2.0, we use the official GPT-4-based pairwise comparison framework [Conover et al., 2023], where each model's output is evaluated against a GPT-4 reference response, and the win rate is computed as the final metric. A detailed description of some evaluation metrics for FuseEval is as follows: + +- BLEU (B-n) [Papineni et al., 2002] and ROUGE (R-n) [Lin, 2004] compare a generated response with a reference by calculating $n$ -gram overlap. For the Chinese results, we use Jieba² to split the text into words before calculating these two scores. +- BERTScore [Zhang et al., 2019] (comprising Precision, Recall, and F1-score) measures the similarity between two texts based on the contextualized embedding from BERT [Devlin et al., 2019]. In this paper, we report the F1 score of BERTScore. + +- BARTScore [Yuan et al., 2021] is a unified evaluator which evaluates with the average likelihood of the pretrained encoder-decoder model, BART [Lewis et al., 2019]. It can predict different scores depending on the formats of the source and target. +- The GPT4-Rank [OpenAI et al., 2024] evaluation utilizes the GPT-4o-2024-11-20 model to compare two different responses against a ground-truth response. The model will select the better of the two responses. For each test sample, we pair the responses generated by different models and have GPT-4 determine which one is superior. Since the MBR and PairRank methods do not generate new responses, we do not re-rank the responses they select from the base LLMs. Instead, we use the average rankings of the responses they select from the base LLMs to represent their GPT4-Rank. Once all comparisons are complete, we count the number of wins for each model. Based on these win counts, we rank the responses from the different models. The average ranking of each model across all data in the dataset is the value reported in our table. The evaluation instructions for GPT-4 are shown in Table 5. +- The win rate comparisons between models in this study were conducted using GPT-4o-2024-11-20 as the evaluator. Both Table 3 and Table 4 employ GPT-4o-2024-11-20 to compare outputs from different models, with the evaluation instructions for GPT-4 shown in Table 5. In Table 3: The "English FuseEval winrate" and "Chinese FuseEval winrate" metrics compare the outputs of various base models (and ensemble methods) against those generated by Owen2_72b instruct. + +# B.3Baselines + +Since our approach has not undergone any additional training, we select several types of untrained baseline models for comparison with our method: + +- PairRank: An English reward model introduced in the LLM-Blender [Jiang et al., 2023b] which compares candidate results generated by different LLMs and selects the best candidate as the ensemble output. +Majority Voting [Davani et al., 2022]: each model provides a choice, and the final result is determined by the option with the most votes. +- Minimum Bayes Risk (MBR) [Freitag et al., 2023]: Selects the answer with the highest lexical similarity to other candidate answers. In this paper, we use the SimCSE [Gao et al. 2022] model to calculate the similarity between candidate responses. +- Generation Fusion (GF) [Jiang et al., 2023b]: Uses the outputs of other models as context passing them to a new model, which generates a response based on this context. In our implementation, Mistral-7B-v0.3 is employed as the final model for the 7B-9B scale integration, and Mistral-24B-instruct-2501 for the 24B-72B scale integration, based on their performance advantages. +- Mixture-of-agents (MOA) [Wang et al., 2024]: Multi-layer fusion is applied, where the outputs of all base models are concatenated and fed back into the models, with an aggregator outputting the final result. In this work, we adopt the stronger-performing Mistral-7B-v0.3 as the aggressor in the 7B-9B scale integration, and the better-performing Mistral-24B-instruct-2501 as the aggressor in the 24B-72B scale integration. The fusion process is repeated three times, consistent with the original MOA methodology. +- Unite [Yao et al., 2025]: Constructs a new union vocabulary by combining the vocabularies of multiple models to include all tokens from each model. + +# C Case Study + +Table 7 presents a case from the SpecEM workflow where the user's request is "Write a simile to describe a person who is hard-working." The reasoning process goes through four iterations, and the verify model's selection of the best candidate is not always from the same model. In the first round, the best candidate is generated by Qwen2. In the second round, Mistral, after receiving Qwen2's output from the previous round, is inspired and generates a response that better meets the user's needs, as using "farmer" to describe a hard-working person is inappropriate. Additionally, the table shows + +Table 5: The template used for GPT-4 compares two models' responses. + +
TemplateInstruction: +${instruction}
Ground-Truth Response: +${Truth response}
Model A: +${A response}
Model B: +${B response}
Given the User's Instruction and Ground-Truth response above, please compare the two Model's responses. +You only have 2 choices to output: +If you think A is better, please output: 1. A is better +If you think B is better, please output: 2. B is better +Output your choice below:
Comparison Option1. A is better
2. B is better
+ +that through mutual inspiration between models, the final response more closely matches the user's expectations, thereby improving the overall quality of the reply. + +# D Discussion on Segment Length + +SpecEM employs fixed-length candidate segments for model evaluation and routing, which simplifies the computation process but may occasionally cause semantic truncation, particularly when segment boundaries intersect with meaningful linguistic or contextual units. This truncation can weaken semantic coherence and reduce the accuracy of optimal segment selection. In contrast, adaptive segmentation approaches such as NUDGING Fei et al. [2025] dynamically determine token boundaries based on uncertainty estimation across model families. However, these methods assume a shared vocabulary and tokenization scheme, which restricts their applicability in heterogeneous large-model scenarios where model architectures and vocabularies differ. Future work could explore adaptive or semantics-aware segmentation mechanisms that maintain context completeness while remaining compatible with multi-model routing frameworks. + +User instruction: Write a simile to describe a person who is hard-working. + +
Input for each roundQwen2 GenerationMistral GenerationGlm-4 GenerationBest Candidate
Round 1:Write a simile to describe a person who is hard-working.A person who is hard-working is like a determinedThis person is as dedicated to their work asA hard-working person is like a wellA person who is hard-working is like a determined
Round 2:Write a simile to describe a person who is hard-working. A person who is hard-working is like a determinedfarmer, tirelessly cultivating the land and nurturing their cropsriver, steadily carving its path through thehorse pulling a heavy cart through a long journey.river, steadily carving its path through the
Round 3:Write a simile to describe a person who is hard-working. A person who is hard-working is like a determined river, steadily carving its path through therockiest of landscapes.<|im_end|>mountains.</s>rocks and obstacles in its way, never giving uprocks and obstacles in its way, never giving up
Round 4:Write a simile to describe a person who is hard-working. A person who is hard-working is like a determined river, steadily carving its path through the rocks and obstacles in its way, never giving upuntil it reaches the sea.until it reaches its destination.</s>.<|user|>until it reaches its destination.
+ +Figure 7: Case study of SpecEM integrating the base LLMs Qwen2, Mistral, and GLM-4. The Best Candidate is the top-ranked option determined by the verify component and is directly presented to the user. $< |\mathrm{im\_end}| > , < / \mathrm{s} >$ , and $< |\mathrm{user}|>$ are special end tokens for the three base LLMs, and generation halts when the best candidate includes any of these end tokens. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. + +Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist: + +- You should answer [Yes], [No], or [NA]. +- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available. +- Please provide a short (1–2 sentence) justification right after your answer (even for NA). + +The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper. + +The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering + +"[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found. + +IMPORTANT, please: + +- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist", +- Keep the checklist subsection headings, questions/answers and guidelines below. +- Do not modify the questions and only use the provided macros for your answers. + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction provide a comprehensive overview of the background and motivation of this study, effectively outlining its main contributions point-by-point, thus accurately reflecting the paper's scope and significance. + +# Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We primarily focused on discussing the limitations associated with this study in Appendix 6. + +# Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. 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For larger models with 24 billion parameters or more, we utilize $4 \times \mathrm{H}200$ GPUs equipped with 140 GB memory. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. +- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. +- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). + +# 9. 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Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +Answer: [NA] + +Justification: This paper focuses on improving response quality and reducing bias via model ensembling, with a societal impact comparable to that of deploying a strong single LLM, while introducing no additional societal risks beyond those already associated with standard high-performing models. + +Guidelines: + +- The answer NA means that there is no societal impact of the work performed. +- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. +- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. +- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. +- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. +- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). + +# 11. Safeguards + +Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? + +Answer: [No] + +Justification: The datasets used in this study, such as AlpacaEval, MMLU, and GSM8K, as well as the pretrained models such as Qwen and Llama, are sourced from open-access platforms like Hugging Face. These resources have undergone comprehensive safety assessments and are widely utilized within the research community. 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Current server-centric systems overlook consumer-grade GPUs at the edge. We introduce SpecEdge, an edge-assisted inference framework that splits LLM workloads between edge and server GPUs using a speculative decoding scheme, exchanging only token outputs over the network. SpecEdge employs proactive edge drafting to overlap edge token creation with server verification and pipeline-aware scheduling that interleaves multiple user requests to increase server-side throughput. Experiments show SpecEdge enhances overall cost efficiency by $1.91 \times$ through achieving $2.22 \times$ server throughput, and reduces inter token latency by $11.24\%$ compared to a server-only baseline, introducing a scalable, cost-effective paradigm for LLM serving. The code is available at https://github.com/kaist-ina/speededge + +# 1 Introduction + +Large language models (LLMs) have become integral to modern applications such as conversational AI, code generation, and real-time content creation [Dubey et al., 2024, Lozhkov et al., 2024, Achiam et al., 2023, Touvron et al., 2023, Jiang et al., 2023, Brown et al., 2020]. However, scaling LLM deployments to meet growing demand remains challenging when balancing operational costs against latency requirements. + +A compelling opportunity exists to dramatically reduce LLM serving costs by leveraging consumer-grade GPUs at the network edge. The GeForce RTX 4090 [Nvidia, 2025a] delivers up to 330.3 TFLOPS for FP16 tensor operations with FP16 accumulate [Nvidia, 2024a], exceeding the 312 TFLOPS of the data-center-class A100 [Nvidia, 2025b], at $14.43\mathrm{x}$ lower cost [GCP, 2025, Vas, 2025]. With the widespread availability of these powerful edge devices [Valve, 2024, Nvidia, 2024b], an edge-assisted inference approach that offloads computation to these cost-effective resources could fundamentally transform the economics of LLM deployment. + +Despite this opportunity, existing inference architectures fail to leverage these edge resources effectively. Current parallelization techniques [Shoeybi et al., 2019, Rasley et al., 2020, Aminabadi et al., 2022] that split computation within data centers break down over public internet conditions, where high latency and limited bandwidth make frequent communication of intermediate results impractical. Mainstream approaches like tensor and pipeline parallelism rely on high-bandwidth, low-latency interconnects such as NVLink or InfiniBand [Nvidia, 2025c,d]. In wide-area networks (WANs), transferring intermediate model states between edge and server GPUs quickly becomes prohibitive, preventing meaningful collaboration between these heterogeneous resources. + +In this paper, we present SpecEdge, the first practical edge-assisted inference framework that fundamentally reduces LLM serving costs by splitting computation between consumer-grade edge GPUs and cloud servers. The core innovation of SpecEdge is its ability to effectively coordinate edge and server resources over wide-area networks—a capability previously unattainable with traditional + +parallelization techniques. To achieve this, SpecEdge adopts a speculative decoding paradigm that divides LLM inference into edge drafting and server verification, exchanging only token outputs rather than full model states. This approach dramatically reduces bandwidth requirements while minimizing communication rounds. + +To make this edge-assisted paradigm practical in real-world deployment scenarios, SpecEdge implements two key enabling techniques. Our Proactive Edge Drafting allows edge GPUs to continue generating tokens while awaiting server verification, effectively masking network and verification latency with local computation. Complementing this, our Server-side Pipeline-aware Scheduling orchestrates verification requests from multiple users through intelligent batching to maintain high GPU utilization. Together, these techniques ensure that the inherent cost advantages of edge-assisted inference are not undermined by network constraints or inefficient resource utilization. + +Our evaluation validates the effectiveness of this edge-assisted approach, demonstrating that SpecEdge achieves $1.91 \times$ better cost efficiency while increasing server-side throughput by $2.22 \times$ and reducing inter token latency by $11.24\%$ . These improvements persist even under challenging wide-area network conditions, outperforming server-only baselines with zero network delays. By effectively harnessing widely available edge GPUs, SpecEdge establishes a new paradigm for scalable and cost-effective LLM serving that addresses the growing computational demands of generative AI applications. + +# 2 Background and Related Work + +LLM serving systems. Modern LLM frameworks address latency, throughput, and resource efficiency challenges through various optimizations. DeepSpeed-Inference [Rasley et al., 2020] and TensorRT-LLM [Nvidia, 2024c] leverage low-level GPU optimizations, model parallelism, and quantization, while vLLM [Kwon et al., 2023] introduces PagedAttention for efficient memory management. Modern parallelization strategies [Shoeybi et al., 2019, Aminabadi et al., 2022] enhance the performance with multiple GPUs. However, these approaches depend on data-center GPUs connected via specialized interconnects (InfiniBand, NVLink) with throughput exceeding hundreds of GB/s [Nvidia, 2025c]—speeds unattainable over wide area networks (WANs). + +Speculative decoding. Speculative decoding [Leviathan et al., 2023, Chen et al., 2023] reduces latency by having a smaller auxiliary model generate multiple candidate tokens for parallel verification by the main model. The process involves three phases: drafting candidates, verification, and reconciliation for generated tokens. Recent advances have focused on more efficient drafting methods, either using lighter auxiliary models or the target model with reduced parameters [Bhendawade et al., 2024, Cai et al., 2024, Li et al., 2024, Stewart et al., 2024, Zhang et al., 2023]. + +Tree-based speculative decoding. Standard speculative decoding suffers from exponentially declining acceptance rates as sequence length increases [Leviathan et al., 2023, Chen et al., 2023]. Tree-based approaches [Miao et al., 2024, Chen et al., 2024, Svirschevski et al., 2024, Cai et al., 2024, Sun et al., 2024] address this by exploring multiple paths simultaneously. Sequoia [Chen et al., 2024] and SpecExec [Svirschevski et al., 2024] further optimize by pruning unpromising branches. + +Distributed LLM serving. Split-inference approaches like Petals [Borzunov et al., 2024] and Helix [Mei et al., 2024] distribute model layers and pipeline requests across multiple devices, improving throughput over memory offloading methods [Ren et al., 2021, Pudipeddi et al., 2020], but introduce network delays that increase latency compared to data center solutions. Several works [Timor et al., 2025, Liu et al., 2025, McDanel, 2024] utilize multiple GPU devices within a server node to overlap draft and verification tasks of speculative decoding for faster inference, yet with an exchange of higher cost as they require additional server devices per query. + +On-device LLM inference. Frameworks like MLC LLM [MLC team, 2023-2024] and Web LLM enable fully on-device inference with smaller or quantized models. While recent approaches [Svirchevski et al., 2024, Song et al., 2024, Xue et al., 2024] showcase the potential of leveraging user devices, they compromise output quality and latency. In contrast, our approach retains the high-quality verification stage on the server—achieving a hybrid solution that outperforms purely centralized or fully local inference. + +# 3 Problem and Motivation + +Serving large language models (LLMs) presents significant computational challenges due to their resource-intensive nature. While data centers rely on expensive H100 and A100 GPUs, abundant computing resources exist at the edge in the form of consumer-grade GPUs. As Figure 1 demonstrates, edge devices like the RTX 4090 and RTX 3090 generate tokens at approximately $30 - 50 \times$ lower cost than server-class GPUs when running small but capable language models like Qwen2-0.5B. Despite this cost advantage and widespread availability, current LLM serving architectures fail to incorporate these edge resources, creating a substantial missed opportunity for distributed inference that could reduce costs while maintaining high-quality LLM service. + +Conventional split computing. One approach to utilizing edge GPUs is split computation across edge and server resources. Split computing has been extensively studied in machine learning literature [Kang et al., 2017, Zhou et al., 2019, Wang et al., 2020], predating transformer architectures. Traditionally, this approach divides neural network layers between server and edge, reducing server computational load or minimizing network communication by transmitting intermediate tensors rather than raw input data. However, this method is poorly suited for LLM serving due to the unique characteristics of transformers [Vaswani, 2017]: + +![](images/1cab33f2334ed53ec50f0784fe26931b9ceeb31e80034415bdd47700f01f4671.jpg) +Figure 1: Token generation cost comparison with Qwen2-0.5B model. + +**Limitation #1: Excessive latency.** While layer-wise splitting has been used in distributed peer-to-peer systems [Borzunov et al., 2024, Mei et al., 2024] to enable serving large models otherwise infeasible on consumer GPUs, these approaches prioritize feasibility over performance. This is because, unlike earlier ML applications [He et al., 2016, Redmon, 2016, Diwan et al., 2023] that typically involve only a few communication rounds, LLM applications require communication for every generated token due to their autoregressive nature. In our distributed network setting ( $\S 5.3$ ), layer-wise splitting increases latency by $2.35\mathrm{x}$ compared to the server-only solution. + +Limitation #2: Ineffectiveness in reducing I/O. LLM inference is inherently constrained by memory I/O [Pope et al., 2023], as generating a single token requires accessing the entire model's parameters. Transformer architectures, with operations like general matrix-vector multiplication (GEMV), have low arithmetic intensity relative to their memory I/O demands. While layer-wise split computing reduces computational load by offloading layers to the edge, it fails to alleviate the I/O-bound nature of inference. A more effective approach is batch-verifying multiple candidate tokens, which significantly reduces the I/O overhead per token and increases GPU throughput. + +# 4 SpecEdge Design + +Our goal is to combine consumer-grade edge GPUs with server resources for cost-efficient LLM serving without compromising quality or latency. + +# 4.1 Disaggregated LLM Decoding + +To overcome conventional split computing limitations, we offload token drafting to edge GPUs while keeping verification on servers. Unlike traditional LLM serving, where servers handle the entire process, SpecEdge disaggregates speculative decoding into two distinct phases: edge-based drafting and server-based verification. + +In our novel design, edge GPUs generate candidate tokens and send them to the server, which verifies them in a single forward pass. The server returns both the verified tokens and one additional token to the edge device, which then updates its sequence, KV cache, and continues drafting. This cycle repeats until an end-of-sequence token is generated. Critically, this approach preserves the exact output distribution of the server model [Leviathan et al., 2023]—even when using different models for drafting and verification, the final output is guaranteed to be sampled from the same distribution as if generated by the server model alone. + +![](images/7642a872cf9e9faea69572251df42aa4dccb1b29f4db1cd2b6efa0144eeff471.jpg) +Figure 2: Abstract timeline of SpecEdge with draft (edgeside) and verify (server-side) inference concept. + +![](images/0cbd66cf159d51299a12d49b6efe16eb8763dd82b3553e28f8a2ea72b61ca695.jpg) +Figure 3: A speculative decoding cycle composed of draft and verify stages. + +As illustrated in Figure 2, this disaggregated approach enables the server to focus solely on verification while edge devices handle the autoregressive drafting process. By shifting much of the drafting time to inexpensive edge GPUs, we significantly reduce the overall serving cost while increasing server throughput. + +Figure 3 quantifies the potential advantage of this approach by showing that draft stages account for the majority of the draft-verify cycle across different batch sizes. By offloading this dominant component to edge devices, SpecEdge can substantially reduce server time and improve overall system efficiency. + +This disaggregated approach promises two fundamental advantages over conventional split computing: + +- Resolving excessive latency (Limitation #1): Unlike layer-wise splitting that requires per-token communication rounds, SpecEdge dramatically reduces network interaction by a speculative decoding approach. This minimizes both the number of client-server round trips and the volume of data transferred, eliminating the excessive latency inherent in traditional split approaches. +- Addressing I/O bottlenecks (Limitation #2): By enabling batch verification of multiple tokens in a single server-side forward pass, SpecEdge amortizes the cost of model parameter access across multiple token verifications. This increases the arithmetic intensity of each operation, directly addressing the I/O-bound nature of LLM inference and substantially improving GPU throughput. + +However, to fully realize these benefits, we must overcome two critical challenges that emerge when separating drafting and verification across different devices: + +- Potential latency increase: In conventional speculative decoding, drafting and verification happen sequentially on the same device with minimal transition overhead. In our disaggregated setting, naively implementing this sequence would add network round-trip delays to each draft-verify cycle, potentially negating our latency advantages and deteriorating user experience. +- Risk of server underutilization: Without careful coordination, server GPUs would remain idle while waiting for edge devices to complete drafting. This inefficiency could severely limit throughput and undermine the cost benefits of our approach, particularly when serving multiple concurrent users with varying workload patterns. + +To unlock the full potential of disaggregated speculative decoding, SpecEdge introduces two key innovations: Proactive edge drafting ( $\S 4.2$ ), which masks network latency by continuously generating token candidates without waiting for verification results, and pipeline-aware scheduling ( $\S 4.3$ ), which maximizes server GPU utilization by intelligently batching verification requests from multiple users. Together, these techniques enable SpecEdge to achieve both low latency and high cost efficiency. + +# 4.2 Proactive Draft Generation at the Edge + +Our key insight is to eliminate the idle time by continuing draft generation during server verification. The edge GPU performs two types of drafting: initial drafting to generate the first batch of candidate tokens, and proactive drafting that continues during server verification. When the edge GPU sends $n$ candidate tokens to the server, it immediately continues drafting additional tokens without waiting for verification results. If any token is rejected during verification, these proactively generated tokens are discarded. However, when all $n$ tokens are accepted and the server's bonus token matches the first proactively drafted token—a scenario we call "complete draft alignment"—these additional tokens can be immediately utilized, effectively hiding network and verification latency. This approach significantly reduces end-to-end latency by overlapping computation with communication, eliminating waiting periods between draft-verify cycles. Simultaneously, it improves server throughput by + +![](images/5d91ca42d6ec5ffc5d06225a31dbfe35452f9139fd8bf4dc766a3dc56a8a7f77.jpg) +Figure 4: Illustrative example of a proactive draft tree expansion. + +![](images/e37b241dd5cfe8d66440c8ab0b813ca1adf77510e6a6d97136bfc2a5c2c29e3b.jpg) +Figure 5: Post-verification update with complete draft alignment and subsequent draft submission. + +ensuring a continuous flow of verification requests, keeping expensive server GPUs highly utilized rather than idle while waiting for edge devices to complete drafting. + +Initial drafting. SpecEdge employs state-of-the-art tree-based drafting [Svirschevski et al., 2024]. However, SpecEdge is designed to be future-proof and not tied to any specific drafting technique. It simply requires a draft phase that produces candidate tokens and is compatible with various speculative decoding approaches, including tree-based methods [Chen et al., 2024, Miao et al., 2024], lossy/lossless methods, and earlier single-sequence schemes [Leviathan et al., 2023, Chen et al., 2023]. As the field evolves, any advances in speculative decoding can be seamlessly integrated. + +Proactive draft tree expansion. Unlike conventional speculative decoding that stops after initial token generation, our approach continues drafting proactively to anticipate future server responses. The expected gain from this proactive expansion can be expressed as: + +$$ +\mathbb {E} (\text {G a i n}) = P _ {\text {a l i g n}} \cdot P _ {\text {m a t c h | a l i g n}} \cdot \left(\frac {T _ {\text {d r a f t}}}{H _ {\text {e x p a n}}} - 1\right) \tag {1} +$$ + +where $P_{\mathrm{align}}$ represents the probability of alignment between verified and drafted sequences, $P_{\mathrm{match|align}}$ is the probability that the server's extra token matches an expansion head given alignment, $T_{\mathrm{draft}}$ is the total number of proactively drafted tokens, and $H_{\mathrm{expan}}$ is the number of expansion heads (leaf nodes from which drafting continues). + +This formulation reveals a fundamental trade-off: increasing $H_{\mathrm{expan}}$ improves alignment probability ( $P_{\mathrm{align}}$ ) but decreases the token preservation ratio ( $\frac{T_{\mathrm{draft}}}{H_{\mathrm{expan}}} - 1$ ), while fewer heads reduce alignment probability but significantly increase preservation when alignment succeeds. Naively applying the initial drafting strategy—treating every leaf node as an expansion head—yields negligible gains despite maximizing $P_{\mathrm{align}}$ . Our empirical results demonstrate a counterintuitive but optimal approach: after generating the initial draft tree, SpecEdge identifies the single path with the highest cumulative log probability and continues drafting exclusively from that node (Figure 4). This focused strategy maximizes expected gain by producing significantly higher returns when alignment occurs, despite the reduced alignment probability. + +Post-verification update. When server verification completes, the edge GPU receives both the accepted tokens and one additional token generated during the server's forward pass. At this point, SpecEdge compares the server's output with its proactively drafted tokens. Also, the edge updates its draft model KV cache according to the accepted tokens. + +If the server's accepted tokens and additional token perfectly match a path in the proactively drafted tree—a complete draft alignment—SpecEdge retains that branch and allows the edge to continue drafting from this advanced position (Figure 5). This approach eliminates the need to regenerate already-drafted tokens, generating a deeper draft for the next drafting round by efficiently reusing prior computations. + +If complete alignment fails—either because the verified path diverges from the proactive tree or doesn't reach a leaf node—SpecEdge discards the proactive work and reverts to initial drafting, rebuilding the tree from the end of the verified sequence. While this scenario doesn't benefit from proactive drafting, the strategy still improves average performance by exploiting the successful alignments when they occur. + +# 4.3 Server-side Pipeline-aware Scheduling + +Unlike conventional speculative decoding, where servers perform both drafting and verification, SpecEdge dedicates server GPUs exclusively to verification. This separation allows the server to focus on batch-based verification rather than token-by-token generation, significantly improving + +resource utilization. However, this disaggregated approach creates a new scheduling challenge: while the edge is busy drafting the next set of tokens, the server would idle if it only awaited verification tasks from that same request. This pattern creates "bubbles" of unutilized compute capacity whenever one part of the system is waiting on the other. + +Pipeline-aware verification scheduling. SpecEdge eliminates computational inefficiencies by interleaving verification tasks across multiple requests processed on separate edge devices. The server continuously verifies completed draft batches from one set of requests while other requests simultaneously generate new drafts on their respective edge devices. This pipelined approach ensures immediate processing of incoming drafts, with verified requests promptly returning to their edge devices for additional drafting, thereby freeing server resources for the next verification batch. By aligning edge device count with server verification capacity, this orchestration effectively doubles server throughput compared to conventional server-only configurations of equivalent batch size, substantially enhancing both cost efficiency and GPU utilization. + +For optimal pipeline efficiency, SpecEdge dynamically calibrates the relationship between server verification time and edge operations using real-time performance measurements. The system adjusts draft depth—the number of forward passes through the draft model—to satisfy the equation: server verification time $\approx$ edge drafting time + network round-trip time. This calibration ensures token batches arrive at the server precisely as it completes verifying previous batches, eliminating computational bubbles and minimizing end-to-end latency while maintaining maximum resource utilization across the distributed system. + +Processing heterogeneous requests. Server-side verification must efficiently handle batches containing requests with varying sequence lengths—a common scenario when multiple users are at different stages in their generation process. SpecEdge addresses this challenge through two complementary techniques. First, it employs custom attention masking for each token sequence in the batch, ensuring the model attends only to valid tokens within that sequence while enabling parallel processing without cross-sequence interference. Second, it implements KV cache padding to match the longest sequence in the batch, avoiding the substantial computational cost of reconfiguring tensor shapes during inference despite minimal overhead for shorter sequences. + +This dual approach allows SpecEdge to process diverse verification requests in unified batches, fully leveraging GPU parallelism while accommodating the asynchronous nature of edge-to-server communication. The result is maximized server throughput without sacrificing the responsiveness essential for interactive applications. + +# 5 Evaluation + +We evaluate SpecEdge in an edge-assisted server configuration against a server-only configuration across various LLMs and datasets. Our findings are summarized as follows: + +- SpecEdge enhances cost efficiency by an average of $1.91 \times$ compared to the server-only environment through increasing server throughput by $2.22 \times$ on average. +- It reduces the inter token latency by an average of $11.24\%$ , even with a 14.07 ms round-trip time between the server and edge, outperforming the server-only configuration with no network delay. + +Implementation and Setup. Our system's edge-assisted configuration utilizes a server-side NVIDIA A100 GPU connected to multiple edge-side NVIDIA RTX4090 GPUs over a wide-area network. The number of RTX4090 GPUs scales with the number of concurrent requests (batch size x 2). In our experiments, we measured an average round-trip time (RTT) of 14.07ms between the local edge node and our Google Cloud instance. We conducted evaluations across various models and datasets under diverse operating conditions. The code is available at https://github.com/kaist-ina/speededge + +Baseline and metrics. Our primary baseline is a server-only configuration employing tree-based speculative decoding, supplemented by autoregressive decoding and a layer-split approach that offloads part of the LLM's layers to an edge device. SpecEdge can leverage either client-side or edge GPUs; in this evaluation, we assume it uses consumer-grade GPUs from edge cloud providers. Based on provider pricing [GCP, 2025, Vas, 2025], the server-side A100 40GB GPU costs $4.05 per hour, while running the 32B model requires an A100 80GB at$ 5.05 per hour. In comparison, SpecEdge adds $0.35 per hour for each RTX 4090 used. Our key metrics include cost efficiency + +Table 1: Throughput and cost efficiency comparison between SpecEdge and server-only method. + +
Target/DraftTaskGen. tokens per verifyServer Throughput (tok/s)Cost Efficiency (1k toks/$)
Server-onlySpecEdgeServer-onlySpecEdgeServer-onlySpecEdge
Qwen3 14B/1.7BMulti-turn bench3.92±1.513.98±1.5731.7866.54 (2.09x)28.2550.60 (1.79x)
Translation3.95±1.474.25±1.4532.2465.25 (2.02x)28.6649.47 (1.73x)
Summarization3.73±1.603.95±1.6129.7067.53 (2.27x)26.4051.22 (1.94x)
QA3.42±1.573.59±1.5627.3062.04 (2.27x)24.2647.09 (1.94x)
Math.4.10±1.484.25±1.4932.8472.93 (2.22x)29.1955.28 (1.89x)
RAG3.73±1.533.83±1.5629.8964.04 (2.14x)26.5748.78 (1.84x)
Qwen3 14B/0.6BMulti-turn bench3.87±1.414.41±2.2533.4569.58 (2.08x)29.7352.97 (1.78x)
Translation3.79±1.484.67±2.3432.8869.00 (2.10x)29.2252.23 (1.79x)
Summarization3.68±1.494.21±2.1631.1768.60 (2.20x)27.7152.16 (1.88x)
QA3.33±1.413.79±1.9428.8961.90 (2.14x)25.6846.98 (1.83x)
Math3.90±1.575.27±2.2733.5383.88 (2.50x)29.8063.56 (2.13x)
RAG3.53±1.524.29±2.1630.0769.51 (2.31x)26.7352.76 (1.97x)
Qwen3 32B/1.7BMulti-turn bench4.22±1.934.71±2.6624.9656.47 (2.26x)17.8035.38 (1.99x)
Translation4.08±1.975.24±2.8424.3358.79 (2.42x)17.3436.83 (2.12x)
Summarization4.19±2.014.52±2.6824.3354.07 (2.42x)17.6533.90 (2.12x)
QA3.62±1.953.93±2.4921.5946.14 (2.14x)15.3928.99 (1.88x)
Math.4.60±1.935.40±2.7827.5264.01 (2.33x)19.6240.18 (2.05x)
RAG3.89±2.054.19±2.6922.6749.46 (2.18x)16.1631.05 (1.92x)
+ +![](images/072d3e26c0459ecaa71f6954afd2e94ba28ba4d9ebf1f8ec44dc24428fb4f21a.jpg) +(a) 14B-1.7B + +![](images/8a023003c5e503acdb36428ac76d951110470a9371d2ceadd20cb564bfd2f9af.jpg) +(b) 14B-0.6B +Figure 6: Per token cost and inter token latency comparison between server-only baselines and SpecEdge with varying batch size (BS) and model pairs. + +![](images/55e71a4c57c0db70b5069ef0c5fa1b2eba51446f7feb9ffa245308d04583f171.jpg) +(c) 32B-1.7B + +(generated tokens per dollar), server throughput (generated tokens per unit time) and inter token latency (user-perceived output latency). All configurations produce identical output distributions as they use the same underlying models. + +Models and data sets. We use four different LLMs: Qwen3-32B/14B [Team, 2025], Vicuna-33B [Chiang et al., 2023] and Llama2-13B-chat-hf [Touvron et al., 2023]. Unless specifically noted, all models are configured with a temperature setting of 0.7. For the draft models, we use five different models: Qwen3-1.7B/0.6B [Team, 2025], Sheared Llama-1.3B [Xia et al., 2023], Tiny Llama-1.1B [Zhang et al., 2024], and JackFram-160M [Miao et al., 2024]. Finally, we use SpecBench [Xia et al., 2024], C4 (en) [Raffel et al., 2020], OpenAssistant conversations datasets [Köpf et al., 2024], WikiText-2 [Merit et al., 2016], and MTBench [Zheng et al., 2023]. + +# 5.1 End-to-end Performance and Cost-efficiency + +Table 1 presents a comparison of server-side throughput and cost efficiency between SpecEdge and a server-only speculative decoding baseline on six SpecBench tasks. We use a batch size of 1, which shows the lowest inter token latency for both SpecEdge and server-only baseline, appropriate for the latency-sensitive interactive LLM serving. Throughout the end-to-end evaluation, SpecEdge achieves $1.91 \times$ better cost efficiency on average through $2.22 \times$ throughput gain compared to the server-only setup. Despite the slight cost increase of employing consumer-grade edge GPUs, + +![](images/c7690d20b478e6e8ed93cf851a149861f2d32bb91e0a206d27e49d7803f4d43b.jpg) +Figure 7: Server run time between server-only and server with edge drafting. + +![](images/dac8649f43e9c301ab2653dec99c80c5b9c6d294aa8654d4eafb78a125e00ef4.jpg) +Figure 8: Performance comparison with SpecEdge components ablation. + +![](images/e509e2dbb6abe1dc57eacc2cb05c79bba52d131ada441303e25d673efe170a78.jpg) +Figure 9: Generated tokens per verification between server-only and SpecEdge. + +![](images/b26d2d852c6818ec2b372ab0713b31f4f08cf02675fc5732d85405932015734d.jpg) +Figure 10: Server throughput according to draft depth under various network latencies. + +![](images/aa8c2adef402af2d4dee0cf0cb55de86ed1ae3788def5faf97a163b44d32d4ed.jpg) +Figure 11: Inter token latency according to network round-trip time. + +![](images/228e19104376e232d59d264cb1554acbab7d26d7383f414874f842aacaee42f1.jpg) +Figure 12: SpecEdge speedup gain with various edge devices upon autoregressive decoding. + +SpecEdge's significant increase in server-side throughput ultimately leads to greater cost efficiency. This throughput gain is driven by server-side pipeline-aware scheduling (§4.3), which interleaves multiple requests that effectively increase server utilization, and by proactive edge drafting (§5.2), which increases the average number of generated tokens per verification. We also report the mean and standard deviation of tokens produced per verification cycle. + +Beyond throughput improvements, Figure 6 demonstrates that SpecEdge also reduces inter token latency by an average of $11.24\%$ compared to server-only speculative decoding across various batch sizes. This dual benefit—higher throughput with lower latency—is particularly valuable for interactive LLM applications where both resource efficiency and user experience are critical. + +We extend our evaluation to other models, using Vicuna-33B and Llama2-13B-Chat as target LLMs across diverse datasets including C4, OAsst, WikiText-2, and MTBench. The results show that proactive edge drafting SpecEdge consistently improves performance, with greater benefits observed when the drafting models generate deeper drafts with better alignment toward the target LLMs. This alignment quality directly correlates with improvements in server throughput and reduced inter token latency. The complete results for all combinations of models and sets are provided in the Appendix B.1. + +# 5.2 Component-wise Benefit + +As illustrated in Figure 7, the server-only baseline devotes resources to both drafting and verification, causing prolonged server occupancy with each draft-verify cycle. In contrast, SpecEdge dedicates the server to verification alone while offloading drafting to more cost-effective edge devices, reducing server runtime by approximately $40 - 50\%$ in all model configurations. + +However, simple disaggregation alone can increase latency and leave the server underutilized ( $\S 4.1$ ). Figure 8 compares three progressive implementations with a 14B/1.7B model pair: basic disaggregation, disaggregation with proactive edge drafting ( $\S 4.2$ ), and complete SpecEdge with pipeline-aware scheduling ( $\S 4.3$ ). Basic disaggregation achieves only 32.76 tokens/s throughput with higher latency, while adding proactive drafting reduces inter token latency to $28~\mathrm{ms}$ . The complete SpecEdge with pipeline-aware scheduling dramatically increases server throughput to 67.89 tok/s ( $2.07\times$ improvement) while maintaining latency, demonstrating the complementary benefits of each component. + +Generated tokens per verification. Figure 9 compares the number of generated tokens per verification cycle between server-only speculative decoding and SpecEdge across three target-draft model pairs. On average, SpecEdge achieves $13.21\%$ higher tokens per verification round. For the 32B-1.7B configuration, SpecEdge produces 4.5 tokens per verification compared to 4.02 with the server-only approach, while the 14B-0.6B pair sees similar gains (4.45 vs. 3.66). This efficiency gain stems from + +proactive draft tree expansion, where each complete draft alignment allows for deeper draft trees in subsequent rounds, significantly enhancing verification efficiency and overall system performance. + +Pipeline-aware draft depth adjustment. We demonstrate that dynamically adjusting the number of forward passes for edge drafting, as outlined in (§4.3), aligns the drafting phase with both server verification and network round-trip times, achieving optimal throughput in practice. Figure 10 shows how server-side throughput varies with draft tree depth under different RTTs using a 32B/1.7B model pair. On average, verification takes $94.2\mathrm{ms}$ , while each draft model forward pass needs about $11\mathrm{ms}$ . When RTT is $15\mathrm{ms}$ , SpecEdge sets the draft depth to seven; at $40\mathrm{ms}$ RTT, it sets the depth to five; and at $50\mathrm{ms}$ RTT, it decreases further to four. These results show that SpecEdge adapts draft depth for peak throughput across a range of network conditions. + +# 5.3 System Sensitivity Analysis + +Network RTT sensitivity. We evaluate the average inter token latency of SpecEdge against layer-split and server-only configurations across varying network round-trip times (RTTs), using Qwen32B on SpecBench. Layer-split configuration runs autoregressive decoding, where one-quarter of model layers run on an edge RTX4090, with the remainder on a server-side A100. The server-only configuration, unaffected by network RTT, runs tree-based speculative decoding entirely on an A100. Figure 11 shows that SpecEdge provides lower inter token latency than the server-only baseline below $50\mathrm{ms}$ RTT, with a $13.90\%$ gain at $15\mathrm{ms}$ RTT (36.47 ms vs. 42.36 ms). Even at $65\mathrm{ms}$ , SpecEdge's latency rises by only $22.00\%$ over its $15\mathrm{ms}$ RTT performance (to 44.47 ms), remaining competitive. By contrast, layer-split is much slower: at $15\mathrm{ms}$ RTT, it is $2.73\times$ slower than SpecEdge (99.63 ms vs. 36.47 ms), increasing to $3.35\times$ slower performance at $50\mathrm{ms}$ RTT (134.90 ms vs. 40.26 ms). This resilience stems from SpecEdge's less frequent communication rounds and proactive edge drafting, which offset network latency more effectively than layer-split approaches. + +Performance with varying edge devices. Figure 12 presents the speedup achieved by SpecEdge when the server is assisted by different edge devices (RTX 4090, 4070 Ti Super, 3090, and 2080 Ti) across three target-draft model combinations. The speedup is measured relative to default autoregressive decoding using only the A100 server GPU. The consistent speedup across all model combinations confirms the architecture's robustness to different hardware configurations. As expected, more powerful edge GPUs like the RTX 4090 deliver greater speedups, while even more affordable options like the RTX 2080 Ti still provide significant acceleration. This demonstrates that SpecEdge's approach remains effective across a spectrum of edge hardware capabilities, allowing deployment flexibility. Additional results with lighter GPUs (3060 Ti, and 2080 Ti) are available in Appendix B.2. + +Performance with alternative drafting approaches. We also evaluated SpecEdge with a non-tree speculative decoding approach [Leviathan et al., 2023] to demonstrate versatility beyond tree-structured methods. Using this alternative architecture, SpecEdge achieved up to $1.96 \times$ higher server throughput (Llama2 13B/TL 1.1B pairing) and $1.67 \times$ better cost efficiency compared to server-only deployments. Performance gains remained consistent across different model combinations, with even the smallest draft model (JF 160M) delivering a $1.52 \times$ throughput improvement while preserving end-user speedup. Complete results are available in Appendix B.3. + +Performance with batch drafting method. We explore an alternative drafting configuration where a single edge GPU serves concurrent requests through batching. This approach enables operators to reduce the number of edge GPUs. Experiment with various batch sizes revealed a trade-off between better cost efficiency and increased latency. This configuration could be advantageous in budget-constrained deployments where latency tolerance is higher. Full results across model combinations are available in Appendix B.4. + +Performance under reasoning mode. Modern LLMs provide reasoning mode for enhancing output quality, where reasoning tokens might influence speculative decoding efficiency [Wei et al., 2023, Team, 2025]. To explore SpecEdge performance under reasoning mode, we measured accepted tokens, server throughput, and cost efficiency with and without reasoning enabled. Results in Appendix B.5 show consistent improvements across all three metrics when reasoning is active, implying that our system inherently benefits from the redundancy in reasoning processes, which enhances speculative decoding performance. + +Cost analysis with various GPU Providers. To ensure SpecEdge's cost-efficiency findings generalize beyond a single provider, we validated SpecEdge's performance across diverse cloud environments. + +We compared results using GPUs from multiple GPU providers (Vast.ai [Vas, 2025], Runpod [run, 2025], and TensorDock [Ten, 2025]) and Cloud Service Providers (Google Cloud Platform [GCP, 2025], Amazon Web Services [AWS, 2025], and Microsoft Azure [Azu, 2025]), accounting for the pricing variations. Across all tested configurations, SpecEdge consistently delivered cost efficiency improvements, confirming that our architectural benefits persist regardless of the specific cloud infrastructure. Detailed cross-provider comparisons are presented in Appendix C. + +Detailed Case Study. Complementing our quantitative evaluation, Appendix D offers a visualization of SpecEdge in action. Using Llama models responding to a query about Dyson Spheres, we trace the complete token generation lifecycle—from initial drafting through verification and subsequent accelerated generation. The case study specifically highlights two key operational advantages: (1) how edge GPUs remain productive during server verification phases through proactive expansion, and (2) how successful draft alignments lead to deeper draft candidates in subsequent rounds. + +# 6 Conclusion + +We have presented SpecEdge, an edge-assisted LLM inference framework that leverages user-side consumer-grade GPUs for drafting candidate tokens, while the server focuses on final verification. By transmitting only finalized outputs, SpecEdge efficiently operates under typical wide-area network conditions. Proactive edge drafting on the user side maximizes the utilization of edge GPUs and reduces end-to-end latency, while pipeline-aware verification scheduling at the server ensures high throughput by efficiently aggregating and processing verification requests. Our experimental results demonstrate that SpecEdge significantly reduces operational costs by $1.91 \times$ through delivering $2.22 \times$ higher server throughput, and achieves a modest reduction in latency compared to server-only baselines. Overall, SpecEdge unlocks the untapped potential of powerful consumer GPUs at the edge, offering a scalable and cost-effective approach for future LLM serving deployments. + +# Broader Impact + +This work redefines the division of labor in large language model (LLM) serving by integrating edge-side draft token generation with server-side verification, moving beyond the conventional centralized paradigm. This paradigm shift not only boosts server throughput without requiring additional data center infrastructure but also enables a novel business model for LLM services. By harnessing edge GPUs—whether through user-owned devices or edge cloud providers—our approach reduces reliance on expensive centralized servers, allowing service providers to deliver scalable, high-performance inference at a fraction of the cost. This decentralized architecture empowers businesses to adapt their infrastructure to edge resources, unlocking more flexible and cost-effective deployment strategies. Furthermore, SpecEdge alleviates the cost constraints associated with drafting, opening new avenues for speculative decoding research. By enabling the development of richer and more precise draft token generation methods, it advances the performance and capabilities of LLM services. + +# Limitation and Future Work + +SpecEdge is designed with flexibility in mind, supporting scenarios where users may contribute their own GPUs for the edge drafting phase. Our cost-efficiency analysis focuses on deployments using consumer-grade GPUs rented from edge cloud providers, but user-owned GPUs could further enhance cost savings and scalability. Extending SpecEdge to fully leverage user-operated hardware opens up exciting opportunities for decentralized and community-driven inference. At the same time, such scenarios raise new challenges in areas such as fault tolerance and security when untrusted devices participate in computation. While SpecEdge already supports a distributed multi-user environment, exploring these broader system and security aspects is an important direction for future work. + +# Acknowledgment + +We thank the anonymous reviewers for providing helpful feedback and suggestions to improve our work. This work was supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) of the Korea government (MSIT) (No. RS-2024-00398157). + +# References + +Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Amy Yang, Angela Fan, et al. The llama 3 herd of models. arXiv preprint arXiv:2407.21783, 2024. +Anton Lozhkov, Raymond Li, Loubna Ben Allal, Federico Cassano, Joel Lamy-Poirier, Nouamane Tazi, Ao Tang, Dmytro Pykhtar, Jiawei Liu, Yuxiang Wei, et al. 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The llama 3 herd of models. arXiv preprint arXiv:2407.21783, 2024. + +# A Additional Details on Evaluation Setup + +For both SpecEdge and the server-only speculative decoding baseline, we adopt a tree construction algorithm from recent tree-based speculative decoding [Chen et al., 2024, Svirschevski et al., 2024]. Given a specified tree size, each forward pass of the draft model generates multiple parallel candidate tokens, which are then pruned based on cumulative log probabilities so that the total number of tokens remains within the tree budget. The total number of forward passes (the draft tree depth) is set as a hyperparameter. In our main experiments, we use the draft tree size to 32 for each request. For the server-only baseline, we find and select the optimal draft tree depth through exhaustive search, while SpecEdge determines its draft depth based on verification and network latencies as described in Section 4.3. + +Throughout our evaluation, we use off-the-shelf target LLMs and draft models without any additional fine-tuning. For the main experiments, we employ Qwen3 32B/14B as the target LLM and Qwen3 1.7B/0.6B as the draft models, using the SpecBench dataset. To demonstrate broader applicability, the appendix includes experiments with Vicuna 33B and Llama2-13B-Chat-hf as target LLMs, alongside Sheared Llama 1.3B, Tiny Llama 1.1B, and JackFram 160M as draft models, evaluated on C4, OAsst, WikiText-2, and MTBench. For each query, we generate up to 256 output tokens. Since the prefetch stage in SpecEdge can involve parallel processing on both server and edge, the prefetch latency is determined by max(server_time, edge_time) which makes metrics such as time to first token comparable to the baseline; therefore, our reported metrics focus on the output tokens after prefetch. Unless otherwise specified, SpecBench (spanning six different tasks) is used throughout the experiments. + +# B Comprehensive Performance Analysis + +# B.1 Performance with Diverse Models and Datasets + +In Table 2, we showcase SpecEdge's adaptability across different target-draft model pairs and datasets, using a batch size of 1. We measure server throughput and cost efficiency gains, as well as the speedup achieved in inter token latency. Table 3 compares these speedups for both SpecEdge and a server-only speculative decoding baseline against an autoregressive decoding reference. + +Our evaluation demonstrates the effectiveness of SpecEdge across several models including Qwen, Vicuna, and Llama with parameters up to 33B. While exploration with even larger models is planned as future work, we anticipate that no fundamental design changes will be needed for scaling. The + +Table 2: Throughput and cost efficiency comparison between SpecEdge and server-only method. + +
Target/DraftDatasetGen. tokens per verifyServer Throughput (tok/s)Cost Efficiency (1k toks/$)
Server-onlySpecEdgeServer-onlySpecEdgeServer-onlySpecEdge
Vicuna 33B /SL 1.3BC43.47±1.414.41±2.0328.1249.96 (1.78x)24.9938.23 (1.53x)
OAsst3.55±1.374.35±2.0628.7249.94 (1.74x)25.5337.89 (1.48x)
WikiText-23.42±1.454.19±1.9827.2647.94 (1.76x)24.2336.42 (1.50x)
MTBench3.62±1.724.57±2.1329.6252.70 (1.78x)26.3340.21 (1.53x)
Llama2 13B /TL 1.1BC43.40±1.353.48±1.3436.8070.76 (1.92x)32.7253.72 (1.64x)
OAsst3.57±1.403.67±1.3938.5569.49 (1.80x)34.2752.75 (1.54x)
WikiText-23.52±1.373.61±1.4037.6970.21 (1.86x)33.5053.30 (1.59x)
MTBench3.67±1.433.75±1.4640.9676.69 (1.87x)36.4158.21 (1.60x)
Llama2 13B /JF 160MC43.03±1.343.13±1.3742.7266.20 (1.55x)37.9850.36 (1.33x)
OAsst2.79±1.372.85±1.3739.1956.52 (1.44x)34.8442.95 (1.23x)
WikiText-22.72±1.372.72±1.3937.8755.39 (1.46x)33.6642.08 (1.25x)
MTBench2.80±1.392.80±1.4340.8060.42 (1.48x)36.4245.92 (1.26x)
+ +Table 3: Speedup of server-only, SpecEdge compared to autoregressive. + +
Target/Draft ModelDatasetITL (ms)Speedup
Server-onlySpecEdgeServer-onlySpecEdge
Vicuna 33B/SL 1.3BC433.22930.6121.86x2.02x
OAsst34.49630.4601.76x2.00x
WikiText-234.7730.6311.69x1.92x
MTBench33.94831.1681.82x1.98x
Llama2 13B/TL 1.1BC423.68721.8511.63x1.76x
OAsst24.66122.3771.54x1.70x
WikiText-223.88321.3381.59x1.78x
MTBench23.13621.4631.66x1.79x
Llama2 13B/JF 160MC420.619.0071.87x2.03x
OAsst23.71122.3781.60x1.70x
WikiText-226.22224.5791.45x1.54x
MTBench23.45822.1991.64x1.73x
+ +core principles of our split computing paradigm—offloading partial decoding workloads from server to edge—naturally extend to models of any size through the draft-verify speculative decoding scheme. Our upcoming research will quantify these benefits across the full spectrum of model scales. + +# B.2 Performance with Lighter GPUs + +We use the RTX 4090 as representative of consumer-grade GPUs, now widespread in both edgecloud providers and user devices. However, to demonstrate broader generalizability, we conducted additional experiments with lighter consumer-grade GPUs. We measured SpecEdge performance using the RTX 3060 Ti and the RTX 2080 Ti. Table 4 shows the inter token latency and throughput with target/draft model pairs configured as Qwen3-14B/1.7B and Qwen3-14B/0.6B. Compared to the server-only approach, SpecEdge still attains meaningful throughput improvements even with less powerful GPUs. + +Table 4: Performance comparison of lighter edge GPUs. + +
Target/Draft ModelEdge GPUPeak FP16 TFLOPSMemory Bandwidth (GB/s)Inter token Latency (ms)Server Throughput (tok/s)
Qwen3-14B/1.7BRTX 3060 Ti16.20448.036.81850.297
RTX 2080 Ti26.90616.034.40954.657
Server-only (A100 40GB)312155532.45130.816
Qwen3-14B/0.6BRTX 3060 Ti16.20448.036.81856.135
RTX 2080 Ti26.90616.034.40954.657
Server-only (A100 40GB)312155532.32630.935
+ +# B.3 Performance with Non-Tree-based Speculative Decoding Method + +To demonstrate the versatility of our approach beyond tree-structured methods, we implemented the speculative decoding technique from [Leviathan et al., 2023], which uses a linear candidate sequence rather than exploring multiple branching paths. This implementation allows us to evaluate whether SpecEdge's core innovation—disaggregating drafting and verification between edge and server—generalizes effectively across different speculative decoding paradigms. + +Tables 5 and 6 present the results of this evaluation, comparing SpecEdge against the conventional server-only deployment. The throughput measurements in Table 5 demonstrate SpecEdge's efficiency + +Table 5: Server throughput and cost analysis on SpecEdge with non-tree based speculative decoding method. + +
Target/Draft ModelServer Throughput (tokens/s)Cost Efficiency (1k tokens/$)
Server-onlySpecEdgeServer-onlySpecEdge
Vicuna 33B/SL 1.3B24.52739.370 (1.61x)17.48424.649 (1.41x)
Llama2 13B/TL 1.1B33.04464.851 (1.96x)29.37249.150 (1.67x)
Llama2 13B/JF 160M32.75549.639 (1.52x)29.11637.621 (1.29x)
+ +Table 6: Inter token latency (ITL) comparison on SpecEdge with non-tree based speculative decoding method. + +
Target/Draft ModelITL (ms)Speedup
Server-onlySpecEdgeServer-onlySpecEdge
Vicuna 33B/SL 1.3B40.77138.0671.61x1.72x
Llama2 13B/TL 1.1B30.26326.0851.40x1.62x
Llama2 13B/JF 160M30.53030.8181.38x1.37x
+ +advantages, while Table 6 quantifies the relative speedup over server-only autoregressive decoding. These results confirm that our disaggregated architecture delivers consistent improvements regardless of the underlying speculative decoding strategy, reinforcing the broad applicability of our approach. + +# B.4 Performance with Batch Drafting Method + +While our main deployment architecture assumes that each concurrent request utilizes a dedicated edge GPU, we investigate an alternative architecture where a single edge GPU generates draft tokens for multiple concurrent requests. This alternative method offers trade-off for scenarios where operators want to utilize fewer consumer-grade edge GPUs. We evaluate this configuration by batching multiple requests on the RTX 4090 drafter with batch sizes from 2 to 4, measuring both inter token latency and cost efficiency across different target/draft model pairs. + +The results show a trade-off between cost efficiency and latency. As shown in Table 7, cost efficiency improves with the alternative method, ranging from $4.4\%$ to $29.5\%$ better across different configurations due to better edge GPU utilization. However, this comes at the expense of increased inter token latency ( $5.9\%$ to $19.0\%$ slower), primarily caused by contention from batching multiple requests and longer draft-to-verify cycles. This alternative deployment method could be preferable in cost-sensitive scenarios where higher latency is acceptable. + +Table 7: Performance comparison of alternative deployment method. + +
Target/Draft ModelBatch SizeInter Token Latency (ms)Cost Efficiency (1k ticks/\\()
Qwen3-14B/1.7B231.777 (6.8% slower)67.299 (14.7% better)
336.617 (13.0% slower)71.369 (29.5% better)
440.099 (16.7% slower)85.662 (24.5% better)
Qwen3-14B/0.6B230.494 (8.5% slower)67.680 (5.8% better)
334.028 (15.2% slower)80.252 (12.0% better)
439.035 (19.0% slower)82.376 (20.0% better)
Qwen3-32B/1.7B243.539 (5.9% slower)51.245 (4.4% better)
348.889 (3.9% slower)59.635 (21.3% better)
458.402 (9.5% slower)57.857 (25.1% better)
+ +# B.5 Performance under Reasoning Mode + +Many modern LLMs support reasoning capabilities to enhance their inference performance. Reasoning tokens often exhibit highly repetitive patterns, which can impact the acceptance rate in speculative decoding. To investigate the inference performance with reasoning capabilities, we measured the accepted tokens, server throughput, and cost efficiency with and without reasoning mode enabled. + +Table 8 presents the accepted tokens, server throughput, and cost efficiency for each target-draft model pair. When generating reasoning tokens, we observe consistent improvements across all three metrics: accepted tokens, server throughput, and cost efficiency. These results indicate that SpecEdge can inherently benefit from improved speculative decoding performance coming from redundancy in reasoning processes. + +Table 8: Performance comparison of SpecEdge with and without reasoning mode. + +
Target/Draft ModelAccepted TokensServer Throughput (tok/s)Cost Efficiency (1k toks/€)
Non-ReasoningReasoningNon-ReasoningReasoningNon-ReasoningReasoning
Qwen3-14B/1.7B3.80 ± 1.544.17 ± 1.3964.34372.24948.88354.898
Qwen3-14B/0.6B3.68 ± 1.504.10 ± 1.3670.70381.89053.69362.157
Qwen3-32B/1.7B4.09 ± 1.954.62 ± 1.9024.88027.61739.43043.373
+ +# C Cost Efficiency with Various GPU Providers and Cloud Service Providers + +We anchored our cost estimates to widely available public pricing: A100 GPUs from Google Cloud Platform and RTX 4090 GPUs from Vast.ai [Vas, 2025]. To validate the robustness of our cost-efficiency claims, we conducted additional experiments across multiple cloud environments, including various GPU providers (Vast.ai, Runpod [run, 2025], and TensorDock [Ten, 2025]) and major cloud service providers (Google Cloud Platform [GCP, 2025], Amazon Web Services [AWS, 2025], and Microsoft Azure [Azu, 2025]). Table 9 and 10 show the GPU pricing from different providers. Table 11 demonstrates that SpecEdge maintains cost efficiency improvements consistently across all tested configurations. + +Table 9: Edge GPU Pricing. + +
GPUGPU ProviderCost ($/hr)
RTX 4090RunPod0.69
Vast.ai0.35
TensorDock0.359
RTX Pro 6000Vast.ai1.08
TensorDock1.15
+ +Table 10: Server GPU Pricing. + +
GPUCloud Service ProviderCost ($/hr)
A100 40GBGoogle Cloud Platform4.05
Amazon Web Services4.10
A100 80GBGoogle Cloud Platform5.05
Amazon Web Services5.12
Microsoft Azure3.673
+ +# D Case Study: Proactive Edge Drafting in Action + +This section illustrates SpecEdge's proactive drafting mechanism through an illustrative example with a sample query. We demonstrate how the system handles the complete drafting lifecycle: initial + +![](images/87dedc7d61f21d8bf3b77832d225a44bd9884c4170a29bff17266368d6ae2b27.jpg) +Figure 13: Timeline view showing parallel operations at edge and server, with proactive drafting occurring during server verification. + +Table 11: Cost efficiency comparison of GPU Providers and Cloud Service Providers. + +
Target/Draft ModelServer/Edge GPUGPU ProviderCost Efficiency (1k toks/$)
Google Cloud PlatformAmazon Web ServiceMicrosoft Azure
Qwen3-14B/1.7BA100 40GB / RTX 4090Vast.ai51.018 (1.87×)50.485 (1.87×)-
RunPod44.721 (1.64×)44.311 (1.65×)-
TensorDock50.829 (1.87×)50.30(1.87×)-
Baseline (Server-only)27.22226.890-
Qwen3-14B/1.7BA100 40GB / RTX Pro 6000Vast.ai47.68 (1.75×)47.30 (1.76×)-
TensorDock46.64 (1.71×)46.27 (1.72×)-
Baseline (Server-only)27.22226.890-
Qwen3-14B/0.6BA100 40GB / RTX 4090Vast.ai52.938 (1.86×)52.386 (1.88×)-
RunPod46.382 (1.63×)45.957 (1.65×)-
TensorDock52.741 (1.86×)52.192 (1.88×)-
Baseline (Server-only)28.41527.802-
Qwen3-32B/1.7BA100 80GB / RTX 4090Vast.ai31.619 (1.82×)31.332 (1.83×)41.757 (1.75×)
RunPod28.453 (1.64×)28.144 (1.65×)36.280 (1.52×)
TensorDock31.619 (1.82×)31.239 (1.83×)41.591 (1.75×)
Baseline (Server-only)17.33017.09023.822
+ +draft generation, proactive expansion, server verification, and subsequent drafting. Figure 13 shows the timeline view of each operation. + +![](images/edcc244fe1ff045aaf72ad951359896bfd7047185684d70cb1d6b73174732293.jpg) +Figure 14: Example of proactive draft tree expansion after initial drafting. After creating the initial draft tree (left), the edge device identifies the most probable leaf token and proactively generates additional tokens (right) while awaiting server verification. + +Experimental Setup. For this demonstration, we use a sample from the OAsst dataset with the query "What is a Dyson Sphere?". We employ Llama-3.2-3B [Grattafori et al., 2024] as the target model and Llama-3.2-1B as the draft model. + +Initial Draft and Proactive Expansion. Figure 14 illustrates the transition from initial drafting to proactive expansion. Once the edge device constructs the initial draft tree, it submits this tree to the server for verification. Simultaneously—rather than remaining idle—the edge identifies the expansion head with the highest cumulative log probability and begins generating additional tokens proactively. + +Server Verification and Alignment. When the verification results arrive from the server, the edge device compares its locally expanded tree with these results. Figure 15 demonstrates a case of complete draft alignment, where the server's verified tokens ("A Dyson Sphere is") match both the initial draft tree path and the selected expansion head. + +![](images/88869a86cb4a96d6c3bfa0bc004b972a083a0b147196dbfc894f856b4a24aa7d.jpg) +Figure 15: Server verification and complete draft alignment. The server verification results (showing "A Dyson Sphere is") perfectly align with a path in the draft tree, validating both the initial draft and the chosen expansion head. + +![](images/0404a34f9ff0719a2a9c938af1990ffbde8897d92b708656dd2dba4ee0d41bbf.jpg) +Figure 16: Deeper token drafting with proactively drafted tokens. The previously generated proactive tokens tree following the extra token ("is") can be immediately used in the next drafting cycle without additional computation. + +Accelerated Token Drafting. Following successful verification, SpecEdge leverages the proactively drafted tokens to benefit the next drafting cycle. Figure 16 shows how these pre-generated tokens contribute to the next draft submission, eliminating the need to regenerate these tokens and thus building deeper draft candidates. + +Performance Implications. This example demonstrates how SpecEdge's proactive drafting strategy provides tangible performance benefits: + +- Reduced idle time: The edge GPU remains productive during server verification periods. +- Deeper subsequent drafting: Pre-generated tokens allow additional draft forward passes for the next drafting cycle. +- Effective resource utilization: Computational resources on both edge and server are maximized. + +The complete alignment case shown here represents the optimal scenario, though SpecEdge handles partial or misaligned drafts as well (Section 4.2). + +# NeurIPS Paper Checklist + +# 1. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/NeurIPS/2025/SpecEdge_ Scalable Edge-Assisted Serving Framework for Interactive LLMs/images.zip b/NeurIPS/2025/SpecEdge_ Scalable Edge-Assisted Serving Framework for Interactive LLMs/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..021348c0d427408fc2491dfe8f193e62e064ef88 --- /dev/null +++ b/NeurIPS/2025/SpecEdge_ Scalable Edge-Assisted Serving Framework for Interactive LLMs/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3b3c7c78db84cd86a846d120691f850b2163264c6e929286887dc4340586a039 +size 1004165 diff --git a/NeurIPS/2025/SpecEdge_ Scalable Edge-Assisted Serving Framework for Interactive LLMs/layout.json b/NeurIPS/2025/SpecEdge_ Scalable Edge-Assisted Serving Framework for Interactive LLMs/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..989e071ecf3439f76fd899caca3a1f4c804ad05a --- /dev/null +++ b/NeurIPS/2025/SpecEdge_ Scalable Edge-Assisted Serving Framework for Interactive LLMs/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5a8e569300d1c43b4fbc3e5be38d8ec11ae1070fe9f2aed61f14288b1ad24b02 +size 746916 diff --git a/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_content_list.json b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5f62c560893fae636b5863d587e330225bf1e63d --- /dev/null +++ b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c0bf695704eead5a85f4d174675a7f7fa0a71e6dbac0acafd7c903b56c4f78ce +size 113300 diff --git a/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_model.json b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_model.json new file mode 100644 index 0000000000000000000000000000000000000000..481dab9f35edb012eef0b80677a54919e9721475 --- /dev/null +++ b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:59e098db0b33b4d5a6ba6861bc0c46f8feb988f64a9eb42dd7fa6233bd07f3b0 +size 148706 diff --git a/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_origin.pdf b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f27c1d1cf32370bbaf5b2f8e0e0c8feb16aab588 --- /dev/null +++ b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/40dcd91b-7824-4baf-b57f-f948f5f70fd3_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:215842634cda6ed4b3fc5e6ba62215d222beff10aa55d7f10dc671e2b39d10ac +size 3020667 diff --git a/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/full.md b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/full.md new file mode 100644 index 0000000000000000000000000000000000000000..b6b6fcc0bb3ca58149662a61dfcbd939b8e3e6ad --- /dev/null +++ b/NeurIPS/2025/SpecMAS_ A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking/full.md @@ -0,0 +1,522 @@ +# SpecMAS: A Multi-Agent System for Self-Verifying System Generation via Formal Model Checking + +Rishabh Agrawal $^{1*}$ Kaushik Tushar Ranade $^{1*}$ Aja Khanal $^{1}$ Kalyan S. Basu $^{2}$ Apurva Narayan $^{1}$ + +$^{1}$ Department of Computer Science, University of Western Ontario, London, Ontario $^{2}$ ICASSSD, Cambridge, Ontario {ragrawa9, kranade, akhanal3, apurva.narayan}@uwo.ca ks.basu@gmail.com + +# Abstract + +We present SpecMAS, a novel multi-agent system that autonomously constructs and formally verifies executable system models from natural language specifications. Given a Standard Operating Procedure (SOP) describing a target system, SpecMAS parses the specification, identifies relevant operational modes, variables, transitions, and properties, and generates a formal model in NuSMV code syntax, an industry-standard symbolic model checker. A dedicated reasoning agent extracts both explicit and implicit properties from the SOP, and verification is performed via temporal logic model checking. If any properties fail to verify, an autonomous debugging agent analyzes counterexamples and iteratively corrects the model until all properties are satisfied. This closed-loop system design guarantees provable correctness by construction and advances the state of the art in automated, interpretable, and deployable verification pipelines. We demonstrate the generality, correctness, and practical feasibility of SpecMAS across a set of representative case studies and propose a new benchmark dataset for the evaluation and comparison of model checking performance. We release our code and benchmark dataset github.com/Idsl-group/SpecMAS. + +# 1 Introduction + +The increasing complexity and autonomy of modern software systems have made it critical to develop frameworks that not only generate such systems from specifications but also provide formal guarantees about their behavior. Formal specifications define pre-/post-conditions, invariants, and other assertions that must hold throughout system execution [1, 2]. In mission-critical domains such as healthcare, aviation, and industrial automation, informal reasoning and empirical testing are not enough [3, 4]. This is especially true when AI agents are used to generate and execute the system. In such cases, a proper testing suite or expert-defined system properties may not be available to verify correctness of the system while it is being generated [5]. This necessitates approaches that integrate both system synthesis and formal verification in a closed-loop manner. + +In this paper, we propose SpecMAS, a novel multi-agent framework that autonomously ingests system specifications in the form of a Standard Operating Procedure (SOP) and constructs a complete formal system model using temporal logic and symbolic representations (via NuSMV [6]). We extract system states, transitions, and property specifications, and systematically verify their correctness using model checking [7]. To ensure comprehensive verification, our framework includes a dedicated reasoning agent [8] that identifies and encodes all properties. Explicit properties are directly extracted from the SOP, while implicit ones are inferred through a property exploration process driven by semantic analysis and contextual reasoning. If any specification properties fail during verification, + +our system leverages a dedicated Model Checker Debugging Agent that interprets counterexample traces to iteratively revise and correct the system model or properties until all generated specifications pass verification checks. + +# 1.1 Objectives and Contributions + +1. Automated Model Synthesis: Design a multi-agent system capable of extracting operational modes, variables, and transitions from natural language SOPs and translating them into formal NuSMV models. +2. End-to-End Formal Verification: Use symbolic model checking to ensure that all temporal logic properties (e.g., invariants, safety, reachability) are satisfied by the synthesized model. +3. Recursive Self-Correction: Introduce an agentic feedback loop that interprets NuSMV counterexamples and autonomously revises faulty specifications or transitions until full verification is achieved. +4. Theoretical Guarantees: Provide a foundation for building agentic systems that are correct by construction, thus bridging AI generated systems with formal logic and safety. +5. Scalability and Generality: Demonstrate the generalisability of SpecMAS across diverse domains through empirical validation and analysis on MiniSpecBench dataset with both synthetic and expert-designed formal models. + +A key contribution of SpecMAS is its direct use of NuSMV, an industry-standard tool for symbolic model checking. Unlike prior work that relies on custom languages or proprietary compilers, our framework generates models in NuSMV's native specification language, enabling seamless integration with existing verification toolchains. This design choice reduces adoption barriers, eliminates the need for custom infrastructure, and leverages a mature ecosystem with robust support for temporal logic, counterexample generation, and scalability. By merging model synthesis, formal verification, and self-correction within a unified multi-agent framework, SpecMAS represents a step toward trustworthy autonomous systems capable of reasoning about the correctness of their own design. + +# 2 Previous Work + +Model checking is a formal verification technique that systematically explores the state space of a system model to determine whether it satisfies specified properties, typically expressed in temporal logic such as Computation Tree Logic (CTL) or Linear Temporal Logic (LTL). It has been widely adopted in safety-critical domains due to its ability to provide exhaustive guarantees about system behavior under all possible execution paths. Tools like NuSMV have established themselves as industry standards by offering symbolic and explicit-state verification capabilities. However, despite its rigor, model checking is traditionally a manual process that requires expert knowledge to construct accurate models and formal specifications. Recent research has explored automating parts of the process but poses a significant barrier to scalability and real-world deployment. + +# 2.1 Model Translation and DSL-Based Methods + +Several model translation and domain-specific language (DSL) approaches have been proposed to facilitate formal verification. Preibusch and Kammüller [9] presented a framework for translating Object-Z specifications to SMV, preserving object-oriented behavior but requiring manual specification. Zhongsheng et al. [10] compared SPIN, UPPAAL, and NuSMV, finding NuSMV more suited for distributed systems due to its support for CTL and LTL. DSL-based methods like MoSt [7] and NL2CTL [11] improve model structure by translating annotated or natural language inputs into temporal logic, though they rely on structured inputs or synthetic datasets. Liu et al. [12] introduced a Property Specification Language (PSL) tailored to smart contracts, offering formal rule generation but with limited generalizability. While these methods enhance automation and clarity, they are often domain-specific and depend on controlled input formats or prior modeling expertise. + +# 2.2 LLM-Assisted Specification Synthesis and Verification + +Wen et al. [13] introduced AutoSpec, a hybrid framework combining static analysis and program verification with LLM-based specification generation. The system incrementally synthesizes pre + +Table 1: Addressing limitations of prior work with the proposed SpecMAS method. + +
Limitation CategorySpecMAS Solution
Fragmented ScopeIntegrates system model generation, property extraction, and recursive verification into a single multi-agent pipeline.
Domain Lock-inDesigned to be domain-agnostic, able to operate across SOPs from various fields (e.g., manufacturing, logistics, control systems).
Post-hoc VerificationEnsures correctness-by-construction by embedding verification directly into the synthesis loop.
No Transition ModelExplicitly constructs full NuSMV transition systems (states, variables, transitions) based on SOPs.
Shallow Property CoverageExtracts both explicit and implicit properties using reasoning agents, covering invariants, liveness, and safety conditions.
No Debugging or CorrectionIncludes a self-corrective debugging agent that uses counterexamples to iteratively fix unverifiable models.
Non-interpretable OutputsOutputs models in NuSMV, an industry-standard formal language, ensuring transparency and adoption feasibility.
+ +conditions, postconditions, and loop invariants for C programs and validates them using theorem provers. Similarly, Ma et al. [14] presented SpecGen, a two-phase approach for formal specification generation in Java Modeling Language using LLMs. The first phase engages the LLM in a conversational prompt-driven loop, refining outputs based on verifier feedback. When this fails, a second mutation-based mechanism generates specification variants, which are heuristically selected for verification. SpecGen outperformed traditional tools [15, 16] which utilize templates defined by human experts to generate candidate specifications. However, such models are fine-tuned for specific domains (e.g., C/Java programs, smart contracts), limiting their adaptability. + +Loop invariants are critical for verifying programs with iterative behavior, yet generating them remains a long-standing challenge. Wu et al. [17] introduced a neuro-symbolic framework that leverages LLMs to generate candidate invariants in a "query-filter-reassemble" loop. Bounded Model Checking (BMC) [18] is used to filter invalid predicates, and surviving clauses are recombined into new invariant candidates. Pirzada et al. [19] proposed a novel extension of BMC that replaces loop unrolling with LLM-generated invariants, verified using a first-order theorem prover [20]. This method was able to prove correctness for programs that other tools [21, 22] failed to verify. + +# 3 Proposed Method + +We propose SpecMAS, a multi-agent system that automates the synthesis and verification of formal models from natural language SOP documents describing a system. At the core of our framework lies the formal verification of system behavior using model checking, grounded in the semantics of Kripke structures. Formally, we represent the synthesized system as a finite-state transition system (Kripke structure) defined by the tuple: + +$$ +\mathcal {M} = \langle S, S _ {0}, R, L \rangle \tag {1} +$$ + +where, $S$ is a finite set of states, $S_0 \subseteq S$ is the set of initial states, $R \subseteq S \times S$ is a total transition relation capturing all valid system behaviors, and $L: S \to 2^{AP}$ is a labeling function mapping each state to a set of atomic propositions that hold in that state. Given a temporal logic specification $\varphi$ , the verification task reduces to checking whether the system satisfies the property, denoted as: + +$$ +\mathcal {M} \vDash \varphi . \tag {2} +$$ + +SpecMAS constructs $\mathcal{M}$ directly from the SOP by identifying system states, variables, and transition conditions through a structured parsing and reasoning pipeline. Properties $\varphi$ are either explicitly extracted from the SOP or inferred through a property exploration agent. The model is then passed to a symbolic model checker (NuSMV), which determines the satisfaction of $\varphi$ . LTL properties are + +interpreted over paths; CTL combines path quantifiers $A$ (for all paths) and $E$ (there exists a path) with temporal operators $X, F, G, U$ . + +Input: Standard Operating Procedure document SOP + +Output: Machine-verified NuSMV model $\mathcal{M}_{0\mathrm{K}}$ + +Pipeline I - Kripke Model & Specification Synthesis; +begin // Phase 1 - Information Gathering +TaskInfo $\leftarrow$ EXTRACTTASKINFORMATION(SOP, KB); +end +begin // Phase 2 - Kripke Model Generator + $\mathcal{K}\gets$ KRIPKEXTRACTOR(TaskInfo); + $\mathcal{K}\gets$ KRIPKESOLVER( $\kappa$ ,ReACT,KB); + $\mathcal{K}\gets$ SELFCORRECT( $\kappa$ ,ReACT); +M_raw $\leftarrow$ SERIALIZETONUSMV( $\kappa$ ); +end +begin // Phase 3 - Specification Generator + $\varphi \leftarrow$ PROPERTYEXTRACTOR(SOP,KB); + $\varphi \leftarrow$ PROPERTYSOLVER( $\varphi$ ,ReACT); +Specs_raw $\leftarrow$ SERIALIZESpeCS( $\varphi$ ); + +# end + +Pipeline II - Model-Checking and Debugging; +$\mathcal{M}\gets \mathcal{M}_{\text{raw}}$ Specs $\leftarrow$ Specs_raw; RevHist $\leftarrow \{\}$ // initial revision history; repeat // coding-debugging loop if STATUS $=$ Verified then return (M, Specs) // all properties satisfied else if STATUS $=$ SyntaxError then $(\mathcal{M},\mathrm{Specs})\gets$ FIXSYNTAX(M, Specs,Refine,KB,RevHist); else if STATUS $=$ CounterExample then $(\mathcal{M},\mathrm{Specs})\gets$ GATHERCONTEXT&REFINE(M, Specs,Plan,KB,RevHist); end else // unsupported feature $(\mathcal{M},\mathrm{Specs})\gets$ FIXSEMANTICS(M, Specs,feedback,KB,RevHist); end RevHist $\leftarrow$ RevHistU $\{(M,\mathrm{Specs})\}$ until (STATUS, feedback) $\leftarrow$ RUNNUSMV(M, Specs); + +Algorithm 1: SPECMAS- End-to-End Generation and Verification + +In both Linear Temporal Logic (LTL) and Computation Tree Logic (CTL) model checking, a counterexample is a witness path (or tree) that demonstrates why a temporal-logic specification $\varphi$ is false for a Kripke structure $\mathcal{M} = \langle S, S_0, R, L \rangle$ . For LTL, the violation is witnessed by an infinite ultimately-periodic path: + +$$ +\pi = s _ {0} s _ {1} \dots s _ {k} (s _ {k + 1} \dots s _ {k + \ell}) ^ {\omega}, \quad s _ {0} \in S _ {0}, (s _ {i}, s _ {i + 1}) \in R, \tag {3} +$$ + +whose finite prefix of length $k$ is followed by a loop of length $\ell > 0$ ; by construction $\pi \models_{\mathrm{LTL}} \neg \varphi$ . Because $\mathcal{M}$ is finite, every counter-example returned by a model checker can be represented in this $\alpha \beta^{\omega}$ form, where $\alpha = s_0 \dots s_k$ and $\beta = s_{k+1} \dots s_{k+\ell}$ . + +For CTL, a property failure is shown by a finite path: + +$$ +\pi = s _ {0} s _ {1} \dots s _ {m}, \quad s _ {0} \in S _ {0}, (s _ {i}, s _ {i + 1}) \in R, \tag {4} +$$ + +or, for universal violations, a finite tree, that ends in a state $s_m$ violating the required sub-formula. Each position $i$ can be annotated with the set: + +$$ +\mathcal {F} _ {i} = \left\{\psi \subseteq \varphi \mid \mathcal {M}, s _ {i} \right. \nmid \psi \}, \tag {5} +$$ + +so the complete counter-example is the pair $(\pi, (\mathcal{F}_0, \ldots, \mathcal{F}_m))$ and the non-empty set $\mathcal{F}_m$ pinpoints exactly which obligations are violated. Thus, an LTL counter-example is mathematically an ultimately-periodic path $\alpha \beta^\omega$ with $\pi \models \neg \varphi$ , whereas a CTL counter-example is a finite, transition-consistent + +![](images/0899267f2a134f73e6382ccca8db1501f442955300319d66e66220664e231ed6.jpg) + +![](images/b37548bd7960ef21e247746ea44c4979cb8aa9fa5b40251f00eccba9c5b3771c.jpg) +Figure 1: Overview of the SPECMAS framework, consisting of two pipelines: (I) Model and Specification Synthesis, where task information is extracted from the SOP to generate a Kripke structure and formal properties; and (II) Model Checker Coding and Debugging, which verifies the synthesized NuSMV model using temporal logic and iteratively refines it based on counterexamples or syntax errors until all properties are satisfied. + +path (or tree) annotated with the falsified sub-formulas at the point of failure. So, if the verification fails, counter-examples are used by the proposed debugging agent to recursively revise the model until the relation $\mathcal{M} \models \varphi$ holds for all specified properties, thus ensuring correctness by construction. + +# 3.1 Architecture Overview + +As shown in Fig. 1, the proposed architecture is organised around two tightly-coupled agentic pipelines. The first, Kripke-Model and Specification Synthesis, processes the raw Standard Operating Procedure (SOP) corpus, extracts states, variables and transitions, and packages the resulting Kripke structure together with automatically derived CTL/LTL properties into two artefacts: a .smv model file and a companion specification file. This pipeline unfolds over three successive phases—Information Gathering, Kripke Model Generation, and Specification Generation. Throughout these phases every agent can query a shared knowledge base for retrieval-augmented prompts and invoke a reasoning LLM that performs on-the-fly symbolic checks, validating both the global plan and the intermediate outputs of each task. + +The second pipeline, Model-Checking and Debugging, takes these intermediate artefacts as input, executes the NuSMV engine, and inspects any syntax errors or counter-example traces that NuSMV returns. Guided by this feedback, the pipeline iteratively adjusts both the state-transition model and the property set until the model checker confirms that every specification is satisfied. Its output is a final, machine-verified NuSMV file, thereby closing the loop from informal SOP text to formally validated code. During each refinement cycle we expose the complete code-revision history to the reasoning LLM, enabling it to explore alternative bounds, variable domains and transition guards when repairing syntax or localisation faults. Once these structural issues are resolved, the verifier performs a semantic pass; if any SOP requirement remains unmet, the agent replans and the debugging loop repeats until full compliance is achieved. + +Table 2: LLM-Judge evaluation results across five benchmark systems. Each cell shows a metric or compliance score, with models grouped by task. Green highlights indicate the top three performers; red marks the bottom two. * represents that the .smv artifact failed to execute after debugging. + +
ModelStructural AlignmentProperty FidelitySemantic FidelityCode BonusCode ComplianceExecution ComplianceFinal Compliance Score
BRP Protocol
SpecMAS5.666.845.500.380.55 ± 0.020.9 ± 0.140.76 ± 0.08
Claude [25]6.668.166.840.10.60 ± 0.080.38 ± 0.040.46 ± 0.05
DeepSeek [26]5.005.165.340.20.45 ± 0.010.5 ± 0.140.48 ± 0.08
Gemini [27]7.348.848.170.520.75 ± 0.040.0* ± 00.30 ± 0.01
Qwen [24]6.678.007.17-0.140.55 ± 0.040.16 ± 0.040.32 ± 0.04
Shuttle Autopilot Guidance System
SpecMAS4.845.175.660.190.45 ± 0.10.97 ± 0.050.76 ± 0.01
Claude5.666.506.000.250.53 ± 0.080.34 ± 0.120.42 ± 0.1
DeepSeek4.334.335.000.030.37 ± 0.070.45 ± 0.170.42 ± 0.07
Gemini4.835.665.00-0.230.36 ± 0.040.6 ± 00.51 ± 0.01
Qwen5.165.165.330.200.45 ± 0.040.04 ± 0.050.20 ± 0.04
Mutual Exclusion
SpecMAS8.57.847.000.190.66 ± 0.090.85 ± 0.070.77 ± 0.08
Claude9.669.59.340.020.76 ± 0.130.95 ± 0.070.88 ± 0.09
DeepSeek8.59.167.160.030.66 ± 0.131.0 ± 00.86 ± 0.05
Gemini89.846.50.380.73 ± 0.080.85 ± 0.070.80 ± 0.08
Qwen9.6610.09.50.060.79 ± 0.060.1 ± 00.38 ± 0.02
Robot Controller
SpecMAS4.04.503.500.280.38 ± 0.060.90 ± 0.070.69 ± 0.07
Claude6.838.348.00.730.76 ± 0.040.0 ± 00.30 ± 0.02
DeepSeek5.166.165.50-0.060.44 ± 0.040.64 ± 0.060.56 ± 0.02
Gemini8.849.348.340.220.75 ± 0.030.0* ± 00.30 ± 0.01
Qwen6.667.175.660.340.58 ± 0.050.0* ± 00.24 ± 0.02
GigaMax Cache Coherence Protocol
SpecMAS5.05.674.670.460.50 ± 0.040.84 ± 0.060.70 ± 0.05
Claude6.58.846.840.270.65 ± 0.010.06 ± 0.040.29 ± 0.01
DeepSeek6.167.166.500.140.56 ± 0.010.66 ± 0.050.62 ± 0.02
Gemini8.59.347.660.400.76 ± 0.010.0* ± 00.30 ± 0.01
Qwen6.57.166.84-0.080.53 ± 0.040.61 ± 0.010.57 ± 0.01
+ +# 4 Experimental Setup + +We report results on MiniSpecBench, a curated suite of 10 Standard-Operating-Procedure (SOP) documents paired with the reference expert NuSMV model files with CTL/LTL specifications. All expert written .smv files were taken from the canonical NuSMV example suite distributed with version 2.5.1 and later, covering a broad spectrum of industrial design motifs. Because the original distribution provides code but no documentation, we automatically synthesised a natural-language SOP for every NuSMV model using a customised variant of SYNTHIA [23], supplying the ground-truth .smv files as the sole input. SpecMAS queries the SOP documents and the available knowledge base (consisting of documents on formal verification; more information in supplementary material) during agent execution through a retrieval-augmented generation framework. + +Results are averaged over two independent LLM-as-a-Judge evaluation runs. Full results on the MiniSpecBench dataset will be included in the supplementary material. The SpecMAS framework uses 8-bit-quantised Qwen3-32B model [24] as its backbone reasoning LLM. All experiments are conducted on a workstation equipped with an AMD Ryzen Threadripper PRO 7955WX, 256GB RAM and two NVIDIA RTX A6000 GPUs. + +# 5 Evaluation + +We evaluate SpecMAS against baseline LLM models using an LLM-as-a-Judge framework [28] for evaluation. Each generated .smv is assessed across three Code Quality metrics- Structural Alignment, Property Fidelity, Semantic Fidelity, along with a Bonus metric Code Bonus, and Compliance Scores. + +Structural alignment captures how faithfully the agent reproduces the topology of the expert model and consists of the following components: (i) role coverage of critical variables, (ii) accuracy of transition logic, and (iii) consistency of the chosen module/define decomposition. + +Property fidelity measures the logical equivalence and syntactic similarity between the generated properties and the expert-authored reference properties. It quantifies the quality of the generated CTL / LTL suite and comprises of: (i) coverage of the SOP- or expert-specified requirements, (ii) logical equivalence to expert formulas, and (iii) syntactic operator correctness. Semantic Fidelity evaluates behavioural soundness: (i) behaviour match under the execution semantics, (ii) treatment of edge cases such as recovery modes or fairness constraints, and (iii) clarity of identifier naming. Property and Semantic fidelity metrics are essential as generated properties can be logically sound and semantically well-formed, yet still lead to counterexample traces during model checking due to mismatches in assumptions, missing state conditions, or incorrect variable scoping. + +Compliance Scores are mixtures of penalty and reward values that are used to measure the Model's adherence to the input SOP. (i) Code Compliance measures the model's overall coverage of input SOP requirement to NuSMV syntax and module definitions, (ii) Execution Compliance denotes the Model's functional coverage of input SOP in terms of property specifications and state variables and the ability to produce executable code without the presence of counter-example traces. The Bonus Metric evaluates a model's ability to generate a NuSMV model that surpasses expert-written files in quality by exploring additional edge cases, tightening specification bounds, and improving structural modularity beyond the reference implementation. Code Quality metrics are reported on a scale of 10, the Bonus and Compliance scores are in range $[-0.5, 1]$ and $[0, 1]$ respectively. + +Table 2 presents a detailed evaluation of five systems from Mini SpecBench- Mutual Exclusion for two processes, BRP Message Protocol, Shuttle Autopilot Guidance System, Real-time robot controller, GigaMax Cache Coherence Algorithm. Each system was assessed across a total of 17 metrics, which were aggregated into higher-level categories for analysis as described above (implementation details are provided in supplementary material). The Final Compliance Scores are then calculated as the weighted average of the Code Compliance $(40\%)$ and Execution Compliance $(60\%)$ scores. + +The evaluation results highlight significant variations in performance across all models and metrics, reflecting the diverse strengths and limitations of each system in the task of generating executable and semantically aligned NuSMV code from SOP documents. Among all models tested, SpecMAS, Claude, and DeepSeek consistently perform better overall, particularly in terms of structural and semantic fidelity. Claude (3.7-Sonnet) performs competitively, particularly in structural alignment and property fidelity, closely trailing SpecMAS on several benchmarks. Both Claude and SpecMAS achieve high Semantic fidelity score, revealing the better instruction following and CoT-based exploration capabilities of these methods. However, Claude's performance drops significantly on execution-sensitive tasks, as seen in its failure to execute the Robot Controller model. + +DeepSeek (R1-Distill-Llama-70B) maintains balanced, moderate performance across metrics, ranking third overall, but occasionally struggles with semantic precision in complex safety-critical systems. Gemini (Flash-2.5), despite scoring well on some logical and semantic aspects, suffers from poor execution reliability and fails to generate executable models in three out of five tasks. This reflects structural inconsistencies and weaker domain alignment. Surprisingly, Qwen (3.0-Chat-235B), despite having a much larger parameter count than SpecMAS's Qwen3-32B 8bit model, performs poorly across most tasks particularly in execution. This suggests that SpecMAS's advantage does not come from model size or raw reasoning power alone. Instead, its strength lies in the integration of model and specification synthesis with an iterative code debugging pipeline. + +SpecMAS demonstrates the most stable and robust performance across the five benchmark systems, achieving the highest Final Compliance Score for four out of five SOPs evaluated in Table 2. It combines strong formal reasoning, structured synthesis, and execution reliability in a single unified framework. It further maintains a strong balance across all evaluation axes: structural alignment, property fidelity, semantic fidelity, and execution correctness. This indicates its superior ability to generalise system behaviours from SOPs into valid formal models. Notably, SpecMAS excels in Execution Compliance, outperforming all other models on this critical dimension. + +Table 3: Comparison of lowest Debug-Step models vs. highest Final Compliance score models. + +
SOPMin Debug-StepsMax Final-Compliance ScoreΔ Debug StepsΔ Score
BRP ProtocolDeepSeek1SpecMAS0.76+5+0.28
Shuttle AutopilotDeepSeek0SpecMAS0.76+5+0.34
Mutual ExclusionClaude0SpecMAS0.77+6-0.11
Robot ControllerClaude1SpecMAS0.69+4+0.39
GigaMax CacheDeepSeek2SpecMAS0.7+3+0.08
+ +![](images/78042550545ecee38fa0c62713dd726f79d22518c132817dff2cf3ee978d34f4.jpg) +(a) Comparison of debugging/errorm correction steps required to execute generated . smv files. + +![](images/cf99c825ede16396e396982fee8e922d0390812c22e361acf817406c47192615.jpg) +(b) Comparison of compliance scores for .smv files of different SOP system domains in MiniSpecBench. +Figure 2: Evaluation of generation performance across debugging effort and compliance quality. + +# 5.1 Qualitative Analysis of Code Generation and Execution Compliance + +For all models except SpecMAS, we measure the number of manual debug steps-defined as the minimal edits required to produce a syntax-error-free file. For SpecMAS, we instead track the number of iterations performed by our Debugger Agent, which first resolves syntax errors and then iteratively refines counterexamples until a model-checked artifact is produced. Figure 2a compares the number of manual debug steps and agent-led repair steps across ten benchmark SOPs. Intuitively, fewer debug steps suggest that an LLM's initial draft is closer to a "compile-and-verify" SMV artifact. + +However, as shown in Tables 2 and 3, once all files are repaired, SpecMAS consistently outperforms all baselines on key metrics such as Structural Alignment, Property Fidelity, and Compliance Scores. This highlights a critical insight: low debug step counts do not necessarily translate into high-quality or high-compliance .smv files. A plausible explanation is that baseline models may produce files that are syntactically valid with minimal edits (or none at all), but structurally incomplete or semantically shallow. For instance, DeepSeek requires zero syntax fixes but achieves very low Final Compliance Scores. This suggests that, while its code is "runnable", it often omits critical state variables, transition logic, or property specifications. In contrast, SpecMAS generates drafts that are more detailed and semantically rich-capturing edge cases, nuanced transitions, and full property coverage-which, while increasing the likelihood of syntax errors. But this results in much higher semantic fidelity once agentic debugging is applied as shown in Figure 2b. + +The tendency of baseline LLMs to produce error-prone .smv code is likely linked to NuSMV being a low-resource, domain-specific language, where token-level exposure during pretraining is minimal. This leads to higher hallucination rates, a consistent pattern observed especially in Gemini and Qwen. These issues reinforce the value of providing LLMs with tools and domain-specific context during generation. Our evaluation strongly supports this hypothesis: tool-augmented, context-aware models (like SpecMAS) produce higher-quality artifacts, even if they initially require more debugging effort. + +![](images/139a3abe684328181ec94f0be9d1f0e12cefc5570a2a3414cd0a521f024903ed.jpg) +(a) Ablation study showing the impact of removing core agent pipelines on final compliance score across selected MiniSpecBench SOPs. + +![](images/8da51b578345a08adf7b14551bc1bef68ce0e4484e53fbb2b8e934d524807517.jpg) +(b) Case study comparing average specification quality (left axis) and execution compliance (right axis) across models for the BRP Protocol SOP. +Figure 3: (a) Contribution of individual agents in SpecMAS to overall model-checking performance. (b) Analysis of the trade-off between Specification quality (SPEC) and Execution Compliance. + +# 5.2 Ablation Studies for Evaluation of Agent Contributions + +The ablation study results shown in Figure 3a evaluate the incremental contribution of key components within the SpecMAS pipeline, specifically the impact of: (i) The baseline Qwen3 32B 8-bit LLM without planning or verification model, (ii) Model and specifications synthesis alone (iii) the full pipeline including symbolic model checking and debugging—on the Final Compliance score across five benchmark SOPs from Table 2. Each axes is decomposed into two stacked segments representing the weighted contributions of Code Quality and Execution Compliance to the final score. + +The ablation results confirm that the key performance gains of SpecMAS are not solely from LLM-based synthesis, but from its integrated verification and debugging loop. While high-quality initial generation is important, reliable execution, particularly in formal verification tasks, requires post-hoc error correction and planning. This is a feature that the full SpecMAS pipeline provides. Synthesis-only pipeline improves code compliance over baseline Qwen model and often fails to produce executable models. For instance, in the Robotics Controller and GigaMax Cache benchmarks, execution performance without debugging counterexample traces remains low. + +We observe a drop in performance for the Mutual Exclusion SOP when using only the model and specification synthesis pipeline. The intermediate artefact generated at this stage was not executable due to syntax errors and a missed explicit state transition condition. Additionally, the property exploration plan introduced a novel property related to turn alternation which was not explicitly required by the SOP. Once the agentic debugging loop was activated, it successfully identified and corrected both the structural and property-level issues. + +# 5.3 Case Study: Evaluating the Impact of Proposed Agents + +We observe that although SpecMAS occasionally scores lower than baseline LLMs like Claude and Gemini on structural alignment, property fidelity, and semantic fidelity, it still achieves the highest Final Compliance scores. This raises a question about how lower Code Compliance artifacts can outperform more aligned model files in Execution Compliance. We attribute this to two main factors: (1) the execution-centric design of SpecMAS and (2) the brittleness of one-shot generation in baseline models. To illustrate this, we examine the BRP protocol SOP from MiniSpecBench. + +Figure 3b highlights this paradox. It plots Execution Compliance against Average Specification (SPEC) quality for BRP Protocol SOP, calculated as the mean of structural alignment, property fidelity, and semantic fidelity. Although Qwen achieves the highest mean SPEC quality score, it fails almost completely in Execution Compliance similar trend can be observed for Gemini as well. In contrast, SpecMAS and Claude have more modest average SPEC quality scores, but maintain a high Execution Compliance. + +This discrepancy reveals a critical flaw in one-shot generation approaches. While their outputs may appear structurally or logically correct, they lack post-generation verification, counterexample-guided refinement, and SOP-grounded planning. As a result, even minor syntax or semantic issues that are undetectable by surface-level scoring can lead to failure during NuSMV execution. In contrast, + +SpecMAS deliberately trades alignment and fidelity scores to incorporate multi-agent planning and automated debugging pipelines that corrects errors before finalizing the model file. This result in lower Code Compliance score, but a substantially higher Execution Compliance—an outcome that is far more valuable for formal verification tasks. + +# 6 Conclusion + +In this work, we introduced SpecMAS, a multi-agent framework for generating formally verified systems from natural language SOPs using model checking. Through evaluations on the MiniSpecBench suite, SpecMAS consistently outperformed state-of-the-art LLMs across key metrics. Our results highlight the value of an execution-centric, multi-agent approach over one-shot code generation. By combining planning, retrieval, and automated debugging, SpecMAS improves the reliability of generated NuSMV models. This work marks a step forward in applying agentic LLMs to system synthesis and verification, paving the way for more robust LLM based formal verification agents. + +# 7 Limitations and Future Directions + +While SpecMAS demonstrates strong potential for formal self-verification of agentic systems, some limitations remain that highlight valuable opportunities for future research and refinement. + +First, handling underspecified or inconsistent SOPs is inherently challenging due to the variability and ambiguity of real-world documentation. SpecMAS already mitigates this through reasoning-capable LLMs, hierarchical planning, and SCMRAG [29] based retrieval to infer missing information and align transitions with the logical intent of the SOP. However, extreme ambiguity may still require human-in-the-loop assistance. Integrating guided expert feedback into the verification loop is an extension that would enhance semantic fidelity and interpretability without compromising automation. + +Second, while the system's property extraction and symbolic relevance checks substantially reduce vacuous or trivial properties, there remains limited risk that certain verified properties may not fully capture the intended operational semantics. The built-in entailment classification and counterexample-based debugging already provide interpretability and grounding, but future versions can incorporate richer semantic reasoning and coverage metrics to strengthen this safeguard further. + +Third, SpecMAS currently relies on NuSMV for symbolic model checking, which restricts full generalization to hybrid or probabilistic systems. Nonetheless, our results on the traffic collision avoidance and shuttle autopilot controller tasks show that the framework can model continuous behaviors through discrete abstractions. Future expansions will integrate complementary verification backends such as SPIN, extending the approach to hybrid and stochastic systems. + +Finally, scalability is an ongoing consideration. Symbolic model checking can encounter state-space explosion in high-complexity systems, occasionally increasing reasoning time and debugging iterations. Current results show strong convergence, with an average of only 18 additional steps on the most complex tasks. Future work will explore component-based and modular verification, allowing larger systems to be decomposed and verified more efficiently. Also, the current implementation of SpecMAS focuses on generated SOPs rather than human-authored documents. This choice was made because existing NuSMV model files lacked corresponding natural-language specifications, making aligned SOP-model pairs unavailable. SYNTHIA was designed to generate semantically faithful yet syntactically diverse SOPs from human-authored code artifacts. We plan to incorporate human-written SOPs in future work to assess robustness under natural linguistic variability. + +In summary, these limitations represent growth avenues rather than shortcomings, each pointing toward richer semantic understanding, broader generalizability, and improved scalability. Addressing them will further strengthen SpecMAS as a reliable, interpretable, and extensible framework for formal self-verification of LLM-generated system models. + +# 8 Acknowledgments and Disclosure of Funding + +This research was funded through the MITACS Accelerate Program (Grant IT40058) in partnership with the International Center for Applied Systems Science for Sustainable Development(ICASSSD), Cambridge, Ontario. + +# References + +[1] Gianpiero Cabodi, Sergio Nocco, and Stefano Quer. Strengthening model checking techniques with inductive invariants. 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In Proceedings of the 24th International Conference on Autonomous Agents and Multiagent Systems, AAMAS '25, page 50-58, Richland, SC, 2025. International Foundation for Autonomous Agents and Multiagent Systems. ISBN 9798400714269. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Sections 3, 5, and 5.3 cover the major claims we've made. Section 5.2 provides an explanation for our claims. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Walton + +Georgia Institute of Technology + +twalton42@gatech.edu + +Aryan Musharaf + +Georgia Institute of Technology + +amusharaf3@gatech.edu + +Darin Tsui + +Georgia Institute of Technology + +darint@gatech.edu + +Amirali Aghazadeh + +Georgia Institute of Technology + +amiralia@gatech.edu + +# Abstract + +Autoregressive models have transformed protein engineering by enabling the generation of novel protein sequences beyond those found in nature. However, their sequential inference introduces significant latency, limiting their utility in high-throughput protein screening. Speculative decoding accelerates generation by employing a lightweight draft model to sample tokens, which a larger target model then verifies and refines. Yet, in protein sequence generation, draft models are typically agnostic to the structural and functional constraints of the target protein, leading to biologically implausible outputs and a shift in the likelihood distribution of generated sequences. We introduce SpecMER (Speculative Decoding via k-mer Guidance), a novel framework that incorporates biological, structural, and functional priors using k-mer motifs extracted from multiple sequence alignments. By scoring candidate sequences in parallel and selecting those most consistent with known biological patterns, SpecMER significantly improves sequence plausibility while retaining the efficiency of speculative decoding. SpecMER achieves $24 - 32\%$ speedup over standard autoregressive decoding, along with higher acceptance rates and improved sequence likelihoods. + +# 1 Introduction + +Designing proteins with enhanced or novel biological functions is an important problem with wideranging applications in therapeutics, sustainability, and drug discovery [1]. Protein design is challenging due to the astronomically large space of possible protein sequences, of which only a tiny fraction is likely to exhibit the desired functions [2]. Generative models trained on the vast landscape of evolved protein sequences have emerged as powerful tools for navigating this combinatorial sequence space. Among them, autoregressive models have been especially successful in the design of proteins with functional properties comparable to natural proteins [3]. Their autoregressive nature integrates well with textual information [4], ensures training stability [5], enables variable-length sequence generation [5], and effectively captures long-range dependencies that encode complex amino acid interactions [4]. Despite these advantages, autoregressive models struggle with pathological repetition [6-8], early termination [7], out-of-distribution generalization [8], and most consequential, high computational cost during inference, resulting in slow sequence generation. This computational cost is a debilitating limitation for high-throughput protein screening, where thousands of protein sequences need to be generated and screened to create large-scale libraries of functional and structurally stable proteins [9]. For instance, generating 20,000 protein sequences of length 200 amino acids using ProGen2-XL, a 6.4-billion-parameter transformer-based autoregressive model, takes approximately 65 hours using a single NVIDIA A6000 GPU. Such slow sequence generation can + +![](images/d4ff1031a4e0b48aa1f0da62feae501b402e688199e9d4d4e31d57be522d329b.jpg) + +![](images/982f6dcb455332c9879557f40422c450289773d10954cb05f30127f690436b3b.jpg) + +![](images/1cda1eb7850f9d3d1c9570591ff42800fa0caead691f17b5bc6c6399478fab91.jpg) +Figure 1: Overview of speculative decoding and SpecMER. a, Given a context sequence, speculative decoding uses a draft model to sample tokens. The target model verifies generated tokens, accepting or rejecting them. In this example, two tokens were accepted from the draft model, and the token “ $E$ ” was corrected to “ $F$ ”. b, SpecMER augments the drafting process, generating multiple sequences which are then filtered by k-mers at near-zero cost. K-mers are formed through a multiple sequence alignment, capturing evolutionary patterns through homologous protein sequences. In this example, “ $VLL$ ” was selected from the proposed candidates, where two tokens were accepted and the token “ $L$ ” was corrected to “ $K$ ”. c, Speculative decoding undersamples high log-likelihood sequences, missing the tail distribution of the target model. K-mers act as guidance for SpecMER, resulting in a distribution of generated sequences that more adequately samples high-likelihood, high-plausibility protein sequences. Additionally, SpecMER retains nearly the same generation speed as speculative decoding, with only a $3\%$ overhead observed at $c = 3$ . + +![](images/e2b45140cea3a7b9f2197d4e762c2a8730de334e5a6fe1822b990db1d4f32e34.jpg) + +delay high-throughput design workflows by days, not accounting for the additional time required for iterative design processes. + +Recently, speculative decoding [10] has proven effective in accelerating autoregressive generation in natural language processing. Speculative decoding utilizes a lightweight draft model to propose tokens, which are then verified by a larger, more expressive target model. This speculate-then-verify framework of generation has demonstrated wall-time speedup factors exceeding $2 \times$ by selecting outputs consistent with the target model distribution [11]. + +In protein sequence generation, however, target-consistent outputs may not align with the most biologically plausible proteins, particularly those that best satisfy structural or functional constraints. This misalignment is amplified in protein design workflows, where satisfying biochemical constraints is essential. While draft models may capture broad structural patterns, small deviations in local structure can lead to misfolded or nonfunctional proteins. Among target-consistent outputs, some may result in proteins with more plausible biological properties; however, such distinctions are not considered during speculative decoding. To address this, incorporating biologically meaningful local priors offers a principled way to guide generation. Notably, k-mers, or short contiguous amino acid subsequences, encode local structural motifs that correlate with both folding and function, and can be extracted with minimal computational cost. + +We hypothesized that incorporating structural cues from k-mers could guide speculative decoding toward more biologically plausible proteins, while retaining or even enhancing generation speed. To test this hypothesis, we develop speculative decoding via k-mer guidance (SpecMER), a novel decoding framework that leverages structural motifs from homologous protein sequences to improve both generation speed and quality. By incorporating structural priors into the decoding process, SpecMER provably selects tokens that are not only consistent with the target model but also biased toward completions likely to yield structured, biologically plausible proteins. This motif-informed + +Algorithm 1 Token-level maximal coupling [11] +Input: Distributions $p, q$ , Draft sample $X \sim p$ . +Compute the residual distribution $p_{\mathrm{res}}$ where $\forall x \in \mathcal{V}$ , $p_{\mathrm{res}}(x) = \frac{q(x) - \min\{p(x), q(x)\}}{1 - \sum_{x' \in \mathcal{V}} \min\{p(x'), q(x')\}}$ . +Sample $\eta \sim U(0, 1)$ +if $\eta \leq \min\left(1, \frac{q(X)}{p(X)}\right)$ then +Return $Y = X$ . {Accept the draft token.} +else +Return $Y \sim p_{\mathrm{res}}$ . {Correct the token by sampling from the residual distribution.} +end if + +guidance increases the likelihood that generated sequences will fold into stable structures. Our contributions are as follows: + +- We develop the first speculative decoding framework for protein generation. Using ProGen2-S as the draft model and ProGen2-M as the target model, we demonstrate an average increase in generation speed by $32\%$ . +- We develop SpecMER, a novel framework that incorporates k-mer frequencies from natural proteins to guide draft token selection. SpecMER produces sequences more likely to resemble natural proteins while retaining increased generation speed from speculative decoding, enabling fast generation of plausible proteins. Software for SpecMER is available at https://github.com/amirgroup-codes/SpecMER.git. +- We establish theoretical bounds on wall-time speedup and characterize how guidance from k-mers improves draft token selection, offering practical guidance for estimating speedups under varying system configurations and sequence constraints. + +# 2 Preliminary + +Notation. Let draft model $P$ and target model $Q$ be autoregressive models with probability density functions $p$ and $q$ . A protein sequence $x \sim P$ is denoted by sequence $x(i), x(i + 1), \ldots, x(j)$ , where $x(i)$ is the $i^{th}$ token sampled from $p$ . Tokens are defined by the shared vocabulary $\mathcal{V}$ of $P$ and $Q$ , consisting of amino acids and special tokens. + +# 2.1 Speculative Decoding + +Speculative decoding has recently emerged as a highly effective technique for accelerating autoregressive generation. Speculative decoding employs a smaller draft model to propose candidate tokens, which are then refined by a larger, more expressive target model [10, 11]. This approach generates tokens consistent with the target model distribution while significantly reducing decoding latency. Various extensions to speculative decoding have explored architectural modifications [12], tree-based verification [13], distillation of the draft model [14], and optimized scheduling or reference-free inference [15]. + +Given a context sequence $x(t)$ and a draft model $P$ , speculative decoding efficiently samples $L$ tokens, $\tilde{x}(t+1), \ldots, \tilde{x}(t+L)$ . Speculative decoding can be broken down into three steps: 1) Draft construction, 2) Conditional probability computation, and 3) Draft selection (Figure 1a). + +1. Draft construction. The draft model first samples $L$ tokens, $\tilde{x}(t + 1), \ldots, \tilde{x}(t + L)$ . For every token $i < L$ , we use the draft model to compute the conditional probability of the next token $y$ , $P(y \mid x(t), \tilde{x}(t + 1 : t + i))$ , where $y \in \mathcal{V}$ . +2. Conditional probability computation. For every token $i < L$ , we use the target model $Q$ to compute the conditional probability of the next token $Q(y \mid x(t), \tilde{x}(t + 1 : t + i))$ . +3. Draft selection. We validate the first $L'$ out of $L$ tokens based on the conditional probabilities of the draft and target models. We set $x(t + i) = \tilde{x}(t + i)$ for $i \leq L'$ . For the rejected token $\tilde{x}(t + L' + 1)$ , sample from a residual distribution and correct it. + +After steps 1 through 3, we use $x(t + L' + 1)$ as the next context and continue the speculative decoding process iteratively. Algorithm 1 [11] details the acceptance and rejection process for each + +token generated by the draft model. Rejected tokens are corrected by sampling from the residual distribution $p_{res}(x)$ . + +The correctness guarantees of speculative decoding are formally defined for ancestral sampling, where outputs are drawn directly from the full target distribution. In protein language models, however, ancestral sampling is rarely used in practice, as it often produces implausible or repetitive sequences [8]. Instead, decoding with nucleus sampling (top- $p$ ) is common to balance plausibility and diversity, and is utilized with both ProGen and ProGen2 [4, 8, 16, 17]. When paired with speculative decoding, nucleus sampling truncates the low-probability tail of the target distribution, as observed in Figure 1. Since this tail accounts for a small fraction of probability mass ( $p = 0.95$ in this study), outputs remain closely aligned with the target model, with differences confined to rare tail events. + +Speculative decoding has demonstrated impressive speedups in natural language generation, where structural constraints are minimal. However, existing methods are agnostic to domains like protein design, where sequences must satisfy intricate biochemical and structural requirements. In this setting, small changes can disrupt folding, stability, or function, and distributional drift between the draft and target models can accumulate over time, degrading generation quality. To the best of our knowledge, no prior work in speculative decoding incorporates sequence-level structural information to guide draft generation, limiting its applicability in this domain. This work addresses that gap by introducing a structure-aware drafting strategy that leverages local sequence motifs to guide speculative decoding toward more plausible protein sequences. + +Which tokens are optimal? Assume that the acceptance ratio of token $x(t + i)$ sampled from $p$ is given by $\beta (t + i)$ . If we assume each $\beta (t + i)$ is i.i.d., then $\mathbb{E}[\beta ] = \alpha$ , where $\alpha$ indicates the overall acceptance ratio for a given configuration of speculative decoding [10]. It follows from [11] that $\alpha$ is the complement to the expected total variation distance between $p$ and $q$ . Therefore, $\alpha$ describes exactly how well $p$ approximates $q$ . If the generation cost coefficient $c_{e} = \frac{M_{p}}{M_{q}}$ is known, where $M_{p}$ and $M_{q}$ represent the time it takes for $P$ and $Q$ to generate one full-length sequence, the wall-time speedup for drafts of length $\gamma$ can be computed as: + +$$ +\frac {1 - \alpha^ {\gamma + 1}}{(1 - \alpha) (\gamma c _ {e} + 1)}. \tag {1} +$$ + +Therefore, an effective speculative decoder should aim to maximize $\alpha$ or minimize $c_{e}$ . Importantly, the set of tokens that increase $\alpha$ is not limited to the highest likelihood token under the target model. Rather, there exists a set of completions $E$ which satisfy the requirements of Algorithm 1. While speculative decoding excels at proposing any given token from $E$ , only a subset of these completions may align with biologically meaningful constraints. For instance, two candidate sequences "EEL" and "VLL" (as in Figure 1) might both pass the maximal coupling acceptance check, yet "VLL" may correspond to a conformation with higher predicted structural stability. Current speculative decoding frameworks lack a mechanism to prioritize such candidates, as they do not account for auxiliary objectives, which are essential in protein design but not explicitly modeled during generation. + +# 2.2 K-mers + +K-mers are contiguous substrings of length $k$ extracted from biological sequences. In proteins, k-mers often correspond to recurring structures such as alpha-helices or beta-strands, capturing local features that are critical to folding and function. Compared to analogous representations in other domains, protein k-mers encode a dense combination of physicochemical properties, structural motifs, and functional sites, all at minimal computational cost. In contrast, n-grams in natural language primarily reflect syntactic or statistical patterns without necessarily encoding physical or functional constraints. Due to their high information content, k-mers have been used across many applications in computational biology, including tasks such as classification, motif discovery, or alignment-free sequence comparison [18]. Notably, Tranception [19] utilizes k-mer attention to better capture structural dependencies during the generation process. + +# 3 Methods + +# 3.1 SpecMER: K-mer Guided Protein Generation + +SpecMER leverages k-mer statistics to guide protein sequence generation. Figure 1 details the modified sampling procedure of SpecMER. Instead of sampling one completion at a time from + +the draft model, SpecMER batch generates multiple completions, which we refer to as candidate sequences. A k-mer scoring function then assesses each length $L$ candidate sequence, providing a score based on the selected values of $k$ . The highest-scoring candidate is then scored by the draft model and verified by the target model. To summarize, SpecMER decodes protein sequences using the following steps, which repeat until generation is complete: + +1. Candidate construction. The draft model first samples $L$ tokens $c$ times, producing a batch with dimensions $c \times L$ of candidate sequences. +2. K-mer scoring. A k-mer scoring function then selects the candidate sequence $[\tilde{x}^{(1)}(t + 1),\dots ,\tilde{x}^{(c)}(t + L)]$ which yields the highest score. +3. Conditional probability computation. For every token $i < L$ , the selected candidate sequence is scored using the draft and target models. +4. Draft selection. Each proposed token in the candidate sequence is accepted or rejected in accordance with Algorithm 1. The new context is set to $x(t + i) = \tilde{x}(t + i)$ , where $i$ is the number of accepted tokens to add. + +# 3.2 K-mer Scoring + +After the draft model samples $c$ candidate sequences, each sequence is evaluated based on its k-mer frequency. K-mers are computed prior to generation using a multiple sequence alignment (MSA) constructed for a target wild-type protein sequence. MSA data consists of homologous sequences identified based on similarity to the wild-type protein. For each wild-type sequence, we retrieve its corresponding MSA from ProteinGym [20]. The resulting alignment encapsulates evolutionary and functional constraints, making MSA-derived data particularly well-suited for k-mer analysis. + +We extract k-mers by applying a sliding window of size $k$ across all sequences in the MSA. We selected $k$ values of 1, 3, and 5, in accordance with [19]. We chose not to exceed $k = 5$ as the number of possible k-mers can grow exponentially relative to the vocabulary size, increasing access cost during inference. K-mers are tracked over all sequences in the MSA as they appear, enabling better memory and computational efficiency during retrieval. The resulting k-mer frequencies serve as a proxy for biological plausibility and are normalized to form a probability distribution. During the decoding process, k-mers are used to score candidate sequences $s$ : + +$$ +\operatorname {S c o r e} (s) = \frac {1}{L} \sum_ {k \in K} \sum_ {i = 0} ^ {L - k} \mathbb {P} _ {k} (s (i: i + k)), \tag {2} +$$ + +where $L$ is the length of the candidate sequence, $K$ is the set of k-mers to evaluate, and $\mathbb{P}_k$ is the probability of a k-mer given the normalized distribution from the MSA. The scoring is additive rather than multiplicative to avoid a zero score in the case of unseen k-mers and to promote exploration of sequences with partially formed motifs. The candidate with the highest k-mer score is selected to continue the decoding process. A significant advantage of k-mer scoring is that it can be precomputed and is easy to access during generation, contributing negligible computational overhead. Consequently, SpecMER has the same computational complexity as speculative decoding: $\mathcal{O}(L^2)$ . + +# 3.3 Performance Gain from K-mers + +Vanilla speculative decoding drafts individual sequences and verifies each token in the sequence independently. The maximal coupling acceptance algorithm (Algorithm 1) ensures that selected tokens do not diverge from the target model distribution, increasing generation speed with no sacrifice to the quality of the generated sequences. However, there could be many tokens that are consistent with the target distribution at each verification step, passing the acceptance criteria $\eta \leq \min(1, \frac{Q(x)}{P(x)})$ . To quantify the looseness of this criterion and the efficacy of external scoring functions, we describe the conditions for selecting tokens consistent with the target model and optimal under the scoring function. + +Proposition 4.4. The expected batch-and-select acceptance satisfies: + +$$ +\mathbb {E} \left[ A ^ {*} \right] = 1 - (1 - \alpha) ^ {m} - \epsilon , \tag {3} +$$ + +where $\mathbb{E}[A^{*}]$ is the acceptance ratio of the speculative decoder, $\alpha$ is the acceptance ratio of vanilla speculative decoding, and $m$ is the number of batch generated candidate sequences. + +Proof. Let $\mathrm{P}$ be the draft model and $\mathbf{Q}$ the target as before. For a drafted (candidate) sequence $s$ , let $M(s)$ be an indicator of the acceptance or rejection of $s$ . That is, $M(s) = 1$ if $\mathbf{Q}$ accepts $s$ , and $M(s) = 0$ otherwise. For a given context sequence $x$ , + +$$ +\mathbb {P} [ M (s) = 1 \mid s ] = r (s) = \min \left(1, \frac {Q (s \mid x)}{P (s \mid x)}\right). \tag {4} +$$ + +If $s$ is a single draw, as in vanilla speculative decoding, then the acceptance rate $\alpha$ is defined as $\mathbb{P}_{s\sim P}[M(s) = 1] = \mathbb{E}_{s\sim P}[r(s)]$ . In the case of batch generation, $m^{-1}$ i.i.d. candidates are selected for each draft: $s_1,\ldots ,s_m\sim P$ . Following drafting, the scoring function $S$ selects the highest scoring candidate $s^* = \operatorname{argmax}_i S(s_i)$ . Let acceptance indicator $A^{*} = M(s^{*})$ , and define $E = \{\exists i:M(s_i) = 1\}$ as the event in which at least one $s_i$ is an acceptable candidate under $Q$ . The probability that an acceptable candidate $s_i$ exists but is not selected by $S$ is defined by the misranking loss $\epsilon = \mathbb{P}[E\wedge (A^{*} = 0)]$ . Consider the case where no candidates satisfy the acceptance criteria of $Q$ , $\mathbb{P}[E^c ] = \mathbb{P}[M(s_1),\dots ,M(s_m)] = (1 - \alpha)^m$ . Intuitively, $(1 - \alpha)^{m}$ is the probability that among $m$ candidates, none pass the criteria of $Q$ . The probability that a selected candidate will be rejected is then defined by $\mathbb{P}[A^{*} = 0] = \mathbb{P}[E^{c}] + \mathbb{P}[E\wedge (A^{*} = 0)] = (1 - \alpha)^{m} + \epsilon$ . The overall acceptance of the decoder is then: + +$$ +\mathbb {E} \left[ A ^ {*} \right] = 1 - \mathbb {P} \left[ A ^ {*} = 0 \right] = 1 - (1 - \alpha) ^ {m} - \epsilon . \quad \square \tag {5} +$$ + +In practice, $\mathbb{E}[A^{*}]$ is measured empirically as the average acceptance ratio of the speculative decoder, and $\alpha$ is the average acceptance ratio of vanilla speculative decoding (no external scoring). $\epsilon$ is the probability that the scoring function chooses the wrong candidate, given that there was an acceptable candidate to choose. A good scoring function should yield $\epsilon \ll 1$ , indicating that it excels in selecting candidates that will be accepted by $Q$ . Additional bounds on expected speedups for given hardware configurations can be found in Appendix A. + +# 4 Experiments + +In this section, we detail experiments across seven different proteins. We tested SpecMER with ProGen2 [4], an autoregressive protein language model. For each experiment, we conditioned generation on a fixed context length from a given wild-type protein as detailed in Table 1. We selected proteins with varying molecular functions and lengths to ensure robustness of testing. Each experiment consisted of generating 200 sequences on an NVIDIA RTX A6000 GPU. Sequences were generated up until the length of the wild-type protein or until a stop token was generated. We swept across a set of hyperparameters for each decoding method to determine the configuration with the highest acceptance ratio and generation speed. Results of a full hyperparameter sweep are located in Appendix B.3. + +# 4.1 Datasets + +We focus on the conditional generation task, where the objective is to generate starting from a context sequence. We selected seven proteins with varying functions and collected their MSA from ProteinGym [20]. For each protein, we set the context length to roughly $10\%$ of the wild-type sequence, balancing exploration of sequence space while avoiding common pitfalls in autoregressive generation such as pathological repetition. Table 1 details each protein and its respective molecular function. + +Table 1: Summary of proteins and context length used. + +
ProteinDescriptionMolecular FunctionLengthContextMSA SequencesCitation
GFPGreen fluorescent proteinFluorescence23820396[21]
RBP1RalA-binding protein 1Stability5210135922[22]
ParD3Antitoxin ParD3Growth enrichment931538613[23]
GB1IgG-binding domain of protein GBinding561044[24]
Bg13β-glucosidaseEnzyme function50150105913[25]
ADRB2Beta-2 adrenergic receptor (GPCR)Receptor activity41340204722[26]
CBSCystathionine beta-synthaseGrowth5515019563[27]
+ +# 4.2 Experimental Setup + +Hyperparameters. Each experiment consisted of generating 200 protein sequences given the starting context sequence (Table 1). We swept over the following hyperparameters: draft tokens $\gamma \in \{5,10,15\}$ , temperatures $T \in \{0.7,1,1.4\}$ , and k-mers $k \in \{(1),(3),(1,3),(1,3,5)\}$ . Sequences were sampled using nucleus (top- $p$ ) sampling, setting $p = 0.95$ . We opted for a large value of $p$ to enable diverse sampling, controlling how deterministic the output is instead with temperature. + +Evaluation metrics. We measured four categories across all experiment configurations: acceptance ratio, negative log-likelihood (NLL), top-20 NLL, and top-5 NLL. Acceptance ratio $\alpha$ is defined as: + +$$ +\alpha = \frac {\text {N o . a c c e p t e d t o k e n s}}{\text {N o . a c c e p t e d t o k e n s} + \text {N o . r e j e c t e d t o k e n s}}, \tag {6} +$$ + +where a higher acceptance ratio is better. After generation, we computed NLL, top-20 NLL, and top-5 NLL under the target model, normalizing for length. We assessed the top-20 and top-5 NLL, as protein design experiments often create a library of sequences and filter for only the most plausible candidates. Sequences with higher likelihoods are more likely to resemble natural proteins, making likelihood a proxy for sequence quality [28, 29]. In addition to likelihood, we computed predicted local-distance difference test (pLDDT) scores [30] using ESMFold [31]. The pLDDT score measures how confident the structure prediction is for a given protein sequence, and has been demonstrated to correlate with the likelihood that a protein sequence will fold into a stable structure [30]. We evaluated biological plausibility using both sequence likelihood and pLDDT, under the assumption that sequences achieving high scores on both metrics are more likely to adopt stable, protein-like structures consistent with natural biological constraints. Sequence likelihoods and pLDDT scores for each wild-type sequence can be found in Appendix B.3. + +We generated embeddings for both MSA and sampled sequences $(c = 1,2,3,5)$ using ESM2 [31], and used PCA to qualitatively assess alignment between natural and generated sequences. For each decoding method, we selected the top three configurations by average NLL, further filtering for the 100 sequences with the highest pLDDT scores. Bgl3, ADRB2, and CBS were excluded from the pLDDT score analysis due to sequence length limitations of ESMFold. + +Generation speed experiments were conducted using ProGen2-S and ProGen2-M as baselines. We tested vanilla speculative decoding ( $c = 1$ ) and SpecMER ( $c = 2,3,5$ ) against these baselines, tracking tokens per second. We swept over a range of hyperparameters to find the fastest configuration, averaging over 20 generated sequences. + +# 4.3 Results + +Acceptance and log-likelihood. The best results per experimental configuration are reported in Table 2. We observed that acceptance ratios were marginally higher on average for SpecMER compared to speculative decoding and increased with $c$ . Sequences generated with SpecMER had significantly lower NLL, including a lower top-20 and top-5 NLL. That is, while k-mer scoring increased acceptance for some configurations, it nearly unanimously produced protein sequences with higher likelihood. + +We observed that the acceptance ratio was sensitive to $T$ and $k$ . Specifically, $k$ strongly influenced the acceptance ratio for some proteins, while others were relatively unaffected. For example, GFP exhibited the best performance with $k = \{1,3\}$ , whereas RBP1 had no preference toward $k$ . A full discussion on important hyperparameters can be found in Appendix B.3. + +Generation of plausible sequences. SpecMER yielded sequences with higher likelihoods compared to speculative decoding, indicating that sequences generated by this framework are more likely to be biologically plausible. To corroborate this claim, we computed pLDDT scores, a measure that correlates with the likelihood that a protein will fold into a stable conformation. Table 3 details the average pLDDT scores across 300 sequences per configuration. We observed that on average, SpecMER produces sequences with higher pLDDT scores. GFP was an exception to this trend, exhibiting higher scores for vanilla speculative decoding. However, top-5 pLDDT scores for GFP demonstrated that top-end performance was comparable between all decoding methods (Table 10 in Appendix D). The pLDDT scores did not increase linearly with $c$ ; instead, the best value of $c$ varied across proteins. This suggests that the optimal value of $c$ reflects a trade-off between exploration of diverse sequence space and adherence to the likelihood distribution of the target model. Furthermore, this trade-off may depend on structural characteristics of the target protein. + +Table 2: Decoding results using ProGen2. Each metric is averaged over 200 generated sequences. The value of $c$ chosen for SpecMER experiments represents the number of batch-generated candidate sequences, with $c = 1$ indicating speculative decoding. For each method, a sweep over hyperparameters is conducted as described in Section 4. The best results in each category are reported. A full summary of results is located in Appendix B.3. + +
Decoding MethodProteinAccept Ratio ↑NLL ↓Top-20 NLL ↓Top-5 NLL ↓
Speculative DecodingGFP0.911 ± 0.0292.45 ± 0.421.38 ± 0.320.98 ± 0.11
RBP10.938 ± 0.0432.73 ± 0.192.43 ± 0.192.01 ± 0.44
ParD30.902 ± 0.0361.93 ± 0.590.80 ± 0.210.50 ± 0.09
GB10.927 ± 0.0402.79 ± 0.152.60 ± 0.252.28 ± 0.33
Bgl30.852 ± 0.0250.91 ± 0.110.76 ± 0.060.67 ± 0.08
ADRB20.866 ± 0.0791.90 ± 0.651.18 ± 0.300.78 ± 0.16
CBS0.910 ± 0.0512.42 ± 0.422.06 ± 0.441.43 ± 0.38
SpecMER (c=3)GFP0.937 ± 0.0761.23 ± 0.680.44 ± 0.040.35 ± 0.04
RBP10.926 ± 0.0382.58 ± 0.322.08 ± 0.391.48 ± 0.21
ParD30.919 ± 0.0931.56 ± 0.490.75 ± 0.120.54 ± 0.06
GB10.924 ± 0.0432.71 ± 0.182.43 ± 0.262.06 ± 0.57
Bgl30.869 ± 0.0300.81 ± 0.140.65 ± 0.070.54 ± 0.08
ADRB20.857 ± 0.0621.33 ± 0.500.77 ± 0.130.61 ± 0.04
CBS0.902 ± 0.0812.17 ± 0.661.47 ± 0.500.84 ± 0.16
SpecMER (c=5)GFP0.945 ± 0.0841.09 ± 0.640.41 ± 0.070.33 ± 0.02
RBP10.926 ± 0.0472.41 ± 0.401.72 ± 0.301.25 ± 0.12
ParD30.942 ± 0.0631.33 ± 0.410.67 ± 0.120.52 ± 0.05
GB10.925 ± 0.0382.61 ± 0.272.20 ± 0.311.74 ± 0.15
Bgl30.867 ± 0.0220.80 ± 0.170.63 ± 0.110.46 ± 0.05
ADRB20.869 ± 0.0821.03 ± 0.600.57 ± 0.110.43 ± 0.04
CBS0.908 ± 0.0661.87 ± 0.681.14 ± 0.320.74 ± 0.07
+ +Table 3: Average pLDDT scores across four different proteins. Sequences are collected from the three best configurations for each decoding method, determined by the highest average log-likelihood under the target model (ProGen2-M). + +
ProteinSpeculative Decoding (c=1)SpecMER (c=2)SpecMER (c=3)SpecMER (c=5)
GFP (↑)0.493 ± 0.1560.449 ± 0.1600.479 ± 0.1910.426 ± 0.176
RBP1 (↑)0.571 ± 0.1160.664 ± 0.1270.700 ± 0.1150.740 ± 0.105
ParD3 (↑)0.638 ± 0.2060.650 ± 0.2020.584 ± 0.2010.519 ± 0.176
GB1 (↑)0.465 ± 0.0840.464 ± 0.0710.477 ± 0.0750.504 ± 0.09
+ +When compared to speculative decoding ( $c = 1$ ), SpecMER ( $c = 5$ ) generated sequences that more closely aligned with homologous sequences from an MSA, while also exploring beyond the MSA centroid. Figure 2a illustrates this result for RBP1, where SpecMER captures both MSA-like sequences and novel variants. Notably, many of these divergent sequences exhibited high likelihood under the target model and achieved higher pLDDT scores on average (Figure 2 and Table 3), indicating a greater degree of biological plausibility. Beyond structural plausibility, we also assessed sequence diversity and found that both speculative decoding and SpecMER generated novel sequences far from the wild-type while maintaining high inter-sequence diversity (see Appendix D.1). Taken together, these results suggest that k-mer guidance preserves evolutionary plausibility while facilitating exploration of high-quality regions of sequence space. + +To investigate the role of MSA-derived k-mers, we conducted two ablation experiments (see Appendix C for full details). First, we tested SpecMER under cross-protein mismatches, conditioning on GFP while using GB1-derived k-mers to select continuations, and conditioning on GB1 while using Bgl3-derived k-mers. In both cases, likelihoods dropped relative to protein-specific k-mers, indicating that observed gains depend on the correct evolutionary context. Second, we reduced the + +![](images/d48797b8f93653cccd1e3ea4f65341b6c97a36831c5471fe06c0b5442868a6fd.jpg) +Figure 2: Impact of candidates $c$ on biological plausibility of sequences generated by vanilla speculative decoding ( $c = 1$ ) versus SpecMER for RBP1. a, PCA of embeddings from sequences generated by speculative decoding ( $c = 1$ ) (highlighted on the left plot) and SpecMER ( $c = 5$ ) (highlighted on the right plot), compared to embeddings from the MSA of RBP1. Each sequence is shaded by its likelihood under the target model (ProGen2-M). Sequences generated by SpecMER cluster more closely to the MSA embeddings and demonstrate a higher likelihood, indicating a higher degree of biological plausibility. b, pLDDT distributions for different configurations of SpecMER, with $c = 1$ representing vanilla speculative decoding. SpecMER demonstrates higher structural confidence scores, indicating that generated sequences are more likely to fold. + +![](images/f60b29aebe65e54bbf6fc3aab2a2b51ed07e85088c2b40600776b848e76e4d8e.jpg) + +![](images/1347b0a7f466ed9da08e248ffe246f9e1cb32f2f876f18f874bda935d32a57d9.jpg) + +depth of the Bgl3 MSA from $\sim 105\mathrm{k}$ to $1\mathrm{k}$ sequences. The resulting likelihoods steeply declined, supporting our conclusion that MSA quality impacts the performance of SpecMER. + +While vanilla speculative decoding ( $c = 1$ ) routinely undersamples the high-likelihood tail of the target distribution (Figure 1c; see Section 2.1), our experiments demonstrate that SpecMER overcomes this limitation. By using k-mers extracted from the MSA to bias the draft model toward structurally grounded completions, SpecMER achieves improved coverage of biologically plausible sequences, often exceeding the likelihoods obtained by target-only decoding. Table 4 compares the top-20 NLL values from SpecMER ( $c = 5$ ) compared to those from the target model, highlighting the improved coverage of high-likelihood sequences. + +Table 4: Comparison of top-20 NLL from sequences generated by the target model (ProGen2-M) versus SpecMER using the same temperature values. Sequences are scored using ProGen2-M. + +
MethodBgl3GFPRBP1GB1ParD3
Target0.78 ± 0.020.51 ± 0.041.62 ± 0.152.27 ± 0.240.69 ± 0.11
SpecMER (c = 5)0.63 ± 0.110.41 ± 0.071.72 ± 0.302.20 ± 0.310.67 ± 0.12
+ +Wall-time speedups. We observed an increase in sequence likelihoods as the number of candidates increased (Table 2a). This improvement, however, incurred a slight cost to generation speed, indicating a trade-off between sequence quality and computational efficiency. As demonstrated in Table 5, generating more candidates slowed decoding, though the decline in speed was minimal. Speculative decoding ( $c = 1$ ) achieved the fastest generation speed with an average speedup of $32\%$ , while SpecMER ( $c = 5$ ) yielded the slowest configuration, achieving a modest $24\%$ speedup. Notably, $c = 3$ yielded the largest increase in sequence log-likelihood for the smallest decrease in speed. We noticed an increase in variance as $c$ increased due to batch generation, which we discuss further in Appendix B.1. + +We also tested the generation speed with ProGen2-XL as the target model and ProGen2-M as the draft, finding that with $c = 3$ , SpecMER demonstrated a speed increase of $38\%$ over target-only decoding. In reference to the example from the introduction, SpecMER ( $c = 3$ ) would have taken 40 hours to generate 20,000 protein sequences of length 200 amino acids, saving an entire day of computing. + +Table 5: Generation speed (tokens/sec) over the best configuration per method. Results are averaged over GFP, RBP1, and GB1. The draft and target models are ProGen2-S and ProGen2-M, and the baseline is speculative decoding ( $c = 1$ ). Speedup is calculated with respect to the target model. + +
-DraftTargetBaselineSpecMER (c = 2)SpecMER (c = 3)SpecMER (c = 5)
Toks/sec74.1131.4841.62 ± 0.8841.43 ± 1.2440.57 ± 2.8339.11 ± 4.50
Speedup--32%32%29%24%
+ +While increasing the number of candidates decreased tokens per second, it resulted in selecting better candidates the majority of the time. SpecMER $(c = 5)$ had the lowest misranking error (Equation 3 and Figure 3), selecting an acceptable sequence from a pool of candidates $92\%$ of the time. Of the set of tokens consistent with the target distribution, SpecMER was able to correctly discern between tokens that increased acceptance and tokens that increased both acceptance and biological plausibility. To summarize the trade-off space, increasing $c$ decreases $\epsilon$ , tokens per second, and negative log-likelihood, while increasing acceptance ratio and pLDDT scores. Further discussion on this trade-off space can be found in Appendix B.2. + +# 5 Conclusion + +Discussion. In this study, we developed the first speculative decoding framework for protein generation. Using ProGen2-S as the draft model and ProGen2-M as the target model, we achieved an average speedup of $32\%$ across functionally diverse proteins. Building on this foundation, we developed SpecMER, a novel framework that incorporates k-mer guidance derived from multiple sequence alignments to guide generation toward biologically meaningful regions of sequence space. + +SpecMER drafts multiple candidate sequences and uses evolutionary patterns from k-mers to select completions that are more likely to fold into stable, biologically plausible proteins. As illustrated in Figure 1, speculative decoding often undersamples high-likelihood sequences, despite proposing tokens consistent with the target model distribution. SpecMER addresses this deficiency by guiding generation toward sequences that both align with MSA-derived motifs and exhibit improved log-likelihood and pLDDT scores. In doing so, it preserves the speed advantages of speculative decoding while producing sequences that are more likely to fold into stable conformations, achieving the best of both efficiency and quality. + +Beyond empirical gains, we also provide bounds on wall-time speedups and define the criteria for selecting tokens that conform to guidance and acceptance objectives. Furthermore, we offer practical guidance for tailoring hyperparameter configurations to maximize performance improvements for protein design workflows. As speculative decoding techniques continue to evolve and more protein data becomes available, we anticipate larger speedups in automated protein generation. SpecMER marks a step forward in accelerating protein design, paving the way for high-throughput design workflows and beyond. + +Limitations. The effectiveness of SpecMER depends on the quality of the input alignment (MSA); performance may degrade when informative motifs are sparse or unavailable. For example, target proteins with extensive disordered regions or lacking discernible structural patterns may benefit less from k-mer guidance. While k-mers impose minimal computational overhead, increasing the number of drafted candidates $c$ increases compute and therefore energy cost, which may limit scalability in certain settings. SpecMER utilizes batch generation to efficiently draft multiple candidates, though this is not strictly parallel. In practice, a fully parallel implementation could yield even greater speedups, but was not attainable given the limitations of our hardware setup. + +Future work. While our method was developed for conditional generation of protein sequences, its principles could extend to natural language, where low-cost scoring functions can guide generation under constraints such as style or safety. + +# Acknowledgments and Disclosure of Funding + +This work was supported by the Parker H. 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Let $c_{e}$ be the cost coefficient for one iteration of SpecMER. Then, + +$$ +c _ {e} = \frac {c M _ {p} + M _ {k}}{M _ {q}}, \tag {7} +$$ + +where $M_{k}$ is the time it takes the k-mer scoring function to score $c$ candidates. Assuming $M_{p}$ can batch generate candidates, then the cost of $M_{p}$ grows sublinearly with respect to $c$ . Additionally, $M_{k} \ll M_{p}$ , and it follows that: + +$$ +c _ {e} = \frac {\xi M _ {p}}{M _ {q}}, \tag {8} +$$ + +where $1 \leq \xi < c$ is the batch generation cost. Under true parallelism, $\xi = 1$ and the expected improvement factor remains the same as in Equation (1). + +Proposition A.2. The expected (batch) wall-time speedup factor of batch generation is + +$$ +S (\gamma) \approx \frac {1 - \alpha^ {\gamma + 1}}{(1 - \alpha) \left[ c _ {e} + 1 \right]}. \tag {9} +$$ + +Proof. Let $T_{\mathrm{iteration}}(\gamma)$ be the time it takes for one iteration of draft selection and evaluation to complete on $\gamma$ draft tokens. Similarly, let $T_p(\gamma)$ , $T_q(\gamma)$ , and $T_k$ be the time it takes for the draft model (single candidate), target model, and k-mer scoring function to complete one iteration. + +Since $T_{p}(\gamma)$ can batch generate, $T_{p}(\gamma) = \xi T_{p}(\gamma)$ , whereas in the serial case, $T_{p}(\gamma) = cT_{p}(\gamma)$ . The total time for an iteration is then $T_{\mathrm{iteration}}(\gamma) = [\xi T_p(\gamma) + T_k] + T_q(\gamma)$ , with an expected speedup of: + +$$ +S (\gamma) = \frac {1 - \alpha^ {\gamma + 1}}{(1 - \alpha) \left(\frac {T _ {\text {i t e r a t i o n}} (\gamma)}{T _ {q} (\gamma)}\right)}. \tag {10} +$$ + +Let us define the cost coefficient $c_{\mathrm{draft}}(\gamma) = \frac{\xi T_p(\gamma) + T_k}{T_q(\gamma)}$ . By Equation (8), $c_{\mathrm{draft}}(\gamma) \approx c_e$ , simplifying the ratio between total iteration time and target iteration time $\frac{T_{\mathrm{iteration}}(\gamma)}{T_q(\gamma)} \approx c_e + 1$ . Consequently, + +$$ +S (\gamma) \approx \frac {1 - \alpha^ {\gamma + 1}}{(1 - \alpha) [ c _ {e} + 1 ]}. \tag {11} +$$ + +In the case that generation is serial, the wall-time speedup results in a similar equation. + +Corollary A.3. The expected serial wall-time speedup factor is: + +$$ +S (\gamma) \approx \frac {1 - \alpha^ {\gamma + 1}}{(1 - \alpha) [ \frac {c}{\xi} c _ {e} + 1 ]}. \tag {12} +$$ + +We note that $T_{p}(\gamma)$ and $T_{q}(\gamma)$ are functions of $\gamma$ as the time it takes to decode a sequence of length $\gamma$ does not scale linearly. In practice, generating short sequences approximately scales linearly with $\gamma$ , in which near exact equality is achieved in Equation (9) and Equation (12). Increasing the acceptance ratio leads to large gains in time saved across all configurations of $\gamma$ . + +# B Hyperparameters + +# B.1 Implementation Details + +Code for SpecMER is publicly available here: https://github.com/amirgroup-codes/SpecMER.git. + +ProGen2 [4] is a decoder-only Transformer architecture with four different parameter sizes: small - 151M; medium/base - 764M; large - 2.7B; xlarge - 6.4B. No model training is performed in this work; inference is run on a server with eight NVIDIA A6000 GPUs. + +We conducted extensive testing to find the correct parallel implementation of candidate sequence generation. Batch generation, while not exactly parallel, yielded the best results in regard to tokens per second. We estimate that batch generation took about $1.2 - 1.3 \times$ longer to generate $c = 5$ sequences as compared to $c = 1$ sequences, incurring additional latency due to larger batch operations. This discrepancy in generation time resulted in a larger variance in tokens per second as $c$ increased. When a speculated token is rejected, decoding takes longer, and the added cost of batch generation appears in the variance. Due to the non-deterministic nature of sampling, some generations have higher acceptance rates than others; therefore, occasional rises and dips in acceptance have a larger impact on tokens per second than producing a single sequence at a time. We implemented SpecMER on top of the sample method implementation provided by ProGen2. We considered attempting a fully parallel implementation of batched sequence generation; however, we ultimately decided to defer to the default implementation. This was due to hardware limitations and practical considerations. A fully parallel multi-GPU system, while likely to yield a higher tokens per second, is significantly more demanding for practitioners who wish to use SpecMER. We implemented SpecMER with single GPU instances in mind. + +In addition to testing configurations for parallel implementations, we tested KV caching versus full rescoring during each iteration of SpecMER. In a normal setting, KV caching would almost always be preferable. However, due to the maximal coupling acceptance logic, full rescoring versus KV caching depended on the expected performance of the decoder. We stored KV caches for the draft model and target model, updating at each iteration; however, if any token was rejected and resampled using the logic in Algorithm 1, that token would require a full forward pass. Since correction tokens are resampled from the residual distribution, the corresponding logits must be recomputed with a full forward pass. This poses a tradeoff: either compute a full uncached forward pass for both the draft and target model at each iteration, or compute the cached forward pass at the risk of incurring an extra forward pass for every rejected token. We observed that as long as the speculative decoder achieved an acceptance ratio of $\approx 80\%$ or higher, the KV cache implementation was faster. For this reason, we chose to use a custom implementation of KV caching for SpecMER. We recognize that our implementation of SpecMER is not fully optimized and could be improved to realize even faster speedups. We leave this task for future work, or for specialized implementations with access to better computing hardware. + +# B.2 Trade-off Space + +Selecting the number of candidates to draft $c$ comes at the cost of a trade-off between generation speed and generation quality. Figure 3 details how $c$ influences generation speed and misranking error $\epsilon$ . Setting $c = 3$ yielded the best tradeoff in this space, demonstrating the largest gain in likelihood improvements for the least decrease in tokens per second. We observed an improvement in misranking error for SpecMER, indicating that structural cues from SpecMER not only increase structural plausibility but also help close the distribution gap between the target and draft model. The target model in protein generation often exhibits higher correlation with functional and structural benchmarks, learning these auxiliary objectives implicitly from millions of protein sequences. This correlation increases as a function of the number of parameters. SpecMER illustrates that this auxiliary gap between the target and draft can be supplanted with external functional or structural information. + +# B.3 Results From Hyperparameter Sweep + +This section details how each hyperparameter impacted acceptance and likelihood. As stated in the main text, we swept over the following hyperparameters: draft tokens $(\gamma)$ , temperature $(T)$ , and k-mers $(k)$ . We tested every combination of the following configurations: $\gamma \in \{5,10,15\}$ , $T \in \{0.7,1,1.4\}$ , $k \in \{(1),(3),(1,3),(1,3,5)\}$ . We chose this consolidated list to encompass realistic hyperparameters used for protein generation. For each configuration, we generated 200 protein sequences, stopping generation at max sequence length (length of the wild-type for the context being used) or upon generation of a stop token ("2" in the case of ProGen2). Depending on the protein selected, testing a single configuration could last anywhere between 2 - 30 minutes to generate 200 full-length sequences. The final hyperparameter set used to report results in Table 2 is listed in Table 6. + +![](images/344f85254cc6a32a9509a36064869ba1275bfa90a468c5b0c6a8bf0cd2386514.jpg) +Figure 3: Trade-off space between number of candidates, misranking error, and negative log-likelihood (NLL). a, Generation speed versus NLL. As $c$ increases, NLL drastically increases at the penalty of reduced generation speed. b, Number of candidates versus misranking error $\epsilon$ . As $c$ increases, misranking error decreases, indicating the influence of k-mers on selecting candidates that maximize acceptance while also maximizing the k-mer score. + +![](images/27a09efcb2f93c3767fae379448c4fc7e47ac069de079ee5cdc15262a64efecf.jpg) + +Table 6: Final hyperparameter configurations for each protein corresponding to results in Table 2. + +
ProteinTemperatureDraft TokenskvalueCandidates
Bgl31.0535
GFP0.751,35
RBP11.01035
GB11.4101,3,55
ParD31.051,3,55
CBS0.751,3,55
ADRB20.751,35
+ +We observed variance in generated sequence lengths, largely dependent on temperature. Sharper temperatures $(T = 0.7)$ led to longer sequences on average, shifting probability mass away from producing a stop token early. Sequence likelihoods were heavily influenced by $T$ , with likelihood increasing on average when temperature decreased. This is not an unexpected result: decreasing temperature with top- $p$ sampling results in a more deterministic sampling procedure. With this in mind, we chose to generate sequences over a range of temperature values, as increased sequence likelihood does not always result in more plausible structures. + +We noticed that each protein had subtle preferences toward the selection of $\gamma$ . Shorter proteins, such as ParD3 and GB1, preferred $\gamma = 5$ , while longer proteins, such as GFP, saw improvements with $\gamma = 15$ . Bgl3 and RBP1 showed no significant preference toward this hyperparameter. Plots pertaining to the effect of $T$ , $k$ , and $c$ are available at the conclusion of the appendix. + +For additional context to the main results, we include the pLDDT and NLL measurements for each protein in Table 7. + +Table 7: Negative log-likelihood (NLL) and pLDDT scores for each wild-type sequence. Missing values indicate proteins for which structural predictions were not available. + +
ProteinNLLpLDDT
CBS0.75-
Bgl30.92-
ADRB21.31-
ParD32.110.79
GB12.520.82
RBP12.630.83
GFP2.930.42
+ +# C MSA Ablations + +We performed two ablation studies to test the efficacy of k-mers in SpecMER. + +Cross-protein k-mers. We tested SpecMER using mismatched evolutionary signals: (1) generate conditioned on GFP with GB1-derived k-mers, and (2) generate conditioned on GB1 with Bgl3-derived k-mers. The results are shown in Table 8. In both cases, we observed a drop in likelihoods, both on average and for the top-20 highest-likelihood sequences, compared to the protein-specific results in Table 4. This suggests that protein-specific k-mers are responsible for the increase in likelihood observed with SpecMER. + +Table 8: Cross-protein k-mer ablation results. + +
ConditionMean NLLTop-20 NLL
GFP + GB1 k-mers2.52 ± 0.271.78 ± 0.23
GB1 + Bgl3 k-mers2.79 ± 0.102.59 ± 0.11
+ +MSA depth. We next tested the importance of MSA depth by generating Bgl3 proteins using only 1,000 sequences from the MSA instead of the full set of $\sim 130\mathrm{k}$ sequences. This reduction limited the number of k-mers available to guide generation. The top-20 sequences yielded an average NLL of $1.56 \pm 0.20$ , compared to $0.63 \pm 0.11$ for SpecMER ( $c = 5$ ) with the full-depth MSA. These results support our conclusion that SpecMER's performance can degrade when informative motifs are sparse or unavailable. + +# D Structural Scores and Embeddings + +This section describes experiments pertaining to structural scores, diversity metrics, and embeddings for generated protein sequences. For each section, metrics are computed over all tested configurations of vanilla speculative decoding and SpecMER. + +# D.1 Sequence Diversity + +Novelty is a key metric for determining how well generative models sample from the space of learned protein sequences. We computed two diversity metrics to assess how SpecMER affects sequence novelty: WT Hamming distance and inter-sequence Hamming distance. WT Hamming distance illustrates how many edits away the generated protein is from the wild-type. Inter-sequence illustrates how similar sequences generated under the same configuration are. Results for both metrics are reported in Table D.1. + +Table 9: Wild-type (WT) distance and inter-sequence (Inter-Seq) distance across proteins. SpecMER generates sequences further from the WT while maintaining inter-sequence diversity, thereby exploring sequence space while producing plausible protein sequences. + +
ProteinWT Dist. (SpecMER)WT Dist. (Spec. Dec.)Inter-Seq (SpecMER)Inter-Seq (Spec. Dec.)
GFP208.35 ± 5.76208.49 ± 4.91181.78 ± 27.14184.56 ± 27.14
RBP141.27 ± 3.4842.81 ± 3.4342.60 ± 3.8744.88 ± 4.00
ParD375.97 ± 3.0178.68 ± 2.4167.39 ± 6.8770.00 ± 5.41
GB144.70 ± 3.2545.27 ± 3.2746.47 ± 4.4646.99 ± 3.86
Bgl3324.88 ± 30.88333.12 ± 33.96261.02 ± 42.28284.97 ± 40.44
CBS378.64 ± 140.60431.00 ± 106.05291.84 ± 161.60457.27 ± 40.98
ADRB2263.80 ± 120.22290.18 ± 109.01270.82 ± 93.46340.28 ± 47.66
+ +We observed that SpecMER followed the same trends as speculative decoding for both WT and inter-sequence distance. Both methods generate many dissimilar designs far away from the WT, ensuring that sequence generation explores novel sequence space, a key advantage of autoregressive PLM generation. + +# D.2 Embeddings + +We assessed embeddings of sequences generated from vanilla speculative decoding and SpecMER. Embeddings were generated using ESM-2 8M [31] with mean pooling and unaligned sequences. ESM-2 is a protein language model that embeds sequences that are functionally or structurally similar closer together. For each protein used as context for conditional generation, we computed embeddings for its respective MSA. Sequences from the MSA are expressed in nature, providing an anchor point for which sequences are likely to be biologically plausible. We used PCA to compare embeddings over two principal components, finding that sequences generated by SpecMER were closer in proximity to the MSA sequences than vanilla speculative decoding. We deduced from this analysis that k-mers formed from the MSA did guide generation toward biologically plausible sequences. However, this analysis alone does not provide enough evidence to support this claim, as embedding distance and PCA are approximate metrics for sequence plausibility. To further corroborate claims of plausibility, we computed structural confidence scores as detailed in the next section. Embedding plots for each protein can be found at the conclusion of the appendix. + +# D.3 pLDDT Scores + +We assessed pLDDT [30] scores as a structural confidence measure. pLDDT ranges from 0 - 100 at each amino acid site, with 0 indicating a structure very unlikely to fold, and 100 indicating a structure very likely to fold. We computed pLDDT scores using ESMFold [31], a structure prediction model that provides per-residue pLDDT scores. To assess the overall structural confidence of generated proteins, we averaged the pLDDT scores across all amino acid positions. Figure 2b details pLDDT scores from top configurations of vanilla speculative decoding and SpecMER for different $c$ values, with generation conditioned on RBP1. For each candidate $c$ , we identified the three best configurations for each method, determined by average log-likelihood. Furthermore, we selected the top-100 most likely sequences from these three configurations, resulting in a set of 300 data points for each $c$ value. This was done to ensure diversity in the set of sampled sequences. We observed that as $c$ increased, so did the average pLDDT score. This indicates a clear relationship between increasing the influence of k-mers and generating sequences with higher biological plausibility. + +We computed average (Table 3) pLDDT scores for all proteins with the exception of Bgl3. This is due to the sequence length, which exceeds 400, and is therefore incompatible with ESMFold. For the remaining proteins, we selected the top 300 sequences across the three best configurations for each method as described before. We observed that sequences generated using SpecMER yielded the highest pLDDT scores, with the exception of GFP, which demonstrated the highest pLDDT scores under vanilla speculative decoding. This was a surprising observation, as conditional generation using GFP as the context sequence yielded higher log-likelihoods when using SpecMER. However, top-5 statistics indicated higher-end scores were comparable for GFP. A summary of top-5 pLDDT scores can be found in Table 10. We observed a similar trend with top-5 and average pLDDT scores. That is, increasing $c$ led to higher pLDDT scores, with the exception of GFP, which was competitive across all values of $c$ . + +Table 10: Top-5 pLDDT scores across four different proteins. Sequences are collected from the three best configurations for each decoding method, determined by the highest average log-likelihood under the target model (ProGen2-M). + +
ProteinSpeculative Decoding (c=1)SpecMER (c=2)SpecMER (c=3)SpecMER (c=5)
GFP0.892 ± 0.0200.838 ± 0.0620.888 ± 0.0080.850 ± 0.035
RBP10.864 ± 0.0190.884 ± 0.0160.876 ± 0.0160.888 ± 0.011
ParD30.912 ± 0.0040.918 ± 0.0080.922 ± 0.0040.926 ± 0.005
GB10.790 ± 0.0300.704 ± 0.0100.760 ± 0.0590.794 ± 0.019
+ +# E Multiple Sequence Alignment and K-mer Generation + +We collect MSAs of target proteins from ProteinGym [20]. As described in Section 3.2, k-mers are formed by applying a $k$ length sliding window over all sequences in the MSA of a respective protein. Gap characters “-” are ignored. + +We observed that the best $k$ value for each protein varied substantially. Some proteins demonstrated little to no preference toward $k$ , such as RBP1 and GB1, whereas GFP was heavily influenced by $k = \{1,3\}$ and $k = \{1,3,5\}$ . It is possible that for proteins comprised of large disordered regions, SpecMER would show reduced effectiveness, or even be detrimental to generation. Plots describing how $k$ values affected log-likelihood can be found at the conclusion of the appendix. + +![](images/bf1e9a3fd2aac9a744bc318e02d719d76eeee84fa299d5749e1460e3b5df8616.jpg) +Figure 4: Log-likelihood versus selection of $k$ for ParD3. + +![](images/2fa15dd3edff04534191a447ec178d5eae4c3c1bd9eff593d1342ee3047da6e5.jpg) +Figure 5: Log-likelihood versus selection of candidates $c$ for ParD3. + +![](images/e4c24f42a1dda3fe62d10e67cc367b532abd73ed4ad0deb3cc4969a5933fffba.jpg) +Figure 6: Log-likelihood versus selection of temperature for ParD3. + +![](images/24f59ccb234bee8bb6057a7a4ac3220ea52db029bfeb4f740c3620c800490930.jpg) +Figure 7: Likelihood distribution of ParD3 sequences generated using SpecMER compared to the likelihood distribution of its collected MSA (scored using ProGen2-M). + +![](images/b913fac01b238c5a407a6c5c5e78198da9cbd6c79cd2c595060727f5187b6bc7.jpg) +Figure 8: PCA plot for sequence embeddings generated with varying $c$ , with $c = 1$ denoting speculative decoding, and $c > 1$ denoting SpecMER. Sequences are generated using the context of ParD3. + +![](images/ae74c65680090ad451d747d49a4581b0b2d4a2df465d348aea3a235a44a053a1.jpg) +Figure 9: Log-likelihood versus selection of $k$ for GB1. + +![](images/e2fb00f0db8bbda0cb5744935162c2115c73e6e5762919ec66a9515ce6e9b931.jpg) +Figure 10: Log-likelihood versus selection of candidates $c$ for GB1. + +![](images/615ac55464b0b139718cca8704f0d28a972e76891d90687db0b22c252c1eaf52.jpg) +Figure 11: Log-likelihood versus selection of temperature for GB1. + +![](images/4064034837f2fd8f23de83c33509d8b4a0041841fafcb7dcae6475504ee5a50f.jpg) +Figure 12: Likelihood distribution of GB1 sequences generated using SpecMER compared to the likelihood distribution of its collected MSA (scored using ProGen2-M). + +![](images/c2940fed525aec22019a8d89470edc834ba34d4d48801f282fd09227b690bc03.jpg) +Figure 13: PCA plot for sequence embeddings generated with varying $c$ , with $c = 1$ denoting speculative decoding, and $c > 1$ denoting SpecMER. Sequences are generated using the context of GB1. + +![](images/0d8d3a120afed02a0525013bf6697aa16cc4f856dd4561cf82f48115dadfb81b.jpg) +Figure 14: Log-likelihood versus selection of $k$ for GFP. + +![](images/f9b5afdac95112741f07eb1d85647165348819c1bb30cceb9021de65d8e01ac3.jpg) +Figure 15: Log-likelihood versus selection of candidates $c$ for GFP. + +![](images/9d18a85e412e981a716eecfacbc9a97485ba4eace6365c65252fccc0e2b1cea7.jpg) +Figure 16: Log-likelihood versus selection of temperature for GFP. + +![](images/3ff6028bcae224ecf2bb44f027f7b20baee9a97ad98be127a351d273bf9009e0.jpg) +Figure 17: Likelihood distribution of GFP sequences generated using SpecMER compared to the likelihood distribution of its collected MSA (scored using ProGen2-M). + +![](images/d7bc378e460fd3ea4130aadd738b0703f7859a36fd05b17dd6b4f3a891324df3.jpg) +Figure 18: PCA plot for sequence embeddings generated with varying $c$ , with $c = 1$ denoting speculative decoding, and $c > 1$ denoting SpecMER. Sequences are generated using the context of GFP. + +![](images/71330715c4e114432c9a9172fbce5388983c9e5acc8244fe496ff8d365e2639f.jpg) +Figure 19: Log-likelihood versus selection of $k$ for Bgl3. + +![](images/6b14c6c1f18f95092156a2739791b38765f475a3ab668cd9531486bb5bfcb763.jpg) +Figure 20: Log-likelihood versus selection of candidates $c$ for Bgl3. + +![](images/208202b27c3b8961493d7417579e3e3717d118bd40f21630e9b6a63f0e5a6f33.jpg) +Figure 21: Log-likelihood versus selection of temperature for Bgl3. + +![](images/87de1e2a93861438d0ff5f6eb9967be0b0870490345f59886fe662a1817b9220.jpg) +Figure 22: Likelihood distribution of Bgl3 sequences generated using SpecMER compared to the likelihood distribution of its collected MSA (scored using ProGen2-M). + +![](images/073189ad3469ae98f5306aa3101a34d11fce068aaf04e5389bdbfbc9a87aff1d.jpg) +Figure 23: PCA plot for sequence embeddings generated with varying $c$ , with $c = 1$ denoting speculative decoding, and $c > 1$ denoting SpecMER. Sequences are generated using the context of Bgl3. + +![](images/37be5f43420056c47f70864c0ada50b0f61eb8197cc63974497dfda1727a1771.jpg) +Figure 24: Log-likelihood versus selection of $k$ for RBP1. + +![](images/4620789dbe79ca3d55a11f89940c496762fa00d3ef55b75e02241de79809e195.jpg) +Figure 25: Log-likelihood versus selection of candidates $c$ for RBP1. + +![](images/1809ae85483f62c3db1e5b2275634eb022010c9a5f50cd2b7994c2edfd5a5e2a.jpg) +Figure 26: Log-likelihood versus selection of temperature for RBP1. + +![](images/a38871f5d7a960b86c6ea373fbe3866d77bcb0ec57db7b346c6590731553704b.jpg) +Figure 27: Likelihood distribution of RBP1 sequences generated using SpecMER compared to the likelihood distribution of its collected MSA (scored using ProGen2-M). + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Generation speed claims are supported in Section 4.3 and Tables 2 and 5. Claims of sequence plausibility are corroborated in Section 4.3, Figure 2, and Tables 2 and 3. Theoretical results are proven in Section 3.3, with additional results proven in Appendix A. The scope of claims is centered around protein generation, which is supported by over five different functionally diverse protein sequences. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: Method development did not involve LLMs. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/NeurIPS/2025/SpecMER_ Fast Protein Generation with K-mer Guided Speculative Decoding/images.zip b/NeurIPS/2025/SpecMER_ Fast Protein Generation with K-mer Guided Speculative Decoding/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..77271e6647200edd3c67e3a0af31bf92d3bea718 --- /dev/null +++ b/NeurIPS/2025/SpecMER_ Fast Protein Generation with K-mer Guided Speculative Decoding/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1337f02ff74e5413ab75c7c8187a328ef33019c733bcf85a21c5c91c0d07fded +size 1840591 diff --git a/NeurIPS/2025/SpecMER_ Fast Protein Generation with K-mer Guided Speculative Decoding/layout.json b/NeurIPS/2025/SpecMER_ Fast Protein Generation with K-mer Guided Speculative Decoding/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..b023014cdd3702646c2999cbf0840ecf7d922c0d --- /dev/null +++ b/NeurIPS/2025/SpecMER_ Fast Protein Generation with K-mer Guided Speculative Decoding/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6aae22ca525e5b9fd0e233ed04f0d63f5567654a2ee86cdc3c83502b9f2ce436 +size 970114 diff --git a/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_content_list.json b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b060d4a40acfc95de81cc230e9637f466235a14b --- /dev/null +++ b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aec535df71e8b30182bf5c6f4edf572873859e85c639611cb812760cc94d5b79 +size 108268 diff --git a/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_model.json b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_model.json new file mode 100644 index 0000000000000000000000000000000000000000..71dfa3b3961306434ddd79a6ce877c08dcccb64e --- /dev/null +++ b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5e7cd493678e090e2be973e0eff7f80c157080b1b37a7e47a0535b18f44e2b0c +size 145059 diff --git a/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_origin.pdf b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8725f305f242614dc101888dc8fd38d174f17118 --- /dev/null +++ b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/433c4dde-2ce7-449e-9723-1d5cd1cd6950_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8d8ef3b17a781a71c2b44c4c6112efea72b50c52f707db2924303fbe246e72c6 +size 545735 diff --git a/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/full.md b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/full.md new file mode 100644 index 0000000000000000000000000000000000000000..3dc6f0161b9d6d47566303130b4ce55f52d95198 --- /dev/null +++ b/NeurIPS/2025/SpecReason_ Fast and Accurate Inference-Time Compute via Speculative Reasoning/full.md @@ -0,0 +1,500 @@ +# SpecReason: Fast and Accurate Inference-Time Compute via Speculative Reasoning + +Rui Pan $^{\S}$ Yinwei Dai $^{\S}$ Zhihao Zhang† Gabriele Oliaro† + +Zhihao Jia† Ravi Netravali§ + +$^{\S}$ Princeton University + +{ruipan,yinweid}@princeton.edu,{zhihaoz3,goliaro}@cs.cmu.edu, + +zhihao@cmu.edu, rnetravali@cs.princeton.edu + +# Abstract + +Recent advances in inference-time compute have significantly improved performance on complex tasks by generating long chains of thought (CoTs) using Large Reasoning Models (LRMs). However, this improved accuracy comes at the cost of high inference latency due to the length of generated reasoning sequences and the autoregressive nature of decoding. Our key insight in tackling these overheads is that LRM inference, and the reasoning that it embeds, is highly tolerant of approximations: complex tasks are typically broken down into simpler steps, each of which brings utility based on the semantic insight it provides for downstream steps rather than the exact tokens it generates. Accordingly, we introduce SpecReason, a system that automatically accelerates LRM inference by using a lightweight model to (speculatively) carry out simpler intermediate reasoning steps and reserving the costly base model only to assess (and potentially correct) the speculated outputs. Importantly, SpecReason's focus on exploiting the semantic flexibility of thinking tokens in preserving final-answer accuracy is complementary to prior speculation techniques, most notably speculative decoding, which demands token-level equivalence at each step. Across a variety of cross-domain reasoning benchmarks, SpecReason achieves $1.4 - 3.0 \times$ speedup over vanilla LRM inference while improving accuracy by $0.4 - 9.0\%$ . Compared to speculative decoding without SpecReason, their combination yields an additional $8.8 - 58.0\%$ latency reduction. We open-source SpecReason at https://github.com/ruipeterpan/specreason. + +# 1 Introduction + +Inference-time compute has unlocked a new axis for scaling AI capabilities. Recent advancements in Large Reasoning Models (LRMs) such as OpenAI o1/o3 [Jaech et al., 2024, ope, 2025] and DeepSeek R1 [Guo et al., 2025] have demonstrated state-of-the-art performance across a wide range of complex tasks. Although these LRMs share the architectural backbones as traditional large language models (LLMs), their inference behavior differs significantly: LRMs first "think" by generating internal thinking tokens—tokens that decompose a task into a sequence of composable reasoning steps via a long chain-of-thought (CoT) [Wei et al., 2022] before producing the final tokens that summarize the reasoning process. + +Despite their promise, LRMs incur substantial inference latency due to the length of the reasoning sequences they generate. This challenge is primarily driven by the autoregressive nature of LLMs, where decoding time scales linearly with sequence length. As a result, final output generation can routinely take minutes, if not hours, to answer a single query; such delays far exceed those of typical LLMs and are prohibitively slow for many interactive applications, ultimately degrading user experience [Fu et al., 2024b]. + +Question: Every morning Aya goes for a \(9\)-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of \(\$ 8\)/kilometers per hour, the walk takes her 4 hours, including \(\$ 8\)minutes spent in the coffee shop. When she walks \(\$ 5+\)2\(kilometers per hour, the walk takes her 2 hours and 24 minutes, including \(\$ t\)\(8 minutes spent in the coffee shop. Suppose Aya walks at \(\$ 5+\) \(\backslash\)frac{1}{2}\)kilometers per hour. Find the number of minutes the walk takes her, including the \(\$ t\)\(8 minutes spent in the coffee shop. + +![](images/deea8423b9d8d8040b1873ccbe564eeda45c43155574c10044106489d1999587.jpg) +Figure 1: SpecReason leverages a smaller reasoning model to speculate individual reasoning steps, deferring to the base model only for assessment (and optionally as a fallback), enabling faster yet accurate reasoning. For illustration, we show a math question as an example; our evaluation includes more general reasoning workloads. + +Our approach to tackling reasoning delays—without compromising accuracy—is rooted in two fundamental properties of LRMs: (1) LRMs tackle difficult tasks by generating long CoTs that decompose them into many simpler, sequential steps. For example, in mathematical problem solving, a few key reasoning steps require complex long-term planning and have a major influence on downstream reasoning, while most subsequent steps simply execute the plan through straightforward calculations or case analyses (Fig. 1); (2) The utility of an individual reasoning step hinges less on the exact wording of the thinking tokens but more on the semantic insight it provides. That is, as long as a step contributes meaningfully to advancing the CoT, it remains effective—even if phrased imprecisely or differently (Fig. 2). Moreover, LRMs possess self-reflection capabilities that enable them to revise or correct occasional missteps from earlier steps. + +Taken together, these properties make the decoding of thinking tokens—the dominant source of inference latency in LRMs—inherently more approximation tolerant than typical LLM decoding. A large fraction of intermediate reasoning steps can be effectively handled by lightweight reasoning models, which both align with the nature of these steps and can tolerate minor inaccuracies. As shown in Fig. 3, this opens the door to significantly faster inference without sacrificing output quality. + +Building on these insights, we propose SpecReason, a system for accelerating LRM inference by selectively offloading easier intermediate steps to be speculated by a smaller model without compromising final output accuracy. SpecReason employs a lightweight reasoning model to generate individual reasoning steps, while reserving the slower but more capable base model to efficiently verify these speculated steps (§4.1) and guide the reasoning process along the correct trajectory (Fig. 1). Consistent with prior findings [Song et al., 2025], we observe that base models can be prompted to act as critic models—assessing the utility of intermediate steps and accepting or rejecting them as needed (Fig. 7). + +Speculative reasoning vs. speculative decoding. While SpecReason is conceptually related to speculative decoding [Leviathan et al., 2023], which accelerates LLM inference by using a smaller draft model to predict future tokens, there are key distinctions between the two. Most notably, speculative decoding is an exact optimization: it relies on token-level equivalence between the small and base models, i.e., focusing on typical LLM serving where all generated tokens are part of the final model output being assessed. In contrast, SpecReason explicitly leverages the approximation tolerance inherent in reasoning: it targets thinking tokens—intermediate steps in the reasoning process—where semantic alignment, rather than token-level equivalence, is sufficient. This relaxation enables substantial latency savings during LRM inference, as semantically similar intermediate steps (Fig. 2) are often adequate to preserve end-task accuracy (Fig. 3). In many cases, SpecReason even improves final accuracy over the base model by generating fewer unnecessary tokens (Fig. 4). To further address the high inference cost of LRMs, SpecReason also exposes a user-configurable knob that allows trading off accuracy for latency by adjusting the tolerance level + +![](images/e79de0e1ea818a6bb7e1ce2e9971533cb71d451044711aba9f50fa1cef36ca6e.jpg) +Figure 2: The spectrum of approximations of one example reasoning step (equation 1 in Fig. 1). SpecReason can control the exactness of reasoning approximations by adjusting its acceptance threshold to navigate through the accuracy-latency tradeoff space ( $\S 5.3$ ). + +for speculative approximations. Finally and most importantly, because speculative reasoning and speculative decoding operate at different levels, we show that they are complementary techniques (§4.2), and when combined in a hierarchical speculation framework, achieve even greater reductions in inference latency. + +We evaluate SpecReason across a wide range of reasoning workloads spanning tasks of varying complexity [aim, 2025, Hendrycks et al., 2021, Rein et al., 2024]. Overall, SpecReason reduces end-to-end inference latency by $1.4 - 3.0 \times$ compared to vanilla LRM inference while improving accuracy by $0.4 - 9.0\%$ . Moreover, SpecReason can be combined with speculative decoding to provide an additional $8.8 - 58.0\%$ improvement over speculative decoding alone. + +# 2 Background + +Inference-time scaling. LRMs introduce a structured problem-solving approach that breaks down complex problems into multiple simpler reasoning steps, commonly referred to as a long chain of thought (CoT) [Wei et al., 2022]. This enables the model to generate intermediate reasoning steps before progressing further, reflect, and backtrack to correct errors if needed. LRMs that output long CoTs have been a popular approach to scale inference-time compute [Guo et al., 2025, Jaech et al., 2024, ope, 2025], and there also exist other schemes like Tree of Thoughts [Yao et al., 2023], process-reward-model-guided tree search [Lightman et al., 2023, Qi et al., 2024, Guan et al., 2025], and repeated sampling for scaling inference-time compute [Brown et al., 2024]. + +Speculative decoding. Speculation has long been a classic concept in the literature of computer architecture [Burton, 1985]. Due to the memory-bound nature of LLM decoding, recent work has also leveraged the technique of speculation to accelerate the decoding phase [Stern et al., 2018, Leviathan et al., 2023, Yan et al., 2024] of LLM inference. The speculative decoding process alternates between speculation and verification steps to ensure correctness while achieving speed-ups. The speculation phase usually consists of either a standalone draft model [Leviathan et al., 2023, Miao et al., 2024], a trainable module on top of the base model [Cai et al., 2024, Li et al., 2025], a tree-based token cache [Oliaro et al., 2024, Luo et al., 2024, Zhao et al., 2024], an n-gram lookup table [Fu et al., 2024a], or a retrieval-based data store [He et al., 2023] to make efficient but less accurate speculations. The verification process, on the other hand, is a base model chunked-prefill over the speculation results, which usually consists of either a single sequence of tokens as in [Leviathan et al., 2023] or tree-like structures to further boost the accuracy of speculation [Miao et al., 2024, Cai et al., 2024, Li et al., 2025, Chen et al., 2024]. The verification process then accepts the longest matched sequences on the token level from the speculation results and repeats the process. As a result, the speculation length is usually conservative to maintain an optimal trade-off between the speculation overhead and accuracy. + +Existing approaches for reducing latency. Sky-T1-Flash [Team, 2025] reduces unnecessary thinking tokens by fine-tuning models to curb overthinking, thereby reducing the length of reasoning chains and, consequently, latency. Dynasor-CoT [Fu et al., 2024b, 2025] takes a different approach by probing intermediate model confidence and terminating the reasoning process early when the model exhibits sufficient confidence in its current output. LessIsMore [Yang et al., 2025] selects tokens for sparse attention by leveraging cross-head attention patterns in reasoning and reduces the decoding latency. + +# 3 Motivation + +In this work, we show that reasoning workloads executed by LRMs exhibit unique opportunities for latency reduction due to their inherent tolerance to approximation—setting them apart from traditional generation tasks in LLMs. We illustrate these properties using a representative example from the AIME dataset, selected for its clarity and ease of exposition. + +Intermediate steps are easier than end-to-end reasoning. A key observation in LRM behavior is that reasoning difficulty is not uniform across the steps in a long chain-of-thought (CoT). As shown in Fig. 1, while the overall task might be too challenging for a small model to solve end-to-end, only a few critical steps—such as problem analysis, decomposition through formulations or case analyses, and high-level planning—are critical to the overall reasoning progress. In contrast, many other steps are significantly easier. + +This behavior is intentional by design: LRMs are often trained with reinforcement learning to generate CoTs that decompose complex problems into sequences of simpler, more tractable reasoning steps. These intermediate steps often include routine reasoning such as arithmetic calculations, case enumeration, or basic logical deductions—operators that are much easier to decode than synthesizing a full solution directly. This heterogeneity in step difficulty and importance creates an opportunity for lightweight models to handle a substantial portion of the reasoning process both efficiently and accurately. + +Reasoning progress depends on insights, not exact tokens. Another key takeaway from our work is that the utility of a reasoning step lies in the semantic contribution it makes to the overall reasoning process, rather than the precise tokens it uses. Unlike tasks like translation in traditional LLM inference, where fidelity to exact combinations of tokens matters more, reasoning CoTs within LRM's thinking tokens care more about the information that advances the reasoning chain. As illustrated in Fig. 2, a spectrum of valid phrasings often exists for a given step: semantically equivalent or similar expressions can convey the same insight and lead to the same downstream reasoning trajectory. This semantic flexibility is a key enabler for approximation-tolerant inference. + +Occasional mistakes can be corrected via self-reflection. LRMs exhibit strong self-reflection capabilities, enabling them to recover from earlier reasoning errors. Even when an earlier step contains a factual or logical mistake, the model often revises its trajectory in subsequent steps, marked by tokens like "Wait" or "Hmm". Moreover, unlike LLM inference where all output tokens contribute to the final answer, in LRM inference, only the tokens generated after the thinking tokens determine the final outcome. Therefore, LRM inference can tolerate occasional mistakes during the reasoning phase, as the model can often identify and correct these mistakes during self-reflection. This inherent fault tolerance further underscores the viability and effectiveness of approximation-based acceleration. + +In summary, compared to traditional LLM inference, LRM inference is inherently more tolerant of approximations that do not require token-level equivalence as long as the overall reasoning trajectory is preserved. This property is not limited to a single, linear CoT; rather, it extends naturally to more general inference-time compute scaling paradigms such as tree-based search strategies and other structured reasoning approaches. + +# 4 Method + +# 4.1 Speculative Reasoning + +Due to its reliance on autoregressive decoding, LRM inference incurs significantly higher latency than typical LLMs—often to the point of being prohibitively slow for interactive applications and degrading user experience [Fu et al., 2025]. Existing approaches for latency reduction include using a distilled version of the base model [Guo et al., 2025], limiting the number of thinking tokens via a predefined token budget, or disabling the reasoning process altogether by omitting the thinking tokens ( and ) during generation [qwe, 2025]. However, these approaches impose a harshly trade-off between accuracy for latency: they either limit the model's capacity to reason or apply a lower-quality model uniformly across all reasoning steps. In contrast, SpecReason takes a more fine-grained and adaptive approach. Instead of explicitly restricting output length, it selectively + +offloads only the easier reasoning steps to a lightweight model, preserving overall reasoning quality while substantially reducing inference latency. + +The approximation-tolerant nature of LRM reasoning enables a new form of speculative execution: tentatively carrying out reasoning steps using a lightweight model, assessing their utility with a stronger base model, and selectively accepting them. SpecReason leverages this flexibility to reduce decoding latency while preserving output quality. To achieve this goal, SpecReason offloads easier or less critical reasoning steps—defined as semantically self-contained units such as complete sentences or logical steps—to a smaller, faster speculator model. Each step is decoded in two stages: (1) the lightweight speculator proposes the next reasoning step based on the current context, and (2) the base model evaluates the proposed step for semantic utility. If the step is accepted, SpecReason proceeds to the next step; otherwise, SpecReason falls back to the base model to regenerate the step. While our implementation uses a simple static-threshold mechanism for verification, the framework supports richer, customizable decision strategies. We outline key design principles below. + +Navigating the Pareto frontier of the latency-accuracy tradeoff. SpecReason expands the Pareto frontier of the latency-accuracy tradeoff by exposing fine-grained control knobs to navigate through this space. The key knob SpecReason employs is the acceptance threshold: after each speculated reasoning step, the base model is prompted to generate a single-token utility score (e.g., an integer from 0 to 9) indicating the quality of the step. If the utility score is above a static acceptance threshold (e.g., score $\geq 7$ ), the speculated reasoning step is accepted; otherwise, it is discarded and regenerated by the base model. + +Adjusting this threshold allows users to control the strictness of speculation (Fig. 5): a higher threshold requires speculated steps to be closer to token-level equivalence on the equivalence spectrum (Fig. 2), improving accuracy but reducing the acceptance rate and thereby increasing latency. Conversely, a lower threshold increases speculation efficiency at the cost of potential accuracy degradation. + +An additional knob involves forcing the first $n$ reasoning steps to be decoded by the base model. Since LRMs often use the initial steps to analyze the problem and formulate a high-level plan, assigning these initial steps to the base model can steer the overall reasoning trajectory toward higher quality. We show in Fig. 6 that this knob also allows SpecReason to manage the latency-accuracy tradeoff, though with less impact than the acceptance threshold knob. + +While our current implementation uses a simple, discrete threshold-based scoring scheme—offering only a coarse-grained configuration space—it establishes a lower bound on verification quality. Future work can explore more sophisticated strategies, such as logprob-based confidence estimates or dynamic thresholds, to enable finer-grained tradeoffs without incurring additional runtime cost, and may further improve overall performance. + +Efficient verification. Because each step requires verification by the base model, it's crucial to keep verification overhead low to avoid compounding latency. Instead of autoregressively decoding or reranking multiple candidate steps, SpecReason evaluates each speculated step in a single prefetch-only pass of the base model. The verification prompt is templated to reuse most of the CoT prefix, so each verification requires prefilling only $\sim 70$ new tokens. Since short-prefix forward passes are memory-bound, the overhead is comparable to decoding just 1-2 tokens, making verification highly efficient in practice. + +Implementation details. Since the small model is lightweight, we colocate both the small and base models on the same GPU. The memory reserved for Key-Value caches [Kwon et al., 2023] is statically partitioned between the two models. They do not share any internal model states-only the token IDs of the generated reasoning steps are managed and shared by SpecReason. If a speculative step is rejected, the corresponding KV cache entries are discarded. + +Inference is performed sequentially: the small and base models take turns, avoiding kernel-level interference. In future work, we plan to explore pipelining to overlap the small model's decoding with the base model's inference. While this may introduce mild resource contention, it could further reduce end-to-end latency. + +# 4.2 Hierarchical Speculation across Semantic Similarity and Token Equivalence + +At a high level, SpecReason's speculative reasoning resembles the philosophy behind traditional speculative decoding, but differs in two important ways. First, speculative decoding guarantees + +token-level equivalence between draft and verified outputs, making it a form of exact acceleration. In contrast, SpecReason targets semantic-level similarity, accepting steps that carry the same insight even if phrased differently, and exposes knobs to control the exactness of reasoning approximations. Second, speculative decoding is typically applied to output generation tasks (e.g., text continuation or translation), where the fidelity of each token matters. SpecReason, on the other hand, is designed specifically for internal thinking tokens in reasoning tasks, where intermediate steps are approximate and interchangeable as long as they preserve the logical progression of thought. + +Further, because SpecReason and speculative decoding operate at different levels (semantic-level similarity vs. token-level equivalence), these two approaches are complementary and can be combined into a unified, hierarchical system - SpecReason+Decode first applies step-level speculative reasoning to draft and verify reasoning steps. If a step is rejected and regenerated by the base model, standard token-level speculative decoding can be applied during the base model regeneration to further accelerate decoding. + +# 5 Evaluation + +The overview of our evaluation results includes: + +- Reducing end-to-end latency. Because many intermediate steps are easier than end-to-end reasoning, many (up to $80\%$ ) of the speculated steps are accepted. SpecReason achieves a $1.4 - 3.0 \times$ speedup over vanilla LRM inference. Additionally, when combined with speculative decoding, SpecReason further reduces latency by $8.8 - 58.0\%$ over speculative decoding alone, highlighting the complementary nature of these optimizations. +- Improving token-budget-aware accuracy. Beyond latency reduction, SpecReason also improves accuracy over the base model by $0.4 - 9.0\%$ under the same token budget. We empirically find that small, lightweight models typically have shorter output sequence lengths – meaning, they need fewer thinking tokens before deriving an answer. Thus, by accepting many small model's speculated reasoning steps, SpecReason reduces the token consumption compared to the base model's vanilla inference. When the token budget is low – a common setup to curb inference cost and latency – SpecReason helps improve accuracy as the base model would need more tokens to get to an answer (Fig. 4). + +# 5.1 Setup + +Models. In our main results, we use two base models: QwQ-32B [qwq, 2025] and Skywork-OR1-Preview-32B [sky, 2025]. We also use two different small models for speculation: DeepSeek-R1-1.5B [Guo et al., 2025] and Zyphra's ZR1-1.5B [zyp, 2025] - both of which are based on Qwen-2.5 [Yang et al., 2024] and embed the capability of reasoning with long CoTs - and evaluate all four different model combinations. We evaluate an additional base model with a different size and architecture, R1-70B [Guo et al., 2025], a distilled version of DeepSeek-R1 onto Llama3.3-70B [Grattafiori et al., 2024], in §A.1. + +Datasets. We evaluate SpecReason on three diverse reasoning benchmarks: AIME [aim, 2025] for high-school competition-level mathematical problems, MATH500 [Hendrycks et al., 2021] for high-school competition-level mathematical problems sampled from AMC 10, AMC 12, and AIME, and GPQA Diamond [Rein et al., 2024] for graduate-level questions in general domains like biology, physics, and chemistry. The accuracy metric we evaluate on is pass@1. Similar to prior work [Guo et al., 2025], we set $k = 16$ when calculating pass@1 – i.e., we generate 16 responses with temperature=0.6 for every query and calculate the average accuracy – and set the token budget to be 8192 tokens to ensure an apples-to-apples comparison between baselines. + +Baselines. We run vanilla inference using the small and base models as the latency and accuracy baseline, respectively. Aside from SpecReason, we also run speculative decoding ("SpecDecode") with the smaller model as the draft model, speculating five tokens at a time. To demonstrate SpecReason's compatibility with speculative decoding, we also run a "SpecReason+Decode" baseline that employs the hierarchical speculation described in §4.2. + +![](images/5dec30634d4083c9ad726ba552874655dfa7038c8da53d8a9801d686752a7357.jpg) +(a) QwQ-32B + R1-1.5B + +![](images/1d5d8302cd41b1c8c06f941e3550427543065eac566de5e9cfcf5da2d74c5303.jpg) +(b) QwQ-32B + Zyphra-1.5B + +![](images/7886504774b13fedec4090c358b86079c20eaeed0f9ecce62ac3553bc3a0913a.jpg) +(c) Skywork-Preview-32B $^+$ R1-1.5B + +![](images/60e0d02cef6627032bd90ebf0d0ffba12775342f55d95c27098f0992c9a89f8a.jpg) +(d) Skywork-Preview-32B + Zyphra-1.5B +Figure 3: Comparison of the accuracy and latency of different schemes on different model combinations. SpecReason significantly reduces latency while improving accuracy over vanilla inference. When combined with speculative decoding, SpecReason outperforms speculative decoding in both latency and accuracy on all datasets and model combinations. + +Hardware. We run our evaluations on two NVIDIA A6000-48GB GPUs. We use vLLM 0.8.2 as the underlying inference engine and enable prefix caching [Kwon et al., 2023, Zheng et al., 2023, Pan et al., 2025]. Both models are served with a tensor parallelism degree of two. + +# 5.2 Main Results + +We compare SpecReason against baseline methods in Fig. 3. Across the four model combinations, SpecReason achieves a $1.5 \times -2.5 \times$ , $1.6 \times -3.0 \times$ , $1.4 \times -2.5 \times$ , $1.7 \times -2.4 \times$ reduction in latency, respectively, compared to vanilla inference with the base model. + +Accuracy improvement. Alongside these efficiency gains, SpecReason also yields modest accuracy improvements of $1.3\% - 3.6\%$ , $4.0\% - 9.0\%$ , $0.4\% - 1.7\%$ , and $1.4\% - 5.0\%$ compared to the base model. The key reason behind this accuracy improvement is the reduction in token consumption required for reasoning. In Fig. 4, we focus on the model combination with the highest overall accuracy improvement, QwQ-32B + Zyphra-1.5B, and compare the average number of thinking tokens needed to derive an answer between the base model, the small model, and SpecReason. As seen in Fig. 4a, the small model is generally less verbose than the base model, and because SpecReason adopts many speculated steps from the small model, its token consumption is also reduced by $1.2 \times -2.0 \times$ . We also focus on the AIME dataset and vary the token budget to study its effect on the difference in accuracy between SpecReason and the base model in Fig. 4b. The effect of token reduction on accuracy is the most significant for tighter output token budgets (16.2% at 4096 tokens) but shrinks as the base + +Figure 4: [QwQ-32B + Zyphra-1.5B] Intuition behind SpecReason's accuracy improvement. See Fig. 9 in §A for the full set of results. +![](images/21552857693f013149f42111a88a60677c38d8ab166ef155fdef57cd69599d6d.jpg) +(a) Output length comparison. SpecReason reduces the token consumption needed to answer queries by adopting speculated steps from small models that are less verbose. + +![](images/ce0d4d0bb0446ac0898acc4c98d8e43580ee80960e25591a47eb7c06de779074.jpg) +(b) [AIME] Accuracy gap under different token budgets. + +![](images/81248256d05e17f5080f251b0d865495406d1a307d92aae740b02d5b1585ac61.jpg) +Figure 5: [QwQ-32B + R1-1.5B] SpecReason allows trading off latency for accuracy via adjusting the acceptance threshold (from left to right, the thresholds are: 3, 5, 7, and 9 out of 9). + +![](images/bfb2fa6c2171b04a10c4be080ec56a7d08b9b85e07308f906ed99d99d672f9cc.jpg) + +![](images/39f3ec225012b64658b55f36d16f26812fae62c72c6b9d10b8ad95509527ee11.jpg) + +model is allowed to generate more thinking tokens (4.7% at 8192 tokens). We also attribute these accuracy gains to SpecReason's explicit judgment and scoring mechanism at each reasoning step, which augments the model's internal self-reflection with more structured assessment. + +When compared with speculative decoding, SpecReason lies on the Pareto frontier of the accuracy-latency tradeoff. More importantly, combining SpecReason with speculative decoding (SpecReason+Decode) results in further latency reductions of $19.4\% - 44.2\%$ , $30.8\% - 58.0\%$ , $8.8\% - 52.2\%$ , and $25.1\% - 51.8\%$ over speculative decoding alone. The most significant performance gains for SpecReason when the base model is QwQ-32B occur on the MATH dataset, where both models achieve relatively high accuracies and the capability gap between the small and base models is the narrowest. This makes intermediate steps easier for the small model to speculate correctly, increasing the acceptance rate of speculated steps and thereby lowering end-to-end latency. In comparison, Skywork-Preview-32B is slightly inferior at instruction following, so SpecReason has to adopt a higher threshold to avoid an accuracy loss, reducing SpecReason's latency wins. + +Finally, when comparing SpecReason+Decode with SpecReason, SpecReason+Decode reduces latency by $1.7 \times -1.9 \times$ , $1.7 \times -1.8 \times$ , $1.6 \times -2.2 \times$ , and $1.6 \times -2.1 \times$ , demonstrating the difference in ease of speculation across varying tasks. On these three datasets, the ratio of steps carried out by small models in SpecReason is $38.1\% - 80.0\%$ , $36.5\% - 71.3\%$ , $39.3\% - 70.2\%$ , and $41.4\% - 66.6\%$ , respectively. + +# 5.3 Controlling the Accuracy-Latency Tradeoff + +In Fig. 5, we illustrate how SpecReason enables flexible control over the accuracy-latency tradeoff, using a representative, randomly selected subdataset from the full datasets in §5.2 on QwQ-32B + R1-1.5B for ease of evaluation. During the base model's evaluation of each reasoning step, we vary the acceptance threshold for the utility score between 3, 5, 7, and 9, and report the resulting accuracy and latency. + +![](images/0cc18ea748776fa3b580c255d50e5a19e3ff0cc9610fef73dd62c3b7b1c0b9c5.jpg) +Figure 6: Effect of the alternative knob: forcing the first $n$ steps for base model decoding. + +![](images/c3c2747ca7dd1d212f321b3036facb9d912b9eda1ddb1d263e26fe2f5e75b1cf.jpg) +Figure 7: The utility scores in SpecReason closely reflect the quality score judgements from a process reward model. $x$ on the x-axis denotes PRM scores in the range $[x, x + 0.1)$ . + +On the MATH subdataset, increasing the acceptance threshold from 3 to 7 results in fewer speculative steps from the small model being accepted. This leads to a latency increase from 35.7s to 69.2s, while accuracy improves from $59.4\%$ to $63.7\%$ , due to tighter control over the approximation level of intermediate reasoning steps. Notably, the gap between SpecReason+Decode and SpecReason widens from 8.1s to 28.8s, since more reasoning steps are delegated to the base model, and SpecReason+Decode reduces only the base model's decoding time compared to SpecReason. + +A similar trend is observed on the AIME and GPQA subdatasets: as the acceptance threshold increases from 3 to 7, latency grows from 109.4s to 261.9s and from 72.7s to 223.0s, and accuracy improves from $22.3\%$ to $39.3\%$ and from $33.1\%$ to $50.7\%$ . However, the accuracy degrades less gracefully as the threshold is relaxed compared to the MATH subdataset. This is because the small model exhibits a larger performance gap relative to the base model on AIME and GPQA, making aggressive acceptance of its speculative steps more costly in terms of accuracy. + +In Fig. 6, we also study the effect of the alternative knob, forcing the first $n$ reasoning steps to be decoded by the base model, on the accuracy-latency tradeoff. As we change $n$ from 0 to 10, 20, 30, and 40, SpecReason's accuracy increases from $33.2\%$ to $37.3\%$ while the latency increases from 270.4s to 292.6s, showcasing an alternative approach to improve accuracy with a slight increase in latency. + +# 5.4 Base Model's Judgement Capability + +The base model's ability to assess the quality of intermediate reasoning steps is a crucial cornerstone of SpecReason's performance. In this experiment, we compare the scores generated by a process reward model (PRM) - which assigns a reward score to each step within the solution to a math problem - with those given by the QwQ-32B base model on the AIME dataset. Specifically, we use Math-Shepherd [Wang et al., 2023], a PRM trained via reinforcement learning from the Mistral-7B base model on math problems, to score each speculated step produced by the R1-1.5B small model. + +In Fig. 7, we bin the reward scores (a float from 0 to 1) into ten bins. Within each bin, we calculate the mean utility score given by the base model in SpecReason. This analysis demonstrates a strong correlation between the base model's and the PRM's assessments, particularly for lower-quality reasoning steps, where both models assign low scores. The results suggest that the base model can effectively approximate the PRM's judgments, making it a viable option for evaluating reasoning step quality in SpecReason. + +# 6 Conclusion + +In this work, we introduce SpecReason, a novel approach that accelerates LRM inference by leveraging speculative reasoning. By offloading simpler intermediate reasoning steps to a smaller, lightweight model and reserving the base model for assessment, SpecReason significantly reduces inference latency while maintaining or even improving accuracy. Our results demonstrate that SpecReason achieves a $1.4 - 3.0 \times$ speedup over vanilla LRM inference, with accuracy improvements ranging from $0.4 - 9.0\%$ . Additionally, when combined with speculative decoding, SpecReason further reduces latency by $8.8 - 58.0\%$ , highlighting the complementary nature of these optimizations. We + +believe this work opens up new angles for efficient LRM inference acceleration, making it especially valuable for scenarios that demand both high accuracy and low latency. + +# Acknowledgments and Disclosure of Funding + +We thank Princeton's Systems for Artificial Intelligence Lab (SAIL) and Princeton Language and Intelligence (PLI) for providing the hardware resources for running experiments. Rui would like to thank Chongyi Zheng for sharing his experience on submitting to NeurIPS. 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Efficiently programming large language models using sglang. 2023. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The main claims in the abstract and introduction accurately reflect the paper's contributions and scope. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: The design section discussed the simple nature of the default implementation of the scoring mechanism. The evaluation section discussed the potential pitfalls, e.g., model assessment's quality degradation when the base model is weak at instruction following. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. 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Given the size of the R1-70B model, we deploy it across four A100-80GB GPUs using a tensor parallelism degree of 4. + +On the AIME subdataset, SpecReason achieves a $1.5 \times$ latency reduction compared to vanilla R1-70B inference. This speedup is smaller than the gains observed with the QwQ-32B model in our main results $(1.9 \times)$ due to two key factors. First, the R1-70B model benefits from both stronger hardware and greater parallelism (4-way TP on A100s), resulting in a $1.5 \times$ lower time-per-token (TPT) compared to QwQ-32B (2-way TP on A6000s). In contrast, the smaller model R1-1.5B sees only a modest $1.1 \times$ TPT improvement on stronger hardware, which narrows the performance gap between base and small models and thus diminishes latency savings. Second, QwQ-32B is empirically a stronger model – outperforming R1-70B across many reasoning benchmarks qwq [2025] – and this performance gap impacts their respective abilities to assess intermediate steps. To maintain accuracy, we adopt a stricter acceptance threshold when using R1-70B as the base model, which reduces the fraction of steps offloaded to the small model (23.2% compared to 40.8% in the main results). + +# A.2 Intuition behind Accuracy Improvement + +![](images/b1d2cc26ad124a156bf158d76f347711cdcf62434b78674002454921cd9f2d2a.jpg) +Figure 9: Intuition behind SpecReason's accuracy improvement on all datasets and model combinations. + +In Fig. 9, we evaluate the average thinking token count of SpecReason and two vanilla inference baselines on a wide range of datasets and model combinations. 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Our approach builds upon recent work that represents symmetric LDSs as convolutions learnable via fixed spectral transformations. We show how to invert this representation, thereby recovering an LDS model from its spectral transform and yielding an end-to-end convex optimization procedure. This distillation preserves predictive accuracy while enabling constant-time and constant-space inference per token, independent of sequence length. We evaluate our method, SpectraLDS, as a component in sequence prediction architectures and demonstrate that accuracy is preserved while inference efficiency is improved on tasks such as language modeling. + +# 1 Introduction + +The emergence of attention-based transformer architectures has revolutionized sequence modeling tasks, particularly in natural language processing [37] and large-scale sequence-to-sequence learning [16, 29]. These transformer models rely on the self-attention mechanism, which allows each token in a sequence to attend to every other token, enabling strong contextual understanding. However, this approach suffers from quadratic complexity in sequence length, making it computationally expensive for longer sequences. Recent research has thus explored alternative, more efficient architectures that preserve expressiveness while reducing computational costs for long sequences. Among these are attention-free approaches such as convolution-based or state-space models (SSMs) [9, 2, 22, 28], which can offer sub-quadratic or even near-linear time generation. + +The most basic SSM, which is the starting point for all aforementioned models, are linear dynamical systems (LDS), a foundational framework for modeling sequential dependencies in control theory, signal processing, and learning theory [20, 13]: + +$$ +x _ {t} = A x _ {t - 1} + B u _ {t}, \quad \hat {y} _ {t} = C x _ {t} + D u _ {t}. \tag {1} +$$ + +Here $u_{t}$ represents the input sequence, $x_{t}$ encodes the past state information, and $\hat{y}_{t}$ approximates the target sequence. By maintaining a fixed latent state, LDSs enable efficient inference and allow for efficient memory utilization. However, gradient-based approaches for learning LDSs suffer from exploding or vanishing gradients, particularly when modeling systems with long-term memory or large hidden state dimension [14]. + +To address these limitations, Agarwal et al. [2] leveraged the spectral filtering method [14] and introduced the Spectral Transform Unit (STU), a convex relaxation that shifts from learning the hidden + +transition matrix $A$ directly to a reparameterization of the LDS impulse response that learns how input signals convolve with fixed spectral filters. This approach provably preserves the expressiveness of an LDS with symmetric $A$ while making the training problem far more tractable. In practice, it has proven robust for systems requiring long-range memory and empirical results show that hybrid STU models—with alternating attention and STU layers—can match or even surpass purely attention-based architectures, as well as other popular hybrid state-space models, on long-context tasks such as language modeling and time-series prediction [22]. Furthermore, unlike self-attention, which requires $O(L^2)$ operations per $L$ tokens during training and identically for inference with KV-caching, the STU operates in $O(L\log L)$ operations per $L$ tokens during training and $O(L\log^3 L)$ operations per $L$ tokens during inference through algorithms based on the Fast Fourier Transform [1, 22]. + +However, although STUs capture LDS-like dynamics using spectral filtering, there has been no straightforward way to convert (or "distill") a trained STU layer back into explicit LDS form. Moreover, it has remained unclear whether every STU layer can be represented as a LDS with provably small error. Such a distillation from an STU layer to an LDS layer with hidden-dimension $h$ would permit recurrent inference in place of convolution operations, reducing the cost of generating $L$ tokens during inference to $O(h \cdot L)$ , while maintaining the training robustness guarantees of the STU. + +Such a distillation is especially desirable in the light of the resurgence of SSMs as scalable alternatives to Transformers in long-context tasks [9, 10, 34]. Thus, our work allows one to retain the STU's performance on long sequences while enabling logarithmic-time per-token generation—providing an appealing alternative to both long convolutions and Transformer-based self-attention caching. + +# 1.1 Our Contribution + +We present a novel technique for distilling STU filters into LDS form, achieving a substantial reduction in operations—from $O(\log^3 L)$ to $O(\log L)$ per token $^1$ during generation—while preserving the STU's expressivity and performance. Moreover, due to the training stability guarantees of the STU architecture, even when learning a marginally stable symmetric LDS or an LDS with large hidden dimension, this distillation procedure provides the first provable method to directly learn the parameters of a symmetric LDS of arbitrarily high effective memory and with bounded noise. Specifically: + +- In Algorithm 2, we show how to convert a learned STU into explicit system matrix parameters whose recurrence can be computed in logarithmic time. +- We provide a theoretical analysis of this distillation in Theorem 1, and empirically demonstrate that the new LDS form incurs negligible degradation in modeling quality while improving the autoregressive generation speed. +- We demonstrate in Section 6 that by applying SpectraLDS to trained STUs, we maintain the same accuracy and attain $O(\log L)$ per-token computational generation cost. +- As a consequence of our theoretical analysis, we show in Section 5.4 how we can efficiently convert a symmetric LDS of arbitrary high dimension, via an intermediate STU learning step, to one of $O(\log L)$ dimension with minimal error. + +We open-source the SpectraLDS code at https://github.com/dshah02/SpectraLDS. + +# 2 Related Work + +State-space models have long been a cornerstone of control theory and signal processing. The simplest variants, linear dynamical systems, provide a succinct way to capture temporal dependencies via a hidden state that evolves with linear transitions. Classical methods include the Kalman filter [20], which remains widely used due to its robust theoretical properties and computational efficiency. In recent years, a new wave of SSMs have emerged as efficient alternatives to attention-based methods for long-sequence tasks, promising sub-quadratic or even near-linear complexity without + +compromising expressive power. Models like S4 [10] and its diagonal variants [11] exploit structured state matrices to learn long-range dependencies, while works such as Hyena, Mega, and Mamba [34, 24, 9] incorporate gating or convolution-based parameterizations to compete against (and sometimes outperform) transformers in language modeling, time-series analysis, and other long-context applications. Their growing popularity and the challenges of training SSMs underscore the need for methods with greater training robustness, efficiency, and performance guarantees. The Spectral Transform Unit (STU) [2] lies squarely in this tradition, offering a powerful convex relaxation for training LDS-like systems that achieves impressive empirical results on long-context tasks. Our work builds directly on this line of research, introducing the first method to distill a learned STU layer into an explicit LDS with provable guarantees, thereby unifying the convex training advantages of spectral filtering with the real-time inference benefits of a recurrent LDS. + +Our contributions also align with the long tradition of system identification for LDSs, where the aim is to learn the hidden transition and output matrices $(A,B,C,D)$ from observed sequences. Early influential approaches—such as the Ho-Kalman algorithm [15], Kung's method [21], and the Eigensystem Realization Algorithm (ERA) [18]—rely on linear-algebraic decompositions (e.g., SVD) of Hankel matrices consistent with observations. Modern variants allow for single-trajectory identification [31], and subsequent refinements like MOESP and N4SID [17] added stochastic noise modeling, while prediction-error and maximum-likelihood methods improved estimation accuracy and statistical efficiency. More recent lines of work incorporate regularization and spectral methods (e.g., stable-spline kernels, sparse identification) to yield more robust or interpretable LDS representations. In the context of distillation, these methods still require matrix decompositions that scale super-linearly in the problem size. + +In contrast to many previous approaches, since the STU's parameterization avoids direct reconstruction of the system matrix $A$ , distillation from a STU layer into an LDS remains agnostic to hidden dimension. Moreover, as our method uses a fixed and abstract Hankel matrix—rather than having to construct it anew from observed data—we can perform a significant part of the distillation computation offline. + +Finally, we highlight the recent closely related work of [28], which distills state-space-like recurrences from convolution-based sequence models. Their Laughing Hyena method accelerates long convolutional filters by approximating them with a diagonal LDS, thus allowing constant-time generation at inference. While this approach generalizes to any convolution-based model (e.g., Hyena [34]), it does not provide formal guarantees on distillation quality. In contrast, we focus on the STU's spectral filters, which have expressive power comparable to a symmetric LDS with real eigenvalues, and present the first theoretical framework to convert the filters into such an LDS with provable bounds on distillation quality. By leveraging the STU's fixed bank of spectral filters, our method preserves long-sequence expressiveness while achieving a symmetric LDS realization with a guaranteed approximation error (see Section 5). + +# 3 Token Generation and Complexity for Language Modeling + +In this section, we summarize the autoregressive generation costs for three model classes, Transformers, Convolutional Models, and RNNs, considering a prompt length of $T$ and the generation of $K$ tokens by each model, with $L$ the length of the convolutional filters (i.e., the maximum sequence length). We show the runtimes and memory requirements for each of the listed models in Table 1. + +Attention. Processing a prompt requires $O(T^2)$ time. However, token-by-token generation can be accelerated to $O(T + K)$ for each generated token via key-value caching, with a total of $O(T^2 + K(T + K)) = O(T^2 + K^2)$ operations and requiring $O(T + K)$ space [37, 28]. + +Convolutional Model. A Convolutional Model with $k$ convolutional filters will require $O(kN)$ operations to autoregressively generate a new output given $N$ inputs, and thus a naive autoregressive convolutional implementation will require $O(k \cdot K \cdot (T + K))$ operations to generate $K$ tokens. A more refined "Epoched Future Fill" algorithm with prompt prefilling can reduce this to + +$O(k \cdot (T \log T + K^{3/2} \sqrt{\log K}))$ to generate $K$ tokens. The "Continuous Future Fill" algorithm has theoretical guarantees of $O(k \cdot (T \log T + K \log^2 K))$ operations, although it suffers from numerical instability and has not been implemented or used in practice [1]. For the guarantees of the STU architecture, we require $k = O(\log L)$ and in practice we choose $k = 24$ [22]. + +RNN (LDS). For an RNN with state dimension $h$ , autoregressive generation requires $O(h)$ operations per token generated and $O(h)$ memory, allowing generation of $K$ tokens with $O(h \cdot (T + K))$ operations. As we will prove, for an LDS with representation capacity comparable to an STU with $k$ filters, we require $h = O(k) = O(\log L)$ . + +
MethodPrefill + Generation RuntimeCache SizeRuntime with K, T = O(L)
Standard Conv(TK + T log T + K2)kT + KL2 log L
Standard Attn.T2 + K2T + KL2
EpochFF(T log T + K3/2√log K)kKL3/2(log L)3/2
ContinuousFF(T log T + K log2 K)kKL log3 L
SpectralDLS (ours)(T + K)hhL log L
+ +Table 1: Comparison of architecture runtime and memory requirements for generating $K$ tokens from a length $T$ prompt with $k, h = O(\log L)$ , where $O(\cdot)$ is omitted for brevity. + +# 4 Problem Background + +In this section, we survey the fundamentals relevant to our approach. First, we discuss linear dynamical systems and the inherent challenges of training them directly on tasks requiring long memory. We then present the main theoretical results of spectral filtering and outline how the Spectral Transform Unit (STU) leverages fixed spectral filters to model linear recurrences without explicitly learning the transition matrix $A$ . Finally, we set the stage for our method of distilling STUs back into linear dynamical systems. + +# 4.1 Linear Dynamical Systems + +Linear dynamical systems (LDS) have been widely used in control theory to represent time-dependent processes, forming the basis of classical state-space formulations and optimal control methods [20, 23, 3, 19, 6]. Concretely, we consider an input sequence $u_{1}, u_{2}, \ldots, u_{t} \in \mathbb{R}^{n}$ , and the corresponding output sequence $y_{1}, y_{2}, \ldots, y_{t} \in \mathbb{R}^{m}$ . The hidden state $x_{t} \in \mathbb{R}^{d}$ summarizes the system's memory of past inputs, with the evolution of the system being represented as + +$$ +x _ {t} = A x _ {t - 1} + B u _ {t}, \quad y _ {t} = C x _ {t} + D u _ {t}. \tag {2} +$$ + +where $A \in \mathbb{R}^{d \times d}$ , $B \in \mathbb{R}^{d \times n}$ , $C \in \mathbb{R}^{m \times d}$ , and $D \in \mathbb{R}^{m \times n}$ . We omit dynamics and observation noise terms for simplicity with this derivation, although we test our methods on signals with noise. + +Expanding the LDS to the Convolutional Form. In a noiseless environment with $x_0 = \vec{0}$ , we can expand the LDS equations as follows: + +$$ +y _ {t} = C x _ {t} + D u _ {t} = C \left(A x _ {t - 1} + B u _ {t}\right) + D u _ {t} = \dots = \sum_ {i = 0} ^ {t - 1} C A ^ {i} B u _ {t - i} + D u _ {t}. +$$ + +Note that if any eigenvalue $|\lambda_i(A)| > 1$ , the system becomes unstable and $y_{t}$ may tend to infinity in magnitude. Even for $|\lambda_i(A)| < 1$ , systems with $\| A\| \approx 1$ are prone to failure due to large $A^i$ powers in backpropagation, as a noisy algorithm may approximate $A$ with spectral radius greater than 1 during training. If $|\lambda_i(A)| < 1 - \delta$ for some spectral gap $\delta >0$ , then + +$$ +y _ {t} = \sum_ {i = 0} ^ {\tau} C A ^ {i} B u _ {t - i} + \varepsilon_ {\tau}, \| \varepsilon_ {\tau} \| \leq \varepsilon , +$$ + +where $\tau = O\left(\frac{1}{\delta}\log \frac{1}{\varepsilon}\right)$ and the effective memory is thus on the order $\frac{1}{\delta}$ [2]. As $\delta \to 0$ , learning $A$ directly becomes unstable for large contexts [5, 32, 30], highlighting the need for methods such as spectral filtering. Since the $D$ matrix serves as a skip connection that does not affect the representation capacity, we fix it to a 0-matrix and omit its consideration for the remainder of this paper. We will sometimes use the shorthand $\mathrm{LDS}(C,A,B)$ to refer to a linear dynamical system with those parameters and $D = 0$ . + +Additionally, for the remainder of this paper, we restrict our attention to systems where $A$ is a symmetric real matrix. An LDS with symmetric $A$ can be diagonalized over the real numbers, making it equivalent to an LDS with a diagonal $A$ . Without loss of generality, we therefore assume $A$ is diagonal with eigenvalues $\alpha_{1},\ldots ,\alpha_{d}$ . + +Spectral Filtering. With initial state $x_0 = \vec{0}$ , defining $\mu(\alpha) = (1, \alpha, \alpha^2, \ldots)$ , and indexing the columns of $C$ as $c_\ell$ and the rows of $B$ as $b_\ell$ , we can extend the convolutional representation: + +$$ +y _ {t} = \sum_ {i = 0} ^ {t - 1} C A ^ {i} B u _ {t - i} = \sum_ {i = 0} ^ {t - 1} C \left(\sum_ {\ell = 1} ^ {d} \alpha_ {\ell} ^ {i} \left(e _ {\ell} \otimes e _ {\ell}\right)\right) B u _ {t - i} = \sum_ {\ell = 1} ^ {d} \left(c _ {\ell} \otimes b _ {\ell}\right) \sum_ {i = 1} ^ {t} \mu \left(\alpha_ {\ell}\right) (i) \cdot u _ {t - i + 1} +$$ + +To circumvent the non-convex optimization problem of finding $\alpha$ that best fit an LDS, [14] propose the spectral filtering algorithm, which learns an approximation of $\mu (\alpha)$ in a convex manner. They prove that given eigenvalue-eigenvector pairs $\{\sigma_j,\phi_j\}_j$ of the Hankel matrix $Z$ + +$$ +Z := \int_ {0} ^ {1} \mu (\alpha) \mu (\alpha) ^ {\top} d \alpha , Z _ {i, j} := \frac {2}{(i + j) ^ {3} - (i + j)} +$$ + +any $\mu (\alpha)$ with $0\leq \alpha \leq 1$ can be approximated by the top $k$ eigenvectors $\{\phi_1,\ldots ,\phi_k\}$ of $Z$ with an exponentially decreasing error in $k$ . Thus, if $y_{t}$ is generated by a PSD linear dynamical system, we have the following result: + +$$ +y _ {t} \approx \sum_ {\ell = 1} ^ {d} \left(c _ {\ell} \otimes b _ {\ell}\right) \sum_ {i = 1} ^ {t} \tilde {\mu} \left(\alpha_ {\ell}\right) (i) \cdot u _ {t - i + 1} = \sum_ {j = 1} ^ {k} M _ {j} \left(\sum_ {i = 1} ^ {t} \phi_ {j} (i) \cdot u _ {t - i + 1}\right) +$$ + +where we define $\tilde{\mu} (\alpha)\coloneqq \sum_{i = 1}^{k}\langle \mu (\alpha),\phi_{i}\rangle \phi_{i}$ and learn suitable parameters $M_j\in \mathbb{R}^{m\times n}$ . Rather than depending on powers of $A$ , learning an LDS with this parameterization remains convex in $\{M_j\}$ , since eigenvectors $\{\phi_j\}$ from the matrix $Z$ are computed offline. + +The Spectral Transform Unit. To account for learning negative eigenvalues of $A$ , the spectral filtering construction can be adapted by introducing positive and negative sets of feature maps. If $\{\sigma_j, \phi_j\}$ are the eigenvalue-eigenvector pairs of $Z$ , then for each time $t$ and each respective filter $\phi_j$ , we define the projections of the inputs onto the spectral basis: + +$$ +U _ {t, j} ^ {+} = \sum_ {i = 1} ^ {t} u _ {t - i + 1} \cdot \phi_ {j} (i), \quad U _ {t, j} ^ {-} = \sum_ {i = 1} ^ {t} u _ {t - i + 1} \cdot (- 1) ^ {i - 1} \cdot \phi_ {j} (i). +$$ + +One then forms the output by learning linear combinations of both $U_{t,k}^{+}$ and $U_{t,k}^{-}$ and an optional autoregressive term [12, 2]: + +$$ +y _ {t} ^ {\mathrm {S F}} = \underbrace {\sum_ {j = 1} ^ {k} M _ {j} ^ {\phi +} U _ {t - 2 , j} ^ {+} + \sum_ {j = 1} ^ {k} M _ {j} ^ {\phi -} U _ {t - 2 , j} ^ {-}} _ {\text {S p e c t r a l F i l t e r i n g c o m p o n e n t}} + \underbrace {\hat {y} _ {t - 2} + \sum_ {i = 1} ^ {3} M _ {i} ^ {u} u _ {t + 1 - i}} _ {\text {A R c o m p o n e n t}} \tag {3} +$$ + +Without the autoregressive component we compute $y_{t}^{\mathrm{SF}} = \sum_{j=1}^{k} M_{j}^{\phi+} U_{t,j}^{+} + \sum_{j=1}^{k} M_{j}^{\phi-} U_{t,j}^{-}$ . The above expression is considered the Spectral Transform Unit, where $\{M_{j}\}$ is the set of parameters to be learned using a differentiable algorithm. Following [22], we consider the STU without the autoregressive component. Empirical evidence [22] shows that hybrid STU models can compete with or even outperform purely attention-based architectures. + +Error Bounds for Spectral Approximation. We repeat the result from [14] (see also [2, 26, 27]) stating that given any LDS parameterized by $A, B, C$ where $A$ is a symmetric matrix with $\| A \| \leq 1$ , there exist matrices $M_1^{\phi+}, \ldots, M_k^{\phi+}, M_1^{\phi-}, \ldots, M_k^{\phi-}$ such that for all $L$ and for all input sequences $u_1, \ldots, u_L$ , with $\| u_t \| \leq 1$ , the following holds for all $t \in [L]$ : + +$$ +\left\| y _ {t} ^ {\mathrm {L D S}} - y _ {t} ^ {\mathrm {S F}} \right\| \sim e ^ {- \frac {k}{\log L}}. +$$ + +where $k$ is the number of spectral filters, $y_{t}^{\mathrm{LDS}}$ is the sequence generated by the LDS, and $y_{t}^{\mathrm{SF}}$ is the sequence generated by spectral filtering. + +Therefore, one can approximate any LDS that meets these specifications up to error $\varepsilon$ by selecting $k = O\left(\log L\log \left(\frac{1}{\varepsilon}\right)\right)$ spectral filters. Thus, the STU can capture LDS dynamics with only a logarithmic number of filters $k$ , providing a compact and stable representation even for systems with high effective memory ( $\| A\| \approx 1$ ). + +# 4.2 Distilling STU into an LDS + +While the STU avoids direct learning of $A$ , it still implements LDS-like dynamics via spectral filtering. A natural question is whether we can recover explicit parameters $(\widetilde{A},\widetilde{B},\widetilde{C})$ from the learned STU, enabling an equivalent recurrence rather than $O(L)$ costs when convolving spectral filters with input sequences. This can be achieved by approximating the convolution kernel of the STU by the implicit convolution kernel of an LDS. Such a distillation would bridge the gap between the stable convex training of the STU and the fast inference of a recurrent LDS. + +# 5 Algorithm and Main Result + +We now present our main theoretical result and accompanying algorithm, which shows how to recover an accurate LDS representation from the learned spectral filters. Concretely, we demonstrate that each STU filter $\phi_j$ can be approximated by a linear combination of geometrically decaying LDS impulse response filters (i.e., the LDS implicit convolutional kernel). + +# 5.1 A General Transformation from Spectral Filters to LDS + +Our main result is given in Algorithm 2. For the dynamics $\mathrm{LDS}(1 - \alpha, \alpha, 1)$ , we denote the impulse response filter by + +$$ +\mu_ {L} (\alpha) = (1 - \alpha) \left[ \begin{array}{c c c c c} 1 & \alpha & \alpha^ {2} & \ldots & \alpha^ {L - 1} \end{array} \right]. +$$ + +Note that $\mu_L(\alpha) * u_{1:L} = \sum_{i=1}^L (1 - \alpha) \alpha^{i-1} \cdot u_{L-1} = \mathrm{LDS}(1 - \alpha, \alpha, 1)(u_{1:L})$ , and thus inference with the LDS is the same as convolution with its impulse response filter. Let $\phi_1, \ldots, \phi_k \in \mathbb{R}^L$ be the spectral filters of length $L$ . We write the first $k$ filters in matrix form as $\Phi_{1:k} \in \mathbb{R}^{k \times L}$ , such that the $i$ -th row is $\phi_i$ . + +A first observation is that we can write any LDS impulse filter approximately in the spectral basis. This is a direct consequence of the spectral filtering methodology for learning an LDS [14]. + +# Algorithm 1 FindSpectralRepresentation + +1: Input: Scalar LDS parameter $\alpha \in \mathbb{R}$ , representation size $k$ . +2: Output: Spectral parameters $m \in \mathbb{R}^k$ . +3: Construct the impulse response vector $\mu_L(\alpha) \in \mathbb{R}^L$ as $\mu_L(\alpha) = (1 - \alpha)[1, \alpha_i, \alpha_i^2, \dots, \alpha_i^{L-1}]$ . +4: return best spectral fit of the system over random inputs $u \in \mathbb{R}^L$ using gradient descent + +$$ +m = \arg \min _ {m \in \mathbb {R} ^ {k}} \underset {u \in \mathbb {R} ^ {L}} {\mathbb {E}} \left[ \left| m ^ {\top} \Phi_ {1: k} u - \mu_ {L} (\alpha) ^ {\top} u \right| ^ {2} \right]. +$$ + +As a consequence of results from spectral filtering for learning LDSs, we show in Appendix A.4 that the procedure FindSpectralRepresentation returns a vector $m$ for which, for some constant $c > 0$ , + +$$ +\left\| m ^ {\top} \Phi_ {1: k} - \mu_ {L} (\alpha) \right\| \leq c e ^ {- \frac {k}{\log L}}. +$$ + +We proceed to use this subroutine to find a distillation to the spectral filters. + +Our main performance guarantee is given in the following theorem. + +Theorem 1. As long as $h \geq k$ , Algorithm 2 returns w.h.p. a matrix $\widetilde{M}$ such that + +$$ +{\left\| \Phi_ {1: k} - \widetilde {M} \mu_ {L} (\alpha_ {1}, \ldots , \alpha_ {h}) \right\|} \leq {c \lambda_ {\max} h e ^ {- \frac {k}{\log L}},} +$$ + +where $\lambda_{\mathrm{max}}$ is the largest eigenvalue of the Penrose-Moore pseudo inverse of the matrix $M$ . + +# Algorithm 2 Spectral Filters to LDS Filters + +1: Input: The first $k$ spectral filters matrix $\Phi_{1:k} \in \mathbb{R}^{k \times L}$ ; and parameter $h > k$ . +2: Output: The transformation matrix $\widetilde{M}$ . +3: Sample $h$ randomly chosen independent scalars $\alpha_{1},\ldots ,\alpha_{h}$ with each $\alpha \in [0,1]$ . +4: Construct the matrix $\mu_L(\alpha_{1:h})\in \mathbb{R}^{h\times L}$ with $i$ th row $\mu_L(\alpha_i) = (1 - \alpha_i)[1,\alpha_i,\alpha_i^2,\dots ,\alpha_i^{L - 1}]$ . +5: For each scalar impulse response, find the spectral representation by + +$$ +m _ {i} = \operatorname {F i n d S p e c t r a l R e p r e s e n t a t i o n} \left(\alpha_ {i}\right). +$$ + +6: Let $M$ be the $h \times k$ matrix whose rows are $m_i$ . + +7: return $\widetilde{M} := M^{-1} \in \mathbb{R}^{k \times h}$ + +The significance of Theorem 1 is that it allows us to translate between the representation of spectral filters and linear dynamical systems. + +We note that it is not immediate to upper bound $\lambda_{\mathrm{max}}$ . Indeed, for $h \sim k$ , this can be exponentially large in $k$ , as it corresponds to the condition number of a Vandermonde matrix [4]. However, we note experimentally, under the distribution described in the practical algorithm below, that as $h$ grows, $\lambda_{\mathrm{max}}$ quickly becomes smaller. This is an overparametrization effect, which we show experimentally in Appendix A.3. We provide analysis of Theorem 1 in Appendix A.4. + +# 5.2 Practical Algorithm + +For improved practical performance we adopt the following procedure. Fix $H \gg h$ . As above, let $\mu_{L}(\alpha_{1:H}) \in \mathbb{R}^{H \times L}$ denote $H$ impulse responses and let $\Phi_{1:k} \in \mathbb{R}^{k \times L}$ denote the first $k$ spectral filters. To obtain a representation of size $h$ , we select a small index set $S \subseteq \{1, \dots, H\}$ , $|S| = h \ll H$ , by running multi-target Orthogonal Matching Pursuit (OMP) [25, 36] on the regression + +$$ +\mu_ {L} \left(\alpha_ {1: H}\right) ^ {\top} X \approx \left(\Phi_ {1: k}\right) ^ {\top}. +$$ + +OMP greedily sets one coefficient per step from 0 to reduce the regression loss, with the selected coefficients after $h$ steps providing the indices for $S$ . Given $S$ , we solve the unregularized least-squares problem + +$$ +X ^ {\star} = \arg \min _ {X} \left\| \mu_ {L} (\alpha_ {S}) ^ {\top} X - \left(\Phi_ {1: k}\right) ^ {\top} \right\| _ {F} ^ {2}, +$$ + +to yield $\Phi_{1:k} \approx \widetilde{M}\mu_L(\alpha_{1:H})$ , where $\widetilde{M} = (X^{*})^{\top}$ . + +Another change is the distribution from which the $\alpha_{i}$ are drawn. We employ symmetric, near-unit-radius sampling to emphasize long-memory behavior: for each $i$ , draw $U_{i} \sim \mathrm{Unif}[0,1]$ and $S_{i} \sim \mathrm{Unif}(\{-1,1\})$ , and set + +$$ +\alpha_ {i} = S _ {i} \left(1 - U _ {i} ^ {4}\right). +$$ + +Equivalently, $|\alpha_i| \stackrel{d}{=} 1 - U_i^4$ with an independent random sign. This deliberately skews mass toward $|\alpha_i| \approx 1$ , improving reconstruction at a better $h$ to $H$ ratio. We ablate this choice in Appendix A.9. For numerical stability we use f1oat64 arithmetic end-to-end and scale $\Phi_{1:k}$ by $s_{\mathrm{num}} > 0$ so that $\| \mu_L(\alpha_{1:H}) \|_F \approx \| s_{\mathrm{num}} \Phi_{1:k} \|_F$ . To mitigate the impact of magnitude on OMP, we normalize the columns of $\mu_L(\alpha_{1:H})^\top$ . We later undo the scaling. In Figure 1 we report the reconstruction error of $\Phi_{1:k}$ against an LDS of hidden dimension $h$ produced by this practical method. + +# 5.3 Converting $\widetilde{M}$ into an LDS. + +As a result of Theorem 1, with $A \coloneqq \mathrm{Diag}(\alpha_1, \ldots, \alpha_h)$ , $\Gamma \coloneqq \mathrm{Diag}(1 - \alpha_1, \ldots, 1 - \alpha_h)$ and $\mu \coloneqq \mu_L(\alpha_1, \ldots, \alpha_h)$ , we can replace the costly STU convolutions: + +![](images/7d3debf160b3909d764a3bc2045fdda704fd031966de82cdc115cc95a9d85991.jpg) +Figure 1: Reconstruction error of spectral filters as a function of LDS state dimension. For this experiment, $H = 10000$ , but results are similar for $H = 1000$ to 20000 (see Appendix A.8). + +$$ +U _ {t, j} ^ {+} = \sum_ {i = 1} ^ {t} u _ {t - i + 1} \cdot \phi_ {j} (i) \approx \sum_ {i = 1} ^ {t} \left(\widetilde {M} _ {j} \mu\right) (i) \cdot u _ {t - i + 1} = \widetilde {M} _ {j} \sum_ {i = 0} ^ {t - 1} \Gamma A ^ {i} \mathbf {1} _ {h} u _ {t - i} ^ {\top} = \mathrm {L D S} (\widetilde {M} _ {j} \Gamma , A, \mathbf {1} _ {h}) (u _ {1: L} ^ {\top}) +$$ + +$$ +U _ {t, j} ^ {-} = \sum_ {i = 1} ^ {t} u _ {t - i + 1} \cdot \phi_ {j} (i) \approx \widetilde {M _ {j}} \sum_ {i = 0} ^ {t - 1} \Gamma (- A) ^ {i} \mathbf {1} _ {h} u _ {t - i} ^ {\top} = \mathrm {L D S} (\widetilde {M _ {j}} \Gamma , - A, \mathbf {1} _ {h}) (u _ {1: L} ^ {\top}) +$$ + +This provides the basis of our autoregressive inference advantage: rather than computing $U_{t,j}^{+}$ and $U_{t,j}^{-}$ as pure convolutions, we can maintain the hidden state for LDS $(\widetilde{M}_j\Gamma ,A,\vec{1})$ and LDS $(\widetilde{M}_j\Gamma , - A,\vec{1})$ for the $O(h)$ computation during inference. For practical efficiency, we can compute all $U_{t,1}^{+},\ldots ,U_{t,k}^{+},U_{t,1}^{-}\ldots U_{t,k}^{-}$ simultaneously with a single LDS by leveraging the similarities in the state updates (see Appendix A.7). + +# 5.4 LDS to LDS Distillation + +A direct consequence of our approach is that we can distill any high-dimensional symmetric LDS into a low-dimensional LDS with a bounded error. For an LDS with input dimension $d_{in}$ and output dimension $d_{out}$ , spectral filtering provides an $\varepsilon$ -approximation with only $O(d_{in} \cdot d_{out} \cdot \log L \cdot \log \left(\frac{1}{\varepsilon}\right))$ parameters regardless of the hidden dimension. + +Accepting that $h\lambda_{max}$ is $O(1)$ for $h \gg k$ , as justified in Appendix A.3, we can then convert this spectral representation by application of our distillation procedure into an LDS with state dimension $d_{in} \cdot h$ , where with $h = O(k) = O(\log L\log \left(\frac{1}{\varepsilon}\right))$ we maintain $\varepsilon$ error. Thus, the distilled LDS has $O(d_{in} \cdot d_{out} \cdot \log L \cdot \log \left(\frac{1}{\varepsilon}\right))$ total parameters. In other words, the combination of spectral filtering for LDS learning and the following distillation step yields a practical method to reduce the state dimension while preserving the system's dynamics within tight error bounds. Empirically, for tasks as difficult as modeling language, we only require $k \leq 24$ filters for strong performance, and the state dimension $h \geq 80$ of the distilled LDS suffices to retain performance. + +# 6 Experiments + +To demonstrate the effectiveness of our distillation algorithm, we begin by illustrating that a low-dimensional LDS can effectively approximate the spectral filters in Figure 3 and further examine the eigenvalues of the resulting system in Appendix A.7. Building on this, to further quantify the effectiveness of Algorithm 2 in fitting the spectral filters with practically efficient LDSs, we plot the reconstruction error of the spectral filters across different LDS state dimensions in Figure 1. These results show that our algorithm, guided by Theorem 1 amply reduces the approximation error to a sufficiently small level without an excessive increase in state dimension. + +![](images/4e3823d2428f924d34d16f2130bc36cce20c490dc5c35418e02739a6507b0313.jpg) +Figure 2: Comparison of SpectraLDS and other methods learning an arbitrary symmetric LDS with and without noise. The shaded region shows the $95\%$ confidence interval over 8 runs. Each model leveraged default configurations except the LDS, which required a lower learning rate to converge. More details are available in Appendix A.5. + +![](images/43563a69937b9502cb991fe3ed2aab285dc13a998c8b15aa702ab13fe253c2fd.jpg) + +![](images/7e42ee17d32432013832e779ea3630cbd3886ef75b125bb10541c4e4e00e894f.jpg) +Figure 3: Fit of Spectral Filters by an LDS of state dimension 160, where x-axis represents the time domain. 80 dimensions are for the spectral filters $(U_{t,k}^{+})$ and 80 for the negative features $(U_{t,k}^{-}$ , not shown). Filters are normalized to be comparable. The shading on the middle figure represents a filter quickly alternating (negative $\alpha$ ). A complete comparison for $k = 24$ without normalization is provided in Appendix A.6. These filters have MSE error $7.689 \times 10^{-19}$ . + +![](images/2a8603f5a1322a47aa1b15ef479d4d889e4e60c4b134964f9288379ade14d6f8.jpg) + +![](images/b070e81e9977cb2137aba255d846b797eb0a0d584ad19878dff09339f1bdcde8.jpg) + +To validate our method quantitatively, we perform synthetic experiments comparing SpectraLDS against other benchmarks on learning high-memory LDSs (see Appendix A.10). We find that our method significantly outperforms strong baselines in both sample efficiency and reconstruction accuracy, confirming that our approach is well suited for learning systems with long-range dependencies. + +Turning to the large-scale evaluation, we distill a 340M-parameter FlashSTU model [22] into an LDS-based architecture and compare its performance across a suite of language benchmarks. From the results in Table 2, we point out that despite the change from convolution-based spectral filters to an explicit LDS representation for the STU layers, the performance remains identical across all tasks. This observation supports our claim that the STU can be closely approximated by a low-dimensional LDS without compromising predictive accuracy. We provide details of the experimental setup and hyperparameters for the models used in Appendix A.13. + +
ModelMMLUHella.PIQABoolQWino.CSQAOBQAARC-eARC-cAverage
Flash STU 340M26.5830.4665.3460.1251.8520.4820.6054.0823.2939.20
SpectraLDS 340M26.5830.4565.2960.1250.9920.1520.2054.1723.2939.03
Flash STU Std. Err.0.370.471.110.861.401.162.061.021.24-
SpectraLDS Std. Err.0.370.461.110.861.401.152.071.021.24-
Transformer 340M26.8130.4164.6461.1051.6219.9818.8055.4721.8438.96
+ +Table 2: Evaluation of a 340M-parameter FlashSTU model and its distilled representation, replacing each convolution with an LDS of state dim. 160, on language benchmarks. Despite converting convolutions into an explicit LDS formulation, performance remains statistically equivalent. + +![](images/f5a9cc1f8197d4495766d661ea40afc8bd8787ec307dc8cd675783c29c52f798.jpg) +Figure 4: Runtime for generating sequences of increasing length across STU implementations. The naive convolution approach exhibits quadratic growth, the FutureFill variants show logarithmic growth, and the distilled STU-to-LDS layers achieve linear growth. The STU-Only Epoched Future Fill OOMs for the largest sequence length. As shown in the rightmost figure, the SpectraLDS models have nearly identical runtime despite varied state dimension. Further results are in Appendix A.11. + +Finally, we present various measures of inference speed to illustrate the constant-time per token generation provided by the distilled STU. In Figure 4, we compare the inference speed of a distilled STU model against a naive convolutional approach and a numerically stable FutureFill variant [1]. In a 12 STU layer model, the naive convolution exhibits quadratic runtime growth with sequence length; the FutureFill variants achieve logarithmic growth; and the distilled STU-to-LDS model demonstrates the best performance with linear growth. + +In the hybrid model with 6 attention and 6 STU layers, we find the distilled LDS implementation still provides a significant performance increase. Additionally, we note increases in the LDS state dimension have little impact on the overall runtime, indicating that LDS operations are not a compute bottleneck (see Appendix A.11). For both the hybrid and STU-only models up to 16,384 tokens, the distilled LDS, naive convolution, and Epoched Future Fill all have similar runtimes. All performance benchmarks were conducted on a single H100 GPU, with each generation process evaluated separately to ensure consistent measurements. We provide full experiment details in Appendix A.12. + +Looking ahead, further investigation is warranted to better understand how convolution, LDS, and attention layers interact at the hardware level, and to optimize their coordination for even greater speedup. Additionally, further work is required to determine if the LDS layer is stable below float64. + +To conclude, since optimized transformer implementations suffer from KV-Cache memory bottlenecks rather than compute bottlenecks [9], and the LDS layers have drastically lower memory requirements, we anticipate that, with appropriate optimization, the inference speed of hybrid attention-STU architectures will be independent of the amount of LDS layers. + +# 7 Conclusion and Discussion + +We have provided the first provable technique for learning the parameters of a symmetric LDS with arbitrarily high effective memory. By leveraging their convex-learning approach, we show how spectral filters can be distilled into an explicit LDS representation, enabling the construction of a state-space model with logarithmic-time inference and theoretical guarantees on the loss bounds. + +We have provided a lower bound on how increasing state dimension affects reconstruction loss for a linear dynamical system, and we envision SpectraLDS as a drop-in replacement for autoregressive convolutional layers in certain sequence-to-sequence tasks. + +# 8 Acknowledgments + +EH gratefully acknowledges support from the Office of Naval Research and Open Philanthropy. + +# References + +[1] Naman Agarwal, Xinyi Chen, Evan Dogariu, Devan Shah, Hubert Strauss, Vlad Feinberg, Daniel Suo, Peter Bartlett, and Elad Hazan. Futurefill: Fast generation from convolutional sequence models, 2025. +[2] Naman Agarwal, Daniel Suo, Xinyi Chen, and Elad Hazan. Spectral state space models, 2024. +[3] Brian D.O. Anderson and John B. Moore. Linear Optimal Control. Prentice-Hall, 1971. +[4] Bernhard Beckermann. The condition number of real vandermonde, krylov and positive definite Hankel matrices. Numerische Mathematik, 85(4):553-577, 2000. +[5] Yoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks, 5(2):157-166, 1994. +[6] Dimitri P Bertsekas. Dynamic Programming and Optimal Control, volume 1. Athena Scientific, 2005. +[7] Tri Dao. FlashAttention-2: Faster attention with better parallelism and work partitioning. In International Conference on Learning Representations (ICLR), 2024. +[8] Tri Dao, Daniel Y. Fu, Stefano Ermon, Atri Rudra, and Christopher Ré. FlashAttention: Fast and memory-efficient exact attention with IO-awareness. In Advances in Neural Information Processing Systems (NeurIPS), 2022. +[9] Albert Gu and Tri Dao. Mamba: Linear-time sequence modeling with selective state spaces, 2024. +[10] Albert Gu, Karan Goel, and Christopher Ré. Efficiently modeling long sequences with structured state spaces, 2022. +[11] Albert Gu, Ankit Gupta, Karan Goel, and Christopher Ré. On the parameterization and initialization of diagonal state space models, 2022. +[12] E. Hazan, H. Lee, K. Singh, C. Zhang, and Y. Zhang. Spectral filtering for general linear dynamical systems, 2018. +[13] Elad Hazan and Karan Singh. Introduction to online nonstochastic control. arXiv preprint arXiv:2211.09619, 2022. +[14] Elad Hazan, Karan Singh, and Cyril Zhang. Learning linear dynamical systems via spectral filtering, 2017. +[15] B. L. Ho and R. E. Kalman. Effective construction of linear state-variable models from input/output functions. Automatica, 3(1):49-59, 1966. +[16] A. Huang, S. Subramanian, J. Sum, K. Almubarak, S. Biderman, and S. Rush. The annotated transformer, 2022. +[17] I. W. Jamaludin, N. A. Wahab, N. S. Khalid, S Sahlan, Z. Ibrahim, and M F. Rahmat. N4sid and moesp subspace identification methods. In 2013 IEEE 9th International Colloquium on Signal Processing and its Applications, pages 140-145, 2013. +[18] Jer-Nan Juang and Richard S. Pappa. An eigensystem realization algorithm for modal parameter identification and model reduction. Journal of Guidance, Control, and Dynamics, 8(5):620-627, 1985. +[19] Thomas Kailath. Linear Systems. Prentice-Hall, 1980. +[20] Rudolph Emil Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82.1:35-45, 1960. +[21] S. Y. Kung. A new identification and model reduction algorithm via singular value decompositions. In Proc. 12th Asilomar Conf. Circuits, Systems and Computers, pages 705-714, 1978. + +[22] Y. I. Liu, W. Nguyen, Y. Devre, E. Dogariu, A. Majumdar, and E. Hazan. Flash stu: Fast spectral transform units, 2024. +[23] David Luenberger. An introduction to observers. IEEE Transactions on Automatic Control, 16(6):596-602, 1971. +[24] Xuezhe Ma, Chunting Zhou, Xiang Kong, Junxian He, Liangke Gui, Graham Neubig, Jonathan May, and Luke Zettlemoyer. Mega: Moving average equipped gated attention, 2023. +[25] S.G. Mallat and Zhifeng Zhang. Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41(12):3397-3415, 1993. +[26] Annie Marsden, Evan Dogariu, Naman Agarwal, Xinyi Chen, Daniel Suo, and Elad Hazan. Provable length generalization in sequence prediction via spectral filtering, 2024. +[27] Annie Marsden and Elad Hazan. Universal sequence preconditioning. arXiv preprint arXiv:2502.06545, 2025. +[28] Stefano Massaroli, Michael Poli, Daniel Y. Fu, Hermann Kumbong, Rom N. Parnichkun, Aman Timalsina, David W. Romero, Quinn McIntyre, Beidi Chen, Atri Rudra, Ce Zhang, Christopher Ré, Stefano Ermon, and Yoshua Bengio. Laughing hyena distillery: Extracting compact recurrences from convolutions, 2023. +[29] OpenAI. Gpt-4 technical report, 2024. +[30] A. Orvieto, S. L. Smith, A. Gu, A. Fernando, C. Gulcehre, R. Pascanu, and S. De. Resurrecting recurrent neural networks for long sequences, 2023. +[31] Samet Oymak and Necmiye Ozay. Non-asymptotic identification of lti systems from a single trajectory. In 2019 American control conference (ACC), pages 5655-5661. IEEE, 2019. +[32] R. Pascanu, T. Mikolov, and Y. Bengio. On the difficulty of training recurrent neural networks, 2013. +[33] Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017. +[34] Michael Poli, Stefano Massaroli, Eric Nguyen, Daniel Y. Fu, Tri Dao, Stephen Baccus, Yoshua Bengio, Stefano Ermon, and Christopher Ré. Hyena hierarchy: Towards larger convolutional language models, 2023. +[35] Ofir Press, Noah A. Smith, and Mike Lewis. Train short, test long: Attention with linear biases enables input length extrapolation, 2022. +[36] Ron Rubinstein, Michael Zibulevsky, and Michael Elad. Efficient implementation of the k-svd algorithm using batch orthogonal matching pursuit. CS Technion, 40, 01 2008. +[37] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need, 2017. + +# A Appendix + +# A.1 Notation throughout the paper + +$x_{t}\in \mathbb{R}^{d}$ state at time $t$ + +$u_{t}\in \mathbb{R}^{n}$ control (input) at time $t$ + +$y_{t}\in \mathbb{R}^{m}$ sequence model outputs at time $t$ + +$A, B, C, D$ system matrices for linear dynamical system + +$h$ Number of $\alpha$ used in final LDS representation + +Number of random samples of $\alpha$ + +Number of spectral filters + +Length of generated sequence + +Length of prefill sequence + +Length of sequence (or of filters) + +# A.2 Experimental details and notation + +All experiments were performed on Nvidia H100-80GB GPUs in PyTorch [33]. All layers except the LDS leverage bfloat16 precision, whereas the LDS layers require float64 precision. All tests, unless otherwise stated, use $k = 24$ spectral filters and replace the $k$ filters with an LDS with state-dimension $h = 80$ . To fit the additional 24 alternating filters of the negative component (i.e., $\phi_j^-$ , where $\phi_j^- [i] = (-1)^{i - 1}\phi_j[i]$ , as is needed to compute $U_{t,j}^-$ purely convolutionally), we expand to an LDS with state-dimension $h = 160$ . + +For the layer-level inference speed benchmarks, we leverage the STU with tensor-dot approximation (STU-T) [2], the same variant used in the FlashSTU language model [22]. It is worth noting that the STU-to-LDS distillation leads to LDS layers with comparable speed regardless of whether the tensor-dot approximation is employed, and thus this approximation only accelerates the STU layers in benchmarks. A formal definition of this approximation is as follows: + +STU Tensor-Dot Approximation: [2] introduced an important optimization, the tensor-dot approximation, wherein each tensor $M \in \mathbb{R}^{d_{\mathrm{in}} \times 2k \times d_{\mathrm{out}}}$ , representing a concatenation of the tensors $M_1^{\phi+}, \ldots, M_k^{\phi+}, M_1^{\phi-}, \ldots, M_k^{\phi-}$ , is learned as $M^{(1)} \times M^{(2)}$ for $M^{(1)} \in \mathbb{R}^{d_{\mathrm{in}} \times 2k}$ and $M^{(2)} \in \mathbb{R}^{d_{\mathrm{in}} \times d_{\mathrm{out}}}$ . This approximation allows for a reduction in convolutions as, with input $x_1, x_2, \ldots, x_\ell$ , we can compute $y_\ell^{\mathrm{SF}} \approx \sum_{i=1}^\ell (x_{\ell-i+1}^\top M^{(2)}) \odot M_{\mathrm{filters}}[i]$ , where $M_{\mathrm{filters}} = [\phi_1, \dots, \phi_k, \phi_1^-, \dots, \phi_k^-]^\top M^{(1)} \in \mathbb{R}^{L \times d_{\mathrm{out}}}$ and $\odot$ refers to the Schur product (i.e. $(x \odot y)_j = x_j \cdot y_j$ ). This allows for only $d_{\mathrm{out}}$ convolutions with the tensor-dot approximation, as opposed to $k \cdot d_{\mathrm{in}}$ convolutions without it. Although this method can reduce expressivity and does not inherit the same marginal-stability guarantees, it maintains competitive empirical performance while yielding significant improvements in efficiency. + +Additionally, we frequently refer to the impulse response of the LDS and STU models. The impulse response of a linear sequence model $f: \mathbb{R}^L \to \mathbb{R}^1$ is the vector or convolutional kernel $\psi \in \mathbb{R}^L$ that is equivalent to $f$ (i.e. $f(x_{[1,\dots L]}) = \psi * x_{[1,\dots L]}$ ), and thus: + +$$ +\psi [ t ] = f (\underbrace {0 , \ldots , 1 , \ldots , 0} _ {1 \text {a t p o s i t i o n} L - t + 1}) +$$ + +We only consider the impulse response of $f: \mathbb{R}^L \to \mathbb{R}^1$ , but the impulse response is closely related to the derivative with respect to the inputs and is generalized identically. For an STU model with $d_{\mathrm{in}} = d_{\mathrm{out}} = 1$ and parameters $M_1^{\phi +}, \ldots, M_k^{\phi +}, M_1^{\phi -}, \ldots, M_k^{\phi -} \in \mathbb{R}$ , the impulse response $\psi_{\mathrm{SF}}$ is thus described by $\psi_{\mathrm{SF}}[t] = \sum_{j=1}^{k} M_j^{\phi +} \phi_j[t] + (-1)^{t-1} \sum_{j=1}^{k} M_j^{\phi -} \phi_j[t]$ . Similarly, for an LDS with $d_{\mathrm{in}} = d_{\mathrm{out}} = 1$ and parameters $a, b, c \in \mathbb{R}$ , the impulse response $\psi_{\mathrm{LDS}}$ is described by $\psi_{\mathrm{LDS}}[t] = ca^{t-1}b$ + +# A.3 Experimental Results on the Condition Number of $M$ , the Spectral Coefficients Matrix + +In Figure 5, we present experimental results on the condition number of $M$ as defined in Section 5. Recall that $M \in \mathbb{R}^{h \times k}$ is the spectral coefficients matrix produced by Algorithm 2; its $i$ -th row $m_i^\top$ stores the coefficients that express the LDS impulse response $\mu_L(\alpha_i)$ (with geometric decay factor $\alpha_i$ ) in the spectral basis $\Phi_{1:k}$ . Starting at $h = k$ with $k = 24$ , we repeatedly add additional independent vectors $\alpha_i$ and measure the maximum singular value of the constructed $M^{-1}$ . The blue central line shows the mean maximum singular value across 5 experiments for a given value of $h$ , with the shaded region showing the maximum and minimum of the largest singular values across experiments. We draw $\alpha$ independently from the distribution in Figure 7. To justify our statement in section 5.2 that $\lambda_{max} \cdot h$ can be considered $O(1)$ , we additionally plot $\lambda_{max} \cdot h$ under the same experimental setup in Figure 6. + +![](images/d1b4428cfabe786d63904e451872ca659f70560a89a5351c9b7b808759973a1f.jpg) +Figure 5: Largest Singular Value as we increase $h$ . + +![](images/f4d0843f228eef68e3f81361d514a1dc23e850c0f71b3aca9d8bfa4f3cf8d7bd.jpg) +Figure 6: $\lambda_{max} \cdot h$ as we increase $h$ . + +# A.4 Analysis of Theorem 1 + +Proof of Theorem 1. Let + +$$ +\mu_ {L} (\alpha) = (1 - \alpha) \left[ \begin{array}{c c c c c} 1 & \alpha & \alpha^ {2} & \ldots & \alpha^ {L - 1} \end{array} \right] \in \mathbb {R} ^ {L}, +$$ + +![](images/d7d3ce4f79bafa7ad13c377c06fd8b6fca0e0f5057277cf07702cef3a8f51b62.jpg) +Figure 7: Distribution of $\alpha$ ( $H = 10,000$ samples) + +be the $L$ -impulse response of the one-dimensional linear dynamical system with parameters $\alpha, b = 1, c = 1 - \alpha$ . Lemma 11.3 from [13] asserts that for any $\alpha \in [0,1]$ , there exist real coefficients $m_1, \ldots, m_k \in \mathbb{R}^k$ and constant $c > 0$ , such that any sequence of inputs $u_{L:1} \in \mathbb{R}^L$ with $\| u_{L:1} \|_{\infty} \leq 1$ , + +$$ +\left| \sum_ {i = 1} ^ {k} m _ {i} \left\langle \phi_ {i}, u _ {L: 1} \right\rangle - \left\langle \mu_ {L} (\alpha), u _ {L: 1} \right\rangle \right| \leq c e ^ {- \frac {k}{\log L}}. +$$ + +The RHS term represents the evolution of the LDS, with system matrices $b$ , $c$ that are assumed to be identity, and $\mu_L(\alpha)$ is the system evolution. + +As the above is true for all $u_{L:1}$ , for each $\alpha_{j}$ , there exist $m_{j}$ such that, + +$$ +\forall j, \left\| m _ {j} ^ {\top} \Phi_ {1: k} - \mu_ {L} (\alpha_ {j}) \right\| \leq c e ^ {- \frac {k}{\log L}}. \tag {4} +$$ + +Moreover, as the optimization problem is convex in $m_j$ and Lipschitz continuous, FindSpectralRepresentation will return such $m_j$ . + +Let $M \in \mathbb{R}^{h \times k}$ be the matrix whose rows are $m^j \in \mathbb{R}^k$ : + +$$ +M = \left( \begin{array}{c} - m ^ {1} - \\ - m ^ {2} - \\ \vdots \\ - m ^ {h} - \end{array} \right). +$$ + +Let $\mathcal{E} = M\Phi_{1:k} - \mu_L(\alpha_1,\dots ,\alpha_h)$ . By the triangle inequality and (4), + +$$ +\left\| \mathcal {E} \right\| _ {1} \leq \sum_ {j = 1} ^ {h} \left\| m _ {j} ^ {\top} \Phi_ {1: k} - \mu_ {L} \left(\alpha_ {j}\right) \right\| \leq c h e ^ {- \frac {k}{\log L}}. \tag {5} +$$ + +Thus, assuming that $M$ is full rank, multiplying $\mathcal{E}$ by the Penrose-Moore pseudo-inverse of $M$ and using Holder's inequality, we get + +$$ +\| \Phi_ {1: k} - M ^ {- 1} \mu_ {L} (\alpha_ {1}, \ldots , \alpha_ {h}) \| = \| M ^ {- 1} \mathcal {E} \| \leq \| M ^ {- 1} \| _ {\star} \| \mathcal {E} \| _ {1} \leq \lambda_ {\max} \cdot c h e ^ {- \frac {k}{\log L}}. +$$ + +It remains to argue that $M$ is full rank. This follows since $\Phi_{1:k}$ is an orthogonal basis, and the matrix $\mu_L(\alpha_{1:h})$ is a Vandermonde matrix. Thus, both matrices are full rank. + +![](images/d3d4c3ad4ee4b8c370893c8e34bc1d0cafc4d4001300aa014d360de5ea0196b7.jpg) + +# A.5 Learning a Symmetric LDS with and without noise + +Figure 2 in the main paper compares SpectraLDS to other system-identification methods on the task of learning an arbitrary symmetric LDS with and without noise. The LDS signal has hidden + +dimension 256, input and output dimension 5, and maximum eigenvalue magnitude 0.99 with Gaussian initialization. Each step provides a Gaussian input sequence of length 100 with variance $1 / d_{in}$ and the final output. For the learning with noise experiment, Gaussian noise with 0.5 variance was added to the hidden state at each step, and Gaussian noise with 5 variance was added to the output. Each architecture was trained with the default PyTorch RMSProp optimizer configuration, except for the LDS, which required a lower learning rate to converge. The Ho-Kalman matrices are recomputed every 20 steps for computational ease, while the other methods were updated every step, and the Ho-Kalman parameters are the maximum for an input of length 100 (that is T Horizon = 99, T1 = 49, T2 = 49, state_dim_est = 48). The shaded region in the figure shows the $95\%$ confidence interval over 8 runs. + +![](images/a38045886b17dc6aace5e214b400caad625490ae872a2daf4c4e6a216d4557ac.jpg) + +![](images/b1520e9e6f634b4940653e4a7971788d39f9a8fcd224cc7489ecf70daa4bf650.jpg) + +![](images/4f554688ab093558c793a263d4efcd83f01c36992d52abd192b1ddcfa79a1e28.jpg) + +![](images/2378b6865bdefd153e3a43cae6c6c97d8f48fe5ba20c9fea972832797748b1e8.jpg) + +![](images/f3f7bac985846144034746bda14e615b0ae2df9b4d963e435369c460afee8100.jpg) + +![](images/79040fd0705d9c76e76d91ed3a6788a3928dd43db377ac869e83d46a8696a7d4.jpg) + +![](images/baceb514385684d496f377eea418d66b890ce20c71dbbfd76d9f6c9845ee39e2.jpg) + +![](images/55d2a618685d9b2a7f1230c4784d0b8700851b41182f36e5c10098b8baec2407.jpg) + +![](images/e1168e3d6326e916444689ebe912216ada8d1363de43b6a2a8ee32ca31baf7e0.jpg) + +![](images/6835bb572eac76c7e8e50c63e7a9a6938f2221541ad282458f9f96872cd077be.jpg) + +![](images/86f25e9f852f1b1e53e5b50e7fb038de5778b1732d0189d3fd52ce144765c3ac.jpg) + +![](images/2bab72e22c1506cad334e6683e71a1e20b0047446e4c7077121c7b02b7bf1731.jpg) + +![](images/8c12073947114425a422816009983709c0c066b6f8b6417a0cb6ede13ea6ea29.jpg) + +![](images/ac440e55eac47c96b7dc046663e079a2c2f27d38d256c56ae50338ed076595e7.jpg) + +![](images/6e6a5edc221ac704bb46966eb3df6cf7bf711beaf491b287f3f76b97de73fae1.jpg) + +![](images/6e00f3df6a1c6cc5c9afb68f7913fecd8962827c29b3b519106de3c4ce9c2e4d.jpg) + +![](images/2070b90a739261b3e992a6a5e9d0e35beb7ff862fc24cc0a0a9051b7ba3471db.jpg) + +![](images/516f2298e23d7d0124c299f871e7c38dbe51c1f47235712913241bc415e3410f.jpg) + +![](images/cddd79ada9121403426e6308f5d3d2bc6d5e1d5b9a0d5ec62a8d48f421fc23d4.jpg) + +![](images/dcac97c3013dcb9fe3656b032ebc8a6b72051e821790f4a9306d33f2ea0e070b.jpg) + +![](images/6b49906e263223a23af6168b9b690c839ae01d1656c45e3760e1a87c6f34686c.jpg) +Figure 8: Visualization of the positive spectral filters and their approximation using an LDS with state dimension 160, as obtained via the practical algorithm. As described previously, only the first 80 dimensions of the state are used for approximate the Spectral Filters, with the remaining used for the alternating filters (negative filter maps), as described in Appendix A.7. The figure illustrates that even with a low-dimensional state, the LDS can accurately fit the spectral filters, confirming the efficacy of our distillation process. + +![](images/9d2194ed90df4b61c3a4f3a719f36422003b31089195eb34666f22373078ddff.jpg) + +![](images/2f25c240fa1b2fcb0f3d78e923938cfab9d2aff679dac21e3a39ea1f4a6db11c.jpg) + +![](images/2349e3ed49a7b112196360f27b0809415a6a1784c8cf9319e7dbf85ea8600550.jpg) + +# A.7 State Dynamics of the Linear Dynamical System + +We train a linear dynamical system with state dimension 160 to fit the Spectral Filters and alternating filters using Algorithm 2 and examine the eigenvalues of the resulting system in Figure 9. We only plot the 80 eigenvalues corresponding to the Spectral Filters, as the remaining 80 corresponding to the alternating filters are identical except multiplied by $-1$ (see below the plot). The reconstruction error of the spectral filters for the trained LDS is $7.69 \times 10^{-19}$ . We transform the eigenvalues to visualize the full range of the values as they approach 1 and differentiate the negative and positive eigenvalues as blue and red (e.g., 0.97 and $-0.97$ would both map to $\approx 3.5$ , although the first would join a red column and the second blue). + +![](images/10eaa7103efed020882111f4210616f8d74232a9e8c009352b0dfef497ece52e.jpg) +Distribution of Transformed Eigenvalues $f(x) = -\ln (1 - |x|)$ +Figure 9: Distributions of Eigenvalues for LDS corresponding to the first 24 Spectral Filters. + +For practical efficiency, we can compute all $U_{t,1}^{+}, \ldots, U_{t,k}^{+}, U_{t,1}^{-} \ldots U_{t,k}^{-}$ simultaneously with a single LDS parameterized by $\left\{ \begin{bmatrix} \widetilde{M}\Gamma & \mathbf{0} \\ \mathbf{0} & \widetilde{M}\Gamma \end{bmatrix}, \begin{bmatrix} A & \mathbf{0} \\ \mathbf{0} & -A \end{bmatrix}, \mathbf{1}_{2h} \right\}$ . This LDS has a state dimension of $2hm$ , where each of the $m$ input dimensions have a hidden state updated by $\begin{bmatrix} A & \mathbf{0} \\ \mathbf{0} & -A \end{bmatrix}$ . The hidden state dimension differs from $2h$ as we treat $m$ as a batch dimension, resulting in $Bu_{t} \in \mathbb{R}^{2h \times m}$ rather than $\mathbb{R}^{2h}$ . This mimics an LDS acting independently across each of the $m$ dimensions, equivalent to treating $m$ as a batch-axis, and this is the analog to how each of the $k$ spectral filters convolve along each input dimension independently. The above Figure 9 showcases the eigenvalues of matrix $A$ , whereas for practical deployment, the state matrix is $\begin{bmatrix} A & \mathbf{0} \\ \mathbf{0} & -A \end{bmatrix}$ . + +# A.8 Reconstruction Error of the Spectral Filters varying LDS State Dimension and $H$ + +We demonstrate the efficacy of the practical algorithm in fitting the Spectral Filters with a low initial LDS State Dimension. In Figure 10, we ablate the choices of $H$ and $h$ for the practical algorithm. We do not fully understand why larger $H$ leads to worse fits for small state dimension. Notably, the error quickly becomes no longer human perceptible. In this procedure we only aim to fit the $k$ positive spectral filters. To fit both the $k$ positive spectral filters and $k$ negative spectral filters, we can achieve the same error with double the state dimension, as described in Appendix A.7. With our implementation of sparsity, the resulting hidden dimension will be near but not exactly that requested, which is why each line has different reconstruction dimensions. + +![](images/905d239fab4f9fb91dc721a139b1b81202540fdff1859c7c98e476e06327cd30.jpg) +Reconstruction Error vs $h$ +Figure 10: Reconstruction Error varying $H$ and LDS state dimension $h$ . + +# A.9 Reconstruction Error of the Spectral Filters with the Practical Algorithm and Uniform $\alpha$ Distribution + +In Figure 11, we illustrate that if rather than choosing $\left( |\alpha_i| \stackrel{d}{=} 1 - U^4 \right)$ with $U \sim \mathrm{Unif}[0,1]$ with an independent random sign, we instead choose $\alpha_i \in [-1,1]$ , we require larger $H$ to achieve similar performance. At sufficiently large $H$ , uniform sampling will have sufficient coverage of $\alpha$ near 1 and $-1$ . Thus, we instead prefer a distribution with a natural bias towards these extreme points, as $\alpha \in [-0.9,0.9]$ decay towards 0 quickly. + +# A.10 Experiments on Synthetic Tasks + +We evaluate SpectraLDS (state dimension 160) against a strong baseline on a synthetic task, reporting results in Table 3 and Figure 12. A symmetric linear dynamical system with input dimension $d_{\mathrm{in}}$ and state dimension $d_h$ is initialized by drawing all entries from a standard normal distribution, after which the update matrix $A$ is rescaled so its largest eigenvalue has magnitude $1 - \delta$ . We then distill a SpectraLDS model from an STU model trained with AdaGrad (learning rate 1.0) for 2000 steps; each step minimizes the mean-squared-error (MSE) between the STU output and the ground-truth LDS output over batches of 32 sequences of length seq_len. For the baseline, we fit a randomly initialized symmetric LDS with AdaGrad (learning rate 0.0001) under the same loss. To keep training times practical, most baseline runs use a shorter sequence length, and several runs were stopped early due to computational limits, marked with an asterisk. When the state dimension satisfies $d_h \geq 1000$ , baseline training with a learning rate of 0.01 becomes unstable and often diverges, whereas the STU remains stable even with a learning rate of 1.0. + +![](images/29097a086242853147fc25d5298b4d58c2688788787218a61af2f57383c6bfff.jpg) +Reconstruction Error under Uniform $\alpha$ Distribution +Figure 11: Reconstruction Error varying $H$ and LDS state dimension $h$ when choosing $\alpha \in \mathrm{Unif}[-1,1]$ . + +
Performance Results
TypeLen.Deltad_ind_hAvg MSEMSE StdStep 10 LossStep 100 LossRunsTime (s)Time Std (s)
LDS GD81921 × 10-2101002.15 × 10-21.32 × 10-33.49 × 10-11.43 × 10-14*2153.1214.16
LDS GD81921 × 10-3101002.53 × 10-23.18 × 10-34.53 × 10-11.80 × 10-13*2147.6118.64
LDS GD81921 × 10-4101003.03 × 10-26.09 × 10-35.99 × 10-12.29 × 10-13*2150.7328.05
LDS GD10241 × 10-2101009.54 × 10-30.00 × 1001.18 × 10-16.72 × 10-21*248.330.00
LDS GD10241 × 10-21010001.90 × 10-55.93 × 10-65.25 × 10-43.39 × 10-55252.241.64
LDS GD10241 × 10-3101007.34 × 10-31.02 × 10-31.06 × 10-14.97 × 10-22*248.680.13
LDS GD10241 × 10-31010002.40 × 10-51.71 × 10-64.00 × 10-43.71 × 10-55250.240.87
LDS GD10241 × 10-4101007.72 × 10-32.52 × 10-41.13 × 10-15.31 × 10-22*248.320.38
LDS GD10241 × 10-41010002.01 × 10-54.53 × 10-65.29 × 10-43.53 × 10-55250.581.40
LDS GD10241 × 10-5101006.77 × 10-36.43 × 10-41.01 × 10-14.84 × 10-25250.800.17
LDS GD10241 × 10-51010001.94 × 10-54.92 × 10-64.79 × 10-43.39 × 10-55251.510.67
SpectraLDS81921 × 10-2101003.35 × 10-49.84 × 10-58.27 × 10-33.74 × 10-4551.8510.36
SpectraLDS81921 × 10-21010001.86 × 10-52.27 × 10-67.99 × 10-35.48 × 10-5551.140.04
SpectraLDS81921 × 10-3101003.73 × 10-41.57 × 10-49.32 × 10-34.12 × 10-4563.9224.06
SpectraLDS81921 × 10-31010001.73 × 10-52.27 × 10-68.46 × 10-35.49 × 10-5551.931.55
SpectraLDS81921 × 10-4101004.73 × 10-41.91 × 10-49.81 × 10-35.11 × 10-4552.2310.66
SpectraLDS81921 × 10-41010001.84 × 10-54.53 × 10-69.30 × 10-35.54 × 10-5551.150.07
SpectraLDS81921 × 10-5101002.94 × 10-48.78 × 10-51.18 × 10-23.30 × 10-4551.6910.37
SpectraLDS81921 × 10-51010001.66 × 10-54.47 × 10-69.00 × 10-35.27 × 10-5551.150.07
SpectraLDS81921 × 10-6101004.05 × 10-43.61 × 10-51.07 × 10-24.46 × 10-4546.530.03
SpectraLDS81921 × 10-61010001.96 × 10-51.27 × 10-68.55 × 10-35.63 × 10-5551.951.63
SpectraLDS81921 × 10-7101004.50 × 10-41.73 × 10-49.63 × 10-34.87 × 10-4546.510.06
SpectraLDS81921 × 10-71010001.77 × 10-51.95 × 10-69.45 × 10-35.46 × 10-5551.961.64
+ +Table 3: SpectraLDS performance on learning synthetic linear dynamical systems with maximum eigenvalue $1 - \delta$ against a strong baseline. + +Loss Comparison on learning Symmetric LDSs +Figure 12: Comparison of learning performance on synthetic LDS tasks between SpectraLDS and a gradient-descent-updated LDS baseline. The plot shows the loss at different training steps under varying $\delta$ , input/output dimensions, and sequence lengths. SpectraLDS consistently achieves lower loss with significantly reduced runtime, especially at larger output dimensions. Full experimental details are provided in Table 3. +![](images/e5c512191560f65e19a50d7147fcefce8c986b9e9a82152c62a08fd8571d0563.jpg) +Execution Times: LDS 8192 (2150s) - LDS 1024 (248s) - SpectraLDS (56s) + +# A.11 Layer Speeds of SpectraLDS and STU + +We benchmark the inference speed of a single SpectraLDS layer across several state dimensions and compare it with two accelerated schemes for computing the STU convolution: Epoched Future Fill [1] and an STU that employs the tensor-dot approximation [22]. The SpectraLDS layer itself is produced by applying STU-to-LDS distillation to the tensor-dot STU. For the timings reported in Table 4, each model is evaluated on $L$ (Seq. Len.) autoregressive convolutions on inputs of dimension 128, and the mean and standard deviation over five runs are recorded. All STU variants use a filter length of 1,048,576 to accommodate the longest sequences. Although every architecture begins in the linear-scaling regime, only SpectraLDS continues to scale favorably as the sequence length increases. The resulting layer-speed comparison is visualized in Figure 13; benchmarks of SpectraLDS layers embedded in large language-model architectures appear in Appendix A.12. + +
Inference Time Performance (ms)
Seq. Len.STU Future FillSTU Tensor-DotSTUSpectraLDS (SD 100)SpectraLDS (SD 800)
MeanStdMeanStdMeanStdMeanStdMeanStd
3230.861.2031.876.43134.9622.1321.273.8618.980.73
6436.520.7228.920.42123.162.5422.400.1422.270.29
12849.870.5240.206.63135.942.2829.490.1829.720.58
25675.730.3545.981.12162.742.9043.540.2743.620.47
512136.677.9070.240.61211.662.9971.400.5673.010.88
1024234.106.99122.891.19317.914.79127.521.25128.980.97
2048441.093.69226.478.02525.655.09241.164.43252.588.78
4096863.818.72426.437.96936.946.53472.158.40477.968.42
81921856.63191.17837.462.031772.3020.46921.5510.23929.069.76
163843377.2933.161812.1716.473434.1930.831839.6325.211846.0417.43
327687139.35515.404286.8624.746721.8319.653620.8443.713686.9026.45
6553613485.25234.2211614.4328.0413478.5389.727181.1335.967362.7378.96
13107227252.83240.3436427.1014.5626703.77323.9714356.9274.9114649.56162.32
26214463437.26123.25117796.38268.4855775.49275.1628573.69119.0929156.42113.55
524288177502.89651.96365168.31460.23142896.29621.7257083.23404.9158027.04276.39
1048576576129.571425.121145862.831805.50451607.11721.73114044.791215.07115999.77670.78
+ +Table 4: Autoregressive Inference Time (ms) across model architectures (5 runs). + +![](images/59bbbb40fd2d9b596eaa50ff5502742d8810c78ecc4a2afb48f379ca831725ec.jpg) +Figure 13: Autoregressive Inference Time (ms) across model architectures. + +# A.12 FlashSTU Ablations + +# A.12.1 Implementation Architecture Details + +To perform the FlashSTU performance evaluations with STU-T, we employ the architecture depicted in Figure 14. We test token generation efficiency with a hybrid architecture, alternating between layers using STU-T and sliding window attention (SWA), and we additionally test with each layer using STU-T only. Each layer consists of RMS-Norm, followed by STU-T or SWA, followed by RMS-Norm, and then followed by an MLP. Inputs are tokenized by the o200k_base tokenizer and the FlashSTU model begins with an embedding layer, which is weight-tyed to the output unembedding layer. To start generation, we add special tokens such as $<|endoftext|>$ and $<|endofprompt|>$ . + +The sliding window attention layers leverage Flash Attention v2 [8, 7] and ALiBi position embeddings [35]. The tested model has input dimension 896 and 12 layers, which has 550.31 million parameters for the hybrid model, and 535.99 million parameters for the STU-only model. All layers are run in bfloat16 except for the LDS distilled layers, which require float64 precision. The Flash STU-T leverages the STU with tensor-dot [22] approximation rather than the base STU layer for faster inference, and thus we perform the STU-to-LDS distillation on the STU with tensor-dot approximation. For tests with generation length up to 131072, we leverage STU filter length of 131072. For the generation length of 262144, we leverage an STU filter length of 262144. For each runtime measurement, we first run a warmup generation, before reporting the mean of two generations of that length. All benchmarks include only inference time and not model setup or filter computation costs. Additionally, each MLP layer has hidden dimension $12 \times$ the input dimension (MLP expansion factor). Other configuration details are identical to those in Table 7. + +![](images/dab35998ec696d03a9f81f9e478eedf7a3dffe2f3cf6a43a4053b3ff29c7059c.jpg) +Figure 14: FlashSTU architecture [22]. + +# A.12.2 Implementation Efficiency + +Using the setup of Fig. 14, we time autoregressive generation for sequence lengths 4,096-262,144 tokens under two architectures—(i) a hybrid network that interleaves STU-T and sliding-window attention layers and (ii) an STU-only network in which every layer is STU-T. Tables 5-6 report mean runtimes (over two runs after a warmup) for SpectraLDS with state dimensions 100-800 alongside the STU-T with naive convolutions (Base STU) and the Epochched FutureFill method. We note three main results. First, SpectraLDS runtimes grow nearly linearly with sequence length and are virtually independent of the chosen state dimension. Second, while the STU-T with naive convolutions is competitive at $4 - 8\mathrm{k}$ tokens, it becomes progressively slower, so that SpectraLDS is $\approx 2\times$ faster by $65\mathrm{k}$ tokens and over $4\times$ faster at $262\mathrm{k}$ tokens in the STU-only setting (and $2\times$ faster in the hybrid setting). Third, Epochched FutureFill narrows the gap at medium lengths but is still outpaced by SpectraLDS beyond $131\mathrm{k}$ tokens and, in the STU-Only architecture, exhausts memory (OOM) at 262 k tokens, whereas SpectraLDS completes the run. Together these results demonstrate that SpectraLDS delivers the most favorable long-context scaling and remains robust across model hyper-parameters. + +
Seq. Len.LDS SD 100LDS SD 200LDS SD 400LDS SD 800Base STUFutureFill
409620.3420.7120.4020.4018.9119.92
819240.4841.3240.6940.8238.0938.39
1638480.7882.9080.9581.6376.4974.89
32768161.43163.92162.13163.30164.70141.16
65536323.40327.31323.90325.92389.72290.68
131072646.66653.96648.99651.351014.67666.24
2621441588.811639.241591.501536.203113.452498.89
+ +Table 5: Hybrid Model Runtime (seconds) for generation across SpectraLDS with different state dimensions and Baseline and Epoched FutureFill implementations + +
Seq. Len.LDS SD 100LDS SD 200LDS SD 400LDS SD 800Base STUFutureFill
409614.0413.8713.8713.7412.1419.35
819228.1527.7427.6727.4224.8435.86
1638455.7155.5155.3454.7357.3267.96
32768110.85110.88110.57108.68149.12135.75
65536220.11221.55221.13215.00429.20329.91
131072439.87442.65442.04428.361352.40952.70
2621441104.971101.761132.181107.494672.60OOM
+ +Table 6: STU-Only Runtime (seconds) for generation across SpectraLDS with different state dimensions and Baseline and Epoched FutureFill implementations + +# A.13 Details of STU Model In Experiments + +We summarize in Table 7 all relevant details for the FlashSTU model used in the language evaluations in Table 2. The distilled LDS layer used in the language benchmarking experiments was obtained by Algorithm 2 and has a state dimension of 160, incorporating both positive and negative spectral components. The weights for the distilled model were directly transferred from the FlashSTU model described below. The FlashSTU architecture is further described in Appendix A.12 and graphically shown in Figure 14. + +Benchmarks were conducted with task-specific few-shot configurations. For each task, if the few-shot value is not set to -1, that number of examples is provided during evaluation; otherwise, the few-shot setting is left unchanged. Specifically, the few-shot settings are as follows: HellaSwag: 0 shots, MMLU: 5 shots, PIQA: 0 shots, BoolQ: 0 shots, Winogrande: -1, Commonsense_QA: 7 shots, OpenBookQA: -1, ARC-Easy: -1, ARC-Challenge: -1. The model employs bfloat16 precision except for the LDS computations, where it requires float64 precision, and a global batch size of 3 examples per task. All computations were performed on a single H100 GPU. + +# A.14 Example Generation + +The following is an example completion from the FlashSTU 340M model after LDS distillation with temperature 1.0: + +Prompt: After school each day, Deidre Smith would look over the essays her students submitted at the end of class and, without fail, each day Alex would surprise her. Today, he wrote about + +Generated Text: After school each day, Deidre Smith would look over the essays her students submitted at the end of class and, without fail, each day Alex would surprise her. Today, he wrote about 40 essays from his freshman class, all of which were collected over the past three years for The Book of Sentiments: What Students Should Do When Reading Literature – First published in 1962, this book was very well received. What people who read this book would not immediately accept or appreciate the author's thought-provoking message of what it means to read literature. + +What exactly are you doing with your newfound ability to read? You are reading? What emotions do you feel at the time? What do you do with your newfound ability to read? These are just a few examples of how much reading students think they can gain from the experience. It is no accident, then, that each and every level of student will have difficulty with the text in this book. Students are given a choice: choose an assignment that you are comfortable with and one that you would like to read more deeply and critically. As the task to read becomes more difficult and the pressure for it is much greater, students will struggle to put their best minds together when reading literature. To help students overcome this hurdle, I have devised a book that makes the task a bit easier by providing all the information they need from a reader. If a reader is too afraid to read, I have designed a page that provides guidance from the textbook that will... + +
Model Architecture
DescriptionFlash STU
Parameter CountTotal number of parameters340M
Embedding DimensionDimensionality of embedding space1024
Number of HeadsAttention heads4
Number of LayersTransformer + STU layers12
ALiBi AttentionAttention scores modification using linear biasesNo
RoPE ThetaRoPE scaling factor for rotary embeddings10,000
Sliding Window SizeSliding window attention context lookback size512
Sequence Length (Training)Input sequence length during training4,096
Sequence Length (Inference)Input sequence length during inference via position interpolation131,072
Vocabulary SizeSize of the model's vocabulary200,064
MLP Expansion FactorExpansion factor in MLP layers4
BiasUse of bias terms in linear layersNo
DropoutDropout rate0.0
Number of FiltersNumber of filters24
Training and Optimization
EpochsNumber of training epochs1
Global Batch SizeNumber of tokens processed per step524,288
Micro Batch SizeBatch size per GPU8
Gradient Accumulation StepsNumber of steps before performing a gradient update8
Warmup StepsNumber of warmup steps1,907
Evaluation PeriodEvaluation frequency (steps)50
Max Grad NormMaximum gradient norm for clipping1.0
Optimizer Configuration
OptimizerOptimizer typeAdamW
Learning Rate ScheduleLR scheduling strategyLinear decay with warmup
Max Learning RateMaximum learning rate4.0 × 10-4
Min Learning RateMinimum learning rate4.0 × 10-5
Torch DtypeData type for PyTorch tensorsbfloat16
BetasOptimizer betas(0.9, 0.999)
EpsilonOptimizer epsilon1.0 × 10-8
Weight DecayWeight decay factor1.0 × 10-2
AMSGradUse AMSGrad variantNo
FusedUse fused optimizerYes
Optimization Techniques
Activation CheckpointingEnable activation checkpointingYes
Use Flash FFTEnable Flash FFTNo
Use Tensordot Approx.Enable tensordot approximationYes
Use AttentionEnable attention mechanismYes
SoftcapSoftcap threshold50.0
Torch CompileEnable Torch compile optimizationYes
+ +Table 7: Model and training configuration details for the 340M Flash STU model. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We claim that we have identified a method to distill spectral filters to linear dynamical system form and demonstrate in the experiments section that our method successfully works. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: At the end of the experiments section, we mention the current limitations of our distillation. Additionally, throughout we demonstrate understanding of the limitations of our approach, such as restricting our results to linear dynamical systems with symmetric system matrices. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: Our main theoretical result is Theorem 1, in which we mention complete assumptions made and demonstrate when and why they would hold both theoretically and experimentally in the main paper and in the Appendix. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: Yes, we describe our algorithm completely in 1 and 2, and elaborate on the practical implementation in Section 5.2. Our code repository is also open sourced. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [Yes] + +Justification: A link to the code is provided. + +# 6. Experimental setting/details + +Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? + +Answer: [Yes] + +Justification: Experiment details are expansively explained in 6, A.13, A.3, A.10, and in many other instances throughout the paper. + +# 7. Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [Yes] + +Justification: 3, 2, and other graphs where relevant include error bars and statistical significance. + +# 8. Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +Answer: [Yes] + +Justification: In 6, A.13, and in all places where relevant, the paper describes completely the computer resources needed and used. Furthermore, since many of our results relate to inference speed, we explain the hardware used and time of execution/memory extensively. + +# 9. Code of ethics + +Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? + +Answer: [Yes] + +Justification: The paper conforms simply in every respect with the NeurIPS Code of Ethics as explained. + +# 10. Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +Answer: [NA] + +Justification: [NA] + +# 11. Safeguards + +Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? + +Answer: [NA] + +Justification: [NA] + +# 12. Licenses for existing assets + +Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? + +Answer: [Yes] + +Justification: All previous work is fully credited and cited throughout. + +# 13. New assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [Yes] + +Justification: Our new assets are well documented and the documentation is provided. + +# 14. Crowdsourcing and research with human subjects + +Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? + +Answer: [NA] + +Justification: [NA] + +# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +Justification: [NA] + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? 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Over the years, advanced architectures and sampling methods have been developed to make these models practically usable. However, certain synthesis process decisions still rely on heuristics without a solid theoretical foundation. + +In our work, we offer a novel analysis of the DM's inference process, introducing a comprehensive frequency response perspective. Specifically, by relying on Gaussianity assumption, we present the inference process as a closed-form spectral transfer function, capturing how the generated signal evolves in response to the initial noise. We demonstrate how the proposed analysis can be leveraged to design a noise schedule that aligns effectively with the characteristics of the data. The spectral perspective also provides insights into the underlying dynamics and sheds light on the relationship between spectral properties and noise schedule structure. Our results lead to scheduling curves that are dependent on the spectral content of the data, offering a theoretical justification for some of the heuristics taken by practitioners. + +# 1 Introduction + +Diffusion Models (DMs) have become powerful tools for generating high-quality, diverse signals, with applications in image, audio, and video synthesis. Alongside their practical success and the ability to handle complex distributions, some aspects of the diffusion processes still rely on heuristics rooted in empirical experimentation. A key example is choosing an appropriate noise schedule for the inference phase. Developing theoretical foundations for these heuristics may provide valuable insights into the diffusion process itself, and enable greater adaptation to different setups. Our work aims to provide such a theoretical backbone from a spectral perspective, as outlined below. + +Our starting point lies in a well-established observation; while the continuous description of DMs via SDE or the associated flow ODE [28] are mathematically well-founded, their practice necessarily deviates from these theoretical foundations, introducing various errors [4, 24]. A major source of error is discretization, which replaces the analytical formulations by their discrete-time counterparts. Another source of error is the approximation error, originating from the gap between the ideal denoiser and its neural network realization. + +Significant efforts have been made to minimize these associated errors, with advanced numerical solvers [28, 12, 40, 20, 21, 42, 41] offering various methods for better treating the discretization of the differential equations. A fundamental aspect is the decision on the time point discretization, which directly affects the synthesis quality. We refer to these anchor points as the noise schedule, highlighting their role in setting the noise variance introduced at each diffusion step. Realizing their importance, researchers have recently shifted their focus from custom-tailored heuristics [10, 23, 13, 5] to the development of optimized noise schedules [26, 29, 34, 32, 37, 3, 38, 36]. More on these methods and their relation to this paper's contributions is detailed in Section 5. + +In this work, we analyze the reverse diffusion process in the spectral domain, presenting the generated output signal as the outcome of a linear transfer function operating on the i.i.d. Gaussian input noise. This analysis is enabled by assuming that the destination distribution to sample from is Gaussian, providing an explicit spectral expression of the generated signal and insight on the evolving dynamics throughout the process. Our analysis, applied to the discretized inference procedure, encompasses both DDPM [10] and DDIM [27] frameworks, thereby revealing the effects of deviating from the continuous formulation. In addition, we extend our study for both variance preserving (VP) and variance exploding (VE) numerical schemes [28]. + +Posing the derived explicit expressions of the transfer systems as functions of the noise scheduling parameters, we can optimize a noise schedule tailored to a given dataset, its resolution, and the specified number of sampling steps required. We demonstrate how effectively solving these optimization problems numerically yields a noise schedule that accounts for these data characteristics, discuss the relation between the noise schedule structure and spectral properties, and validate our approach with existing works. Finally, we apply our method to publicly available datasets, including CIFAR-10 [18], AFHQv2 [6], MUSIC [22] and SC09 [33], examining the relation to heuristic choices in past work. + +In summary, our contributions are the following: (i) Assuming the target signal is drawn from a Gaussian distribution, we present a novel spectral analysis on the discrete diffusion reverse process and derive a closed-form expression for its spectral transfer function. (ii) We formulate an optimization problem to find an optimal noise schedule that aligns with the data characteristics. Our approach provides an effective solution without relying on bounds or constraints on the number of diffusion steps. (iii) We analyze the evolving spectral properties of the signal throughout the diffusion process and at its output, demonstrating how handcrafted noise scheduling decisions and related diffusion phenomena are often well-predicted by our approach. (iv) Our spectral analysis examines various setups, including DDIM and DDPM procedures, VP and VE formulations, the selection of loss functions, and additional features such as expectancy drift. + +# 2 Background + +We introduce the notations and the framework of diffusion probabilistic models, which are designed to generate samples $\mathbf{x}_0\in \mathbb{R}^d$ from an underlying, unknown probability distribution $p(\mathbf{x}_0)$ . While the sampling procedure can be described as a Stochastic Differential Equation (SDE) or by its associated flow, an Ordinary Differential Equation (ODE), these formulations lack a general analytical solutions and are instead discretized and solved using numerical methods [28]. Accordingly, we turn to describe the discrete formulations - DDPM [10] and DDIM [27] - which stand as the basis for our work. + +The diffusion process is a generative procedure constructed from two stochastic paths: a forward and a reverse trajectories in which data flows [10]. Each process is defined as a fixed Markovian chain composed of $T$ latent variables. During the forward process, a signal instance is gradually contaminated with white additive Gaussian noise as follows: + +$$ +\mathbf {x} _ {t} = \sqrt {\alpha_ {t}} \mathbf {x} _ {t - 1} + \sqrt {1 - \alpha_ {t}} \boldsymbol {\epsilon} _ {t}, \tag {1} +$$ + +where $\alpha_{t}$ for $t\in [1,T]$ is referred to as the incremental noise schedule and $\epsilon_t\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ . Under the assumption that $\alpha_{T}$ is close to zero, we get that the final latent variable becomes $x_{T}\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ . A direct consequence of the above equation is an alternative relation of the form: + +$$ +\mathbf {x} _ {t} = \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0} + \sqrt {1 - \bar {\alpha} _ {t}} \boldsymbol {\epsilon} \quad \boldsymbol {\epsilon} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}), \tag {2} +$$ + +for $\bar{\alpha}_t = \prod_{i=1}^t \alpha_i$ . Based on the above relationships, the reverse process aims to reconstruct $\mathbf{x}_0$ from the noise $\mathbf{x}_T$ by progressively denoising it. This can be written as + +$$ +\mathbf {x} _ {t - 1} = \frac {1}{\sqrt {\alpha_ {t}}} \left(\mathbf {x} _ {t} - \frac {1 - \alpha_ {t}}{\sqrt {1 - \bar {\alpha} _ {t}}} \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} _ {t}, t\right)\right) + \sigma_ {t} \mathbf {z} _ {t}, \tag {3} +$$ + +where $\epsilon_{\theta}$ is the estimator of $\epsilon$ for a given $\mathbf{x}_t$ at time $t$ with a neural network parameterized by $\theta$ , $\sigma_t = \sqrt{\frac{1 - \bar{\alpha}_{t-1}}{1 - \bar{\alpha}_t} (1 - \alpha_t)}$ and $\mathbf{z}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ . Alongside this stochastic formulation, The authors of [27] introduce DDIM, a deterministic framework for the inference process, which can be utilized to enable faster sampling. While DDIM relies on a non-Markovian forward process, it preserves the marginal distribution as in (2). As a result, the sampling trajectory can be expressed by + +$$ +\mathbf {x} _ {s - 1} = \sqrt {\bar {\alpha} _ {s - 1}} \left(\frac {\mathbf {x} _ {s} - \sqrt {1 - \bar {\alpha} _ {s}} \cdot \boldsymbol {\epsilon} _ {\theta} (\mathbf {x} _ {s} , s)}{\sqrt {\bar {\alpha} _ {s}}}\right) + \sqrt {1 - \bar {\alpha} _ {s - 1}} \cdot \boldsymbol {\epsilon} _ {\theta} (\mathbf {x} _ {s}, s), \tag {4} +$$ + +where $s \in \{0,1,\dots,S\}$ is the time index in the DDIM formulation. Throughout the rest of the paper, we denote by $\bar{\alpha}$ the set of noise schedule parameters, $\{\bar{\alpha}_t\}_{t=0}^T$ for DDPM and $\{\bar{\alpha}_s\}_{s=0}^S$ for DDIM. + +# 3 Analysis of Diffusion processes + +We consider the reverse process as a system that takes as input a noisy signal $\mathbf{x}_T$ and outputs $\mathbf{x}_0$ . In this section, we develop the transfer function, which characterizes the relationship between these inputs and outputs in the spectral domain. To do so, we assume that the output signals are vectors drawn from a Gaussian distribution, + +$$ +\mathbf {x} _ {0} \sim \mathcal {N} \left(\boldsymbol {\mu} _ {0}, \boldsymbol {\Sigma} _ {0}\right), \tag {5} +$$ + +where $\pmb{\mu}_0\in \mathbb{R}^d$ and $\Sigma_0\in \mathbb{R}^{d\times d}$ . A similar assumption was used in previous work [24, 26, 31]. While this model simplifies the signal distribution, we will demonstrate that it facilitates spectral analysis throughout the diffusion process and allows us to design a noise scheduling mechanism for different objectives. + +# 3.1 The optimal denoiser for a Gaussian input + +The following theorem states a well-known fact: under the above Gaussianity assumption, the Minimum Mean-Squared Error (MMSE) denoiser operating on $\mathbf{x}_t$ to recover $\mathbf{x}_0$ is linear. It is the Wiener Filter [35] and can be expressed in a closed form. + +Theorem 3.1. Let $\mathbf{x}_0\sim \mathcal{N}(\boldsymbol {\mu}_0,\boldsymbol {\Sigma}_0)$ and let $\mathbf{x}_t$ be defined by (2). Then, the denoised signal obtained from the MMSE (and the MAP) denoiser is given by: + +$$ +\mathbf {x} _ {0} ^ {*} = \left(\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + \left(1 - \bar {\alpha} _ {t}\right) \mathbf {I}\right) ^ {- 1} \left(\sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + \left(1 - \bar {\alpha} _ {t}\right) \boldsymbol {\mu} _ {0}\right). \tag {6} +$$ + +A detailed proof is given in Appendix A. Here, we outline the main steps under the assumption of Gaussian distributions, where applying the Maximum A Posteriori (MAP) estimator is equivalent to minimizing the MSE. The MAP estimator seeks to maximize the posterior probability: + +$$ +\max _ {\mathbf {x} _ {0}} \log p (\mathbf {x} _ {0} | \mathbf {x} _ {t}) = \min _ {\mathbf {x} _ {0}} - \log p (\mathbf {x} _ {t} | \mathbf {x} _ {0}) - \log p (\mathbf {x} _ {0}). \tag {7} +$$ + +By substituting the explicit density functions $p(\mathbf{x}_0)$ and $p(\mathbf{x}_t|\mathbf{x}_0)$ according to (2) into the above, and differentiating with respect to $\mathbf{x}_0$ we obtain the desired result. + +# 3.2 The reverse process in the time domain + +We now turn to analyze the discrete sampling procedures, as introduced in [10, 27] and described in Section 2. We begin by focusing on the DDIM formulation presented in [27], as it highlights the fundamental principles more clearly and facilitates the analysis of faster sampling techniques. The lemma below describes the relationship between two adjacent time steps during the inference process. + +Lemma 3.2. Assume $\mathbf{x}_0\sim \mathcal{N}(\pmb {\mu}_0,\pmb {\Sigma}_0)$ and let $\bar{\alpha}$ be the noise schedule parameters, we have + +$$ +\mathbf {x} _ {s - 1} = \left(a _ {s} \mathbf {I} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} \boldsymbol {\Sigma} _ {0}\right) \mathbf {x} _ {s} + b _ {s} \left(1 - \bar {\alpha} _ {s}\right) \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} \boldsymbol {\mu} _ {\mathbf {0}}, \tag {8} +$$ + +where $\bar{\mathbf{S}}_{0,s} = \bar{\alpha}_s\mathbf{S}_0 + (1 - \bar{\alpha}_s)\mathbf{I}$ , and the coefficients are + +$$ +a _ {s} = \frac {\sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}} , b _ {s} = \sqrt {\bar {\alpha} _ {s - 1}} - \frac {\sqrt {\bar {\alpha} _ {s}} \sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}}. +$$ + +The above is obtained by plugging the optimal denoiser into the DDIM reverse process in (4). The derivation is given in Appendix B. This lemma establishes an explicit connection between adjacent diffusion steps, incorporating the characteristics of the destination signal density function as expressed by $\mu_0$ and $\bar{\Sigma}_0$ , along with the chosen noise schedule parameters $\bar{\alpha}$ . + +# 3.3 Migrating to the spectral domain + +Analyzing the diffusion process in the time domain can be mathematically and computationally challenging, particularly in high-dimensional spaces ( $d \gg 1$ ). To address this, we leverage the spectral decomposition of the covariance matrix, simplifying the analysis and enabling the examination of the signal behavior along its eigen-directions. + +Consider a destination signal $\mathbf{x}_0$ drawn from a multivariate Gaussian distribution with a given mean $\pmb{\mu}_{0}$ and a covariance matrix $\pmb{\Sigma}_{0}$ , and let $\mathbf{U} \in \mathbb{R}^{d \times d}$ be a unitary matrix whose columns are the eigenvectors of $\pmb{\Sigma}_{0}$ . The projection of $\mathbf{x}_0$ onto this eigenbasis, denoted by $\mathbf{v}_0 = \mathbf{U}^\top \mathbf{x}_0$ , also follows a Gaussian distribution. Specifically, $\mathbf{v}_0 \sim \mathcal{N}(\pmb{\mu}_0^{\mathrm{u}}, \pmb{\Lambda}_0)$ , where $\pmb{\mu}_0^{\mathrm{u}} = \mathbf{U}^\top \pmb{\mu}_0 \in \mathbb{R}^d$ is the transformed mean vector, and $\pmb{\Lambda}_0 \in \mathbb{R}^{d \times d}$ is a positive semi-definite diagonal matrix, containing the eigenvalues of $\pmb{\Sigma}_{0}$ , denoted $\{\lambda_i\}_{i=1}^d$ . Based on Lemma 3.2, we derive the following property: + +Lemma 3.3. Assume $\mathbf{x}_0\sim \mathcal{N}(\boldsymbol {\mu}_0,\boldsymbol {\Sigma}_0)$ . The covariance matrices associated with all intermediate steps $\{\mathbf{x}_s\}_{s = 0}^S$ in the diffusion process are jointly diagonalizable using the eigenbasis $\mathbf{U}$ of $\boldsymbol{\Sigma}_0$ . + +This lemma holds since all the diffusion steps in (8) involve only $\Sigma_0$ , its inverse, or the addition of the identity matrix, leading to covariance matrices that share the same eigenbasis. We now turn to describe the diffusion reverse process in the spectral domain. By projecting both sides of (8) onto the eigenbasis $\mathbf{U}$ , we obtain the following result. + +Lemma 3.4. Assume $\mathbf{x}_0\sim \mathcal{N}(\boldsymbol {\mu}_0,\boldsymbol {\Sigma}_0)$ and let $\bar{\alpha}$ be the noise schedule parameters. The subsequent step in the reverse process can be expressed in the spectral domain via + +$$ +\mathbf {v} _ {s - 1} = \mathbf {G} (s) \mathbf {v} _ {s} + \mathbf {M} (s) \boldsymbol {\mu} _ {\mathbf {0}} ^ {\boldsymbol {u}}, \tag {9} +$$ + +where $\mathbf{v}_s$ denotes the projection of the signal $\mathbf{x}_s$ onto the eigenbasis $\mathbf{U}$ , + +$$ +\mathbf {G} (s) = \left[ a _ {s} \mathbf {I} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \left[ \bar {\alpha} _ {s} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} \pmb {\Lambda} _ {0} \right] a n d \mathbf {M} (s) = b _ {s} (1 - \bar {\alpha} _ {s}) \left[ \bar {\alpha} _ {s} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1}. +$$ + +The lemma is proven in Appendix C. Equation (9) describes the relationship between two consecutive steps in the reverse process. Note that both matrices, $\mathbf{G}(s)$ and $\mathbf{M}(s)$ , are diagonal, and thus, the reverse process in the spectral domain turns into a system of $d$ independent scalar equations. Based on the above relationship, we may derive an expression for the generated signal in the spectral domain, denoted as $\hat{\mathbf{v}}_0$ . The complete derivation of the following result can be found in Appendix C as well. + +Theorem 3.5. Assume $\mathbf{x}_0\sim \mathcal{N}(\boldsymbol {\mu}_0,\boldsymbol {\Sigma}_0)$ and let $\bar{\alpha}$ be the noise schedule parameters. The generated signal in the frequency domain $\hat{\mathbf{v}}_0$ can be described as a function of $\mathbf{v}_S$ via + +$$ +\hat {\mathbf {v}} _ {0} = \mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {\mathbf {0}} ^ {\boldsymbol {u}}, \tag {10} +$$ + +where $\mathbf{v}_S\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ , $\mathbf{D}_1 = \prod_{k = 1}^{S}\mathbf{G}(k)$ and $\mathbf{D}_2 = \sum_{i = 1}^{S}\left(\prod_{j = 1}^{i - 1}\mathbf{G}(j)\right)\mathbf{M}(i)$ + +Moreover, $\hat{\mathbf{v}}_0$ follows Gaussian distribution: + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {\mathbf {0}} ^ {\boldsymbol {u}}, \mathbf {D} _ {1} ^ {2}\right), \quad \hat {\mathbf {v}} _ {0} \in \mathbb {R} ^ {d}. \tag {11} +$$ + +Equation (10) provides a novel view of the generated signal in the spectral domain. Specifically, we can view (10) as a transfer function which models the relationship between the input signal $\mathbf{v}_S$ and + +the output $\hat{\mathbf{v}}_0$ . Furthermore, since the matrices $\mathbf{D}_1$ and $\mathbf{D}_2$ are diagonal, the expression simplifies to a set of $d$ scalar transfer functions, with the only parameters being the noise schedule, $\bar{\alpha}$ . + +So far, we examined DDIM. Similarly, a closed-form expression for the stochastic DDPM method [10] is presented in the following Theorem and is proven in Appendix E. + +Theorem 3.6. Assume $\mathbf{x}_0\sim \mathcal{N}(\boldsymbol {\mu}_0,\boldsymbol {\Sigma}_0)$ and let $\bar{\alpha}$ be the noise schedule parameters. The signal $\hat{\mathbf{v}}_0$ generated by DDPM can be expressed as a function of $\mathbf{v}_T$ via + +$$ +\hat {\mathbf {v}} _ {0} = \mathbf {D} _ {1} \mathbf {v} _ {T} + \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\boldsymbol {u}} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\boldsymbol {u}} \tag {12} +$$ + +where $\mathbf{v}_T\sim \mathcal{N}(\mathbf{0},\mathbf{I}),c_i = \sqrt{\frac{1 - \bar{\alpha}_{i - 1}}{1 - \bar{\alpha}_i}(1 - \alpha_i)}$ . The terms $\mathbf{D}_1,\mathbf{D}_2$ and the matrices $\mathbf{G}(j)$ are defined in Appendix E. Moreover, $\hat{\mathbf{v}}_0$ follows a Gaussian distribution: + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {\mathbf {0}} ^ {\boldsymbol {u}}, \mathbf {D} _ {1} ^ {2} + \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} ^ {2} (j)\right) c _ {i} ^ {2} \mathbf {I}\right). \tag {13} +$$ + +# 4 Optimal spectral schedules + +With the closed-form expressions in (10) and (12), we can now explore different aspects of the diffusion process and examine how subtle changes in its design affect the output distribution. More specifically, a key aspect in this design is the choice of the noise schedule. In the discussion that follows we demonstrate how the proposed scheme enables optimal scheduler design. + +We start by focusing on the direct dependence between the generated distribution and the noise schedule coefficients, $\bar{\alpha}$ . We define the spectral domain probability density function of the diffusion process output as $p(\hat{\mathbf{v}}_0;\bar{\alpha})$ . Our objective is to bring this distribution to become as close as possible to the original distribution, $p(\mathbf{v}_0)$ . Specifically, given a dataset with a covariance matrix, $\Sigma_0$ , defined by the eigenvalues $\{\lambda_i\}_{i=1}^d$ and $S$ diffusion steps, our goal is to identify the coefficients $\bar{\alpha}$ that minimize some distance $\mathcal{D}$ between these two distributions. This results in the following optimization problem with a set of specified constraints: + +$$ +\bar {\alpha} ^ {*} = \arg \min _ {\bar {\alpha}} \mathcal {D} (p (\hat {\mathbf {v}} _ {0}; \bar {\alpha}), p (\mathbf {v} _ {0})) \tag {14} +$$ + +subject to $\bar{\alpha}_{0} = 1 - \varepsilon_{0}$ , $\bar{\alpha}_{S} = \varepsilon_{S}$ + +$$ +\bar {\alpha} _ {s - 1} \geq \bar {\alpha} _ {s} \text {f o r} s = 1, \dots , S. +$$ + +The equality constraints ensure compatibility between the training and the synthesis processes. This involves ending the diffusion process with white Gaussian noise and starting it with very low noise to capture fine details in the objective distribution accurately [19]. The inequality constraints align with the core principles of diffusion models and their gradual denoising process [10]. The distance $\mathcal{D}$ between the probabilities can be chosen depending on the specific characteristics of the task. In this work, we consider the Wasserstein-2 and Kullback-Leibler divergence, but other distances can also be used. The theorems presented below are detailed in Appendix D. + +The Wasserstein-2 distance (or Earth Mover's Distance) measures the minimal cost of transporting mass to transform one probability distribution into another. In the case of measuring a distance between two Gaussians, this has a closed-form expression. + +Theorem 4.1. The Wasserstein-2 distance $\mathcal{D}_{W_2}$ between $p(\hat{\mathbf{v}}_0;\bar{\alpha})$ and $p(\mathbf{v}_0)$ is given by: + +$$ +\mathcal {D} _ {W _ {2}} ^ {2} \left(p (\hat {\mathbf {v}} _ {0}; \bar {\boldsymbol {\alpha}}), p (\mathbf {v} _ {0})\right) = \sum_ {i = 1} ^ {d} \left(\sqrt {\lambda_ {i}} - [ \mathbf {D} _ {1} ] _ {i}\right) ^ {2} + \sum_ {i = 1} ^ {d} \left[ \boldsymbol {\mu} _ {\mathbf {0}} ^ {\boldsymbol {u}} \right] _ {i} ^ {2} \left([ \mathbf {D} _ {2} ] _ {i} - 1\right) ^ {2}, \tag {15} +$$ + +where $\{\lambda_i\}_{i = 1}^d$ denote the d eigenvalues of $\pmb{\Sigma}_0$ + +The KL divergence $D_{\mathrm{KL}}(P\| Q)$ assesses how much a model probability distribution $Q$ differs from a reference probability distribution $P$ . Note that this divergence is not symmetric. As with the Wasserstein-2 case, here as well we obtain a closed-form expression for the two Gaussians considered. + +Theorem 4.2. The Kullback-Leibler divergence between the generated distribution $p(\hat{\mathbf{v}}_0; \bar{\alpha})$ and the true distribution $p(\mathbf{v}_0)$ is given by + +$$ +D _ {\mathrm {K L}} \left(p \left(\mathbf {v} _ {0}\right) \| p \left(\hat {\mathbf {v}} _ {0}; \bar {\boldsymbol {\alpha}}\right)\right) = \sum_ {i = 1} ^ {d} \log \left[ \mathbf {D} _ {1} \right] _ {i} - \frac {1}{2} \left(\sum_ {i = 1} ^ {d} \log \lambda_ {i} - d + \sum_ {i = 1} ^ {d} \frac {\lambda_ {i} + \left([ \mathbf {D} _ {2} ] _ {i} - 1\right) ^ {2} \left(\boldsymbol {\mu} _ {\mathbf {0}} ^ {\boldsymbol {u}}\right) _ {i} ^ {2}}{\left[ \mathbf {D} _ {1} \right] _ {i} ^ {2}}\right) \tag {16} +$$ + +For solving the resulting optimization problems, we have employed the Sequential Least Squares Programming (SLSQP) method [17], a well-suited method for minimization problems with boundary conditions, and equality and inequality constraints. + +Before turning into the empirical evaluation of the proposed optimization, and the implications of the obtained noise scheduling on the various schemes, we pause to describe related work in this field. + +# 5 Related work + +Recent work has acknowledged the importance of the noise scheduling in diffusion models, and the need to shift the focus from custom-tailored heuristics [10, 23, 13, 5] to the development of optimized alternatives. For instance, the authors of [26] introduced the KL-divergence Upper Bound (KLUB), which minimizes the mismatch between the continuous reverse-time SDE and its linearized approximation. However, this approach, along with others [34, 32, 37, 29], aim to minimize the estimation error as well, hence requires retraining a denoiser or entailing substantial computation time and resources when solving the optimization problem. + +While pursuing the same goal of optimizing the noise schedule, the work reported in [3, 38, 36] made notable strides in simplifying the induced optimization problem. Such efficiency is enabled in [38] by introducing an upper bound on the truncation error and assuming the data-dependent component to be negligible during optimization. Yet, despite this improved scalability, a direct relationship between data characteristics and the derived noise schedule still remains vague. + +A potential approach to bridge this gap is spectral analysis, a fundamental tool in signal processing, which frequently serves to associate design choices and properties of diffusion models with data characteristics [25, 2, 8, 39, 7]. The application of frequency representation is diverse, ranging from introducing the coarse-to-fine behavior in the reverse process [25], to uncover unique architectural fingerprints and identify biases in modeling different frequency ranges [7, 39]. However, to the best of our knowledge, no existing approach has connected spectral analysis with noise scheduling design. + +From a complementary perspective, several works have explored diffusion models by assuming a Gaussian target distribution [24, 31]. In [24], the authors leveraged a centered distribution and commutativity properties to derive an exact solution for both the reverse SDE and ODE. Similarly, a closed-form solution to the probability flow ODE provided in [31], which also demonstrated the relation between learned neural scores and their single Gaussian score approximation. Although these analytical solutions provide valuable insights into diffusion dynamics, they are derived in the continuous domain via integration, potentially neglecting discretization effects and missing an explicit connection to the full traversal of the noise schedule. + +# 6 Experiments + +We turn to empirically validate the schedules obtained by solving the optimization problem, referred to as the spectral schedule or spectral recommendation. In addition, we examine spectral phenomena arising from the diffusion process and their relation to the schedule structure. + +# 6.1 Synthetic Gaussian distribution + +In the first set of experiments, we assume a Gaussian data distribution, $\mathbf{x}_0\sim \mathcal{N}(\pmb {\mu}_0,\pmb {\Sigma}_0)$ , where $\mathbf{x}_0\in \mathbb{R}^d$ and $\boldsymbol{\Sigma}_{0}$ is a circulant covariance matrix. Although circularity is not essential, it facilitates a direct examination of the frequency components, as $\boldsymbol{\Sigma}_{0}$ is diagonalized by the DFT, with eigenvalues $\{\lambda_i\}_{i = 1}^d$ corresponding to its DFT coefficients [9]. Specifically, the covariance is chosen to satisfy $\pmb {\Sigma}_0 = A^T A$ where $A$ is a circulant matrix whose first row is $a = [-l, - l + 1 / (d - 1),\dots ,l - 1 / (d - 1),l]$ . The mean vector, $\pmb{\mu}_{0}$ , is chosen to be a constant-value vector, following the stationarity assumption. + +![](images/1f1bea7f4616cc7853afbbe6ad9ac1224fc1a17d55154d02e687ffee8f73716d.jpg) +(a) + +![](images/a1c203470071bf606e093773c7904c45dd6e9943cba25439348ea5ba7b3ba6d1.jpg) +(b) +Figure 1: Figure 1a presents the optimized spectral schedules for $\pmb{\Sigma}_0 = A^T A$ with $d = 50$ , $l = 0.1$ , and $\pmb{\mu_0} = 0.05\cdot \mathbf{1}_d$ , obtained by minimizing the Wasserstein-2 distance for various numbers of diffusion steps $S\in [10,28,38,60,90,112,250,334]$ . Figure 1b compares the spectral schedule (dotted red) for $S = 112$ with various heuristic alternatives, including linear, EDM ( $\rho = 7$ ), Cosine-based schedules such as Cosine ( $s = 0,e = 1,\tau = 1$ ) as in [23, 5] and Sigmoid-based schedules like Sigmoid ( $s = -3,e = 3,\tau = 1$ ) from [11, 5]. Parametric estimations for the Cosine and Sigmoid schedules appear in brown and cyan, respectively. Figure 1c compares the Wasserstein-2 distance of the spectral recommendation with that of the baseline schedules across different step counts. + +![](images/a13c939b0ee5cd9b92ccf345a9a22355946d992be5dfebbdd24cf0c0c6d824e8.jpg) +(c) + +Finding the optimal noise-schedule scheme $\bar{\alpha}^*$ depends on the target signal characteristics $\{\lambda_i\}_{i=1}^d$ , the resolution $d$ , and the number of diffusion steps applied $S$ . Figure 1a shows the resulting noise schedules for $d = 50, l = 0.1$ and $\pmb{\mu_0} = 0.05 \cdot \mathbf{1}_d$ , obtained by minimizing the Wasserstein-2 distance for different diffusion steps. $^2$ Further examples involving different forms of $\pmb{\Sigma_0}$ and $\pmb{\mu_0}$ , as well as the use of the KL divergence, are provided in Appendix H. + +At first glance, the optimization-based solution produces a noise schedule that aligns with the principles of diffusion models. Specifically, it exhibits a monotonically decreasing behaviour, with linear drop in $\bar{\alpha}^*$ in the middle of the process and minimal variation near the extremes. Although each schedule was independently optimized for a specified number of diffusion steps, it can be observed that the overall structure remains the same. + +Interestingly, solving the optimization problem in (14) while altering the initial conditions or removing the inequality constraints yields the same optimal solution. This suggests that these constraints are passive, and that known characteristics of noise schedules, such as monotonicity, naturally arise from the problem formulation, as demonstrated in Appendix G.3. + +A key aspect is how the spectral schedule aligns with the existing heuristics. Figure 1b provides a comparison with the Cosine [23], the Sigmoid [11], the linear [10] and the EDM $(\rho = 7)$ [13] schedules, along with a parametric approximation of the spectral recommendation. To achieve this, Cosine and Sigmoid functions were fitted to the optimal solution by minimizing the $l_{2}$ loss, identifying the closest match. An interesting outcome from Figure 1b is that the spectral recommendation provides a partial retrospective justification for existing noise schedule heuristics, as the parametric estimation resembles Cosine and Sigmoid functions, when their parameters are properly tuned. + +To validate the optimization procedure, Figure 1c compares the Wasserstein-2 distance of various noise schedules with that of the spectral recommendation across different diffusion steps. While the spectral schedule consistently achieves the lowest Wasserstein-2 distance, the optimization is most effective with fewer diffusion steps, where discretization errors are higher. As the number of steps increases, the gap between the different noise schedules narrows. + +We also compare our derived schedules with those proposed in prior work. Specifically, the authors in [26] present a closed-form solution for the optimal noise schedule in a simplified setting, assuming an initial isotropic Gaussian distribution with standard deviation $C$ , i.e $\mathbf{x}_0 \sim \mathcal{N}(\mathbf{0}, C^2\mathbf{I})$ . To enable a fair comparison, we frame our optimization problem using the $D_{\mathrm{KL}}$ loss (4.2) and adopt the Variance-Exploding (VE) framework, following the setup in [26]. Appendix G.2 demonstrates the equivalence between the solutions and highlight their relation to commonly used heuristics. + +Frequencies components and schedule structure: We turn to explore the relationship between the eigenvalue magnitudes and the structure of the derived schedule. To do so, we solved the optimization problem for each eigenvalue individually, with the contributions from the other eigenvalues set to + +zero. It can be observed from Figure 2a that the solution becomes more concave as the magnitude of the eigenvalue decreases (yellow curve) and more convex as the magnitude increases (blue curve). + +Notably, under the shift-invariance conditions, the eigenvalues directly correspond to the system's frequency components. In scenarios where this components follow a monotonically decreasing distribution (e.g., the $1/f$ trend in speech [30]), the first eigenvalues correspond to the low frequencies, having larger amplitudes, while the last correspond to high frequencies and smaller amplitudes. This pattern, along with the previous observation, aligns with the well-known coarse-to-fine signal construction behavior of diffusion models. farther details are provided in Appendix K.1 + +Relative error evolution: While the optimization problem primarily focuses on the output signal $\hat{\mathbf{v}}_0$ , each diffusion step $\mathbf{v}_s$ can be also expressed as a function of the initial noise as described in (29) in Appendix C. This enables the analysis of spectral properties across all diffusion steps, rather than only at the output signal. Figure 2b presents the relative error of the 10 largest eigenvalues (sorted, largest on the right) over the final 20 steps of a 60-step diffusion process using the Cosine (0,1,1) [23, 5] schedule. While the error consistently decreases for high-amplitude eigenvalues, it increases near the final diffusion steps for those with smaller magnitudes. Additionally, the final relative error (yellow) tends to be smaller for large eigenvalues (i.e., low frequencies). This behavior aligns with the observed bias in mid-to-high frequencies reported in prior works [39, 7]. Further discussion, including an illustration of the eigenvalue and Wasserstein-2 error dynamics, appears in App. K.2. + +Mean drift: The explicit expressions in (10) and (12) offer a further insight into the diffusion process. A notable consideration is whether this process introduces a bias, i.e. drifting the mean component during synthesis. To study this, we analyze the mean bias expression for DDIM, $(\mathbf{D}_2 - \mathbf{I})\boldsymbol{\mu}_0^u$ derived from the difference between $\mathbb{E}[\mathbf{v}_0]$ and $\mathbb{E}[\hat{\mathbf{v}}_0]$ . In Appendix L, we further explore the relationship between the target signal characteristics $\{\lambda_i\}_{i = 1}^d$ , the noise schedule $\bar{\alpha}$ , and the expression $|\mathbf{D}_2 - \mathbf{I}|$ . It appears that different choices of the noise schedule influence the bias value, with some choices effectively mitigat- + +![](images/cfb8aa3a32f62a0e309457db2057642e36596dd0d2218c0ce9ba2ae29407c048.jpg) +(a) + +![](images/e2a185cffde0b92d9af107c8454d47d81c8d85014b90c2c323494c135e869527.jpg) +(b) +Figure 2: Figure 2a shows the spectral schedules obtained by solving the optimization problem individually for each eigenvalue, with other contributions set to zero (note that the reverse process proceeds from right to left). Figure 2b illustrates the relative error of the 10 largest eigenvalues over the final 20 steps of a 60-step diffusion process using the Cosine $(0,1,1)$ schedule. + +ing it. Additionally, as the depth of the diffusion process increases, the bias value tends to grow, regardless of the selected noise schedule. + +DDPM vs DDIM: The explicit formulations of DDPM and DDIM in (10) and (12) enable a comparison of their losses across varying diffusion depths and noise schedules. In Appendix M, we present such an evaluation using the Wasserstein-2 distance. The results clearly show that DDIM sampling is faster and yields lower loss values, aligning with the empirical observations in [27]. + +# 6.2 Empirical Gaussian distribution + +We now shift towards a more practical scenario in which we refer to real data, while still maintaining the Gaussianity assumption. To assess the generality of our approach, we consider different signal types. Specifically, we use CIFAR-10, and the MUSIC dataset [22], which comprises recordings of various musical instruments down-sampled to $16\mathrm{kHz}$ . We fit a Gaussian distribution for each dataset by estimating their mean and covariance matrix, while for the MUSIC dataset, we extracted piano-only recordings. These datasets are referred to hereafter as the Gaussian MUSIC piano dataset and the Gaussian CIFAR-10 dataset + +![](images/92193f7e2e6e26520f30af981b89e8fc0c61f1dc8b770c526e80978e7c16b878.jpg) +Figure 3: Comparison of the spectral schedules for CIFAR-10 and MUSIC Datasets with various heuristic noise schedules, using 112 diffusion steps. + +![](images/a85769773c171855e09a6f5bc1006200bfc1ead2e05ebef41fecdb86a48d84f5.jpg) +(a) + +![](images/92471a086ee90ee7f0d47bebc7cb611d84c74faa33eede7ba0cac04346078947.jpg) +(b) +Figure 4: Figures 4b and 4c show the Wasserstein-2 distance of the proposed spectral noise schedule (in red) compared to existing heuristics, evaluated across various diffusion step sizes, for CIFAR-10 and MUSIC. Figure 4a presents the corresponding FID scores for CIFAR-10. Across all comparisons, the spectral schedule generally outperforms the heuristics, with a wider gap at lower step counts. For CIFAR-10, the approximation error is less pronounced, with results showing greater stability. + +![](images/3d1a223ad179a6d7a97cb8f5b521fb74a34eb748256b4d5c5773249ae03bb502.jpg) +(c) + +[24]. The estimation procedure for temporal signals, including the role of the window size $(d)$ and silence threshold $(th)$ , is detailed in Appendix I.4. + +Figure 3 presents the spectral recommendations obtained by minimizing the Wasserstein-2 distance for each estimated covariance matrix, alongside several heuristic noise schedules. While both spectral schedules retain some resemblance to the hand-crafted approaches, the one derived for the MUSIC dataset, introduces a somewhat different design of slower decay, adapting to the unique data properties. Consequently, adopting a spectral analysis perspective enables the design of noise schedules tailored to specific needs and data characteristics. The spectral recommendations for the SC09 [33] dataset, with the Wasserstein-2 evaluations, are provided in Appendix I. + +# 6.3 Empirical distribution + +We now aim to evaluate whether the optimized noise schedule from Sec. 6.2 remains effective when the Gaussianity assumption is removed and a trained neural denoiser is employed within the diffusion process. Notably, in this setting, approximation error arises due to suboptimality of the denoiser [24]. + +Our approach towards performance assessment of different noise schedules is the following: We run the diffusion process using a trained denoiser, and the DDIM inference procedure (4) to generate a large corpus of samples. We then compare the statistics of the synthesized signals with those of real data by measuring the distance between their second moments, applying the Wasserstein-2 distance to the empirical covariance matrices. For CIFAR-10, we also use the standard FID score. + +Figure 4 performs such a comparison with various heuristic noise schedules. For each schedule and number of diffusion steps tested, 50,000 signals are synthesized using a trained model. For CIFAR-10, we used a pretrained denoiser from [13] which is based on a continuous noise schedule. For the MUSIC dataset, we trained a model based on the architecture presented in [16, 1], which employs a linear noise schedule with $T = 1000$ diffusion steps during training. + +Figure 4 demonstrates that the spectral schedules consistently outperform most heuristic alternatives across both metrics, with the performance gap narrowing as the number of diffusion steps increases and discretization error diminishes. Notably, the Cosine (0,0.5,1) and EDM heuristics, whose structures closely resemble the spectral schedules derived for the CIFAR-10 and MUSIC datasets in Figure 3, also perform competitively (Figures 4a, 4c). These suggests that the proposed spectral recommendation effectively preserves the core data properties and may support the broader design of scheduling strategies. A more detailed comparison, with results for SC09, appears in Appendix J. + +# 7 Discussion + +By recognizing how the eigenvalues shape the noise schedule, we can refine the loss term to better align with our needs. In Appendix K.3, we introduce a weighted-ll loss, which promotes low frequencies while sacrificing high ones, ultimately yielding a solution consistent with the well-known Cosine $(0,1,1)$ heuristic. This relationship can be further explored and may help in mitigating + +phenomena such as bias in certain frequency ranges [7, 39]. Another potential research direction is leveraging the evolution of the stepwise distribution for enabling accelerated sampling (29). + +We analyze the optimization times of our method. For small step counts, where discretization error is more pronounced, the solutions are typically obtained within a few seconds. However, for higher step counts or high-resolution signals, the computation time may vary. By prioritizing the most significant eigenvalue components during schedule design or initializing the optimization with solutions from smaller step counts, the runtime can be reduced to a few tens of seconds, as detailed in Appendix N. + +Limitations: While our approach demonstrates the benefits of a principled scheduler design and outperforms heuristic methods in FID across multiple datasets, a natural next step is to extend the framework to settings closer to real data distributions, such as Gaussian mixture models. + +This paper presents a spectral perspective on the inference process in diffusion models. Under the assumption of Gaussianity, we establish a direct link between the input white noise and the output signal. Our approach enables noise schedule design based on data characteristics, diffusion steps count, and sampling methods. 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The probability density function $f(\mathbf{x}_0)$ can be written as: + +$$ +f (\mathbf {x} _ {0}) = \frac {1}{\sqrt {(2 \pi) ^ {d} | \boldsymbol {\Sigma} _ {0} |}} \cdot \exp \left\{- \frac {1}{2} \left(\bar {\mathbf {x}} _ {0} - \bar {\boldsymbol {\mu}} _ {0}\right) ^ {T} \boldsymbol {\Sigma} _ {0} ^ {- 1} \left(\bar {\mathbf {x}} _ {0} - \bar {\boldsymbol {\mu}} _ {0}\right) \right\} +$$ + +Through the diffusion process, the signal undergoes noise contamination, leading to the following marginal expression for $\mathbf{x}_t$ : + +$$ +\mathbf {x} _ {t} = \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0} + \sqrt {1 - \bar {\alpha} _ {t}} \boldsymbol {\epsilon} \quad \boldsymbol {\epsilon} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}) \tag {17} +$$ + +For the Maximum A Posteriori (MAP) estimation, we seek to maximize the posterior distribution: + +$$ +\max _ {x _ {0}} \log p (\mathbf {x} _ {0} | \mathbf {x} _ {t}) +$$ + +Using Bayes' rule, this can be written as: + +$$ +\min _ {x _ {0}} - \log \left[ \frac {p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {0}\right) p \left(\mathbf {x} _ {0}\right)}{p \left(\mathbf {x} _ {t}\right)} \right] = \min _ {x _ {0}} - \log p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {0}\right) - \log p \left(\mathbf {x} _ {0}\right) \tag {18} +$$ + +The conditional log-likelihood $\log p(\mathbf{x}_t|\mathbf{x}_0)$ is given by: + +$$ +p (\mathbf {x} _ {t} | \mathbf {x} _ {0}) = \frac {1}{\sqrt {(2 \pi) ^ {d} | \Sigma_ {1} |}} \exp \left\{- \frac {1}{2} \left(\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0}\right) ^ {T} ((1 - \bar {\alpha} _ {t}) \mathbf {I}) ^ {- 1} (\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0}) \right\} +$$ + +$$ +\log p (\mathbf {x} _ {t} | \mathbf {x} _ {0}) = - \frac {1}{2} \log (2 \pi) ^ {d} | \Sigma_ {1} | - \frac {1}{2 (1 - \bar {\alpha} _ {t})} (\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0}) ^ {T} (\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0}) +$$ + +The log-likelihood $\log p(\mathbf{x}_0)$ is given by: + +$$ +p (\mathbf {x} _ {0}) = \frac {1}{\sqrt {(2 \pi) ^ {d} | \boldsymbol {\Sigma} _ {0} |}} \exp \left\{- \frac {1}{2} (\mathbf {x} _ {0} - \boldsymbol {\mu} _ {0}) ^ {T} \boldsymbol {\Sigma} _ {0} ^ {- 1} (\mathbf {x} _ {0} - \boldsymbol {\mu} _ {0}) \right\} +$$ + +$$ +\log p (\mathbf {x} _ {0}) = - \frac {1}{2} \log (2 \pi) ^ {d} | \boldsymbol {\Sigma} _ {0} | - \frac {1}{2} (\mathbf {x} _ {0} - \boldsymbol {\mu} _ {0}) ^ {T} \boldsymbol {\Sigma} _ {0} ^ {- 1} (\mathbf {x} _ {0} - \boldsymbol {\mu} _ {0}) +$$ + +We will differentiate the given expression in (18) with respect to $x_0$ and equate it to zero: + +$$ +\frac {d \log p (\mathbf {x} _ {t} | \mathbf {x} _ {0})}{d \mathbf {x} _ {0}} = \frac {2 \sqrt {\bar {\alpha} _ {t}} (\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0})}{2 (1 - \bar {\alpha} _ {t})} +$$ + +$$ +\frac {d \log p (\mathbf {x} _ {0})}{d \mathbf {x} _ {0}} = - \frac {2 \boldsymbol {\Sigma} _ {0} ^ {- 1} (\mathbf {x} _ {0} - \boldsymbol {\mu} _ {0})}{2} +$$ + +$$ +\frac {- 2 \sqrt {\bar {\alpha} _ {t}} (\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0})}{2 (1 - \bar {\alpha} _ {t})} + \frac {2 \boldsymbol {\Sigma} _ {0} ^ {- 1} (\mathbf {x} _ {0} - \boldsymbol {\mu} _ {0})}{2} = 0 +$$ + +This simplifies to: + +$$ +\frac {- \sqrt {\bar {\alpha} _ {t}} (\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0})}{(1 - \bar {\alpha} _ {t})} + \boldsymbol {\Sigma} _ {0} ^ {- 1} (\bar {\mathbf {x}} _ {0} - \boldsymbol {\mu} _ {0}) = 0 +$$ + +Resulting in: + +$$ +- \sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + \bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {x} _ {0} - (1 - \bar {\alpha} _ {t}) \boldsymbol {\mu} _ {0} = 0 +$$ + +Thus: + +$$ +\left(\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + \left(1 - \bar {\alpha} _ {t}\right) \mathbf {I}\right) \mathbf {x} _ {0} = \sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + \left(1 - \bar {\alpha} _ {t}\right) \boldsymbol {\mu} _ {0} +$$ + +Finally: + +$$ +\mathbf {x} _ {0} ^ {*} = \left(\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + \left(1 - \bar {\alpha} _ {t}\right) \mathbf {I}\right) ^ {- 1} \left(\sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + \left(1 - \bar {\alpha} _ {t}\right) \boldsymbol {\mu} _ {0}\right) \tag {19} +$$ + +# B The Reverse Process in the Time Domain + +Here, we present the reverse process in the time domain for the DDIM [27], as outlined in Lemma 3.2. + +Let $\mathbf{x}_0$ follow the distribution: + +$$ +\mathbf {x} _ {0} \sim \mathcal {N} (\pmb {\mu} _ {0}, \pmb {\Sigma} _ {0}), \quad \mathbf {x} _ {0} \in \mathbb {R} ^ {d} +$$ + +Using the procedure outline in [27], the diffusion process begins with $\mathbf{x}_S\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ , where $\mathbf{x}_S\in \mathbb{R}^d$ and progresses through an iterative denoising process described as follows:3 + +$$ +\mathbf {x} _ {s - 1} (\eta) = \sqrt {\bar {\alpha} _ {s - 1}} \left(\frac {\mathbf {x} _ {s} - \sqrt {1 - \bar {\alpha} _ {s}} \cdot \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} _ {s} , s\right)}{\sqrt {\bar {\alpha} _ {s}}}\right) + \sqrt {1 - \bar {\alpha} _ {s - 1} - \sigma_ {s} ^ {2} (\eta)} \cdot \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} _ {s}, s\right) + \sigma_ {s} (\eta) \mathbf {z} _ {s} \tag {20} +$$ + +where + +$$ +\sigma_ {s} (\eta) = \eta \sqrt {\frac {1 - \bar {\alpha} _ {s - 1}}{1 - \bar {\alpha} _ {s}}} \sqrt {1 - \frac {\bar {\alpha} _ {s}}{\bar {\alpha} _ {s - 1}}} \tag {21} +$$ + +Substituting the marginal property from (2): + +$$ +\boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} _ {s}, s\right) = \frac {\mathbf {x} _ {s} - \sqrt {\bar {\alpha} _ {s}} \hat {\mathbf {x}} _ {0}}{\sqrt {1 - \bar {\alpha} _ {s}}} \quad \hat {\mathbf {x}} _ {0} = \frac {\mathbf {x} _ {s} - \sqrt {1 - \bar {\alpha} _ {s}} \cdot \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} _ {s} , s\right)}{\sqrt {\bar {\alpha} _ {s}}} \tag {22} +$$ + +$$ +\mathbf {x} _ {s - 1} (\eta) = \sqrt {\bar {\alpha} _ {s - 1}} \hat {\mathbf {x}} _ {0} + \sqrt {1 - \bar {\alpha} _ {s - 1} - \sigma_ {s} ^ {2} (\eta)} \left(\frac {\mathbf {x} _ {s} - \sqrt {\bar {\alpha} _ {s}} \hat {\mathbf {x}} _ {0}}{\sqrt {1 - \bar {\alpha} _ {s}}}\right) + \sigma_ {s} (\eta) \mathbf {z} _ {s} \tag {23} +$$ + +For the deterministic scenario, we choose $\eta = 0$ in (21) and obtain $\sigma_s(\eta = 0) = 0$ . Therefore: + +$$ +\begin{array}{l} \mathbf {x} _ {s - 1} (\eta = 0) = \sqrt {\bar {\alpha} _ {s - 1}} \hat {\mathbf {x}} _ {0} + \sqrt {1 - \bar {\alpha} _ {s - 1}} \left(\frac {\mathbf {x} _ {s} - \sqrt {\bar {\alpha} _ {s}} \hat {\mathbf {x}} _ {0}}{\sqrt {1 - \bar {\alpha} _ {s}}}\right) \\ = \frac {\sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}} \mathbf {x} _ {s} + \left[ \sqrt {\bar {\alpha} _ {s - 1}} - \frac {\sqrt {\bar {\alpha} _ {s}} \sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}} \right] \hat {\mathbf {x}} _ {0} \tag {24} \\ \end{array} +$$ + +We denote the following: + +$$ +a _ {s} = \frac {\sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}} b _ {s} = \sqrt {\bar {\alpha} _ {s - 1}} - \frac {\sqrt {\bar {\alpha} _ {s}} \sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}} +$$ + +Therefore we get the following equation: + +$$ +\mathbf {x} _ {s - 1} = a _ {s} \mathbf {x} _ {s} + b _ {s} \mathbf {x} _ {0} ^ {*}. +$$ + +Using the result from the MAP estimator: + +$$ +\mathbf {x} _ {0} ^ {*} = \left(\bar {\alpha} _ {s} \boldsymbol {\Sigma} _ {0} + \left(1 - \bar {\alpha} _ {s}\right) \mathbf {I}\right) ^ {- 1} \left(\sqrt {\bar {\alpha} _ {s}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + \left(1 - \bar {\alpha} _ {s}\right) \boldsymbol {\mu} _ {0}\right) +$$ + +we get: + +$$ +\mathbf {x} _ {s - 1} = a _ {s} \mathbf {x} _ {s} + b _ {s} \left(\bar {\alpha} _ {s} \pmb {\Sigma} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I}\right) ^ {- 1} \left(\sqrt {\bar {\alpha} _ {s}} \pmb {\Sigma} _ {0} \mathbf {x} _ {s} + (1 - \bar {\alpha} _ {s}) \pmb {\mu} _ {0}\right) +$$ + +$$ +\mathbf {x} _ {s - 1} = \left(a _ {s} + b _ {s} \left(\bar {\alpha} _ {s} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I}\right) ^ {- 1} \sqrt {\bar {\alpha} _ {s}} \boldsymbol {\Sigma} _ {0}\right) \mathbf {x} _ {s} + \left(b _ {s} \left(\bar {\alpha} _ {s} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I}\right) ^ {- 1} (1 - \bar {\alpha} _ {s}) \boldsymbol {\mu} _ {0}\right) \tag {25} +$$ + +Introduce the notation: + +$$ +\bar {\boldsymbol {\Sigma}} _ {0, s} = \bar {\alpha} _ {s} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} +$$ + +We can rewrite the equation as: + +$$ +\mathbf {x} _ {s - 1} = \left(a _ {s} \mathbf {I} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \left(\bar {\Sigma} _ {0, s}\right) ^ {- 1} \boldsymbol {\Sigma} _ {0}\right) \mathbf {x} _ {s} + b _ {s} \left(1 - \bar {\alpha} _ {s}\right) \left(\bar {\Sigma} _ {0, s}\right) ^ {- 1} \boldsymbol {\mu} _ {\mathbf {0}} \tag {26} +$$ + +# C Migrating to the Spectral Domain + +Here, we demonstrate the application of projecting both sides of Eq. (8) onto the eigenbasis of $\pmb{\Sigma}_0$ as outlined in Lemma 3.4. Since $\pmb{\Sigma}_0$ is symmetric, its eigenvalue decomposition can written as $\pmb{\Sigma}_0 = \mathbf{U}\pmb{\Lambda}_0\mathbf{U}^\top$ , where the columns of $\mathbf{U}$ represent the eigenvectors, and $\pmb{\Lambda}_0$ is the diagonal matrix containing the corresponding eigenvalues. + +$$ +\mathbf {U} ^ {\top} \mathbf {x} _ {s - 1} = \mathbf {U} ^ {\top} \left[ \left(a _ {s} \mathbf {I} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} \boldsymbol {\Sigma} _ {0}\right) \mathbf {x} _ {s} + b _ {s} (1 - \bar {\alpha} _ {s}) \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} \boldsymbol {\mu} _ {\mathbf {0}} \right] \tag {27} +$$ + +$$ +\mathbf {U} ^ {\top} \mathbf {x} _ {s - 1} = a _ {s} \mathbf {U} ^ {\top} \mathbf {x} _ {s} + \mathbf {U} ^ {\top} b _ {s} \sqrt {\bar {\alpha} _ {s}} \bar {\Sigma} _ {0, s} ^ {- 1} \mathbf {U} \mathbf {U} ^ {\top} \boldsymbol {\Sigma} _ {0} \mathbf {U} \mathbf {U} ^ {\top} \mathbf {x} _ {s} + \mathbf {U} ^ {\top} b _ {s} (1 - \bar {\alpha} _ {s}) \bar {\Sigma} _ {0, s} ^ {- 1} \mathbf {U} \mathbf {U} ^ {\top} \boldsymbol {\mu_ {0}} +$$ + +By introducing the notations $\mathbf{v} = \mathbf{U}^{\top}\mathbf{x}$ and $\pmb{\mu}_0^{\mathrm{u}} = \mathbf{U}^{\top}\pmb{\mu}_0$ , we obtain: + +$$ +\mathbf {v} _ {s - 1} = a _ {s} \mathbf {v} _ {s} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \mathbf {U} ^ {\top} \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} \mathbf {U} \mathbf {U} ^ {\top} \boldsymbol {\Sigma} _ {0} \mathbf {U} \mathbf {v} _ {s} + b _ {s} (1 - \bar {\alpha} _ {s}) \mathbf {U} ^ {\top} \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} \mathbf {U} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} +$$ + +By applying eigenvalue decomposition, the symmetric matrix $\Sigma_0$ can be diagonalized using its eigenbasis $\mathbf{U}$ , where $\mathbf{U}^{-1} = \mathbf{U}^{\top}$ Specifically: + +$$ +\begin{array}{l} \bullet \boldsymbol {\Sigma} _ {0} = \mathbf {U} \boldsymbol {\Lambda} _ {0} \mathbf {U} ^ {\top}, \quad \boldsymbol {\Lambda} _ {0} = \mathbf {U} ^ {\top} \boldsymbol {\Sigma} _ {0} \mathbf {U} \\ \bullet a \boldsymbol {\Sigma} _ {0} + b \mathbf {I} = a \mathbf {U} \boldsymbol {\Lambda} _ {0} \mathbf {U} ^ {\top} + b \mathbf {U} \mathbf {I} \mathbf {U} ^ {\top} = \mathbf {U} (a \boldsymbol {\Lambda} _ {0} + b \mathbf {I}) \mathbf {U} ^ {\top} \\ \bullet \boldsymbol {\Sigma} _ {0} ^ {- 1} = \mathbf {U} \boldsymbol {\Lambda} _ {0} ^ {- 1} \mathbf {U} ^ {\top}, \quad \boldsymbol {\Lambda} _ {0} ^ {- 1} = \mathbf {U} ^ {\top} \boldsymbol {\Sigma} _ {0} ^ {- 1} \mathbf {U} \\ \end{array} +$$ + +Therefore, we obtain: + +$$ +\begin{array}{l} \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} = \left[ \bar {\alpha} _ {s} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} = \left[ \bar {\alpha} _ {s} \mathbf {U} \boldsymbol {\Lambda} _ {0} \mathbf {U} ^ {\mathsf {T}} + (1 - \bar {\alpha} _ {s}) \mathbf {U} \mathbf {I} \mathbf {U} ^ {\mathsf {T}} \right] ^ {- 1} = \left[ \mathbf {U} \left(\bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I}\right) \mathbf {U} ^ {\mathsf {T}} \right] ^ {- 1} \\ \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} = \mathbf {U} [ \bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} ] ^ {- 1} \mathbf {U} ^ {\mathsf {T}} \\ \end{array} +$$ + +Incorporating these elements into the main equation, we obtain: + +$$ +\mathbf {v} _ {s - 1} = a _ {s} \mathbf {v} _ {s} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \left[ \bar {\alpha} _ {s} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} \pmb {\Lambda} _ {0} \mathbf {v} _ {s} + b _ {s} (1 - \bar {\alpha} _ {s}) \left[ \bar {\alpha} _ {s} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} \pmb {\mu} _ {0} ^ {\mathbf {u}} +$$ + +$$ +\mathbf {v} _ {s - 1} = \left[ a _ {s} \mathbf {I} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \left[ \bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right] \mathbf {v} _ {s} + b _ {s} (1 - \bar {\alpha} _ {s}) \left[ \bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} +$$ + +We will denote the following: + +$$ +\mathbf {G} (s) = \left[ a _ {s} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \left[ \bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right] +$$ + +$$ +\mathbf {M} (s) = b _ {s} \big (1 - \bar {\alpha} _ {s} \big) \big [ \bar {\alpha} _ {s} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \big ] ^ {- 1} +$$ + +and get: + +$$ +\mathbf {v} _ {s - 1} = \mathbf {G} (s) \mathbf {v} _ {s} + \mathbf {M} (s) \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \tag {28} +$$ + +We can then recursively obtain $\mathbf{v}_l$ for a general $l$ : + +$$ +\mathbf {v} _ {l} = \left[ \prod_ {s ^ {\prime} = l + 1} ^ {S} \mathbf {G} \left(s ^ {\prime}\right) \right] \mathbf {v} _ {S} + \left[ \sum_ {i = l + 1} ^ {S} \left(\prod_ {j = l + 1} ^ {i - 1} \mathbf {G} (j)\right) \mathbf {M} (i) \right] \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \tag {29} +$$ + +specifically for $l = 0$ : + +$$ +\hat {\mathbf {v}} _ {0} = \left[ \prod_ {s ^ {\prime} = 1} ^ {S} \mathbf {G} \left(s ^ {\prime}\right) \right] \mathbf {v} _ {S} + \left[ \sum_ {i = 1} ^ {S} \binom {i - 1} {j = 1} \mathbf {G} (j) \right] \mathbf {M} (i) \Bigg ] \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \tag {30} +$$ + +We will denote the following: + +$$ +\mathbf {D} _ {1} = \prod_ {s = 1} ^ {S} \mathbf {G} (s) = \prod_ {s = 1} ^ {S} \left[ a _ {s} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \left[ \bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + \mathbf {I} \left(1 - \bar {\alpha} _ {s}\right) \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right] \tag {31} +$$ + +$$ +\begin{array}{l} \mathbf {D} _ {2} = \sum_ {i = 1} ^ {S} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) \mathbf {M} (i) \tag {32} \\ = \sum_ {i = 1} ^ {S} \left[ \left(\prod_ {j = 1} ^ {i - 1} \left[ a _ {j} + b _ {j} \sqrt {\bar {\alpha} _ {j}} [ \bar {\alpha} _ {j} \pmb {\Lambda} _ {0} + \mathbf {I} (1 - \bar {\alpha} _ {j}) ] ^ {- 1} \pmb {\Lambda} _ {0} \right]\right) b _ {i} (1 - \bar {\alpha} _ {i}) [ \bar {\alpha} _ {i} \pmb {\Lambda} _ {0} + \mathbf {I} (1 - \bar {\alpha} _ {i}) ] ^ {- 1} \right] \\ \end{array} +$$ + +Substitute $D_{1}$ and $D_{2}$ into the last equation, we get: + +$$ +\hat {\mathbf {v}} _ {0} = \mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \tag {33} +$$ + +The resulting vector from Equation 33 is a linear combination of Gaussian signals, therefore it also follows a Gaussian distribution. We now aim to determine the mean and covariance of that distribution. + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} (\mathbb {E} [ \hat {\mathbf {v}} _ {0} ], \pmb {\Sigma} _ {\hat {\mathbf {v}} _ {0}}), \quad \hat {\mathbf {v}} _ {0} \in \mathbb {R} ^ {d} +$$ + +Mean: + +$$ +\begin{array}{l} \mathbb {E} \left[ \hat {\mathbf {v}} _ {0} \right] = \mathbb {E} \left[ \mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \right] = \mathbf {D} _ {1} \mathbf {F} \mathbb {E} \left[ \mathbf {v} _ {S} \right] + \mathbb {E} \left[ \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \right] = \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \\ \mathbf {v} _ {S} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}) \\ \mathbb {E} \left[ \hat {\mathbf {v}} _ {0} \right] = \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \\ \end{array} +$$ + +Covariance: + +$$ +\begin{array}{l} \boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}} = \mathbb {E} \left[ \left(\mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - \mathbb {E} \left[ \mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \right]\right) \left(\mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - \mathbb {E} \left[ \mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {o} ^ {\mathbf {u}} \right]\right) ^ {T} \right] \\ = \mathbb {E} \left[ \left(\mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) \left(\mathbf {D} _ {1} \mathbf {v} _ {S} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T} \right] \\ = \mathbb {E} \left[ \left(\mathbf {D} _ {1} \mathbf {v} _ {S}\right) \left(\mathbf {D} _ {1} \mathbf {v} _ {S}\right) ^ {T} \right] = \mathbf {D} _ {1} \mathbb {E} \left[ \mathbf {v} _ {S} \left(\mathbf {v} _ {S}\right) ^ {T} \right] \mathbf {D} _ {1} ^ {T} \\ \boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}} = \mathbf {D} _ {1} \mathbf {D} _ {1} ^ {T} = \mathbf {D} _ {1} ^ {2} \\ \end{array} +$$ + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \mathbf {D} _ {1} ^ {2}\right), \quad \hat {\mathbf {v}} _ {0} \in \mathbb {R} ^ {d}. \tag {34} +$$ + +The same procedure used to derive the distribution of $\hat{\mathbf{v}}_0$ in (34) from (30) can similarly be applied to any intermediate step $\mathbf{v}_l$ in (29), enabling exploration of the spectral dynamics throughout the inference process. + +For the data distribution $\mathbf{x}_0\sim \mathcal{N}(\boldsymbol {\mu}_0,\boldsymbol {\Sigma}_0)$ , where $\mathbf{x}_0\in \mathbb{R}^d$ , its first and second moments in the frequency domain are given by: + +$$ +\mathbb {E} \left[ \mathbf {v} _ {0} \right] = \mathbf {U} ^ {\mathrm {T}} \mathbb {E} \left[ \mathbf {x} _ {0} \right] = \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} +$$ + +$$ +\Sigma_ {\mathbf {v} _ {0}} = \boldsymbol {\Lambda} _ {0} +$$ + +$$ +\mathbf {v} _ {0} \sim \mathcal {N} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \boldsymbol {\Lambda} _ {0}\right) \tag {35} +$$ + +# D Evaluating loss functions expressions + +Here, we present selected loss functions based on the derivations provided in Section 3. + +# D.1 Wasserstein-2 Distance: + +The Wasserstein-2 distance between two Gaussian distributions with means $\mu_{1}$ and $\mu_{2}$ , and covariance matrices $\Sigma_{1}$ and $\Sigma_{2}$ , and the corresponding eigenvalues $\{\lambda_1^{(i)}\}_{i = 1}^d$ and $\{\lambda_2^{(i)}\}_{i = 1}^d$ is given by: + +$$ +W _ {2} \left(\mathcal {N} _ {1}, \mathcal {N} _ {2}\right) = \sqrt {\left(\boldsymbol {\mu} _ {1} - \boldsymbol {\mu} _ {2}\right) ^ {T} \left(\boldsymbol {\mu} _ {1} - \boldsymbol {\mu} _ {2}\right) + \sum_ {i} \left(\sqrt {\lambda_ {1} ^ {(i)}} - \sqrt {\lambda_ {2} ^ {(i)}}\right) ^ {2}} \tag {36} +$$ + +Since $\hat{\mathbf{v}}_0\sim \mathcal{N}(\mathbf{D}_2\pmb {\mu}_0^{\mathbf{u}},\mathbf{D}_1^2),\quad \hat{\mathbf{v}}_0\in \mathbb{R}^d\quad \mathrm{and}\quad \mathbf{v}_0\sim \mathcal{N}(\pmb {\mu}_0^{\mathbf{u}},\pmb {\Lambda}_0),\quad \mathbf{v}_0\in \mathbb{R}^d$ + +we obtain: + +$$ +\left(\boldsymbol {\mu} _ {1} - \boldsymbol {\mu} _ {2}\right) ^ {T} \left(\boldsymbol {\mu} _ {1} - \boldsymbol {\mu} _ {2}\right) = \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T} \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) = \left(\left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T} \mathbf {D} _ {2} ^ {T} - \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T}\right) \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) +$$ + +$$ +\begin{array}{l} = \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} - 2 \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T} \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T} \mathbf {D} _ {2} ^ {T} \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathcal {F}} \\ = \sum_ {i = 1} ^ {d} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) _ {i} ^ {2} - 2 \sum_ {i = 1} ^ {d} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) _ {i} ^ {2} \mathbf {D} _ {2} ^ {(i)} + \sum_ {i = 1} ^ {d} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) _ {i} ^ {2} \left(\mathbf {D} _ {2} ^ {(i)}\right) ^ {2} \\ = \sum_ {i = 1} ^ {d} \left(\mathbf {D} _ {2} ^ {(i)} - 1\right) ^ {2} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) _ {i} ^ {2} \\ \end{array} +$$ + +$$ +\left(\boldsymbol {\mu} _ {1} - \boldsymbol {\mu} _ {2}\right) ^ {T} \left(\boldsymbol {\mu} _ {1} - \boldsymbol {\mu} _ {2}\right) = \sum_ {i = 1} ^ {d} \left(\mathbf {D} _ {2} ^ {(i)} - 1\right) ^ {2} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) _ {i} ^ {2} +$$ + +$$ +\sum_ {i} \left(\sqrt {\lambda_ {1} ^ {(i)}} - \sqrt {\lambda_ {2} ^ {(i)}}\right) ^ {2} = \sum_ {i} \left(\sqrt {\lambda_ {0} ^ {(i)}} - \sqrt {\left(\mathbf {D} _ {1} ^ {(i)}\right) ^ {2}}\right) ^ {2} +$$ + +$$ +W _ {2} \left(\mathbf {v} _ {0}, \hat {\mathbf {v}} _ {0}\right) = \sqrt {\sum_ {i = 1} ^ {d} \left(d _ {2} ^ {(i)} - 1\right) ^ {2} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) _ {i} ^ {2}} + \sum_ {i} \left(\sqrt {\lambda_ {0} ^ {(i)}} - \sqrt {\left(\mathbf {D} _ {1} ^ {(i)}\right) ^ {2}}\right) ^ {2} \tag {37} +$$ + +# D.2 Kullback-Leibler divergence: + +The Kullback-Leibler (KL) divergence between two Gaussian distributions with means $\mu_{1}$ and $\mu_{2}$ , and covariance matrices $\Sigma_{1}$ and $\Sigma_{2}$ , and the corresponding eigenvalues $\{\lambda_1^{(i)}\}_{i = 1}^d$ and $\{\lambda_2^{(i)}\}_{i = 1}^d$ is given by: + +$$ +D _ {\mathrm {K L}} \left(\mathcal {N} \left(\boldsymbol {\mu} _ {1}, \boldsymbol {\Sigma} _ {1}\right) \| \mathcal {N} \left(\boldsymbol {\mu} _ {2}, \boldsymbol {\Sigma} _ {2}\right)\right) = \frac {1}{2} \left(\log \frac {\left| \boldsymbol {\Sigma} _ {2} \right|}{\left| \boldsymbol {\Sigma} _ {1} \right|} - d + \operatorname {t r} \left(\boldsymbol {\Sigma} _ {2} ^ {- 1} \boldsymbol {\Sigma} _ {1}\right) + \left(\boldsymbol {\mu} _ {2} - \boldsymbol {\mu} _ {1}\right) ^ {T} \boldsymbol {\Sigma} _ {2} ^ {- 1} \left(\boldsymbol {\mu} _ {2} - \boldsymbol {\mu} _ {1}\right)\right) +$$ + +Given: + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} (\mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \mathbf {D} _ {1} ^ {2}), \quad \hat {\mathbf {v}} _ {0} \in \mathbb {R} ^ {d} \quad \text {a n d} \quad \mathbf {v} _ {0} \sim \mathcal {N} (\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \boldsymbol {\Lambda} _ {0}), \quad \mathbf {v} _ {0} \in \mathbb {R} ^ {d} +$$ + +The $KL$ divergence is given by: + +$$ +\begin{array}{l} D _ {\mathrm {K L}} \left(\mathbf {v} _ {0} \parallel \hat {\mathbf {v}} _ {0}\right) = D _ {\mathrm {K L}} \left(\mathcal {N} \left(\mathbb {E} \left[ \mathbf {v} _ {0} \right], \boldsymbol {\Sigma} _ {\mathbf {v} _ {0}}\right), \mathcal {N} \left(\mathbb {E} \left[ \hat {\mathbf {v}} _ {0} \right], \boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}}\right)\right) \\ = D _ {\mathrm {K L}} \left(\mathcal {N} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \boldsymbol {\Lambda} _ {0}\right), \mathcal {N} \left(\mathbf {D} _ {2}, \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \mathbf {D} _ {1} ^ {2}\right)\right) \\ \end{array} +$$ + +By decomposing the KL divergence elements, we obtain the following terms: + +- $|\mathbf{\Sigma}_2| = |\mathbf{D}_1^T\mathbf{D}_1| = |\mathbf{D}_1^2| = \prod_{i=1}^d \mathbf{D}_1^{(i)^2}$ + +- $|\Sigma_1| = \prod_{i=1}^d \lambda_0^{(i)}$ + +$\mathrm{tr}\left(\pmb{\Sigma}_2^{-1}\pmb{\Sigma}_1\right) = \sum_{i=1}^d \frac{\lambda_0^{(i)}}{\mathbf{D}_1^{(i)2}}$ + +$\bullet (\pmb {\mu}_2 - \pmb {\mu}_1)^T\pmb {\Sigma}_2^{-1}(\pmb {\mu}_2 - \pmb {\mu}_1) = (\mathbf{D}_2\pmb {\mu}_0^{\mathbf{u}} - \pmb {\mu}_0^{\mathbf{u}})^T (\mathbf{D}_1^2)^{-1}(\mathbf{D}_2\pmb {\mu}_0^{\mathbf{u}} - \pmb {\mu}_0^{\mathbf{u}}) =$ + +$$ += \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) ^ {T} \left(\mathbf {D} _ {2} ^ {T} - I\right) \left(\mathbf {D} _ {1} ^ {2}\right) ^ {- 1} \left(\mathbf {D} _ {2} - I\right) \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} = \sum_ {i = 1} ^ {d} \frac {\left(\mathbf {D} _ {2} ^ {(i)} - 1\right) ^ {2}}{\mathbf {D} _ {1} ^ {(i) 2}} \left(\boldsymbol {\mu} _ {0 i} ^ {\mathbf {u}}\right) ^ {2} +$$ + +Applying the substitution, the term results in: + +$$ +\begin{array}{l} D _ {\mathrm {K L}} \left(\mathcal {N} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \boldsymbol {\Lambda} _ {0}\right), \mathcal {N} \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \mathbf {D} _ {1} ^ {2}\right)\right) = \\ = \frac {1}{2} \left[ \sum_ {i = 1} ^ {d} \log {\bf D} _ {1} ^ {(i) ^ {2}} - \sum_ {i = 1} ^ {d} \log \lambda_ {0} ^ {(i)} - d + \sum_ {i = 1} ^ {d} \frac {\lambda_ {0} ^ {(i)}}{{\bf D} _ {1} ^ {(i) ^ {2}}} + \sum_ {i = 1} ^ {d} \frac {\left({\bf D} _ {2} ^ {(i)} - 1\right) ^ {2}}{{\bf D} _ {1} ^ {(i) ^ {2}}} (\pmb {\mu} _ {0 i} ^ {\bf u}) ^ {2} \right] \\ D _ {\mathrm {K L}} \left(\mathbf {v} _ {0} \parallel \hat {\mathbf {v}} _ {0}\right) = \frac {1}{2} \left[ \sum_ {i = 1} ^ {d} 2 \log \mathbf {D} _ {1} ^ {(i)} - \sum_ {i = 1} ^ {d} \log \lambda_ {0} ^ {(i)} - d + \sum_ {i = 1} ^ {d} \frac {\lambda_ {0} ^ {(i)} + \left(\mathbf {D} _ {2} ^ {(i)} - 1\right) ^ {2} \left(\boldsymbol {\mu} _ {0} ^ {\mathbf {u}}\right) _ {i} ^ {2}}{\mathbf {D} _ {1} ^ {(i) ^ {2}}} \right] (3 8) \\ \end{array} +$$ + +# E DDPM Formulation: + +Here, we apply an equivalent procedure to the DDPM scenario, as we did for the DDIM, as outlined in Theorem 3.6. + +# E.1 The Reverse Process in the Time Domain + +Using the procedure outline in [10], the diffusion process begins with $\mathbf{x}_T\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ , where $\mathbf{x}_T\in \mathbb{R}^d$ and progresses through an iterative denoising process described as follows: + +$$ +\mathbf {x} _ {t - 1} = \frac {1}{\sqrt {\alpha_ {t}}} \left(\mathbf {x} _ {t} - \frac {1 - \alpha_ {t}}{\sqrt {1 - \bar {\alpha} _ {t}}} \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} _ {t}, t\right)\right) + \sigma_ {t} \mathbf {z} _ {t} \quad \mathbf {z} _ {t} \sim \mathcal {N} (0, I) \tag {39} +$$ + +Where $\sigma_{t} = \sqrt{\frac{1 - \bar{\alpha}_{t - 1}}{1 - \bar{\alpha}_{t}}(1 - \alpha_{t})}$ . + +Given the marginal property from (2): + +$$ +\pmb {\epsilon} _ {\theta} (\mathbf {x} _ {t}, t) = \frac {\mathbf {x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \hat {\mathbf {x}} _ {0}}{\sqrt {1 - \bar {\alpha} _ {t}}} +$$ + +we can incorporate it into (39): + +$$ +{\bf x} _ {t - 1} = \frac {1}{\sqrt {\alpha_ {t}}} \left({\bf x} _ {t} - \frac {1 - \alpha_ {t}}{\sqrt {1 - \bar {\alpha} _ {t}}} \left(\frac {{\bf x} _ {t} - \sqrt {\bar {\alpha} _ {t}} \hat {{\bf x}} _ {0}}{\sqrt {1 - \bar {\alpha} _ {t}}}\right)\right) + \sigma_ {t} {\bf z} _ {t} +$$ + +$$ +{\bf x} _ {t - 1} = \frac {1}{\sqrt {\alpha_ {t}}} \left({\bf x} _ {t} \left(1 - \frac {1 - \alpha_ {t}}{1 - \bar {\alpha} _ {t}}\right) + \frac {(1 - \alpha_ {t}) \sqrt {\bar {\alpha} _ {t}}}{1 - \bar {\alpha} _ {t}} \hat {{\bf x}} _ {0}\right) + \sigma_ {t} {\bf z} _ {t} +$$ + +$$ +\mathbf {x} _ {t - 1} = \frac {1}{\sqrt {\alpha_ {t}}} \left(\frac {\alpha_ {t} - \bar {\alpha} _ {t}}{1 - \bar {\alpha} _ {t}}\right) \mathbf {x} _ {t} + \sqrt {\frac {\bar {\alpha} _ {t}}{\alpha_ {t}}} \frac {(1 - \alpha_ {t})}{1 - \bar {\alpha} _ {t}} \hat {\mathbf {x}} _ {0} + \sigma_ {t} \mathbf {z} _ {t} \tag {40} +$$ + +We denote the following, where the final term in each equation is represented by $\bar{\alpha}_t$ and $\bar{\alpha}_{t - 1}$ : + +$$ +a _ {t} = \frac {1}{\sqrt {\bar {\alpha} _ {t}}} \left(\frac {\alpha_ {t} - \bar {\alpha} _ {t}}{1 - \bar {\alpha} _ {t}}\right) = \frac {\sqrt {\bar {\alpha} _ {t}}}{1 - \bar {\alpha} _ {t}} \left[ \frac {1}{\sqrt {\bar {\alpha} _ {t - 1}}} - \sqrt {\bar {\alpha} _ {t - 1}} \right] +$$ + +$$ +b _ {t} = \sqrt {\frac {\bar {\alpha} _ {t}}{\alpha_ {t}}} \frac {(1 - \alpha_ {t})}{1 - \bar {\alpha} _ {t}} = \sqrt {\bar {\alpha} _ {t - 1}} \left(\frac {1 - \frac {\bar {\alpha} _ {t}}{\bar {\alpha} _ {t - 1}}}{1 - \bar {\alpha} _ {t}}\right) +$$ + +$$ +c _ {t} = \sigma_ {t} = \sqrt {\frac {1 - \bar {\alpha} _ {t - 1}}{1 - \bar {\alpha} _ {t}} (1 - \alpha_ {t})} = \sqrt {\frac {1 - \bar {\alpha} _ {t - 1}}{1 - \bar {\alpha} _ {t}} \left(1 - \frac {\bar {\alpha} _ {t}}{\bar {\alpha} _ {t - 1}}\right)} +$$ + +Therefore we get the following equation: + +$$ +\mathbf {x} _ {t - 1} = a _ {t} \mathbf {x} _ {t} + b _ {t} \hat {\mathbf {x}} _ {0} + c _ {t} \mathbf {z} _ {t} +$$ + +Using the result for the MAP estimator from (6): + +$$ +\mathbf {x} _ {0} ^ {*} = \left(\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + \left(1 - \bar {\alpha} _ {t}\right) \mathbf {I}\right) ^ {- 1} \left(\sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + \left(1 - \bar {\alpha} _ {t}\right) \boldsymbol {\mu} _ {0}\right) +$$ + +we get: + +$$ +\mathbf {x} _ {t - 1} = a _ {t} \mathbf {x} _ {t} + b _ {t} \left(\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I}\right) ^ {- 1} \left(\sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + (1 - \bar {\alpha} _ {t}) \boldsymbol {\mu} _ {0}\right) + c _ {t} \mathbf {z} _ {t} +$$ + +$$ +\mathbf {x} _ {t - 1} = \left(a _ {t} + b _ {t} (\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I}) ^ {- 1} \sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0}\right) \mathbf {x} _ {t} + \left(b _ {t} (\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I}) ^ {- 1} (1 - \bar {\alpha} _ {t}) \boldsymbol {\mu} _ {0}\right) + c _ {t} \mathbf {z} _ {t} +$$ + +Using the notation from Appendix B: + +$$ +\bar {\boldsymbol {\Sigma}} _ {0, t} = \bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} +$$ + +Thus, we can rewrite the equation as: + +$$ +\mathbf {x} _ {t - 1} = \left(a _ {t} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \left(\bar {\boldsymbol {\Sigma}} _ {0, t}\right) ^ {- 1} \boldsymbol {\Sigma} _ {0}\right) \mathbf {x} _ {t} + b _ {t} \left(1 - \bar {\alpha} _ {t}\right) \left(\bar {\boldsymbol {\Sigma}} _ {0, t}\right) ^ {- 1} \boldsymbol {\mu_ {0}} + c _ {t} \mathbf {z} _ {t} \tag {41} +$$ + +# E.2 Migrating to the Spectral Domain + +Next, we project both sides of the Eq. (41) onto the spectral domain. As previously described in Appendix C, this step relies on the eigenvalue decomposition of the covariance matrix $\pmb{\Sigma}_0 = \mathbf{U}\pmb{\Lambda}_0\mathbf{U}^{\mathrm{T}}$ where the columns of $\mathbf{U}$ represent the eigenvectors, and $\pmb{\Lambda}_0$ is the diagonal matrix containing the corresponding eigenvalues. + +$$ +\mathbf {U} ^ {\top} \mathbf {x} _ {t - 1} = \mathbf {U} ^ {\top} \left[ \left(a _ {t} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \left(\bar {\Sigma} _ {0, t}\right) ^ {- 1} \boldsymbol {\Sigma} _ {0}\right) \mathbf {x} _ {t} + b _ {t} \left(1 - \bar {\alpha} _ {t}\right) \left(\bar {\Sigma} _ {0, t}\right) ^ {- 1} \boldsymbol {\mu} _ {\mathbf {0}} + c _ {t} \mathbf {z} _ {t} \right] \tag {42} +$$ + +$$ +\mathbf {U} ^ {\top} \mathbf {x} _ {t - 1} = a _ {t} \mathbf {U} ^ {\top} \mathbf {x} _ {t} + \mathbf {U} ^ {\top} b _ {t} \sqrt {\bar {\alpha} _ {t}} (\bar {\Sigma} _ {0, t}) ^ {- 1} \mathbf {U} \mathbf {U} ^ {\top} \boldsymbol {\Sigma} _ {0} \mathbf {U} \mathbf {U} ^ {\top} \mathbf {x} _ {t} + \mathbf {U} ^ {\top} b _ {s} (1 - \bar {\alpha} _ {t}) (\bar {\Sigma} _ {0, s}) ^ {- 1} \mathbf {U} \mathbf {U} ^ {\top} \boldsymbol {\mu_ {0}} + c _ {t} \mathbf {U} ^ {\top} \mathbf {z} _ {t} +$$ + +By introducing the notations $\mathbf{v} = \mathbf{U}^{\top}\mathbf{x}$ and $\pmb{\mu}_0^{\mathrm{u}} = \mathbf{U}^{\top}\pmb{\mu}_0$ , we obtain: + +$$ +\mathbf {v} _ {t - 1} = a _ {t} \mathbf {v} _ {t} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \mathbf {U} ^ {\top} (\bar {\boldsymbol {\Sigma}} _ {0, t}) ^ {- 1} \mathbf {U} \mathbf {U} ^ {\top} \boldsymbol {\Sigma} _ {0} \mathbf {U} \mathbf {v} _ {t} + b _ {t} (1 - \bar {\alpha} _ {t}) \mathbf {U} ^ {\top} (\bar {\boldsymbol {\Sigma}} _ {0, t}) ^ {- 1} \mathbf {U} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + c _ {t} \mathbf {z} _ {t} ^ {\mathbf {u}} +$$ + +By applying eigenvalue decomposition, the symmetric matrix $\Sigma_0$ can be diagonalized using its eigenbasis $\mathbf{U}$ , where $\mathbf{U}^{-1} = \mathbf{U}^{\mathrm{T}}$ . Therefore, we obtain: + +$$ +\begin{array}{l} \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} = \left[ \bar {\alpha} _ {s} \boldsymbol {\Sigma} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} = \left[ \bar {\alpha} _ {s} \mathbf {U} \boldsymbol {\Lambda} _ {0} \mathbf {U} ^ {\mathsf {T}} + (1 - \bar {\alpha} _ {s}) \mathbf {U} \mathbf {I} \mathbf {U} ^ {\mathsf {T}} \right] ^ {- 1} = \left[ \mathbf {U} \left(\bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I}\right) \mathbf {U} ^ {\mathsf {T}} \right] ^ {- 1} \\ \bar {\boldsymbol {\Sigma}} _ {0, s} ^ {- 1} = \mathbf {U} \left[ \bar {\alpha} _ {s} \mathbf {\Lambda} _ {0} + (1 - \bar {\alpha} _ {s}) \mathbf {I} \right] ^ {- 1} \mathbf {U} ^ {\mathsf {T}} \\ \end{array} +$$ + +Incorporating these elements into the main equation, we obtain: + +$$ +\begin{array}{l} \mathbf {v} _ {t - 1} = a _ {t} \mathbf {v} _ {t} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \left[ \bar {\alpha} _ {t} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \pmb {\Lambda} _ {0} \mathbf {v} _ {t} + b _ {t} (1 - \bar {\alpha} _ {t}) \left[ \bar {\alpha} _ {t} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \pmb {\mu} _ {0} ^ {\mathbf {u}} + c _ {t} \mathbf {z} _ {t} ^ {\mathbf {u}} \\ \mathbf {v} _ {t - 1} = \left[ a _ {t} \mathbf {I} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \left[ \bar {\alpha} _ {t} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right] \mathbf {v} _ {t} + b _ {t} (1 - \bar {\alpha} _ {t}) \left[ \bar {\alpha} _ {t} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + c _ {t} \mathbf {z} _ {t} ^ {\mathbf {u}} \\ \end{array} +$$ + +Including those elements in the main equation: + +$$ +\begin{array}{l} \mathbf {v} _ {t - 1} = a _ {t} \mathbf {v} _ {t} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \left[ \bar {\alpha} _ {t} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \pmb {\Lambda} _ {0} \mathbf {v} _ {t} + b _ {t} (1 - \bar {\alpha} _ {t}) \left[ \bar {\alpha} _ {t} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \pmb {\mu} _ {0} ^ {\mathbf {u}} + c _ {t} \mathbf {z} _ {t} ^ {\mathbf {u}} \\ \mathbf {v} _ {t - 1} = \left[ a _ {t} \mathbf {I} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \left[ \bar {\alpha} _ {t} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right] \mathbf {v} _ {t} + b _ {t} (1 - \bar {\alpha} _ {t}) \left[ \bar {\alpha} _ {t} \boldsymbol {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + c _ {t} \mathbf {z} _ {t} ^ {\mathbf {u}} \\ \end{array} +$$ + +We will denote the following: + +$$ +\mathbf {G} (t) = \left[ a _ {t} + b _ {t} \sqrt {\bar {\alpha} _ {t}} \left[ \bar {\alpha} _ {t} \mathbf {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} \mathbf {\Lambda} _ {0} \right] +$$ + +$$ +\mathbf {M} (t) = b _ {t} \left(1 - \bar {\alpha} _ {t}\right) \left[ \bar {\alpha} _ {t} \pmb {\Lambda} _ {0} + (1 - \bar {\alpha} _ {t}) \mathbf {I} \right] ^ {- 1} +$$ + +and get: + +$$ +\mathbf {v} _ {t - 1} = \mathbf {G} (t) \mathbf {v} _ {t} + \mathbf {M} (t) \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + c _ {t} \mathbf {z} _ {t} ^ {\mathbf {u}} +$$ + +We can then recursively obtain $\mathbf{v}_l$ for a general $l$ : + +$$ +\mathbf {v} _ {l} = \left[ \prod_ {t ^ {\prime} = l + 1} ^ {T} \mathbf {G} \left(t ^ {\prime}\right) \right] \mathbf {v} _ {t} + \left[ \sum_ {i = l + 1} ^ {T} \left(\prod_ {j = l + 1} ^ {i - 1} \mathbf {G} (j)\right) \mathbf {M} (i) \right] \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + \left[ \sum_ {i = l + 1} ^ {T} \left(\prod_ {j = l + 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \right] +$$ + +where the process iterates over all the steps: $[1,\dots ,T]$ + +$$ +\hat {\mathbf {v}} _ {0} = \left[ \prod_ {t ^ {\prime} = 1} ^ {T} \mathbf {G} \left(t ^ {\prime}\right) \right] \mathbf {v} _ {t} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \right] + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) \mathbf {M} (i) \right] \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} +$$ + +We will denote the following: + +$$ +\mathbf {D} _ {1} = \prod_ {s = 1} ^ {S} \mathbf {G} (s) +$$ + +$$ +\mathbf {D} _ {2} = \sum_ {i = 1} ^ {S} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) \mathbf {M} (i) +$$ + +Substitute $\mathbf{D}_1$ and $\mathbf{D}_2$ into the last equation, we get: + +$$ +\hat {\mathbf {v}} _ {0} = \mathbf {D} _ {1} \mathbf {v} _ {t} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \right] + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \tag {43} +$$ + +The resulting vector from Equation 43 is a linear combination of Gaussian signals, therefore it also follows a Gaussian distribution. We now aim to determine the mean vector and the covariance matrix of that distribution. + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} (\mathbb {E} [ \hat {\mathbf {v}} _ {0} ], \pmb {\Sigma} _ {\hat {\mathbf {v}} _ {0}}), \quad \hat {\mathbf {v}} _ {0} \in \mathbb {R} ^ {d} +$$ + +$$ +\mathbf {x} _ {T} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}) \quad \text {a n d} \quad \mathbf {z} _ {i} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}), \quad \forall i +$$ + +Mean: + +$$ +\mathbb {E} \left[ \hat {\mathbf {v}} _ {0} \right] = \mathbb {E} \left[ \mathbf {D} _ {1} \mathbf {v} _ {t} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \mathbf {I} \right] + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \right] = \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} +$$ + +$$ +\mathbb {E} \left[ \hat {\mathbf {v}} _ {0} \right] = \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} +$$ + +Covariance: + +$$ +\boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}} = \mathbb {E} \left[ \right.\left(\mathbf {D} _ {1} \mathbf {v} _ {t} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \right] - \mathbb {E} \left[ \mathbf {D} _ {1} \mathbf {v} _ {t} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \boldsymbol {\mathbf {z}} _ {i} ^ {\mathbf {u}} \right]\right]\right) +$$ + +$$ +\left. \right.\left(\mathbf {D} _ {1} \mathbf {v} _ {t} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \right] - \mathbb {E} \left[ \mathbf {D} _ {1} \mathbf {v} _ {t} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \boldsymbol {\mathbf {z}} _ {i} ^ {\mathbf {u}} \right]\right]\right) ^ {T} ] +$$ + +$$ +\boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}} = \mathbb {E} \left[ \left(\mathbf {D} _ {1} \mathbf {v} _ {t} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \right]\right) \left(\mathbf {D} _ {1} \mathbf {v} _ {t} + \left[ \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {z} _ {i} ^ {\mathbf {u}} \right]\right) ^ {T} \right] +$$ + +Lemma E.1. Let $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n$ be $n$ independent Gaussian random vectors with mean $\mathbb{E}[\mathbf{x}_i] = \mathbf{0}$ and covariance matrices $\operatorname{Cov}(\mathbf{x}_i) = \boldsymbol{\Sigma}_i$ , for $i = 1, 2, \ldots, n$ . Let the linear combination be defined as: + +$$ +\mathbf {y} = \sum_ {i = 1} ^ {n} a _ {i} \mathbf {x} _ {i}, +$$ + +where $a_1, a_2, \ldots, a_n$ are constants. The covariance matrix of $\mathbf{y}$ , denoted as $\operatorname{Cov}(\mathbf{y})$ , is given by: + +$$ +C o v (\mathbf {y}) = \sum_ {i = 1} ^ {n} a _ {i} ^ {2} \boldsymbol {\Sigma} _ {i}. +$$ + +Applying the result of Lemma E.1 to the expression in Equation 44, where $\mathbf{x}_T\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ and $\mathbf{z}_i\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ are independent Gaussian noises for all $i$ , we have: + +$$ +\mathbb {E} \left[ \mathbf {v} _ {t} \mathbf {v} _ {t} ^ {T} \right] = \mathbb {E} \left[ \mathbf {U} ^ {\top} \mathbf {x} _ {T} \mathbf {x} _ {T} ^ {T} \mathbf {U} \right] = \mathbf {U} ^ {\top} \mathbb {E} \left[ \mathbf {x} _ {T} \mathbf {x} _ {T} ^ {T} \right] \mathbf {U} = \mathbf {I} +$$ + +$$ +\mathbb {E} \left[ \mathbf {z} _ {i} ^ {\mathbf {u}} \mathbf {z} _ {i} ^ {\mathbf {u} T} \right] = \mathbf {I} +$$ + +Thus, the covariance is given by: + +$$ +\boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}} = \mathbb {E} \left[ \hat {\mathbf {v}} _ {0} \hat {\mathbf {v}} _ {0} ^ {T} \right] = \left[ \mathbf {D} _ {1} \right] ^ {2} + \sum_ {i = 1} ^ {T} \left[ \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {I} \right] ^ {2} +$$ + +$$ +\boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}} = \left[ \mathbf {D} _ {1} \right] ^ {2} + \sum_ {i = 1} ^ {T} \left[ \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) c _ {i} \mathbf {I} \right] ^ {2} +$$ + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} \left(\mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \mathbf {D} _ {1} ^ {2} + \sum_ {i = 1} ^ {T} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} ^ {2} (j)\right) c _ {i} ^ {2} \mathbf {I}\right) \tag {44} +$$ + +As discussed in Appendix C, for a data distribution $\mathbf{x}_0\sim \mathcal{N}(\pmb {\mu}_0,\pmb {\Sigma}_0)$ , where $\mathbf{x}_0\in \mathbb{R}^d$ , the first and second moments in the frequency domain are given by: + +$$ +\mathbb {E} \left[ \mathbf {v} _ {0} \right] = \mathbf {F} \mathbb {E} \left[ \mathbf {x} _ {0} \right] = \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} +$$ + +$$ +\boldsymbol {\Sigma} _ {\mathbf {x} _ {0} ^ {T}} = \boldsymbol {\Lambda} _ {0} +$$ + +$$ +\mathbf {v} _ {0} \sim \mathcal {N} \left(\boldsymbol {\mu} _ {0} ^ {\mathrm {u}}, \boldsymbol {\Lambda} _ {0}\right) \tag {45} +$$ + +# F Variance preserving and Variance exploding theoretical analysis + +The paper [28] distinguishes between two sampling methods: Variance Preserving (VP) and Variance Exploding (VE). The primary difference lies in how variance evolves during the process. While VP maintains a fixed variance, VE results in an exploding variance as $t \to T$ . Here, we focus on comparing these approaches within the context of our spectral noise schedule derivation for the DDIM procedure [10]. Throughout this paper, we described our methods based on the Variance Preserving (VP) formulation, given by: + +$$ +p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {0}\right) = \mathcal {N} \sim \left(\sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0}, \sqrt {1 - \bar {\alpha} _ {t}} \mathbf {I}\right) \tag {46} +$$ + +where the only hyperparameters are the noise schedule parameters: $\{\bar{\alpha}_s\}_{s=0}^S$ where $\bar{\alpha}_s \in (0,1]$ . + +In contrast, under the Variance Exploding (VE) method, the hyperparameters are given by $\sigma_t$ where $\sigma_t \in [0, \infty)$ , and the marginal distribution takes the form: + +$$ +p \left(\bar {\mathbf {x}} _ {t} \mid \mathbf {x} _ {0}\right) = \mathcal {N} \sim \left(\bar {\mathbf {x}} _ {0}, \sigma_ {t} ^ {2} \mathbf {I}\right) \tag {47} +$$ + +We used the notation $\bar{\mathbf{x}}_t$ to distinguish it from $\mathbf{x}_t$ , except in the special case where $\mathbf{x}_0 = \bar{\mathbf{x}}_0$ . Applying the reparameterization trick, we obtain: + +$$ +\operatorname {V P}: \mathbf {x} _ {t} = \sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0} + \sqrt {1 - \bar {\alpha} _ {t}} \boldsymbol {\epsilon} \quad \boldsymbol {\epsilon} \sim \mathcal {N} (\mathbf {0}, \boldsymbol {I}) \tag {48} +$$ + +$$ +\mathrm {V E}: \bar {\mathbf {x}} _ {t} = \bar {\mathbf {x}} _ {0} + \sigma_ {t} \epsilon \quad \epsilon \sim \mathcal {N} (\mathbf {0}, I) \tag {49} +$$ + +A key relationship between the VP and VE formulations, as derived in [14], is given by: + +$$ +\bar {\mathbf {x}} _ {t} = \frac {\mathbf {x} _ {t}}{\sqrt {1 + \sigma_ {t} ^ {2}}} \quad \bar {\alpha} _ {t} = \frac {1}{\left(1 + \sigma_ {t} ^ {2}\right)} \tag {50} +$$ + +# F.1 Determining the Optimal Denoiser: + +Following the derivation in A, we obtained the expression for the optimal denoiser in the Gaussian case under the Variance Preserving (VP) formulation: + +$$ +\hat {\mathbf {x}} _ {0} ^ {\text {w i e n e r , V P}} = \left(\bar {\alpha} _ {t} \boldsymbol {\Sigma} _ {0} + I \left(1 - \bar {\alpha} _ {t}\right)\right) ^ {- 1} \left(\sqrt {\bar {\alpha} _ {t}} \boldsymbol {\Sigma} _ {0} \mathbf {x} _ {t} + \left(1 - \bar {\alpha} _ {t}\right) \boldsymbol {\mu} _ {0}\right). +$$ + +Leveraging a similar approach, we derive the corresponding expression for the Variance Exploding (VE) scenario: + +$$ +\hat {\mathbf {x}} _ {0} ^ {\text {w i e n e r , V E}} = \left(\boldsymbol {\Sigma} _ {0} + I \sigma_ {t} ^ {2}\right) ^ {- 1} \left(\boldsymbol {\Sigma} _ {0} \bar {\mathbf {x}} _ {t} + \sigma_ {t} ^ {2} \boldsymbol {\mu} _ {0}\right). \tag {51} +$$ + +# F.2 Evaluating the Inference Process in the time Domain: + +This part can be performed using two equivalent methods: + +# Method 1: + +The ODE for the VE scenario in DDIM, as outlined in [27], is given by: + +$$ +d \bar {x} = - \frac {1}{2} g (t) ^ {2} \nabla_ {\bar {x}} \log p _ {t} (\bar {x}) d t \quad g (t) = \sqrt {\frac {d \sigma^ {2} (t)}{d t}} \tag {52} +$$ + +Additionally, the score expression and the marginal equation are also derived in [27] as follows: + +$$ +\nabla_ {\bar {x}} \log p _ {t} (\bar {x}) = - \frac {\epsilon_ {\theta} ^ {(t)}}{\sigma (t)} \tag {53} +$$ + +$$ +\bar {x} _ {t} = \bar {x} _ {0} + \sigma_ {t} \epsilon \quad w h e r e \quad \epsilon \sim \mathcal {N} (\mathbf {0}, \mathbf {I}) \tag {54} +$$ + +Substituting Equation 53 into Equation 52: + +$$ +d \bar {x} = \frac {1}{2} \frac {d \sigma^ {2} (t)}{d t} \frac {\epsilon_ {\theta} ^ {(t)}}{\sigma (t)} d t +$$ + +$$ +\frac {d \bar {x}}{d t} = \frac {d \sigma (t)}{d t} \epsilon_ {\theta} ^ {(t)} \tag {55} +$$ + +Substituting Equation 54 into Equation 55, we obtain: + +$$ +\bar {x} _ {t} - \bar {x} _ {t - 1} = (\sigma_ {t} - \sigma_ {t - 1}) \frac {\bar {x} _ {t} - \bar {x} _ {0}}{\sigma_ {t}} +$$ + +$$ +\bar {x} _ {t - 1} = \bar {x} _ {t} + \left(\frac {\sigma_ {t - 1}}{\sigma_ {t}} - 1\right) \bar {x} _ {t} - \bar {x} _ {0} +$$ + +$$ +\bar {x} _ {t - 1} = \frac {\sigma_ {t - 1}}{\sigma_ {t}} \bar {x} _ {t} + \left(1 - \frac {\sigma_ {t - 1}}{\sigma_ {t}}\right) \bar {x} _ {0} \tag {56} +$$ + +# Method 2: + +given the inference process in the VP formulation [27]: + +$$ +\mathbf {x} _ {s - 1} (\eta = 0) = \frac {\sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}} \mathbf {x} _ {s} + \left[ \sqrt {\bar {\alpha} _ {s - 1}} - \frac {\sqrt {\bar {\alpha} _ {s}} \sqrt {1 - \bar {\alpha} _ {s - 1}}}{\sqrt {1 - \bar {\alpha} _ {s}}} \right] \hat {\mathbf {x}} _ {0} +$$ + +By leveraging the connections in Equation 50 we can derive the following relationship between the two successive steps, $\mathbf{x}_{s - 1}$ and $\mathbf{x}_s$ , in the inference process: + +$$ +\bar {x} _ {s - 1} = \sqrt {\frac {\sigma_ {s - 1} ^ {2}}{\sigma_ {s} ^ {2}}} \bar {x} _ {s} + \left(1 - \sqrt {\frac {\sigma_ {s - 1} ^ {2}}{\sigma_ {s} ^ {2}}}\right) x _ {0} \tag {57} +$$ + +The Resulted expressions in (57) and (56) are identical. + +By defining the following terms: + +$$ +\bar {a} _ {s} = \frac {\sigma_ {s - 1}}{\sigma_ {s}} +$$ + +$$ +\bar {b} _ {s} = 1 - \frac {\sigma_ {s - 1}}{\sigma_ {s}} = 1 - \bar {a} _ {s} +$$ + +we can express the relationship between $\mathbf{x}_{s - 1}$ and $\mathbf{x}_s$ as: + +$$ +\mathbf {x} _ {s - 1} = a _ {s} \mathbf {x} _ {s} + b _ {s} \hat {\mathbf {x}} _ {0} \tag {58} +$$ + +# F.3 Evaluating the Inference Process in the Spectral Domain + +Since a similar expression to Equation 58 has already been discussed in Appendix C, we can now describe the inference process in the spectral domain as follows: + +$$ +\mathbf {v} _ {s - 1} = \mathbf {G} (s) \mathbf {v} _ {s} + \mathbf {M} (s) \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \tag {59} +$$ + +where: + +$$ +\mathbf {G} (s) = \left[ \bar {a} _ {s} + \bar {b} _ {s} \left[ \boldsymbol {\Lambda} _ {0} + \mathbf {I} \sigma_ {s} ^ {2} \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right] +$$ + +$$ +\mathbf {M} (s) = \bar {b} _ {s} \left[ \boldsymbol {\Lambda} _ {0} + \mathbf {I} \sigma_ {s} ^ {2} \right] ^ {- 1} \sigma_ {s} ^ {2} +$$ + +Following this and in alignment with the same methodology described in Appendix C we obtain: + +$$ +\hat {\mathbf {v}} _ {0} = \mathbf {D} _ {1} \mathbf {v} _ {s} + \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \tag {60} +$$ + +$$ +\hat {\mathbf {v}} _ {0} \sim \mathcal {N} (\mathbb {E} [ \hat {\mathbf {v}} _ {0} ], \pmb {\Sigma} _ {\hat {\mathbf {v}} _ {0}}), \quad \hat {\mathbf {v}} _ {0} \in \mathbb {R} ^ {d} +$$ + +$$ +\mathbb {E} \left[ \hat {\mathbf {v}} _ {0} \right] = \mathbf {D} _ {2} \boldsymbol {\mu} _ {0} ^ {\mathbf {u}}, \quad \boldsymbol {\Sigma} _ {\hat {\mathbf {v}} _ {0}} = \mathbf {D} _ {1} \mathbf {D} _ {1} ^ {T} = \mathbf {D} _ {1} ^ {2} +$$ + +$$ +\mathbf {D} _ {1} = \prod_ {s = 1} ^ {S} \mathbf {G} (s) = \prod_ {s = 1} ^ {S} \left[ a _ {s} + b _ {s} \sqrt {\bar {\alpha} _ {s}} \left[ \bar {\alpha} _ {s} \boldsymbol {\Lambda} _ {0} + \mathbf {I} \left(1 - \bar {\alpha} _ {s}\right) \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right] \tag {61} +$$ + +$$ +\begin{array}{l} \mathbf {D} _ {2} = \sum_ {i = 1} ^ {S} \left(\prod_ {j = 1} ^ {i - 1} \mathbf {G} (j)\right) \mathbf {M} (i) (62) \\ = \sum_ {i = 1} ^ {S} \left[ \left(\prod_ {j = 1} ^ {i - 1} \left[ a _ {j} + b _ {j} \sqrt {\bar {\alpha} _ {j}} \left[ \bar {\alpha} _ {j} \boldsymbol {\Lambda} _ {0} + \mathbf {I} \left(1 - \bar {\alpha} _ {j}\right) \right] ^ {- 1} \boldsymbol {\Lambda} _ {0} \right]\right) b _ {i} \left(1 - \bar {\alpha} _ {i}\right) \left[ \bar {\alpha} _ {i} \boldsymbol {\Lambda} _ {0} + \mathbf {I} \left(1 - \bar {\alpha} _ {i}\right) \right] ^ {- 1} \right] (63) \\ \end{array} +$$ + +# G Clarifications and Validations: + +# G.1 Method Evaluations: Temporal and Spectral Domains + +We evaluated the compatibility between the diffusion process in the time domain, using the DDIM method [10], and its counterpart derived from Equation 10 in the spectral domain. Using an artificial covariance matrix, $\Sigma_0$ , with parameters $l = 0.1$ and $d = 50$ from 6.1, we estimated the covariance of 50,000 signals that were denoised according to Equation 4, using the optimal denoiser from Equation 6, and computed their eigenvalues, denoted as $\{\lambda_i^{time}\}_{i=1}^d$ . In the spectral domain, we applied the formulation from Equation 10 for deriving $D_1^2$ and extracted $\{\lambda_i^{spectral}\}_{i=1}^d$ from its diagonal elements. The results are illustrated in Figure 5. + +![](images/174c5aa1f16e11fcbdc578c5f576e8bc1cba1cdf76bf8cef337948bb350f7eda.jpg) +(a) A comparison of the eigenvalues obtained from each method + +![](images/bb97b891c38a43eeac53f56634ad4e6fa9d731de3e6741058c3fe5f83250c06e.jpg) +(b) The absolute relative error between the estimated eigenvalues and the data eigenvalues. +Figure 5: Figure 5a compares the eigenvalues derived from the spectral and time domain formulations of the DDIM method [10]. The dataset, described in 6.1 with $l = 0.1$ and $d = 50$ , is used for both approaches, involving 112 diffusion steps and following the linear noise schedule proposed in [27]. Furthermore, Figure 5b illustrates the absolute error between the estimated and original eigenvalues for both methods. + +Figure 5a shows that the derived eigenvalues from both procedures align with each other, thus verifying the transition from the time to spectral domain. However, they are not necessarily identical to the properties of the original dataset. Notably, as the number of steps increases, both processes converge toward the original dataset values. Figure 5b allows for an examination of the absolute error in each process relative to the characteristics of the original dataset. It is evident that while the spectral equation exhibits stable behavior, the time-domain equation displays fluctuations that depend on the number of sampled signals. As the number of samples increases, these fluctuations diminish. + +![](images/3638db3ef3dd0b4d01eb1e89a4a58b20fcf3b9a715c22f3acb35412c169245a0.jpg) +Figure 6: Comparison between the closed-form solution from AYS [26] and our numerical solution for the simplified case where $\mathbf{x}_0\sim \mathcal{N}(\mathbf{0},C^2\mathbf{I})$ with $C = [0.1,0.5,1]$ . + +# G.2 Method Evaluation: Comparison with Prior Works + +We also compare our optimal solution with those from previous works. Specifically, The authors in [26] derive a closed-form expression for the optimal noise schedule under a simplified case, where the initial distribution is an isotropic Gaussian with a standard deviation of $C$ , i.e $\mathbf{x}_0 \sim \mathcal{N}(\mathbf{0}, C^2\mathbf{I})$ . To enable a proper comparison, we frame our optimization problem using the Kullback-Leibler divergence $D_{\mathrm{KL}}$ loss (4.2) as done in [26]. + +Figure 6 compares our optimal solution, obtained by numerically solving Equation (14), with the closed-form solution from [26]. It can be observed that both methods align for arbitrary values of $C$ . Notably, for $C = 1$ , both noise schedules converge exactly to the Cosine $(0,1,1)$ noise schedule, which was originally chosen heuristically [23]. + +# G.3 Constraint Omission and Different Types of Initializations + +We explore the influence of the initializations and the inequality constraints in the optimization problem. Figures 7 and 8 illustrate the evolution of the noise schedule parameter, $\{\bar{\alpha}_s\}_{s=0}^S$ , during the optimization process for two different initializations: uniformly random and linearly decreasing schedules, respectively. In both cases, the diffusion process consists of 28 steps, and the Wasserstein-2 distance is used. Each scenario was conducted twice: once with the inequality constraints from 4 and once without. The results are plotted at 15 evenly spaced intervals throughout the process to avoid presenting each individual optimization step. + +![](images/25284ae25ad68f797301283d46289035698705f14c5fa12d127f88f90557b98e.jpg) +(a) Optimization with inequality constraints. + +![](images/70d4f2bda99b4b6a7bb52656429601e5e8f3091be2129b260d6fb4699cb1cf33.jpg) +(b) Optimization without inequality constraints. +Figure 7: A Comparison of the noise schedule parameters, $\{\bar{\alpha}_s\}_{s=0}^S$ , during the optimization process. The optimization was conducted over 28 diffusion steps, with a uniformly random initialization. Figure 7a shows the results with inequality constraints, and Figure 7b presents those without. + +![](images/e3a016bfa0ed3b3b540f44faf99aa0393cc45e5a1cb595ae9240a087beb2220b.jpg) +(a) Optimization with inequality constraints. + +![](images/a8c18b45a36bc88b4d1fdc01adde97d9350829150c0b7135051bf72949b55933.jpg) +(b) Optimization without inequality constraints. +Figure 8: A Comparison of the noise schedule parameters, $\{\bar{\alpha}_s\}_{s=0}^S$ , during the optimization process. The optimization was conducted over 28 diffusion steps, with a Linearly decreasing initialization. Figure 8a shows the results with inequality constraints, and Figure 8b presents those without. + +The results reveal that the optimized schedule is consistent across both initializations and independent of the inequality constraints. This suggests that known characteristics of noise schedules, such as monotonicity, naturally emerge from the problem's formulation itself, even without an explicit demand for inequality constraints. Similar consistency is observed for other initializations, including linear and cosine schedules, demonstrating the stability of the optimization procedure. + +# H Supplementary Experiments for Synthetic Gaussian Distribution: + +In the following sections, we demonstrate the received spectral recommendations for various alternative selections, applied to the matrix $\Sigma_0$ and the vector $\mu_0$ , differing from those presented in Section 6.1. Additionally, we present the solutions obtained for defining the Wasserstein-2 distance and the KL-divergence. Through this, we aim to provide a broader perspective on the behavior and applicability of the proposed approach. + +Figure 9 visualizes the covariance matrices $\Sigma_0$ and $\Lambda_0$ as discussed in Sec. 6.1. + +![](images/ca6a1cf61e7a846bb015993c10f9abdea8dbef873d79ebd52666dc176cdcf6d1.jpg) +(a) $\Sigma_0$ + +![](images/2976033a7c18ef52b933320f3c74968f800fd184a2b3f1289cafd03ddf573164.jpg) +(b) $tr(\Lambda_0)$ +Figure 9: Visualization of $\Sigma_0$ and the trace of $\Lambda_0$ for $d = 50$ and $l = 0.1$ . The covariance matrix $\Sigma_0$ is circulant (9a), while $\Lambda_0$ is diagonal with symmetric diagonal elements (9b). + +# H.1 Wasserstein-2 distance + +Figure 10 presents the resulting noise schedule based on the Wasserstein-2 distance. + +![](images/a55eba05e66eb9c7dcee44dc8c9be3ff6360e918ac001b52bb87a9c11dea429b.jpg) +(a) +Figure 10: Figures 10a and 10b compare the optimized noise schedules from Sec. 6.1, using $d = 50$ , $l = 0.1$ , and $\mu_0 = 0.05$ , with heuristic schedules for 112 diffusion steps, where the optimization is based on the Wasserstein-2 distance. Figure 10a examines the spectral schedule alongside the Linear [10], EDM [13] and Cosine-based schedules, including Cosine ( $s = 0$ , $e = 1$ , $\tau = 1$ ) from [23, 5]. Likewise, Figure 10b compares it to Sigmoid-based schedules [11]. The parametric estimations for the Cosine and Sigmoid functions are highlighted in red. + +![](images/d385dc1f93fc268de6cab7837f0dc789e611d0a3c29161d58641194f063be09d.jpg) +(b) + +# H.2 KL-Divergence + +![](images/acc90d294a590740c70a9ba88f77f4060d623aa59d53060c5880ee3ee50d5bdc.jpg) +Figure 11 presents the resulting noise schedule based on the KL-Divergence. +(a) + +![](images/c677d07d923e63d76135edb449d641c9d36616cbc51ff2f98d8f2e4c363b8ce4.jpg) +(b) +Figure 11: Figures 11a and 11b compare the optimized noise schedules from Sec. 6.1, using $d = 50$ , $l = 0.1$ , and $\mu_0 = 0.05$ , with heuristic schedules for 112 diffusion steps, where the optimization is based on the KL divergence. Figure 11a examines the spectral schedule alongside the Linear [10], EDM [13] and Cosine-based schedules, including Cosine ( $s = 0$ , $e = 1$ , $\tau = 1$ ) from [23, 5]. Likewise, Figure 10b compares it to Sigmoid-based schedules [11]. The parametric estimations for the Cosine and Sigmoid functions are highlighted in red. + +Notably, under the same conditions, the $KL$ divergence results in a more concave spectral recommendation compared to the Wasserstein-2 distance. + +# H.3 Variations in Covariance Matrices and Mean Configurations + +In 6.1, we designed a specific covariance matrix $\Sigma_0$ and a mean vector $\mu_0$ with the intention of resembling characteristics observed in real signals, such as a centered signal with $\mu_0 \approx 0$ . However, the optimization process is not restricted to these particular choices and can be generalized to accommodate various alternative decisions. Figure 12 displays different covariance matrices along with their corresponding $\mu_0$ vectors, followed by the resulting spectral schedules computed using the Wasserstein-2 distance for 60 diffusion steps. + +![](images/5f92a2918b57961923fdf52d04ada03a9023c1d987c49fb09d656cc0f378dcb4.jpg) + +![](images/843604cd98c8edeb928a18ce15ad06b30444d11aa4e9f9668a058aba1267e1d0.jpg) +(b) + +![](images/a7b041f0b884d87305d3e8a780beb2464e4197bd3d61f93f009c502608203992.jpg) +(c) + +![](images/5c163f120543771b1218f2bf84982ac889ad4a4379209a934ba0d525475cb08c.jpg) +(a) + +![](images/b7327ef1eda192d870876c9175e76e98c6e980d2b47d3ea87396443ba0bf0d77.jpg) +(e) + +![](images/8e309ccfaf88be8ce29d195a0ceddcc770b0f9eb210eb9abcecf97d73b0fbc2f.jpg) +(f) + +![](images/49134751d9d17bb80928b3ef026bbf1530db937ab801dcec76f74da2d63ae32a.jpg) +(d) + +![](images/c6a1fc0ee98bebd12de1edb7e8dd95a25801256959741788dcab169405562266.jpg) +(h) + +![](images/4952388a94c0c092d0d0d47fbec348550a2f0db92f97dc987fe79bf923409a52.jpg) +(i) + +![](images/b614a3ff838777f0619e95cf42b578f148a3a40ef216aecbc9ad9f49800567a7.jpg) +(g) +(j) + +![](images/b2648e4ca131425ad4274be38f2a127c1332b6e959e6e292d219fae5ea7a3548.jpg) +(k) +Figure 12: Various covariance matrices (first column) along with their eigenvalues (second column) and the corresponding optimized noise schedules (third column). The first row presents the matrix from 6.1 with $l = 0.05$ , $\pmb{\mu}_0 = 0.01 \cdot \mathbf{1}_d$ . The second row shows the same matrix scaled by a factor of 20 while keeping the same $\pmb{\mu}_0$ . The third row displays a covariance matrix incorporating a Cosine function in the first row $a$ of the circulant covariance, with $\pmb{\mu}_0 = 0.3 \cdot \mathbf{1}_d$ . The fourth row features a circulant matrix derived from a sinusoidal signal with a frequency of 1000 in the first row $a$ of the circulant covariance matrix, scaled by 0.01, and with $\pmb{\mu}_0 = 0.1 \cdot \mathbf{1}_d$ . + +![](images/3f3ce387d8bbea883b247a03694ec09c1097aef532c23d74b95ab515a142543a.jpg) +(1) + +While we cannot cover all possible choices for the covariance matrix $\pmb{\Sigma}_0$ and the vector $\pmb{\mu}_0$ , we aim to provide a broader perspective on the KL-divergence loss. Figure 13 illustrates a circulant covariance matrix whose first row is derived from a sinusoidal signal with a frequency of $1000\mathrm{Hz}$ , along with the corresponding spectral recommendation based on KL-divergence, where $\pmb{\mu}_0 = 0.1\cdot \mathbf{1}_d$ . + +![](images/043031a2b6d1ce71000f8e5ec2765a621f232406482bb34908d3e54da1c5f3bd.jpg) + +![](images/5b87a9f3bef28ca99c24e95fb6266a1986acd169c8f1c8d67472e05fee975108.jpg) + +![](images/156b00c01df9749fe25fbb0d07d27981ed9cf83d31639c5ea8774e276f851f40.jpg) +(a) $\Sigma_0$ +(c) +Figure 13: Figure 13a shows the circulant covariance matrix, $\Sigma_0$ , whose first row is derived from a sinusoidal signal with a frequency of $1000\mathrm{Hz}$ . Figure 13b displays the trace of the corresponding $\Lambda_0$ matrix. Figures 13c and 13d compare the spectral recommendations for $d = 50$ , 112 diffusion steps, using the KL-divergence, with various noise schedule heuristics including Cosine and Sigmoid, respectively. The parametric estimations for the Cosine and Sigmoid are highlighted in red. + +![](images/286d34a10bfb71da8ed84cdd8295e7485cff510f22cc2a7bcd404852f6098d84.jpg) +(b) $tr(\Lambda_0)$ +(d) + +The results above show that modifying the dataset properties, such as the covariance matrix and expectation, along with altering the loss function, leads to noise schedules with a similar overall structure but varying details. In Appendix K, we explore the connection between the dataset properties, the loss function, and the resulting noise schedules. + +# I Supplementary Experiments for Empirical Gaussian Distribution + +We present additional details on the CIFAR-10, Gaussian MUSIC piano and SC09 datasets, along with the spectral noise schedules derived from them [22, 33]. + +# I.1 CIFAR-10 Dataset + +The estimated covariance matrix for CIFAR-10 has a dimension of $d = 3072$ , corresponding to the original image resolution of $32 \times 32$ with three color channels. Figure 14a presents the spectral schedule derived for this covariance matrix using a 112-step diffusion process. Notably, its structure resembles that of the EDM schedule. Figure 14b illustrates the Wasserstein-2 distance evaluated on the Gaussian distribution associated with CIFAR-10, demonstrating the optimality of the spectral schedule across all diffusion steps. This effect is especially pronounced at lower step counts, where discretization error is more significant. Interestingly, heuristic schedules whose structures closely align with the spectral schedule, such as EDM, also achieve comparable performance, further supporting the connection between schedule structure and diffusion quality. + +![](images/455763e9b4c98307b43d5f65a239f4a96ef99e021a0ae1948973ef2b38c57923.jpg) +(a) + +![](images/0e245159a268a038be2c6eb2e848df0ff6bcf6c475e82731f06b3412f487ae1a.jpg) +(b) +Figure 14: Figure 14a compares the spectral schedule with several heuristic noise schedules for a 112-step diffusion process. The best-fitting parametric approximations for the Cosine and Sigmoid schedules are shown in red and brown, respectively. Figure 14b demonstrates the optimality of the spectral recommendation in the Gaussian case, revealing higher effectiveness in diffusion processes with fewer steps, where discretization error is more pronounced. + +# I.2 MUSIC Dataset + +To estimate the covariance matrix for the MUSIC dataset, we apply a sliding window of length $d = 400$ (0.025 seconds) to the original audio, excluding segments with $L_{1}$ energy below a threshold of $th = 0.05$ to reduce the influence of silent regions. The resulting covariance matrix is symmetric and exhibits an approximately Toeplitz structure. + +Figure 15a presents the spectral schedule derived from the estimated covariance matrix using a 112-step diffusion process, while Figure 15b shows the Wasserstein-2 distance comparison for the MUSIC Piano Gaussian dataset. Similar conclusions regarding optimality and the alignment with heuristic schedules hold here as well, with the Cosine $(0,0.5,1)$ schedule being the closest in both structure and performance to the spectral recommendation. + +![](images/9635f8be0aa27f922bcd89ac870b83873ce683a4ccf19a038e0fe94f469b8aec.jpg) +(a) + +![](images/b17cff904637a2d9eb55b688b92e57fac044f00a8e89810235e621926226bf75.jpg) +(b) +Figure 15: Figure 15a compares the spectral schedule with several heuristic noise schedules for a 112-step diffusion process. The best-fitting parametric approximations for the Cosine and Sigmoid schedules are shown in red and brown, respectively. Figure 15b demonstrates the optimality of the spectral recommendation in the Gaussian case, revealing higher effectiveness in diffusion processes with fewer steps, where discretization error is more pronounced. + +# I.3 SC09 Dataset + +In this section, we apply our method to a different dataset, SC09. SC09 is a subset of the Speech Commands Dataset [33] and consists of spoken digits (0-9). Each audio sample has a duration of one second and is recorded at a sampling rate of $16\mathrm{kHz}$ + +Differing from Sec. 6.2, here we use segments of length $d = 16000$ samples (one second) and set $th = 0.05$ in one setting and 0.1 in another. Figure 16 presents the spectral recommendations for $th = 0.05$ and $th = 0.1$ in the left and right columns, respectively. + +![](images/e54d8dc370754d45079cfec1a0bfb0665c7a1bb9e1053e00a2580fb5fd6fdaec.jpg) + +![](images/4f40a07cab4302a27c12dcbcbcec3778a0146a083c0c714df89d286dc04cb8ea.jpg) + +![](images/d2fd283c4d3e9842279566664552400bd99631ae20da846120f40b3d76029393.jpg) +(a) + +![](images/7856ab92b76c691b59a1dbaa729e14ab75e3c87ece4e59f99aa57a0b39e83cdb.jpg) +(b) + +![](images/daee160cdbba62bb529d38fe04ab57201c0072fd52e1c00f103805b434dabce1.jpg) +(c) +(e) + +![](images/89754a1371fe8b8d85d1a2e7d830b4fcf9fec1f6197bb09f3a4538948eaba9ab.jpg) +(d) +(f) +Figure 16: Spectral noise schedules for the $SC09$ dataset with $d = 16,000$ and thresholds $th = 0.05$ (left column) and $th = 0.1$ (right column). The first row shows spectral recommendations for various diffusion steps, while the second and third rows compare the spectral recommendation for 112 steps with heuristic noise schedules, including the Cosine and Sigmoid schedules. The parametric estimations for the Cosine and Sigmoid functions are shown in red, respectively. + +Figures 16a and 16b demonstrate that the spectral recommendation for $th = 0.05$ exhibits a more concave behavior compared to $th = 0.1$ . A more detailed discussion on the influence of the $th$ and $d$ parameters on the covariance matrix and the resulting noise schedule is provided in Appendices I.4 and K. + +We also evaluate the Wasserstein-2 distance for the spectral recommendation on the SC09 Gaussian dataset, using a silence threshold of $th = 0.1$ . As shown in Figure 17, similar conclusions hold: the spectral schedule remains optimal, with Cosine (0, 1, 1) and EDM heuristics most closely matching its structure and performance. + +![](images/df20440f7885d10afa64474fb3871e14e6635ae51e657f4e9bd0f2655f3c7fbf.jpg) +Figure 17: Wasserstein-2 distance comparison across different numbers of diffusion steps and various heuristic noise schedules for the SC09 Gaussian dataset. The results highlight the optimality of the spectral recommendation, showing higher effectiveness in diffusion processes with fewer steps, where discretization error is more pronounced. + +# I.4 Analysis of Different Aspects + +The estimation of the covariance matrix, which is essential for finding the spectral recommendation for a real dataset, relies on the choice of two key parameters: $th$ and $d$ . + +The parameter $d$ represents the dimension of the signals and controls the frequency resolution, which affects the eigenvalues. A smaller $d$ may result in a more generalized eigenvalue spectrum, reducing accuracy by averaging energy across neighboring eigenvalues. In contrast, a larger $d$ improves the precision in capturing frequency details but increases computational time for both estimation and optimization. + +Figure 18 shows that as $d$ increases, the eigenvalue structure becomes more precise, with the maximum eigenvalues magnitude growing larger. Conversely, as $d$ decreases, the eigenvalue structure becomes more generalized, exhibiting a monotonic decrease, as discussed in Appendix K.3. + +![](images/c113517136a12de4493d04af6e0259202e8a78d365109dc52890fdcc3ff50e59.jpg) +(a) $d = 400$ + +![](images/9cbd5a8f4fe98699d001bfb47b6140063b0589c162d3948c5eb1933066f34016.jpg) + +![](images/3d0e93b43e5e43c96ee5030d3c184be9e32873e0a1641c5b4299d7eb575125cc.jpg) +(b) $d = 4000$ +(c) $d = 16000$ +Figure 18: Eigenvalues of the MUSIC dataset using $d = [400, 4000, 16000]$ + +An additional consideration is the choice of the threshold value $th$ . This threshold helps prevent the covariance matrix estimation from being overly influenced by silent regions in the signal, which are characterized by low $L_{1}$ energy. Adjusting $th$ affects both the covariance matrix values and the eigenvalues, i.e. $C \cdot Av = C \cdot \lambda v$ , thereby influencing the resulting noise schedule, as shown in Appendix K.1. + +# J Supplementary Experiments for Empirical Distribution + +In the following section, we present a more comprehensive analysis of the results for the general empirical distribution, where the Gaussianity assumption no longer holds and a neural denoiser is required. + +We begin by providing technical details of the denoiser architectures used for each dataset. For CIFAR-10, we employed a pretrained denoiser from [13], trained with a continuous noise schedule. This approach facilitates a better comparison across noise schedules by reducing approximation error and ensuring alignment between training and testing conditions. + +For the MUSIC dataset, we trained a model based on the architecture proposed in [16, 1], using a linear noise schedule with $T = 1000$ diffusion steps. Training was performed in an unconditional setting on raw waveforms with a batch size of 64. For the SC09 dataset, we adopted the architecture from GitHub, and similarly trained it with a linear schedule of $T = 1000$ steps. + +In the both cases, during synthesis, we mapped the tested noise schedule to the closest matching step in the linear schedule to enable use of the trained denoiser. While this approach is suboptimal compared to training with a continuous noise schedule, introducing additional approximation error, it reflects a practical scenario where a specific denoiser is given, and is therefore worthwhile to examine. + +We assess the performance of the spectral recommendation on the MUSIC and SC09 datasets using second-order metric. In particular, we compute the Wasserstein-2 distance between the covariance matrices estimated from 20,000 generated samples at each diffusion step and those estimated from the real data. This allows us to evaluate how well each noise schedule preserves the core properties of the distribution during synthesis (Figure 20) + +Additionally, for CIFAR-10, we use the standard FID metric to assess whether perceptual properties are preserved with our recommendation (see Figures 19). The FID score is calculated following the method outlined in [13], using 50,000 samples for each computation, with seeds ranging from 0 to 49,999. As for FAD [15] calculation, it is less reliable for the MUSIC dataset due to its short frame lengths, and it provides limited insight for SC09, given its nature as a speech dataset. + +![](images/72145f55c10a7d32ade68ef2f4953f6172eac73099f0f28592bcf6a85b903aa8.jpg) +(a) + +![](images/4e3b5087f5045695fe4957389803e8a699a299e3dfa965132a0d665743a4b703.jpg) +(b) + +![](images/98a4c05720ab2a063ffd0ddc02a3e9d6f7946f02e950fe85fb51dfb317a87aaa.jpg) +(a) +Figure 20: Wasserstein-2 distance on the MUSIC-piano dataset (Figure 20a) and SC09 dataset (Figure 20b), evaluated over diffusion steps: $\{10, 20, 28, 38, 60, 90, 112, 250\}$ . + +![](images/5d0edc97ae08190f9d328407f1befda45c466440ac885273c7cf9eaef1bfa02d.jpg) +Figure 19: Wasserstein-2 distance (Figure 19a) and FID (Figure 19b) results on the CIFAR-10 dataset. +(b) + +The results support the relevance of the spectral recommendation for a broader range of distributions. The improved stability observed on CIFAR-10 may be attributed to factors such as the model architecture, the training procedure, or differences between discrete and continuous noise schedules. Interestingly, heuristics that achieve similar performance, such as EDM for FID on the CIFAR-10 dataset and cosine (0, 0.5, 1) schedules for the Wasserstein distance on the MUSIC dataset, tend to share structural similarities with the spectral recommendation. This suggests a connection between the structure of the noise schedule and overall performance, and points to the potential of the proposed method to inform the design of effective noise schedules. + +![](images/8f43e7a59284dee9295d0a8f207106655482cec33189072ed63ec31ffcd05075.jpg) +(a) + +![](images/7ce0d48052630a5a26253e5861249e52052dc7d1113ccb528fd48d89a9b48508.jpg) +(b) +Figure 21: Wasserstein-2 distance (Figure 21a) and FID results (Figure 21b) on the AFHQv2 dataset, evaluated over diffusion steps: $\{10, 20, 30, 40, 50, 70, 90, 112\}$ . + +We conducted an additional experiment on the higher-resolution AFHQv2 dataset [6], with a covariance matrix of size $12, 288 \times 12, 288$ (corresponding to $64 \times 64 \times 3$ images). Figure 21 presents Wasserstein distance and FID scores for the spectral recommendation and heuristic baselines, demonstrating the superior performance of the spectral schedule. In this case, EDM with $\rho = 5$ most closely approximates the spectral schedule in both structure and FID, reflecting dataset-specific adaptation. + +# Experiment statistical significance + +We compute the FID and Wasserstein-2 distance for CIFAR-10 using three independent runs, each consisting of 50,000 samples generated with different random seeds ([0-4999], [5000-9999], and [10000-14999]), to capture random variability. The results in Tables 1 and 2 report the Wasserstein-2 distance and FID (respectively), along with a $95\%$ confidence interval, calculated using the standard error and the t-distribution. + +Table 1: Wasserstein scores (mean ± 95% CI) across selected diffusion steps for different noise schedules. + +
102028385090
Linear3.28 ± 0.281.61 ± 0.011.17 ± 0.010.90 ± 0.020.73 ± 0.020.54 ± 0.04
EDM3.77 ± 0.021.99 ± 0.011.43 ± 0.011.07 ± 0.010.83 ± 0.020.53 ± 0.04
Cosine (0, 1, 1)6.81 ± 0.013.69 ± 0.012.56 ± 0.011.80 ± 0.011.30 ± 0.010.63 ± 0.01
Cosine (0.2, 1, 2)3.52 ± 0.011.83 ± 0.011.35 ± 0.011.06 ± 0.010.88 ± 0.020.67 ± 0.03
Cosine (0, 0.5, 1)6.44 ± 0.014.97 ± 0.024.31 ± 0.023.76 ± 0.013.29 ± 0.012.46 ± 0.01
Sigmoid (0, 3, 0.7)4.60 ± 0.012.75 ± 0.012.10 ± 0.011.63 ± 0.011.29 ± 0.010.81 ± 0.02
Sigmoid (-3, 3, 1)5.67 ± 0.013.71 ± 0.012.94 ± 0.012.36 ± 0.011.89 ± 0.011.15 ± 0.00
Sigmoid (-3, 3, 0.7)5.06 ± 0.012.89 ± 0.012.13 ± 0.011.60 ± 0.011.21 ± 0.000.66 ± 0.02
Spectral3.21 ± 0.241.59 ± 0.051.12 ± 0.010.81 ± 0.020.63 ± 0.020.46 ± 0.04
+ +Table 2: FID scores (mean $\pm {95}\%$ CI) across selected diffusion steps for different noise schedules. + +
102028385090
Linear17.59 ± 0.269.00 ± 0.177.19 ± 0.256.04 ± 0.265.22 ± 0.254.00 ± 0.26
EDM19.04 ± 0.256.70 ± 0.174.64 ± 0.193.63 ± 0.223.09 ± 0.212.52 ± 0.19
Cosine (0, 1, 1)24.31 ± 0.148.63 ± 0.166.00 ± 0.164.65 ± 0.183.86 ± 0.202.94 ± 0.21
Cosine (0.2, 1, 2)21.92 ± 0.2913.06 ± 0.2111.10 ± 0.209.85 ± 0.258.90 ± 0.257.33 ± 0.29
Cosine (0, 0.5, 1)20.49 ± 0.279.60 ± 0.216.85 ± 0.075.23 ± 0.124.22 ± 0.143.05 ± 0.16
Sigmoid (0, 3, 0.7)23.66 ± 0.1714.10 ± 0.3111.90 ± 0.2310.48 ± 0.269.44 ± 0.277.78 ± 0.29
Sigmoid (-3, 3, 1)16.28 ± 0.068.16 ± 0.236.56 ± 0.235.66 ± 0.215.03 ± 0.224.12 ± 0.24
Sigmoid (-3, 3, 0.7)13.67 ± 0.066.60 ± 0.195.31 ± 0.204.55 ± 0.244.05 ± 0.253.36 ± 0.22
Spectral12.50 ± 0.225.44 ± 0.204.21 ± 0.213.56 ± 0.243.20 ± 0.232.60 ± 0.21
+ +The results presented in Tables 1 and 2 align with the experimental observations in Figure 19, supporting the spectral recommendation and its relation to alternative noise schedules. The optimality with respect to the Wasserstein loss remains robust, while for the FID metric, as the number of diffusion steps increases and discretization error becomes less significant, schedules such as EDM achieve comparable performance. + +# Experimental result reproducibility + +To facilitate easier reproduction of our results, we provide examples of the derived spectral schedules obtained for the CIFAR-10 dataset by minimizing the Wasserstein-2 loss for [10, 20, 28] diffusion steps. The values are rounded to four decimal places. + +10 Diffusion Steps: [0.9999, 0.9899, 0.9177, 0.6796, 0.3398, 0.1271, 0.0429, 0.0135, 0.0030, 0.00004] +20 Diffusion Steps: [0.9999, 0.9981, 0.9914, 0.9735, 0.9340, 0.8591, 0.7389, 0.5814, 0.4162, 0.2751, 0.1724, 0.1048, 0.0627, 0.0371, 0.0216, 0.0121, 0.0063, 0.0028, 0.0008, 0.00004] +28 Diffusion Steps: [0.9999, 0.9989, 0.9961, 0.9899, 0.9778, 0.9562, 0.9204, 0.8656, 0.7882, 0.6889, 0.5747, 0.4572, 0.3487, 0.2570, 0.1849, 0.1309, 0.0917, 0.0638, 0.0442, 0.0304, 0.0207, 0.0138, 0.0089, 0.0055, 0.0031, 0.0015, 0.0005, 0.00004] + +# K Further Discussion + +# K.1 Relationship Between Noise Schedules and Eigenvalues + +To explore the relationship between the optimal spectral noise schedule and the data characteristics, we solved the optimization problem for each eigenvalue individually, with the contributions from the other eigenvalues set to zero. Using the eigenvalues of the covariance matrix from 6.1, Figure 22a shows these eigenvalues, while 22b presents the optimal solutions for 50 diffusion steps, computed using the Wasserstein-2 distance in the optimization problem. Each solution corresponds to a single eigenvalue (considering only positions 2 to 10 for clarity)7 + +![](images/5dea2eab4548383a97bfa4d0939ae81734e5380fde645cbbc1a91449bbdf32c2.jpg) + +![](images/cb4ed4f5f3b1f9fc133f1070597adbf2e9d9da0f53e93921aa80a2b8d83d475d.jpg) +(a) +(b) +Figure 22: Figure 22a shows the eigenvalues for the covariance matrix from Section 6.1. Figure 22b presents the results of solving the optimization problem for each eigenvalue individually, with the contributions from the other eigenvalues set to zero. + +It can be observed from Figure 22b that the solution becomes more concave as the magnitude of the eigenvalue decreases (yellow) and more convex as the magnitude increases (blue). Notably, this behavior is determined by the magnitude of the eigenvalues $\left(\{\lambda_i\}_{i=1}^d\right)$ rather than their indices $(i)$ , as the objective functions are independent of the index itself. + +When the eigenvalues, or equivalently the DFT coefficients, decrease monotonically, a direct relationship emerges between the eigenvalue magnitude and its corresponding frequency (for example, the $1/f$ behavior observed in speech [30]). In such cases, the first eigenvalues correspond to the low frequencies, having larger amplitudes, while the last correspond to high frequencies and smaller amplitudes. This pattern, along with the previous observations8, aligns with the well-known coarse-to-fine signal construction behavior of diffusion models.9 + +Interestingly, by examining the spectral recommendation from Figure 1a, it closely resembles the solutions obtained by emphasizing the highest eigenvalue (22b). This suggests that using the Wasserstein-2 loss tends to favor larger magnitude eigenvalues. This behavior is also reflected in the relative error, $(|\lambda_i - \lambda_{est}|) / (\lambda_i)$ , shown in Figure 23, where larger eigenvalues exhibit smaller relative errors. + +![](images/625c4bf9eeb2a3639392d5075953e377b8541e4f95b26aa858c14a514dec6f54.jpg) + +![](images/685dd7c2d195ea332d3f54b1cf0b3f9f37ea69dfcf02c97f138fe6cc9fcdfe0a.jpg) +(b) + +![](images/587ead4f5d00d11add0cf36c0de2576b71e74e83f0fecd770036655906c3e1c0.jpg) +(a) +(c) + +![](images/bb22b9a47d99965e6854717ff65e20a829bf652ff3e1eb2209cdf4dba326180a.jpg) +(d) +Figure 23: Figures 23a and 23c present the eigenvalues of matrices 12a and 12g, respectively. Figures 23b and 23d illustrate the relative error of the spectral recommendation, using 60 diffusion steps and the Wasserstein-2 distance. Notably, larger eigenvalues exhibit smaller relative errors. In Figures 23b and 23c, the first elements were manually chosen, as they originally had extreme values (1000 and 10 respectively), and we aimed to keep the figure within a reasonable scale. + +The influence of the eigenvalue magnitude, especially the dominant ones, on the resulting schedule is further illustrated through additional examples. Figure 12d displays the covariance matrix from Figure 12a, scaled by a factor of $C = 20$ , which amplifies the dominant eigenvalues, as shown by the relation $C \cdot Av = C \cdot \lambda v$ . Consequently, the spectral recommendation in Figure 12f appears more convex than in Figure 12c. A similar trend is observed in Figure 16, where the spectral recommendation for $th = 0.05$ exhibits a more concave shape compared to $th = 0.1$ . This relationship open up a possibility of designing loss functions which focus on specific frequency ranges of interest. Further discussion is provided in K.3. + +Note: We used the Wasserstein-2 loss. However, alternative measures, such as KL divergence, could also yield similar results. + +# K.2 Temporal Dynamics of the Diffusion Process + +Our analysis has focused so far on the final distribution resulting from the diffusion process, as described by (10). However, the analytical formulation also enables the examination of phenomena occurring throughout the diffusion process. In particular, Equation (29) describes each intermediate state $\mathbf{v}_s$ as a function of the preceding noise schedule components and the initial noise $\mathbf{v}_0$ . This enables the study of spectral properties and dynamics across all diffusion steps, rather than being limited to the final output. + +Figures 24a and 24b show the relative error $|\lambda_i - \lambda_{\mathrm{est}}| / \lambda_i + \epsilon$ for the 10 largest eigenvalues (sorted, largest on the right) over the final 20 steps of a 60-step diffusion process, using the Cosine schedule ( $s = 0$ , $e = 1$ , $\tau = 1$ ) and the spectral schedule respectively. In both figures, eigenvalues are sorted in ascending order, allowing a comparison between high-frequency components (lower-magnitude eigenvalues) and low-frequency components (higher-magnitude eigenvalues). + +![](images/18527b9b8f35d47eee7f0d090a5b1c41548e4a087c91b6aca4d82c213097087f.jpg) +(a) + +![](images/2bd8807bf32968f39add5a56b3b9ac292d6c553ea1a07de38b889811c1518dda.jpg) +(b) +Figure 24: The relative errors of the the Cosine $(s = 0, e = 1, \tau = 1)$ (24a) and the spectral (24b) schedules for the 10 largest eigenvalues (sorted, largest on the right) over the final 20 steps of a 60-step diffusion process. + +In both figures, We observe that the error decreases consistently for large eigenvalues, but increases in the final steps for smaller ones. In addition, the final relative error (yellow curve) is notably lower for high-magnitude eigenvalues (i.e., low frequencies). This behavior aligns with the observed bias toward mid-to-high frequencies reported in prior works [39]. Notably, this effect is more pronounced under the Cosine schedule; while both schedules yield similar errors for large eigenvalues, the Cosine schedule exhibits much higher errors for small eigenvalues (i.e., high-frequency components). This suggests the effectiveness of the proposed optimization procedure in reducing such errors. + +An additional aspect is the evolution of eigenvalues throughout the diffusion process. Figures 25a and 25b illustrate this for the Cosine ( $s = 0$ , $e = 1$ , $\tau = 1$ ) and spectral schedules for a 60-step diffusion process. An interesting observation is that low-magnitude eigenvalues tend to converge to their final values faster than high-magnitude ones. Notably, the convergence rate does not necessarily reflect the accuracy of the final values, as discussed in Figure 24. Comparing convergence rates across different noise schedules or leveraging the intermediate expressions to design improved schedules are additional aspects that can be explored; however, we leave them for future work. + +![](images/0581d4e9e8d4bc5ae5eae033594d9245fb2515a145445f5147c5efda95370fd9.jpg) +(a) +Figure 25: The dynamics of eigenvalues through a 60-step diffusion process for the Cosine ( $s = 0$ , $e = 1$ , $\tau = 1$ ) (25a) and the spectral (25b)schedules. + +![](images/fada21129e9327b406342e370b71214e4d716c50e3cc1cd8101c60143fea6471.jpg) +(b) + +![](images/ee65b8a413c2f7a85b808f464e5dbbb7b140f89bc7953417e9a0d16a61360aca.jpg) +Figure 26 illustrates the evolution of the Wasserstein-2 error over the final 20 steps of a 60-step diffusion process. It can be observed that the Wasserstein-2 error consistently decreases for both the cosine (0,1,1) (Figure 26a) and the spectral noise schedules (Figure 26b), with the spectral schedule converging to a lower final value +(a) + +![](images/dc6e0e52ebbfd500e7a67dae11ecc2f939434ccf56e47e42b277ec6ae88fad71.jpg) +(b) +Figure 26: The dynamics of the Wasserstein-2 distance over the final 20 steps of a 60-step diffusion process for the Cosine ( $s = 0$ , $e = 1$ , $\tau = 1$ ) (26a) and the spectral (26b) schedules. + +# K.3 Relationship Between Noise Schedules and the loss functions + +Building on the spectral monotonic behavior, the loss function can be adjusted to weight different spectral regions in various ways, shaping the noise schedule based on specific objectives. + +We propose a weighted $l1$ loss for the first and the second moments of two Gaussian distributions $P(\hat{\mathbf{v}}_0; \bar{\alpha})$ and $P(\mathbf{v}_0)$ . + +$$ +\mathcal {D} _ {L _ {1}} \left(P \left(\hat {\mathbf {v}} _ {0}; \bar {\boldsymbol {\alpha}}\right), P \left(\mathbf {v} _ {0}\right)\right) = \sum_ {i = 1} ^ {d} \frac {\lambda_ {i}}{\sum_ {j} \lambda_ {j}} \left| \left[ \mathbf {D} _ {1} \right] _ {i} ^ {2} - \lambda_ {i} \right| + \sum_ {i = 1} ^ {d} \frac {\left[ \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \right] _ {i} ^ {2}}{\sum_ {j} \left[ \boldsymbol {\mu} _ {0} ^ {\mathbf {u}} \right] _ {j} ^ {2}} \left(\left[ \mathbf {D} _ {2} \right] _ {i} - 1\right) ^ {2} \tag {64} +$$ + +The first term applies a weighted $l1$ loss to the eigenvalues, while the second term computes a weighted $l2$ norm of the mean vectors. This design ensures that eigenvalues with larger magnitudes and mean components with higher values have greater influence on the overall loss. + +Figure 27 illustrates the spectral recommendation obtained by solving the optimization problem in 14 using the Weighted 11 loss. The results are based on the Gaussian MUSIC-Piano dataset described in 6.2 where $d = 16,000$ and $th = 0.05$ . + +![](images/733ce7d08ee3d1a13f5c45a51b80332f805ce72ff2f1bec2691f9c9046e7792c.jpg) +Figure 27: Comparison of the spectral schedule with various heuristic noise schedules for $S = 112$ diffusion steps. The figure presents the linear, EDM ( $\rho = 7$ ), Cosine-based schedules, including Cosine ( $s = 0$ , $e = 1$ , $\tau = 1$ ) from [23, 5]. The parametric estimation of the cosine function is shown in red. + +Interestingly, using the weighted $l1$ loss results in a spectral recommendation that aligns with established heuristic methods. Specifically, it corresponds to the manually designed cosine (0,1,1) schedule proposed in [23]. This observation could indicate a potential link between the design of widely used noise schedule heuristics and a bias against high-frequency regions, which has been observed in previous research [39]. + +Note: The relationship between the magnitude of the eigenvalues and their corresponding frequencies holds tight only when monotonic behavior is present. In real-world scenarios, as shown in Figure 18c, the eigenvalues' magnitudes generally decrease, but the function is not strictly monotonic. In such cases, an alternative approach is required, one that either analyzes broader spectral regions or considering both the values and indices of the eigenvalues. + +# L Analysis of Mean Bias + +In the following section, we analyze the mean bias expression. To maintain consistency and preserve the connection to the frequency domain, we conduct the analysis on the circulant matrices used in Section 6.1, which can be diagonalized using the discrete Fourier transform (DFT), denoted by $\mathcal{F}\{\cdot\}$ . Naturally, the same approach can be extended to any covariance matrix, $\Sigma_0$ , using its eigenbasis $\mathbf{U}$ . The mean bias expression $(\mathbf{D}_2 - \mathbf{I})\boldsymbol{\mu}_0^{\mathcal{F}}$ arises from the difference between $\mathbb{E}\left[\mathbf{x}_0^{\mathcal{F}}\right]$ and $\mathbb{E}\left[\hat{\mathbf{x}}_0^{\mathcal{F}}\right]$ . In particular, We focus on the absolute magnitude of the expression $|\mathbf{D}_2 - \mathbf{I}||\boldsymbol{\mu}_0^{\mathcal{F}}|$ . The term $\mathbf{D}_2$ , as defined in (10), depends on $\Lambda_0$ and on $\bar{\alpha}$ , the chosen noise schedule. Notably, for stationary signals, $\boldsymbol{\mu}_0$ is deterministic, resulting in the vector $\boldsymbol{\mu}_0^{\mathcal{F}}$ where all entries are zero except for the first element. Specifically, for a mean-centered signal where $\boldsymbol{\mu}_0^{\mathcal{F}} = \mathbf{0}$ , the DDIM process remains unbiased, regardless of $|\mathbf{D}_2 - \mathbf{I}|$ expression. In other cases, for a given $\boldsymbol{\mu}_0^{\mathcal{F}}$ , the primary source of bias originates from the main diagonal of $|\mathbf{D}_2 - \mathbf{I}|$ . + +Figure 28 analyzes the mean bias for two choices of $\Lambda_0$ with $d = 50$ . It compares the values of $|\mathbf{D}_2 - \mathbf{I}|$ across several cosine noise schedule heuristics and illustrates how its behavior depends on the number of diffusion steps. + +![](images/93a5150de86e8bf121e0e93938244981be45173f47d38a6805cd9848c22321fd.jpg) + +![](images/1249c5d16dca982b45202e9b1254751a3546792459aa93409f8ac0a79e28167b.jpg) + +![](images/cba43d62b1ac6186c966b30460327119a7a87ad2cd1fa921aae050ddfa81489f.jpg) +(a) + +![](images/eacfd0f2f97f3d70f569817a07974660072da22b92db329c20b9241d8501eb4d.jpg) +(b) + +![](images/38b34163331914a26c15add50514d66da3db2810a39cf27a99744607af1dcea3.jpg) +(c) + +![](images/025ddf07c915ab6cdbdf473591b7e7ea6ec214b0cf60060d11bff26c718d33f6.jpg) +(d) + +![](images/b28b887fa9f13ec0339d29da91ab99e7cb1eed20d455122301d168bc1a4251a5.jpg) +(e) +(g) +Figure 28: Figures 28a and 28b display the eigenvalues for two choices of $\Lambda_0$ with $d = 50$ . Figures 28c and 28d compare the mean bias values across different parametrization of the Cosine noise schedule using 112 diffusion steps. Figures 28e and 28f show the bias for varying numbers of diffusion steps with the Cosine (0,1,1) noise schedule, while Figures 28g and 28h illustrate the same for the Cosine (0,0.5,1) schedule. + +![](images/060813a77225faa1fe8c97d9f7637bcfa425ed9d014b7353a2000b01c7e3eb06.jpg) +(f) +(h) + +Figures 28c and 28d reveal that for certain heuristics, such as Cosine $(0,1,1)$ , the bias is negligible, while for others, like Cosine $(0,0.5,1)$ , the bias increases. Additionally, the magnitude of the eigenvalues $\{\lambda_i\}_{i=1}^d$ plays a significant role in determining the bias; as the eigenvalues grow larger, the bias also tends to increase. + +Figures 28e and 28f illustrate the bias across various numbers of diffusion steps $\{10,50,100,200,500,750,1000\}$ for the Cosine $(0,1,1)$ noise schedule. Similarly, Figures $28\mathrm{g}$ + +and 28h show the bias for the Cosine $(0,0.5,1)$ . In both cases, increasing the number of diffusion steps leads to a gradual rise in the mean bias $|\mathbf{D}_2 - \mathbf{I}|$ . + +# M DDPM vs DDIM + +The explicit formulations of DDPM and DDIM in (12) and (10) also enable their comparison in terms of loss across varying diffusion depths and noise schedules. Figure 29 presents such a comparison using the Wasserstein-2 distance on a logarithmic scale. The results show that DDIM sampling is faster and yields lower loss values than DDPM for each noise schedule, aligning with the empirical observations in [27]. In addition, it can be seen that the spectral schedule with DDIM consistently maintains optimality with respect to all other experiments. + +![](images/39cabcd3cd58fdf5611b4df4c5f5616e0e5227d3728ad53ef6e38788132d2d19.jpg) +Figure 29: Comparison of the Wasserstein-2 distance between DDIM and DDPM for different noise schedules, including the spectral recommendation, across various diffusion steps. + +![](images/8d26a6efc2fac8fe3f0e64ed7cec00ea0991baf035e0d4f88fac5721fffd33d1.jpg) +Figure 30: Comparison of the spectral noise schedules in DDPM and DDIM, derived from the covariance matrix outlined in Sec. 6.1, with parameters $d = 50$ , $l = 0.1$ , and total diffusion steps set to $\{10, 28, 38, 60, 90, 112, 250, 334, 500, 750, 1000\}$ . + +# N Optimization Time Analysis + +The computational cost of solving the optimization problem for time-dependent signals depends on several factors. As discussed in Appendix I.4, increasing the resolution of temporal signals improves both accuracy and adaptation to the data characteristics, but also increases computational cost (larger covariance matrices allow for higher frequency accuracy). Additionally, factors like the sampling rate and stationary regions influence the results. + +We report the computation times for the covariance matrices of different datasets, with dimensions $400 \times 400$ for MUSIC, $3072 \times 3072$ for CIFAR10, and $12288 \times 12288$ for AFHQv2. For the MUSIC dataset, we applied stopping criteria of 1500 optimization steps or a function tolerance of $10^{-6}$ . For CIFAR10 and AFHQv2, we used 2000 steps or the same tolerance to ensure convergence for a large number of diffusion steps. All computations were performed on a standard CPU. Notably, using slightly more relaxed stopping criteria could achieve the same optimal schedule in less time. + +Table 3: Comparison of the optimization time (in seconds) for a $400 \times 400$ covariance matrix (e.g., MUSIC dataset), a $3072 \times 3072$ covariance matrix (e.g., CIFAR10 dataset) and a $12288 \times 12288$ covariance matrix (e.g., AFHQv2 dataset) + +
Diffusion steps400 × 4003072 × 307212288 × 12288
100.090.363.88
502.948.1993.09
9018.3125.4292.48
13055.21321.761555.87
17069.5558.762316.73
210172.02594.513884.07
250261.69949.007417.46
+ +As shown in the Table 3, for a low number of steps, where discretization error is significant and the spectral scheduler greatly improves performance, the computation time ranges from less than a second to a few tens of seconds. Over approximately 130 diffusion steps, the computation time exceeds one minute for the CIFAR10; however, in this regime, discretization errors become negligible, the overall structure of the scheduler remains unchanged, and its density increases with minimal change. + +To accelerate the optimization process, we leverage this structural similarity by adopting a gradual optimization strategy: we begin with a small number of diffusion steps (e.g., 10) which requires relatively low computational effort and use the resulting solution (interpolated to the appropriate length) to initialize optimization for a larger number of steps. This approach significantly reduces computation time by reusing information across step counts. + +Table 4 compares optimization times on the AFHQv2 dataset using random initialization (left column) versus iterative initialization. For the iterative approach, the third column reports the time for each individual run, while the forth shows the total cumulative time to reach the solution for a given diffusion step count. As observed in Table 4, iterative initialization consistently reduces optimization time while achieving noise schedules and loss values comparable to random initialization. Further improvements may be possible by refining the strategy for transitioning between diffusion step counts during optimization. + +An additional observation from Table 3 is the significant convergence time required for large covariance matrices, especially when a high number of diffusion steps is used. While covariance matrices in audio signals are typically considered in the stationary domain with moderate dimensions, image data often involve higher resolutions, resulting in larger matrices and increased computational cost. A possible approach to reduce computational cost is to assume a low-rank covariance matrix, selecting only the $d$ most significant directions and ignoring near-zero eigenvalues. + +Specifically, we apply PCA to the original dataset and conduct optimization within a reduced-dimensional subspace. For the AFHQv2 dataset, the dimensionality is reduced from $64 \times 64 \times 3$ (12, 288) to $32 \times 32 \times 3$ (3, 072) using PCA, with results shown in the rightmost column of Table 4. As expected, optimization in this reduced subspace is faster than in the original space ( $64 \times 64 \times 3$ , left column). The subspace dimensionality acts as a hyperparameter; here, we set $d = 3072$ , achieving a significant reduction while preserving the dominant spectral components. + +Table 4: Comparison of optimization times (seconds) across proposed acceleration methods on AFHQv2 dataset + +
Diffusion stepsRandom init. (12288 × 12288)Iterative init. (12288 × 12288)Iterative init. (summed) (12288 × 12288)PCA (3072 × 3072)
103.883.923.920.49
5093.0939.6043.5212.71
90292.4821.6165.1332.80
1301555.8746.05111.18350.27
1702316.7320.59131.78548.57
2103884.0732.00163.78605.66
25074117.4768.23232.02919.99
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sha256:b5d38d3fc9217610d0fa3ac5deb1352f35ce87240bb3584a07fca5b3dc5c53fe +size 2853923 diff --git a/NeurIPS/2025/Spectral Analysis of Representational Similarity with Limited Neurons/full.md b/NeurIPS/2025/Spectral Analysis of Representational Similarity with Limited Neurons/full.md new file mode 100644 index 0000000000000000000000000000000000000000..b328e48b89a5aebbcbeab6ac6567eb0b88e51e26 --- /dev/null +++ b/NeurIPS/2025/Spectral Analysis of Representational Similarity with Limited Neurons/full.md @@ -0,0 +1,1368 @@ +# Spectral Analysis of Representational Similarity with Limited Neurons + +Hyunmo Kang* + +Department of Physics Johns Hopkins University Baltimore, MD 21218 hkang56@jh.edu + +Abdulkadir Canatar* + +Center for Computational Neuroscience Flatiron Institute New York, NY 10010 acanatar@flatironinstitute.org + +SueYeon Chung + +Department of Physics + +Harvard University + +Cambridge, MA 02138 + +sueyeonchung@g.harvard.edu + +# Abstract + +Understanding representational similarity between neural recordings and computational models is essential for neuroscience, yet remains challenging to measure reliably due to the constraints on the number of neurons that can be recorded simultaneously. In this work, we apply tools from Random Matrix Theory to investigate how such limitations affect similarity measures, focusing on Centered Kernel Alignment (CKA) and Canonical Correlation Analysis (CCA). We propose an analytical framework for representational similarity analysis that relates measured similarities to the spectral properties of the underlying representations. We demonstrate that neural similarities are systematically underestimated under finite neuron sampling, mainly due to eigenvector delocalization. Moreover, for power-law population spectra, we show that the number of localized eigenvectors scales as the square root of the number of recorded neurons, providing a simple rule of thumb for practitioners. To overcome sampling bias, we introduce a denoising method to infer population-level similarity, enabling accurate analysis even with small neuron samples. Theoretical predictions are validated on synthetic and real datasets, offering practical strategies for interpreting neural data under finite sampling constraints. + +# 1 Introduction + +Understanding how artificial neural networks relate to biological neural activity remains one of the central challenges in computational neuroscience [13, 50, 40]. As deep learning models become increasingly sophisticated at matching human-level performance on complex tasks, there is growing interest in whether these models actually learn representations that mirror those found in the brain [53, 26, 25, 45, 35]. However, a fundamental obstacle stands in the way of making this comparison: while artificial networks can be analyzed in their entirety, neuroscientists can only record from a small subset of neurons in any given brain region [10, 51, 47]. This sampling limitation poses a critical challenge for the field. When we measure the similarity between model and neural representations using standard techniques like Canonical Correlation Analysis (CCA) or Centered Kernel Alignment (CKA), how much does our limited neural sample size distort the true relationship? Addressing this + +issue is critical, given that these metrics increasingly inform model selection and neuroscientific interpretation [42, 39]. + +Our work provides the first rigorous theoretical framework for understanding how neuron sampling affects representational similarity measures. Our analysis reveals that measuring CCA and CKA with a limited number of recorded neurons systematically underestimates the true population-level similarity. This underestimation stems primarily from eigenvector delocalization [1, 15, 4]—a phenomenon where sample eigenvectors become increasingly misaligned with their population counterparts as the number of recorded neurons decreases. + +Our analysis proceeds in two parts. First, in the forward problem, we investigate how neuron subsampling from the full underlying population distorts the population eigencomponents and how this distortion affects the computed similarity measures. Second, in the backward problem, we ask whether observations from a finite number of neurons can be used to reliably infer the population representational similarity. + +# 1.1 Our Contributions + +- Eigencomponent-wise Analysis of Representation Similarity: We show how neuron sub-sampling alters the eigenvalues and eigenvectors of the Gram matrix, leading to a systematic underestimation of CCA/CKA due to eigenvector delocalization. +- Backward Inference via Denoising Eigenvectors: We introduce a denoising method that leverages population eigenvalue priors (e.g., power-law) to infer the true population similarity from limited data, substantially correcting the sampling bias. +- Validation on Real Neural Data: Applying our framework to primate visual cortex recordings confirms that even modest neuron counts can lead to severe underestimation of model-brain similarity and that our method effectively recovers the missing signal. + +# 1.2 Related Works + +Representation similarity measures expressed in terms of eigencomponents were presented in detail by Kornblith et al. [29], who showed that CCA, CKA, and linear regression scores can all be written in terms of the eigenvalues and eigenvectors of the Gram matrices. + +A key question is how these similarity measures behave under different kinds of noise. Broadly, there are two primary noise sources: + +1. Additive noise, which arises from trial-to-trial variability and measurement error. In many studies, repeated trials and averaging can substantially mitigate this type of noise. +2. Sampling noise, which occurs because we can only record from a limited subset of neurons rather than the entire population. Consequently, the sample eigenvectors and eigenvalues differ from their population counterparts. + +In this work, we focus on the latter issue—sampling noise—since we assume trial averaging already reduces the additive noise to a manageable level. + +One approach to address sampling noise is by studying the moments of the Gram matrix [28, 14]. While these methods provide a way to approximate the effect of sampling on the scalar values of certain similarity measures, they do not directly offer an interpretable description of what happens to the underlying eigencomponents. Recent work by [41] provides bounds on representation similarity measures when the number of sampled neurons is limited. However, these bounds are tight only under the assumption of a white Wishart model (i.e., all population eigenvalues are 1). For more realistic data, where eigenvalues often decay according to a power-law(primary visual cortex for mice in [49] and human visual cortex fMRI in [20]), these bounds can become too loose to be practically informative. + +Instead, we directly investigate how sampling noise affects both the eigenvalues and eigenvectors of the sample Gram matrix using random matrix theory [43, 7, 8]. Extensive results exist for white Wishart matrices and low-rank "spiked" models, including the Baik-Ben Arous-Péché (BBP) phase transition [4], which reveals that sample eigenvectors often serve as poor estimators of their population counterparts. These ideas have been extended to canonical correlation analysis [36, 9]. However, the + +power-law-like spectra observed in neural data have not yet received comparable attention. Our work attempts to bridge this gap by studying sampling noise in representations with strongly decaying eigenvalues, which are ubiquitous in neural datasets. + +# 2 Notation & Problem Setup + +We use bold fonts for matrices and bracket notation for vectors2. We use a tilde to denote quantities related to their population values. + +We consider two centered population activations $\tilde{\mathbf{X}}\in \mathbb{R}^{P\times \tilde{N}_x}$ and $\tilde{\mathbf{Y}}\in \mathbb{R}^{P\times \tilde{N}_y}$ with $\tilde{N}_{x}$ and $\tilde{N}_{y}$ neurons, recorded in response to a fixed set of stimuli of size $P$ . Centered means that we subtract the column-wise mean. Their corresponding population Gram matrices are given by $\tilde{\Sigma}_x = \tilde{\mathbf{X}}\tilde{\mathbf{X}}^\top$ and $\tilde{\Sigma}_y = \tilde{\mathbf{Y}}\tilde{\mathbf{Y}}^\top$ with eigendecomposition: + +$$ +\tilde {\Sigma} _ {x} = \sum_ {i = 1} ^ {P} \tilde {\lambda} _ {i} | \tilde {u} _ {i} \rangle \langle \tilde {u} _ {i} |, \quad \tilde {\Sigma} _ {y} = \sum_ {a = 1} ^ {P} \tilde {\mu} _ {a} | \tilde {w} _ {a} \rangle \langle \tilde {w} _ {a} |, \tag {1} +$$ + +where $\tilde{\lambda}_i, \tilde{\mu}_a$ and $|\tilde{u}_i\rangle, |\tilde{w}_a\rangle$ are respectively their eigenvalues and eigenvectors, and the eigenvectors are mutually orthogonal, i.e. $\langle \tilde{u}_i|\tilde{u}_j\rangle = \delta_{ij}$ and $\langle \tilde{w}_a|\tilde{w}_b\rangle = \delta_{ab}$ . + +The sample activations $\mathbf{X} \in \mathbb{R}^{P \times N_x}$ and $\mathbf{Y} \in \mathbb{R}^{P \times N_y}$ are assumed to be generated from the population ones by a random projection $\mathbf{X} = \tilde{\mathbf{X}}\mathbf{R}$ where $\mathbf{R} \in \mathbb{R}^{\tilde{N} \times N}$ is a random matrix with Gaussian i.i.d entries. Their Gram matrices are defined as $\boldsymbol{\Sigma}_x = \mathbf{X}\mathbf{X}^\top$ and $\boldsymbol{\Sigma}_y = \mathbf{Y}\mathbf{Y}^\top$ with eigendecomposition: + +$$ +\boldsymbol {\Sigma} _ {x} = \sum_ {i = 1} ^ {P} \lambda_ {i} | u _ {i} \rangle \langle u _ {i} |, \quad \boldsymbol {\Sigma} _ {y} = \sum_ {a = 1} ^ {P} \mu_ {a} | w _ {a} \rangle \langle w _ {a} |. \tag {2} +$$ + +Random projections serve as an effective approach for sampling high-dimensional data due to their geometry-preserving properties [31] and are a popular method in analyzing neural dynamics from limited recordings [19]. This assumption allows us to treat sample Gram matrices as structured random Wishart matrices (see SI.A.1). + +Noting that both population and sample eigenvectors reside in $\mathbb{R}^P$ , we define self-overlap matrices between sample and population eigenvectors for each representation as + +$$ +Q _ {i j} ^ {x} := \mathbb {E} [ \langle u _ {i} | \tilde {u} _ {j} \rangle^ {2} ], \quad Q _ {a b} ^ {y} := \mathbb {E} [ \langle w _ {a} | \tilde {w} _ {b} \rangle^ {2} ] \tag {3} +$$ + +and cross-overlap matrices between the eigenvectors of two representations as + +$$ +M _ {i a} := \mathbb {E} [ \langle u _ {i} | w _ {a} \rangle^ {2} ], \quad \tilde {M} _ {i a} := \langle \tilde {u} _ {i} | \tilde {w} _ {a} \rangle^ {2} \tag {4} +$$ + +Expectations are over different instances of neuron sampling via random projections. The crossoverlap $\tilde{\mathbf{M}}$ between two population eigenvectors is deterministic, hence does not require averaging. + +# 2.1 Common Representational Similarity Measures + +Here, we review common representational similarity measures and show that these measures can be expressed in terms of the average quantities presented above. + +Canonical Correlation Analysis (CCA) is an algorithm that sequentially finds a set of orthonormal vectors $\{\mathbf{a}_{\alpha}\}$ and $\{\mathbf{b}_{\alpha}\}$ for which the correlation coefficients $\rho_{\alpha} = \mathrm{corr}(\mathbf{X}\mathbf{a}_{\alpha},\mathbf{Y}\mathbf{b}_{\alpha})$ for two matrices $\mathbf{X},\mathbf{Y}$ are maximized [23]. The squared sum of these coefficients gives the CCA similarity (CCA $=$ $\sum_{\alpha}\rho_{\alpha}^{2}$ ), and can be expressed in terms of the overlap matrix $M_{ia}$ [5, 29] + +$$ +\mathrm {C C A} = \sum_ {i = 1} ^ {N _ {x}} \sum_ {a = 1} ^ {N _ {y}} \frac {\left\langle \right. u _ {i} \left. \right| w _ {a} \left. \right\rangle^ {2}}{\min \left(N _ {x} , N _ {y}\right)} = \sum_ {i = 1} ^ {N _ {x}} \sum_ {a = 1} ^ {N _ {y}} \frac {M _ {i a}}{\min \left(N _ {x} , N _ {y}\right)}. \tag {5} +$$ + +CCA is sensitive to perturbations when the condition number of $\mathbf{X}$ or $\mathbf{Y}$ is large [21]. To enhance robustness, Singular Value CCA (SVCCA) performs CCA on the truncated singular values of $\mathbf{X}$ and $\mathbf{Y}$ [44]. In this approach, the sum of the overlap matrix $\mathbf{M}$ is truncated to include only the first few components. To avoid confusion, from now on, we will refer to SVCCA truncated to the top ten components3 for both $\mathbf{X}$ and $\mathbf{Y}$ as CCA, i.e. (SV)CCA = $\frac{1}{10} \sum_{i=1}^{10} \sum_{a=1}^{10} M_{ia}$ . + +Centered Kernel Alignment (CKA) is a summary statistic of whether two representations agree on the (dis)similarity between a pair of examples based on their dot products [16]. CKA is defined as $\frac{\mathrm{Tr}\Sigma_x\Sigma_y}{\sqrt{\mathrm{Tr}\Sigma_x^2\mathrm{Tr}\Sigma_y^2}}$ and essentially measures the angle between two Gram matrices. In terms of spectral components, it can be expressed as: + +$$ +\mathrm {C K A} = \sum_ {i = 1} ^ {P} \sum_ {a = 1} ^ {P} \frac {\lambda_ {i}}{\sqrt {\sum_ {j = 1} ^ {P} \lambda_ {j} ^ {2}}} \frac {\mu_ {a}}{\sqrt {\sum_ {b = 1} ^ {P} \mu_ {b} ^ {2}}} M _ {i a}. \tag {6} +$$ + +Note that CKA is very similar to CCA but with additional (normalized) eigenvalue weighting. CKA will be the main focus of our work. + +Representational Similarity Analysis (RSA) is a popular method in neuroscience used to compare different brain regions in response to the same set of stimuli [30]. It is similar to CKA, except RSA compares pair-wise Euclidean distances instead of pair-wise inner products. Recent work has established its equivalence to CKA when RSA is combined with an extra centering step [52]. Therefore, our analyses are directly applicable to (centered-)RSA. + +# 3 Theoretical Background + +Treating $\Sigma_{x}$ and $\Sigma_{y}$ as random matrices described in Sec. 2, we leverage results from random matrix theory [43] to compute deterministic equivalents of average CCA and CKA in the asymptotic limit. Defining $q_{x} = P / N_{x}$ and $q_{y} = P / N_{y}$ , we consider the limit $P, N_{x}, N_{y} \to \infty$ by keeping $q_{x}, q_{y} \sim \mathcal{O}(1)$ . + +Both similarity measures depend on the cross-overlap between sample eigenvectors $M_{ia}$ defined in Eq. (4). Asymptotically $M_{ia}$ decouples as [7] + +$$ +M _ {i a} = \sum_ {j, b} Q _ {i j} ^ {x} \tilde {M} _ {j b} Q _ {b a} ^ {y}, \tag {7} +$$ + +where the self-overlaps $Q_{ij}^{x}$ and $Q_{ab}^{y}$ can be computed analytically [32]. The self-overlap matrix for $\mathbf{X}$ can be expressed in terms of the resolvent matrix $\mathbf{G}(z) = (z - \boldsymbol{\Sigma})^{-1}$ given by: + +$$ +Q _ {i j} = C \lim _ {\eta \rightarrow 0 ^ {+}} \operatorname {I m} \mathbf {G} _ {j j} \left(\lambda_ {i} - i \eta\right), \tag {8} +$$ + +where $C$ is a constant and the resolvent $\mathbf{G}(z)$ has a deterministic equivalent defined by the following self-consistent equation + +$$ +\mathbf {G} _ {i j} (z) = \frac {\delta_ {i j}}{z - \tilde {\lambda} _ {j} (1 + q (z \mathfrak {g} (z) - 1))}, \quad \mathfrak {g} (z) = \frac {1}{P} \operatorname {T r} \mathbf {G} (z). \tag {9} +$$ + +We provide a detailed derivation of these results in SI.A. Here, we note that the complex function $\mathfrak{g}(z)$ and Eq. (8) can be solved numerically (see SI.D for details). + +Main result. Asymptotically, we obtain an analytical formula for sample CCA (Eq. (5)) and sample CKA (Eq. (6)) by replacing the cross-overlap matrix $M_{ia}$ with its deterministic equivalent (Eq. (7)). + +Several remarks are in order: + +- While the theory for CCA and CKA should generally apply to the cases where both models are sampled, henceforth, we fix one of the models to be deterministic for practical reasons. Often, neural similarity measures are applied to compare biological data with limited neuron recordings to an + +artificial model where the entire population is available. For example fixing model $\mathbf{Y}$ implies that its self-overlap $\mathbf{Q}^y$ is just an identity matrix, hence simplifying Eq. (7) to $\mathbf{M} = \mathbf{Q}^x\tilde{\mathbf{M}}$ . + +- The analytical formula for CCA and CKA depends only on the population quantities. However, since the self-overlap matrix $Q_{ij}$ in Eq. (8) explicitly depends on individual eigencomponents, its deterministic equivalent specifically depends on the expected sample eigenvalue for the $i^{\text{th}}$ component $(\mathbb{E}\lambda_i)$ and the population eigenvalue for the $j^{\text{th}}$ component $(\tilde{\lambda}_j)$ . The latter makes it harder to apply the theory when the population eigenvalues cannot be observed. We discuss this issue further in Sec. 4.2. + +Sample Eigenvalues: Theoretical values of individual sample eigenvalues $\mathbb{E}[\lambda_i]$ can be predicted given the population eigenvalues by solving the following integral equation [43] + +$$ +\int_ {\mathbb {E} \left[ \lambda_ {i} \right]} ^ {\infty} \rho (\lambda) d \lambda = \frac {i}{P}, \quad \rho (\lambda) = \frac {1}{\pi} \lim _ {\eta \rightarrow 0 ^ {+}} \operatorname {I m} \mathfrak {g} (\lambda - i \eta), \tag {10} +$$ + +where $\rho (\lambda)$ is the deterministic equivalent of the empirical eigenvalue density (see SI.A.2). Computing $\mathbb{E}[\lambda_i]$ this way may be problematic due to numerical instabilities. Alternatively, one can exploit the fact that each single-trial eigenvalue concentrates around this mean with trial-to-trial fluctuations of $\mathcal{O}(1 / \sqrt{P})$ [43] and simply replace $\mathbb{E}[\lambda_i]$ with a single-trial observation in the large $P$ limit. We provide a detailed account of this approximation in SI.A.5. + +Sample Eigenvectors: Unlike eigenvalues, the sample eigenvectors $\langle u_i|\tilde{u}_j\rangle^2$ exhibit trial-to-trial fluctuations that persist even as $P\to \infty$ (see SI.A.6). Still, we can compute the mean value of the overlap represented by the squared overlap $Q_{ij}$ in Eq. (3). + +BBP Phase Transition: In addition to inevitable fluctuation in the sample eigenvectors, their mean behavior can still differ markedly from that of the population eigenvectors. A classic example is the Baik-Ben Arous-Péché (BBP) phase transition [4]. Consider a population Gram matrix with one large "spike" eigenvalue and the rest equal to 1. Depending on whether the spike strength exceeds a critical threshold determined by $P / N$ , the sample eigenvector associated with it can either have an $\mathcal{O}_P(1)$ with the true eigenvector (localized) or can be completely uncorrelated (delocalized). We depict this transition in Fig. 1a. + +Numerical Confirmation: Finally, we numerically test the theoretical prediction for self-overlap given by Eq. (8) on the eigenvectors of deep neural network activations. We extract layer activations from a pre-trained ResNet18 on CIFAR-10 images and subsample $N$ neurons through random projection. In Fig. 1b, we show the self-overlap $Q_{ii}$ for the first few eigenvectors of the layer activations and demonstrate a perfect match with theory. As the number of neurons decreases, the number of delocalized eigenvectors increases since fewer eigenvectors have self-overlap $Q_{ii} \approx 1$ . + +The effect of eigenvector delocalization $^{4}$ is reflected in the CKA between the sampled and population layer activations as shown in Fig. 1c. The alignment is completely misleading when small numbers of neurons are sampled, which poses a significant problem for practical purposes. + +![](images/dd6acd47e6029b1486f76b498d4476ca969cb894afe297c010537c48b0ac9887.jpg) +a) +Spike $>\lambda_{c}$ (Localized) + +![](images/754f153a8ceaa7b482d3a61e38e19742bc3f9352fba6b60b7283455f4158d345.jpg) +Spike $< \lambda_{c}$ (Delocalized) + +![](images/8518193b5f6451c999f7a43e10204b44a11c78149a5332bc086a0dda3657957c.jpg) +Index of eigenvector $i$ +Figure 1: a) Illustration of eigenvector delocalization in BBP phase transition. b) Self-overlap $Q_{ii}$ between sample and population eigenvectors for ResNet18 activations. c) CKA between population and sample activations when $N$ neurons are sampled. The gray-shaded region represents the standard deviation of empirical CKA across different random samplings. + +![](images/d205e7fb22e7e8170db2221576952a0aa14b5e843ca3944f6a07bc537b6b3009.jpg) +Neuron size $N$ + +# 4 Applying Theory to Representation Similarity + +# 4.1 Forward Problem: Impact of Neuron Sampling on Similarity + +In the forward problem, we assume that the population eigenvalues and eigenvectors are known. The first step is to obtain the typical sample eigenvalues by running a single-trial numerical simulation. We then move on to the eigenvectors by computing $\mathbf{Q}$ using Eq. (8). Finally, we calculate the overlap between the two systems, $\mathbf{M}$ , using Eq. (7). Having these components allows us to evaluate both CCA and CKA as functions of the number of neurons $N$ . + +As illustrated in Fig. 2, the theoretical predictions obtained from this eigen-decomposition match the observed CCA and CKA across different values of $N$ . Notice that CKA decreases when the number of neurons $N$ is reduced. As discussed above, both of these effects can be explained by the delocalization of eigenvectors. + +![](images/2c62bd96d978a6a6597ea986c4618987ca5e05d0344bc3229377855352e47087.jpg) +Figure 2: Comparison of sample vs population measures for CKA and CCA: Error bars represent empirical sample similarity and dotted lines the theoretical predictions. The black dotted line marks the true population similarity which is set close to 0.5 for both measures. Solid lines indicate inferred true similarity from samples. Sample similarity is lower due to eigenvector delocalization, while our method consistently provides a closer estimate of the true value. + +![](images/01609345e005050a5071649724c6e65ce47b2a17ed6f26b9653046f55df0d0d1.jpg) + +# 4.2 Backward Problem: Inferring Population Similarity from Limited Neurons + +Just like in our earlier analysis, inferring the population representational similarity begins with estimating the eigenvalues of the underlying population. In general, this is difficult because sample eigenvalues can deviate substantially from their population counterparts. Moreover, if $N < P$ , there are $P - N$ zero eigenvalues in the sample covariance matrix, further complicating the problem. + +However, if we adopt a parametric form, we can often achieve significant improvements in accuracy [41]. Here, we assume a power-law spectrum of the form $\tilde{\lambda}_i = i^{-1 - \gamma}$ , and develop a numerical method based on random matrix theory that reliably infers the true decay rate of population eigenvalues based on only the sample eigenvalues (see SI.C for detailed analysis). One can also consider more sophisticated eigenvalue models (e.g. broken power-law [41]) are also possible. + +While it is possible to estimate the population eigenvalues for general spectra [33], they require estimating each eigenvalue individually and are hence computationally expensive. Here, we only consider the power-law spectrum because 1) we only need to estimate a single parameter and 2) we can derive a closed-form expression for the population eigenvalues (see S.I.C for derivation) and 3) it has been shown to be relevant for biological systems [49]. + +After estimating the population eigenvalues $\{\tilde{\lambda}_i\}$ , we address the eigenvectors by computing the self-overlap matrix $\mathbf{Q}$ using Eq. (8). Since every population eigenbasis produces the same mean self-overlap, estimated population eigenvalues $\{\tilde{\lambda}_i\}$ are sufficient to find $\mathbf{Q}$ . + +Our final goal is to estimate the population cross-overlaps $\tilde{\mathbf{M}}$ , which are required to infer the true population similarity between two systems. Here, we propose a constrained optimization problem to invert the forward relationship $\mathbf{M} = \mathbf{Q}\tilde{\mathbf{M}}$ using the estimated $\mathbf{Q}$ and the observed $\mathbf{M}$ , as shown in Alg. 1. + +Two challenges arise in this naive approach. First, eigenvector statistics do not self-average [43], so the empirical cross-overlap $\mathbf{M}$ deviates from its expected value. This discrepancy can be partially mitigated by trial averaging or statistical bootstrapping. Second, the self-overlap $\mathbf{Q}$ is not invertible unless $P \ll N$ . As a result, it is impossible to recover the entire matrix $\tilde{\mathbf{M}}$ . Intuitively, only the first few eigenvectors are well-localized; the rest delocalize and lose information, so we can only reliably retrieve the corresponding columns of $\tilde{\mathbf{M}}$ . + +While the constrained optimization provides point estimates of $\tilde{\mathbf{M}}$ , it does not directly quantify their uncertainty. To assess the reliability of these estimates, we additionally derive confidence intervals under a maximum likelihood estimation (MLE) framework. This allows us to estimate statistical uncertainty for each $\tilde{M}_{ja}$ , as well as for composite quantities such as CKA and CCA (see SI.G). + +
Algorithm 1 Inferring Pop. Cross-Overlap M
Require: {λi}P{i=1}: Sample eigenvalues6: Step 2: Compute Self-overlap matrix
1: P: # of stimuli, N: # of neurons7: Q ← function{λi},P,N)
2: M ∈ RP×P: sample cross-overlap8: Step 3: Optimize Population Similarity
3: Step 1: Estimate Population Eigenvalues9: Mest ← arg minM ||M - Q · M||F
4: Assume power-law ansatz: λi ∝ i-1-γ10: s.t. Mij ∈ [0,1] for ∀i,j
5: Find γ that best explains {λi}P{i=1}return Mest
+ +# 4.2.1 Up to How Many Eigenvectors Can We Resolve for Given $N, P$ ? + +Consider a power-law spectrum, which decays relatively quickly. Under such a spectrum, only the leading sample eigenvectors tend to be well-localized, as shown in Fig. 3(Left). If we run the backward algorithm, we observe that for a given $N$ , $P$ , we can reliably recover only those initial components that remain localized, as shown in Fig. 3(Right). + +Practical implication. For power-law population spectra $\tilde{\lambda}_i\propto i^{-1 - \gamma}$ , the critical localization index scales as $i^{\star}(N)\approx \frac{1 + \gamma}{\sqrt{8}}\sqrt{N}$ (see SI.H). This provides a simple rule of thumb: with $N$ recorded neurons, one can reliably resolve roughly $\frac{1 + \gamma}{\sqrt{8}}\sqrt{N}$ leading eigenvectors. Conversely, to stably recover the top $k$ components, it is sufficient to record $N\gtrsim \frac{8}{(1 + \gamma)^2} k^2$ neurons. + +![](images/530e3ed04548ddd78ceafb8a00e94c4cf40f9d98d55f5f53d7da8c07e2adfbf7.jpg) +Figure 3: Left: Participation ratio (P.R.) of self-overlap $(1 / \sum_{j}Q_{ij}^{2})$ , indicating the onset of eigenvector delocalization, for a power-law spectrum $\tilde{\lambda}_i\sim i^{-1.2}$ . For fixed $N$ , increasing $P$ marginally affects the leading eigenvectors. By contrast, for fixed $P$ , increasing $N$ makes more eigenvectors localized. Only sample eigenvectors below the black horizontal line are localized (P.R. $\approx 1$ ). Heuristically, $\tilde{M}_{ia}$ can be recovered reliably for only indices below this line. Right: Each column shows the 5-trial averaged M, the theoretical prediction of M, the inferred population overlap $\tilde{\mathbf{M}}_{est}$ , and the actual population overlap $\tilde{\mathbf{M}}$ . With fewer neurons $N$ , sample eigenvectors become delocalized, causing large discrepancies. Nevertheless, our inference method successfully recovers the dominant overlaps, which are enough for global similarity measures such as CKA and CCA. + +![](images/e2f95fa603f3a550e201bc0a546e0e20c8c98acc51d812978bdb421f5a3edf7c.jpg) + +We can explicitly truncate these eigenvectors by taking a partial inverse of $\mathbf{Q}$ (see SI.E). However, this approach can be numerically unstable and might produce values of $M_{ij}$ outside the $[0,1]$ range. + +Additionally, Fig. 3(Left) demonstrates that, under a power-law of the same exponent, varying $P$ has a subtler effect on these leading indices than varying $N$ , which significantly affects localization. + +# 4.2.2 Why This Is Sufficient for Inferring Population Similarity + +Although our denoising approach only manages to recover the leading few eigenvectors (those that remain localized), it is precisely these components that matter most for similarity measures like CKA and (SV)CCA. As shown in Fig. 2, these metrics are governed primarily by the initial eigenvalues and eigenvectors. Thus, even with a very limited number of neurons, estimating those leading components is sufficient for practical purposes. + +Note that for CKA (and not CCA), there is an alternative approach to infer population similarity called the moment-based estimator [22, 28, 14], which computes the similarity using unbiased statistics. This method is more suitable for estimating similarity with small datasets but lacks theoretical insight. In contrast, our approach provides an analytical framework for studying how spectral properties precisely alter the observed similarity, but it assumes sufficiently large datasets so that all biases are negligible. + +# 5 Experiments + +# 5.1 Synthetic Data with a Known Population Gram Matrix + +We first evaluate our approach on a synthetic dataset where the population Gram matrix is fully specified, allowing us to directly compare our estimated similarity measures against the ground-truth population values. + +Fig. 2 illustrates that our forward and backward procedures work well. In the forward approach, we show that the eigencomponent-based analysis matches the empirical results closely. In the backward approach, even with an extremely limited number of neurons ( $N \approx 20$ ), our method infers a population similarity close to the actual value, despite the observed sample similarity being substantially lower. + +Since the population eigenvectors are known, we can also verify how well the inferred overlaps match the true overlaps. Specifically, Fig. 3(Right) displays the top-left $10 \times 10$ block of each matrix: the empirical M, the theoretical M (second column), the inferred population overlap $\tilde{\mathbf{M}}_{est}$ (third column), and the actual population overlap $\tilde{\mathbf{M}}$ (fourth column). In this example, we set $\tilde{\pmb{\Sigma}}_x = \tilde{\pmb{\Sigma}}_y$ , and hence the actual population cross-overlap should be the identity matrix. However, with fewer neurons, the sample eigenvectors become more delocalized, as evident in the first column. The theoretical prediction of this phenomenon (second column) aligns closely with the empirical observation. Notably, even with severely limited neurons, our backward-inference method recovers a cross-overlap matrix $\tilde{\mathbf{M}}_{est}$ (third column) much closer to the true identity than the naive observed M. + +# 5.1.1 Sampling Neurons Can Change Representation Similarity Ranking + +Next, we showcase a synthetic example in which sampling can lead to a reversal in the similarity rankings of models. Specifically, we construct two models: + +- Model 1 has significant overlap with the Brain on its first 3 population eigenvectors. +- Model 2 has significant overlap with the Brain on the next 3 eigenvectors. + +We set the total population (SV)CCA of Model 2 to be higher than that of Model 1. However, as neurons are sampled, eigenvectors corresponding to larger indices (smaller eigenvalues) tend to delocalize more. Hence, the empirical cross-overlap $\mathbf{M}$ for Model 2 deteriorates faster, causing its (SV)CCA to drop more than that of Model 1. Eventually, Model 1 overtakes Model 2 in the sample-based (SV)CCA ranking, as illustrated in Fig. 4(Left). + +Fig. 4(Right) presents the empirical and population cross-overlaps of the two models (each compared to the Brain). We set $P = 200$ and $N = 30$ , and all population eigenvalues follow a power-law with exponent $-1.2$ . Model 2's higher-dimensional overlaps delocalize more strongly, producing an apparent discrepancy that flips their observed ranking once neuron sampling is taken into account. + +![](images/31bedad556b90c3842a64e966a75b6b1e72541130393b42a5f338e2d98e061a1.jpg) + +![](images/ea8ef39d25224c0fc80b5443993be038fe67066d2efa9213b80a08f17ea92804.jpg) +Figure 4: Left: Sample-based CCA ranking flips despite Model 2 having a larger population CCA than Model 1. The decrease in Model 2's CCA is more pronounced due to its stronger reliance on higher-indexed eigenvectors, which become more delocalized with limited neuron sampling. Right: Empirical vs. population cross-overlaps for Model 1 vs. Brain and Model 2 vs. Brain. Here, $P = 200$ and $N = 30$ . All three population eigenvalue spectra follow a power-law with exponent $-1.2$ . Although Model 2's true overlap is higher at the population level, it relies on higher-indexed (smaller eigenvalue) components, which delocalize more severely in the sample. + +# 5.2 Brain Data + +Finally, we apply our denoising framework to real neural recordings in the primate visual cortex, comparing them against various computational model predictions. (for experimental details see SI.D) + +In Fig. 5, we illustrate a scatter plot of the representation similarity for different models compared to neural responses from V2 cortex [18, 46], given an artificially limited neuron count of $N = 20$ out of 103 neurons. The $x$ -axis corresponds to the observed sample CKA or CCA, while the $y$ -axis is our inferred population measure. Observe that our inference method consistently produces higher population similarity estimates than the naive sample estimates. In particular, certain models that appear to have lower similarity (when judged by the raw, sample-based metric) can actually exhibit higher true similarity to the brain once sampling effects are taken into account. + +![](images/22ba1f8059c54a5530c76528ae444c78cf0bdaded218c1b25e71f42a6b386e9e.jpg) +Figure 5: Scatter plots of observed sample similarity vs. inferred population similarity for multiple models compared to V2 cortex, using only $N = 20$ neurons (out of a larger set). (Left) CKA results; (Right) CCA results. The dotted line $y = x$ indicates equality. Notice that the inferred population similarity is consistently higher than the naive sample-based measure, demonstrating how limited neuron sampling can lead to underestimation of the true model-brain correspondence. + +# 6 Conclusion and Outlook + +We have presented an eigencOMPponent-based analysis of how sampling a finite number of neurons affects similarity measures, including CCA and CKA. By applying methods from Random Matrix Theory, we established that this limited sampling systematically underestimates similarity because of eigenvector delocalization in the sample Gram matrices. Our framework provides: + +- Forward Analysis: Predicting how Population eigenvalues and eigenvectors will manifest under neuron sampling, thus explaining the observed drop in similarity. +- Backward Inference: Denoising algorithm to infer Population representation similarity from limited data, overcoming the biases introduced by sampling noise. + +We validated our approach on both synthetic and real datasets. In the synthetic experiments, where the population Gram matrices were fully known, we showed that our method reliably recovers the true population overlaps and similarity values, even in regimes with very few neurons. Importantly, we highlighted a striking effect of sampling: under certain configurations, the ranking of two models with respect to the brain can be inverted when only a limited set of neurons is recorded. In real datasets from primate visual cortex, our method consistently produced higher population similarity estimates than naive sample-based methods, underscoring that the observed decrease in similarity is largely a sampling artifact. + +Moreover, for representations with power-law eigenspectra, we identified a universal scaling law: the number of well-localized eigenvectors grows as the square root of the number of recorded neurons. This $\sqrt{N}$ behavior offers a practical rule of thumb—researchers can estimate how many principal components can be reliably resolved for a given neuron count, or conversely, how many neurons are needed to capture a desired number of components (see SI.H). This scaling bridges theoretical predictions with experimental design, guiding how to interpret and plan neural recording studies under finite sampling constraints. + +Future Directions. There are several promising avenues for extending our work. First, it would be valuable to explore more sophisticated spectral priors—such as broken power-law spectra—to account for multiple functional subpopulations in the data, each contributing a distinct spectral structure. Second, while we have focused on sampling noise, future work should incorporate explicit models of additive noise that arise in real-time neurophysiological recordings, relaxing the assumption that trial averaging eliminates most of it. Third, improved denoising methods could be developed by adopting Bayesian approaches to model the joint distribution of sample eigenvectors and population eigenvectors [38], thus allowing more accurate recovery of the population eigenspaces. Finally, as we outline in SI.B, our framework naturally extends to regression settings, where sampling-induced distortions in eigencomponents can adversely affect regression scores, much like their impact on representational similarity measures. + +Overall, our results suggest that practical neuroscience studies must account for sampling-induced eigenvector delocalization when interpreting representational similarity. By unveiling the intrinsic biases introduced by limited neuron sampling and proposing a systematic solution, we aim to provide neuroscientists and machine learning researchers with more reliable tools for comparing computational models and neural data. + +# 7 Limitations + +Our framework assumes that neural responses arise from Gaussian (linear) projections of latent population codes. This yields analytical tractability via random matrix theory, but real data can exhibit non-Gaussian statistics, nonlinearities, and stimulus-dependent covariances, which may reduce the quantitative accuracy of our estimators when higher-order dependencies dominate. + +In addition, the similarity measures we study rely on specific symmetry assumptions; our method currently models rotational invariance only, ignoring other relevant symmetries (e.g., translation, scaling, permutation). It also does not yet cover dynamic (time-resolved/trajectory) or nonlinear similarity metrics. Extending the theory to these richer classes remains a crucial area for future research. + +# Acknowledgments and Disclosure of Funding + +This work was supported by the Center for Computational Neuroscience at the Flatiron Institute of the Simons Foundation, as well as by a Sloan Research Fellowship and a Klingenstein-Simons Award (to S.C.). 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The contributions—eigenvectors, validation on real neural data—are outlined accurately. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: The paper explicitly discusses limitations and future directions in the conclusion section. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [No] + +Justification: LLMs were not used as components of the research methodology. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Detailed Derivation of the Main Result + +# A.1 The Sample Gram Matrix + +Let $\tilde{\mathbf{X}}\in \mathbb{R}^{P\times \tilde{N}_x}$ denote the true population matrix with $P$ samples and $\tilde{N}_x$ neurons. We consider sampling only in the neuron/feature axis. The sample data $\mathbf{X}\in \mathbb{R}^{P\times N_x}$ is obtained by applying an $\tilde{N}_x\times N_x$ random projection matrix $\mathbf{R}_x$ on $\tilde{\mathbf{X}}$ + +$$ +\mathbf {X} = \tilde {\mathbf {X}} \mathbf {R} _ {x}, \quad \left(\mathbf {R} _ {x}\right) _ {i j} \sim \mathcal {N} \left(0, \frac {1}{N _ {x}}\right). \tag {S1} +$$ + +The population and sample Gram matrices and their corresponding eigencomponents are denoted as + +$$ +\tilde {\boldsymbol {\Sigma}} _ {x} = \tilde {\mathbf {X}} \tilde {\mathbf {X}} ^ {\top} = \sum_ {i = 1} ^ {P} \tilde {\lambda} _ {i} | \tilde {u} _ {i} \rangle \langle \tilde {u} _ {i} |, +$$ + +$$ +\boldsymbol {\Sigma} _ {x} = \mathbf {X} \mathbf {X} ^ {\top} = \sum_ {i = 1} ^ {P} \lambda_ {i} | u _ {i} \rangle \langle u _ {i} |. \tag {S2} +$$ + +In Random Matrix Theory (RMT), it is often convenient to consider matrices of the form $\mathbf{M} = \sqrt{\mathbf{C}}\mathbf{W}\sqrt{\mathbf{C}}$ , where $\mathbf{W} = \mathbf{R}\mathbf{R}^{\top}$ is a random Wishart matrix and $\mathbf{C}$ is a deterministic square matrix. We first put $\boldsymbol{\Sigma}_{x}$ into this form to simplify our calculations [27]. The sample Gram matrix can be written in terms of the SVD components of $\tilde{\mathbf{X}} = \mathbf{U}\tilde{\boldsymbol{\Lambda}}_x^{1/2}\mathbf{V}^{\top}$ + +$$ +\boldsymbol {\Sigma} _ {x} = \tilde {\mathbf {X}} \mathbf {R} _ {x} \mathbf {R} _ {x} ^ {\top} \tilde {\mathbf {X}} ^ {\top} = \mathbf {U} \tilde {\boldsymbol {\Lambda}} _ {x} ^ {1 / 2} \left(\mathbf {V} ^ {\top} \mathbf {R} _ {x} \mathbf {R} _ {x} ^ {\top} \mathbf {V}\right) \tilde {\boldsymbol {\Lambda}} _ {x} ^ {1 / 2} \mathbf {U} ^ {\top}, \tag {S3} +$$ + +where $\tilde{\Lambda}_x\in \mathbb{R}^{P\times \tilde{N}_x}$ is a diagonal matrix, and $U\in \mathbb{R}^{P\times P}$ and $V\in \mathbb{R}^{\tilde{N}_x\times \tilde{N}_x}$ orthogonal matrices. Since deterministic orthogonal transformations of Wishart matrices are again Wishart matrices, we get: + +$$ +\boldsymbol {\Sigma} _ {x} = \mathbf {U} \tilde {\boldsymbol {\Lambda}} _ {x} ^ {1 / 2} \mathbf {W} _ {x} \tilde {\boldsymbol {\Lambda}} _ {x} ^ {1 / 2} \mathbf {U} ^ {\top}, \tag {S4} +$$ + +where $\mathbf{W}_x = \mathbf{V}^\top \mathbf{R}_x\mathbf{R}_x^\top \mathbf{V}$ is a random Wishart matrix with aspect ratio $\phi_x = \tilde{N}_x / N_x$ . We divide our discussion into two cases: + +- When $P \geq \tilde{N}_x$ , the eigenvalue matrix can be completed to a $P \times P$ -matrix by zero padding and replacing $\mathbf{W}_x$ with a Wishart matrix with $q_x = P / N_x$ . Using the orthogonality of $\mathbf{U}$ , this allows us to express $\boldsymbol{\Sigma}_x$ as + +$$ +\boldsymbol {\Sigma} _ {x} = \left(\mathbf {U} \tilde {\boldsymbol {\Lambda}} _ {x} ^ {1 / 2} \mathbf {U} ^ {\top}\right) \left(\mathbf {U} \mathbf {W} _ {x} \mathbf {U} ^ {\top}\right) \left(\mathbf {U} \tilde {\boldsymbol {\Lambda}} _ {x} ^ {1 / 2} \mathbf {U} ^ {\top}\right) = \sqrt {\tilde {\boldsymbol {\Sigma}} _ {x}} \mathbf {W} _ {x} \sqrt {\tilde {\boldsymbol {\Sigma}} _ {x}}, \tag {S5} +$$ + +where $\mathbf{W}_x$ is a Wishart matrix with aspect ratio $q_{x} = P / N_{x}$ + +- When $P < N_{x}$ , the eigenvalue matrix and the Wishart matrix can be written as + +$$ +\tilde {\boldsymbol {\Lambda}} _ {x} = \left( \begin{array}{l l} \tilde {\boldsymbol {\Lambda}} _ {x} ^ {\prime} & \mathbf {0} \end{array} \right), \quad \mathbf {W} _ {x} = \binom {\mathbf {R} _ {1}} {\mathbf {R} _ {2}} \left( \begin{array}{l l} \mathbf {R} _ {1} ^ {\top} & \mathbf {R} _ {2} ^ {\top} \end{array} \right), \tag {S6} +$$ + +where the $P \times P$ matrix $\tilde{\Lambda}_x'$ is the non-zero part of $\tilde{\Lambda}_x$ and $\mathbf{R}_1 \in \mathbb{R}^{P \times N_x}$ , $\mathbf{R}_2 \in \mathbb{R}^{(\tilde{N}_x - P) \times N_x}$ are two projection matrices. Plugging these back in, we arrive at the same form as the previous case. + +In both cases, the statistics of $\Sigma_{x}$ does not depend explicitly on $\tilde{N}_{x}$ . + +# A.2 Eigenvalue statistics of sample Gram matrices + +One of the main objectives of RMT is to understand the eigenvalue distribution of random matrices in terms of deterministic quantities [43]. Here, we review some classical results on the eigenvalue statistics of random matrices of the form $\boldsymbol{\Sigma} = \sqrt{\tilde{\boldsymbol{\Sigma}}}\mathbf{W}\sqrt{\tilde{\boldsymbol{\Sigma}}}$ where $\mathbf{W}$ is a $P\times N$ Wishart matrix + +with ratio $q = \frac{P}{N}$ . Here, $\pmb{\Sigma}$ and $\hat{\pmb{\Sigma}}$ are the sample and population Gram matrices, and they have the following eigendecompositions + +$$ +\boldsymbol {\Sigma} = \sum_ {i = 1} ^ {P} \lambda_ {i} | u _ {i} \rangle \langle u _ {i} |, \quad \tilde {\boldsymbol {\Sigma}} = \sum_ {i = 1} ^ {P} \tilde {\lambda} _ {i} | \tilde {u} _ {i} \rangle \langle \tilde {u} _ {i} |. \tag {S7} +$$ + +We denote their (discrete-)eigenvalue distribution by $\rho (\lambda)$ and $\tilde{\rho} (\tilde{\lambda})$ + +$$ +\rho (\lambda) = \frac {1}{P} \sum_ {i = 1} ^ {P} \delta \left(\lambda - \lambda_ {i}\right), \quad \tilde {\rho} (\tilde {\lambda}) = \frac {1}{P} \sum_ {i = 1} ^ {P} \delta \left(\tilde {\lambda} - \tilde {\lambda} _ {i}\right). \tag {S8} +$$ + +We define the resolvent of the random matrix $\mathbf{X}$ and its trace as + +$$ +\mathbf {G} (z) = (z - \boldsymbol {\Sigma}) ^ {- 1} = \sum_ {i = 1} ^ {P} \frac {| u _ {i} \rangle \langle u _ {i} |}{z - \lambda_ {i}}. \tag {S9} +$$ + +The Stieltjes transform of the empirical spectral distribution is defined as + +$$ +\mathfrak {g} ^ {P} (z) := \int \frac {\rho (\lambda)}{z - \lambda} d \lambda = \frac {1}{P} \operatorname {T r} \mathbf {G} (z). \tag {S10} +$$ + +In the large $P$ limit, this quantity is self-averaging and there is a deterministic equivalent $\mathfrak{g}(z) \sim \mathfrak{g}^P(z)$ given by the self-consistent equation + +$$ +\mathfrak {g} (z) = \int \frac {\tilde {\rho} (\tilde {\lambda})}{z - \tilde {\lambda} (1 - q + q z \mathfrak {g} (z))} d \tilde {\lambda}, \tag {S11} +$$ + +which only depends on the deterministic eigenvalues $\tilde{\rho}_x(\tilde{\lambda})$ and the ratio $q = P / N$ [43]. In practical applications, $\tilde{\rho}_x(\tilde{\lambda})$ is often replaced with the uniform measure over the population eigenvalues $\{\tilde{\lambda}_i\}$ as defined in Eq. (S8). This remarkable result was first obtained in [37] for white Wishart matrices (for which $\tilde{\Sigma} = \mathbf{I}$ ). + +Due to the equivalence $\mathfrak{g}(z)\sim \mathfrak{g}^P (z)$ in large $P$ limit, these two integrals are equivalent + +$$ +\int \frac {\rho (\lambda)}{z - \lambda} d \lambda \underset {P \rightarrow \infty} {\rightarrow} \int \frac {\tilde {\rho} (\tilde {\lambda})}{z - \tilde {\lambda} (1 - q + q z \mathfrak {g} (z))} d \tilde {\lambda}, \tag {S12} +$$ + +from which one can obtain the density of the limiting spectral density using the inversion formula [8] + +$$ +\rho (\lambda) = \frac {1}{\pi} \lim _ {\eta \rightarrow 0 ^ {+}} \operatorname {I m} \mathfrak {g} (\lambda - i \eta). \tag {S13} +$$ + +The Stieltjes transform also connects to the effective regularization in ridge regression [6, 24, 11, 2]. We define a new function $\kappa(z)$ as + +$$ +\kappa (z) := - \frac {z}{1 - q + q z \mathfrak {g} (z)}, \quad \mathfrak {g} (z) = z ^ {- 1} - q ^ {- 1} \left(z ^ {- 1} + \kappa (z) ^ {- 1}\right) \tag {S14} +$$ + +and express Eq. (S11) in terms of this quantity: + +$$ +\mathfrak {g} (z) = \frac {\kappa (z)}{z} \int \frac {\tilde {\rho} (\tilde {\lambda})}{\tilde {\lambda} + \kappa (z)} d \tilde {\lambda} = z ^ {- 1} - q ^ {- 1} \left(z ^ {- 1} + \kappa (z) ^ {- 1}\right). \tag {S15} +$$ + +Then, we obtain a new self-consistent equation for $\kappa$ + +$$ +\kappa (z) = - z + \kappa (z) \int \frac {q \tilde {\lambda}}{\tilde {\lambda} + \kappa (z)} \tilde {\rho} (\tilde {\lambda}) d \tilde {\lambda}, \tag {S16} +$$ + +which is also known as the Silverstein equation [48]. Expressing this in terms of the discrete population eigenvalues, and evaluating it at $z = -\lambda$ , we get + +$$ +\kappa = \lambda + \kappa \frac {1}{N} \sum_ {i = 1} ^ {P} \frac {\tilde {\lambda} _ {i}}{\tilde {\lambda} _ {i} + \kappa}, \tag {S17} +$$ + +which is exactly the equation for the renormalized ridge parameter in [11, 2] with the scaling $\tilde{\lambda}_i\rightarrow N\tilde{\lambda}_i$ + +# A.3 Eigenvector statistics of sample Gram matrices and the self-overlap matrix + +This result from Eq. (S11) can also be generalized to the resolvent matrix itself [27, 8], which becomes diagonal in the population eigenbasis: + +$$ +\mathbf {G} (z) = \sum_ {i = 1} ^ {P} \frac {\left| u _ {i} \right\rangle \left\langle u _ {i} \right|}{z - \lambda_ {i}} \sim \sum_ {i = 1} ^ {P} \frac {\left| \tilde {u} _ {i} \right\rangle \left\langle \tilde {u} _ {i} \right|}{z - \tilde {\lambda} _ {i} (1 - q + q z \mathfrak {g} (z))}, \tag {S18} +$$ + +where the integral over eigenvalues is replaced by the discrete measure over population eigenvalues. This allows us to study the eigenvector statistics by analyzing the quantity + +$$ +\langle \tilde {u} _ {j} | \mathbf {G} (z) | \tilde {u} _ {j} \rangle = \sum_ {i = 1} ^ {P} \frac {\left\langle \right. u _ {i} \left. \right| \tilde {u} _ {j} \left. \right\rangle^ {2}}{z - \lambda_ {i}} \sim \frac {1}{z - \tilde {\lambda} _ {j} (1 - q + q z \mathfrak {g} (z))}. \tag {S19} +$$ + +In the large $P$ limit, the sum over empirical eigenvalues becomes an integral: + +$$ +\langle \tilde {u} _ {j} | \mathbf {G} (z) | \tilde {u} _ {j} \rangle \underset {P \rightarrow \infty} {\longrightarrow} \int \frac {Q (\lambda , \tilde {\lambda} _ {j})}{z - \lambda} \rho (\lambda) d \lambda , \tag {S20} +$$ + +where we defined $Q(\lambda_i, \tilde{\lambda}_j) \coloneqq P\langle u_i|\tilde{u}_j\rangle^2$ is the overlap between the $i^{\mathrm{th}}$ sample eigenvector and the $j^{\mathrm{th}}$ population eigenvector. Now, we can obtain $Q(\lambda_i, \tilde{\lambda}_j)$ using the following inversion formula + +$$ +Q \left(\lambda_ {i}, \tilde {\lambda} _ {j}\right) = \frac {1}{\pi \rho \left(\lambda_ {i}\right)} \lim _ {\eta \rightarrow 0 ^ {+}} \operatorname {I m} \langle \tilde {u} _ {j} | \mathbf {G} \left(\lambda_ {i} - i \eta\right) | \tilde {u} _ {j} \rangle . \tag {S21} +$$ + +Using the equivalence in Eq. (S19) and evaluating this expression explicitly: + +$$ +Q \left(\lambda_ {i}, \tilde {\lambda} _ {j}\right) = \frac {q \lambda_ {i} \tilde {\lambda} _ {j}}{\left[ \tilde {\lambda} _ {j} (1 - q) - \lambda_ {i} + q \lambda_ {i} \tilde {\lambda} _ {j} \mathfrak {h} \left(\lambda_ {i}\right) \right] ^ {2} + \left[ q \lambda_ {i} \tilde {\lambda} _ {j} \pi \rho \left(\lambda_ {j}\right) \right] ^ {2}}, \tag {S22} +$$ + +we get an explicit formula for eigenvector overlaps [32, 8], where $\rho(\lambda_i)$ is given by Eq. (S13) and $\mathfrak{h}(z)$ is its Hilbert transform: + +$$ +\mathfrak {h} (z) = \mathrm {p . v .} \int \frac {\rho (\lambda)}{z - \lambda} d \lambda . \tag {S23} +$$ + +and can be obtained from the Stieltjes transform via + +$$ +\lim _ {\eta \rightarrow 0 ^ {+}} \mathfrak {g} (z - i \eta) = \mathfrak {h} (z) + i \pi \rho (z). \tag {S24} +$$ + +# A.4 Overlap formula for two Gram matrices + +Here, we provide a short review of the work by Bun et al. [7] which derives an overlap formula between eigenvectors from random matrices. We consider observations from two representations $\mathbf{X} \in \mathbb{R}^{P \times N_x}$ and $\mathbf{Y} \in \mathbb{R}^{P \times N_y}$ in response to a common set of inputs of size $P$ . Their sample Gram matrices have decompositions: + +$$ +\boldsymbol {\Sigma} _ {x} = \mathbf {X} \mathbf {X} ^ {\top} = \sum_ {i = 1} ^ {P} \lambda_ {i} | u _ {i} \rangle \langle u _ {i} |, \quad \boldsymbol {\Sigma} _ {y} = \mathbf {Y} \mathbf {Y} ^ {\top} = \sum_ {a = 1} ^ {P} \mu_ {a} | w _ {a} \rangle \langle w _ {a} |. \tag {S25} +$$ + +We assume that $\mathbf{X}$ and $\mathbf{Y}$ are observations sampled from the underlying population features $\tilde{\mathbf{X}}\in$ $\mathbb{R}^{P\times \tilde{N}_x}$ and $\tilde{\mathbf{Y}}\in \mathbb{R}^{P\times \tilde{N}_y}$ through independent random projections. The corresponding population Gram matrices are decomposed as: + +$$ +\tilde {\boldsymbol {\Sigma}} _ {x} = \tilde {\mathbf {X}} \tilde {\mathbf {X}} ^ {\top} = \sum_ {i = 1} ^ {P} \tilde {\lambda} _ {i} | \tilde {u} _ {i} \rangle \langle \tilde {u} _ {i} |, \quad \tilde {\boldsymbol {\Sigma}} _ {y} = \tilde {\mathbf {Y}} \tilde {\mathbf {Y}} ^ {\top} = \sum_ {a = 1} ^ {P} \tilde {\mu} _ {a} | \tilde {w} _ {a} \rangle \langle \tilde {w} _ {a} |. \tag {S26} +$$ + +We consider two sample data matrices $\mathbf{X} \in \mathbb{R}^{P \times N_x}$ and $\mathbf{Y} \in \mathbb{R}^{P \times N_y}$ . In Sec. A.1, we showed that the sample Gram matrices can be expressed in terms of the population ones as: + +$$ +\boldsymbol {\Sigma} _ {x} = \sqrt {\tilde {\boldsymbol {\Sigma}} _ {x}} \mathbf {W} _ {x} \sqrt {\tilde {\boldsymbol {\Sigma}} _ {x}}, +$$ + +$$ +\boldsymbol {\Sigma} _ {y} = \sqrt {\tilde {\boldsymbol {\Sigma}} _ {y}} \mathbf {W} _ {y} \sqrt {\tilde {\boldsymbol {\Sigma}} _ {y}}, \tag {S27} +$$ + +where the Wishart matrices $\mathbf{W}_x$ and $\mathbf{W}_y$ have aspect ratios $q_x = P / N_x$ and $q_y = P / N_y$ , respectively. Resolvents of the sample Gram matrices are + +$$ +\mathbf {G} _ {x} (z) \equiv (z - \boldsymbol {\Sigma} _ {x}) ^ {- 1} = \sum_ {i = 1} ^ {P} \frac {| u _ {i} \rangle \langle u _ {i} |}{z - \lambda_ {i}}, \quad \mathbf {G} _ {y} \left(z ^ {\prime}\right) \equiv \left(z ^ {\prime} - \boldsymbol {\Sigma} _ {y}\right) ^ {- 1} = \sum_ {a = 1} ^ {P} \frac {| w _ {a} \rangle \langle w _ {a} |}{z ^ {\prime} - \mu_ {a}}. \tag {S28} +$$ + +We want to compute + +$$ +\psi_ {P} \left(z, z ^ {\prime}\right) = \mathbb {E} \left[ \frac {1}{P} \operatorname {T r} \left[ \mathbf {G} _ {x} (z) \mathbf {G} _ {y} \left(z ^ {\prime}\right) \right] \right] = \mathbb {E} \left[ \frac {1}{P ^ {2}} \sum_ {i, a = 1} ^ {P} \frac {P \left\langle u _ {i} \mid w _ {a} \right\rangle^ {2}}{\left(z - \lambda_ {i}\right) \left(z ^ {\prime} - \mu_ {a}\right)} \right], \tag {S29} +$$ + +where the expectation is over random realizations of sample Gram matrices [7]. In the limit $P\to \infty$ as empirical eigenvalues become continuous, this object approaches a deterministic function + +$$ +\psi_ {P} \left(z, z ^ {\prime}\right) \sim \psi (z, z ^ {\prime}) = \int \frac {\rho_ {x} (\lambda) \rho_ {y} (\mu)}{(z - \lambda) \left(z ^ {\prime} - \mu\right)} M (\lambda , \mu) d \lambda d \mu , \quad M \left(\lambda_ {i}, \mu_ {a}\right) \sim \mathbb {E} \left[ P \left\langle u _ {i} \mid w _ {a} \right\rangle^ {2} \right] \tag {S30} +$$ + +Here, $\rho_{x}(\lambda),\rho_{y}(\mu)$ are the eigenvalue densities of $\pmb{\Sigma}_{x}$ $\pmb{\Sigma}_{y}$ given by Eq. (S13). The function $M(\lambda_i,\mu_a)\sim \mathbb{E}\left[P\langle u_i|w_a\rangle^2\right]$ denotes the expected overlap between two eigenvectors associated with eigenvalues $\lambda_{i}$ and $\mu_{a}$ , and it is the central object for our analysis since it directly appears in CCA and CKA. This quantity can be obtained by computing $\psi (\lambda_i - i\eta ,\mu_a + i\eta ')$ , collecting the term proportional to $\eta \eta^\prime$ and taking the limit $\eta ,\eta^{\prime}\rightarrow 0$ [7]: + +$$ +\begin{array}{l} \psi (\lambda_ {i} - i \eta , \mu_ {a} + i \eta^ {\prime}) = \int \frac {(\lambda_ {i} - \lambda + i \eta) \rho_ {x} (\lambda)}{(\lambda_ {i} - \lambda) ^ {2} + \eta^ {2}} \frac {(\mu_ {a} - \mu - i \eta^ {\prime}) \rho_ {y} (\mu)}{(\mu_ {a} - \mu) ^ {2} + \eta^ {\prime 2}} M (\lambda , \mu) d \lambda d \mu \\ = \int \frac {\eta \rho_ {x} (\lambda)}{(\lambda_ {i} - \lambda) ^ {2} + \eta^ {2}} \frac {\eta^ {\prime} \rho_ {y} (\mu)}{(\mu_ {a} - \mu) ^ {2} + \eta^ {\prime 2}} M (\lambda , \mu) d \lambda d \mu + (\dots) \\ \underset {\eta , \eta^ {\prime} \rightarrow 0} {=} \pi^ {2} \rho_ {x} \left(\lambda_ {i}\right) \rho_ {y} \left(\mu_ {a}\right) M \left(\lambda_ {i}, \mu_ {a}\right) + (\dots) \tag {S31} \\ \end{array} +$$ + +To simplify, we will assume that the population eigenvectors form a complete set of basis: + +$$ +\mathbf {I} = \sum_ {i = 1} ^ {P} | \tilde {u} _ {i} \rangle \langle \tilde {u} _ {i} | = \sum_ {a = 1} ^ {P} | \tilde {w} _ {a} \rangle \langle \tilde {w} _ {a} |. \tag {S32} +$$ + +Then each resolvent in Eq. (S29) can be expressed in these bases: + +$$ +\mathbf {G} _ {x} (z) = \sum_ {i, j} \left| \tilde {u} _ {i} \right\rangle \left\langle \tilde {u} _ {j} \right| \Phi_ {i j} ^ {x} (z), \quad \Phi_ {i j} ^ {x} (z): \left\langle \tilde {u} _ {i} \mid \mathbf {G} _ {x} (z) \mid \tilde {u} _ {j} \right\rangle , +$$ + +$$ +\mathbf {G} _ {y} (z) = \sum_ {a, b} | \tilde {w} _ {a} \rangle \langle \tilde {w} _ {b} | \Phi_ {a b} ^ {y} \left(z ^ {\prime}\right), \quad \Phi_ {a b} ^ {y} \left(z ^ {\prime}\right) := \langle \tilde {w} _ {a} | \mathbf {G} _ {y} (z) | \tilde {w} _ {b} \rangle , \tag {S33} +$$ + +where $\Phi_{ij}^{x}$ and $\Phi_{ab}^{y}$ are the matrix elements of resolvents $\mathbf{G}_x(z)$ and $\mathbf{G}_y(z')$ in their respective deterministic bases. Then, Eq. (S29) simplifies to + +$$ +\psi_ {P} (z, z ^ {\prime}) = \mathbb {E} \left[ \frac {1}{P} \sum_ {i, j, a, b} \Phi_ {i j} ^ {x} (z) \tilde {C} _ {j a} \Phi_ {a b} ^ {y} \left(z ^ {\prime}\right) \tilde {C} _ {b i} ^ {\top} \right] = \frac {1}{P} \sum_ {i, j, a, b} \mathbb {E} \left[ \Phi_ {i j} ^ {x} (z) \right] \tilde {C} _ {j a} \mathbb {E} \left[ \Phi_ {a b} ^ {y} \left(z ^ {\prime}\right) \right] \tilde {C} _ {b i} ^ {\top}, \tag {S34} +$$ + +where we defined the deterministic overlap matrix elements $\tilde{C}_{ia} \coloneqq \langle \tilde{u}_i|\tilde{w}_a\rangle$ . In the second equality, we assumed that the two resolvents are independent, reducing the problem to computing the expected resolvent of a single Gram matrix. + +As discussed around Eq. (S19), the resolvent $\mathbf{G}_x$ has a limiting value for $P\to \infty$ that is diagonal in the corresponding deterministic basis [8], and its matrix elements are given by: + +$$ +\Phi_ {i j} ^ {x} (z) = \frac {\delta_ {i j}}{z - \tilde {\lambda} _ {i} \left(1 - q _ {x} + q _ {x} z \mathfrak {g} _ {x} (z)\right)} + \mathcal {O} \left(P ^ {- 1 / 2}\right), \tag {S35} +$$ + +where $\mathfrak{g}_x(z)$ satisfies the self-consistency condition in Eq. (S11). + +In order to compute the overlap $M(\lambda_i, \mu_a)$ , we use Eq. (S31) and collect the term proportional to $\eta \eta'$ . Thanks to Eq. (S34) and Eq. (S35), this term simplifies to: + +$$ +\pi^ {2} \rho_ {x} (\lambda_ {i}) \rho_ {y} (\mu_ {a}) M (\lambda_ {i}, \mu_ {a}) = \frac {1}{P} \sum_ {j, b} \left(\lim _ {\eta \rightarrow 0} \operatorname {I m} \Phi_ {j j} ^ {x} (\lambda_ {i} - i \eta)\right) \tilde {C} _ {j b} ^ {2} \left(\lim _ {\eta^ {\prime} \rightarrow 0} \operatorname {I m} \Phi_ {b b} ^ {y} (\mu_ {a} - i \eta^ {\prime})\right). \tag {S36} +$$ + +Defining + +$$ +Q _ {x} \left(\lambda_ {i}, \tilde {\lambda} _ {j}\right) := \frac {1}{\pi \rho_ {x} \left(\lambda_ {i}\right)} \lim _ {\eta \rightarrow 0} \operatorname {I m} \Phi_ {j j} ^ {x} \left(\lambda_ {i} - i \eta\right), \quad Q _ {y} \left(\mu_ {a}, \tilde {\mu} _ {b}\right) := \frac {1}{\pi \rho_ {y} \left(\eta_ {a}\right)} \lim _ {\eta^ {\prime} \rightarrow 0} \operatorname {I m} \Phi_ {b b} ^ {y} \left(\mu_ {a} - i \eta^ {\prime}\right) \tag {S37} +$$ + +we get an equation for $M$ as + +$$ +M \left(\lambda_ {i}, \mu_ {a}\right) = \frac {1}{P} \sum_ {j, b} Q _ {x} \left(\lambda_ {i}, \tilde {\lambda} _ {j}\right) \tilde {C} _ {j b} ^ {2} Q _ {y} \left(\mu_ {a}, \tilde {\mu} _ {b}\right). \tag {S38} +$$ + +Here, $Q_{x}$ and $Q_{y}$ were already calculated in Eq. (S22). Identifying the following quantities + +$$ +Q _ {i j} ^ {x} \equiv \mathbb {E} \left\langle u _ {i} | \tilde {u} _ {j} \right\rangle^ {2} = \frac {1}{P} Q _ {x} (\lambda_ {i}, \tilde {\lambda} _ {j}), Q _ {a b} ^ {y} \equiv \mathbb {E} \left\langle w _ {a} | \tilde {w} _ {b} \right\rangle^ {2} = \frac {1}{P} Q _ {y} (\mu_ {a}, \tilde {\mu} _ {b}), +$$ + +$$ +M _ {i a} \equiv \mathbb {E} \left\langle u _ {i} \mid w _ {a} \right\rangle^ {2} = \frac {1}{P} M \left(\lambda_ {i}, \mu_ {a}\right), \quad \tilde {M} _ {i a} := \left\langle \tilde {u} _ {i} \mid \tilde {w} _ {a} \right\rangle^ {2} = \tilde {C} _ {i a} ^ {2}, \tag {S39} +$$ + +we get our main result [7]: + +$$ +\mathbf {M} = \mathbf {Q} ^ {x} \tilde {\mathbf {M}} \mathbf {Q} ^ {y ^ {\top}}, +$$ + +$$ +Q _ {i j} ^ {x} = \frac {1}{P} \frac {q _ {x} \lambda_ {i} \tilde {\lambda} _ {j}}{\left[ \tilde {\lambda} _ {j} (1 - q _ {x}) - \lambda_ {i} + q _ {x} \lambda_ {i} \tilde {\lambda} _ {j} \mathfrak {h} _ {x} (\lambda_ {i}) \right] ^ {2} + \left[ q _ {x} \lambda_ {i} \tilde {\lambda} _ {j} \pi \rho_ {x} (\lambda_ {i}) \right] ^ {2}}, +$$ + +$$ +Q _ {a b} ^ {y} = \frac {1}{P} \frac {q _ {y} \mu_ {a} \tilde {\mu} _ {b}}{\left[ \tilde {\mu} _ {b} \left(1 - q _ {y}\right) - \mu_ {a} + q _ {y} \mu_ {a} \tilde {\mu} _ {b} \mathfrak {h} _ {y} (\mu_ {a}) \right] ^ {2} + \left[ q _ {y} \mu_ {a} \tilde {\mu} _ {b} \pi \rho_ {y} (\mu_ {a}) \right] ^ {2}}. \tag {S40} +$$ + +# A.5 Statistics of sample eigenvalues and its concentration properties + +As we discussed in the main text, the practical usage of Eq. (S22) requires computing the expectation value of individual sample eigenvalues. Eq. (S40), treating $i^{\mathrm{th}}$ biggest sample eigenvalue as deterministic and plugging $\eta = 1 / \sqrt{P}$ + +For sufficient conditions, we can show that the sample resolvent $\mathfrak{g}(z)$ self-averages. In this case, the sample eigenvalue density $\rho(\lambda)$ converges in law. Here, we show that for practical use of Eq. (S21), Eq. (S40), we can treat $i^{\mathrm{th}}$ largest eigenvalue effectively as deterministic in its most probable position. + +Specifically, we demonstrate that for a large number of eigenvalues $P$ , the most probable $i$ -th largest eigenvalue $\lambda_{i}$ satisfies + +$$ +\int_ {\lambda_ {i}} ^ {\infty} \rho (\lambda) d \lambda = \frac {i}{P}, \tag {S41} +$$ + +and that the fluctuations around this most probable is $O(1 / \sqrt{P})$ . + +Consider a set of $P$ eigenvalues $\{\lambda_1, \lambda_2, \dots, \lambda_P\}$ drawn independently from the probability density $\rho(\lambda)$ . We order these eigenvalues in descending order: + +$$ +\lambda_ {(1)} \geq \lambda_ {(2)} \geq \ldots \geq \lambda_ {(P)}, +$$ + +where $\lambda_{(i)}$ denotes the $i^{\mathrm{th}}$ largest eigenvalue. To find the most probable value $\lambda_{i}$ for the $i^{\mathrm{th}}$ largest eigenvalue, we focus on the probability that exactly #i eigenvalues exceed a threshold $\bar{\lambda}$ . If we define + +$$ +F (\bar {\lambda}) = \int_ {\bar {\lambda}} ^ {\infty} \rho (\lambda^ {\prime}) d \lambda^ {\prime}, +$$ + +then the probability that exactly $i$ out of $P$ samples exceed $\bar{\lambda}$ is given by the binomial expression + +$$ +F (\bar {\lambda}, P, i) = \left( \begin{array}{c} P \\ i \end{array} \right) \left[ F (\bar {\lambda}) \right] ^ {i} \left[ 1 - F (\bar {\lambda}) \right] ^ {P - i}. +$$ + +We determine the threshold $\bar{\lambda}_i$ that maximizes $F(\bar{\lambda}, P, i)$ by setting its derivative (with respect to $\bar{\lambda}$ ) to zero. From this calculation, one obtains the simple condition + +$$ +F (\bar {\lambda} _ {i}) = \frac {i}{P}. +$$ + +Equivalently, since $F(\lambda) = \int_{\lambda}^{\infty} \rho(\lambda') d\lambda'$ , the most probable $i^{\text{th}}$ largest eigenvalue $\lambda_i$ satisfies + +$$ +\int_ {\lambda_ {i}} ^ {\infty} \rho (\lambda) d \lambda = \frac {i}{P}. +$$ + +Now we calculate approximations for fluctuation around this most probable position. Let's analyze $F(\lambda, P, i)$ near $\bar{\lambda}_i$ . Write $\bar{\lambda} = \lambda_i + \delta \lambda$ and expand $F(\bar{\lambda})$ in a Taylor series about $\lambda_i$ : + +$$ +F (\bar {\lambda}) = F (\lambda_ {i} + \delta \lambda) \approx F (\lambda_ {i}) + \left. \frac {d F}{d \lambda} \right| _ {\lambda_ {i}} \delta \lambda + \frac {1}{2} \left. \frac {d ^ {2} F}{d \lambda^ {2}} \right| _ {\lambda_ {i}} (\delta \lambda) ^ {2} + \ldots +$$ + +Since $F(\lambda_i) = \frac{i}{P}$ and $\lambda_i$ is determined by maximizing $F(\bar{\lambda}, P, i)$ , the first derivative of $F$ at $\lambda_i$ vanishes: + +$$ +\left. \frac {d F}{d \lambda} \right| _ {\lambda_ {i}} = 0, +$$ + +thus + +$$ +F (\bar {\lambda}) \approx \frac {i}{P} + \frac {1}{2} F ^ {\prime \prime} (\lambda_ {i}) (\delta \lambda) ^ {2}. +$$ + +(We expect $F''(\lambda_i) < 0$ since $F(\lambda)$ decreases with $\lambda$ .) + +Substituting this expansion back into $\binom{P}{i}\left[F(\bar{\lambda})\right]^i\left[1-F(\bar{\lambda})\right]^{P-i}$ , we find that the dominant dependence on $\delta \lambda$ appears in a Gaussian-like factor + +$$ +\exp \Bigl (- \frac {1}{2} \left| F ^ {\prime \prime} (\lambda_ {i}) \right| P (\delta \lambda) ^ {2} \Bigr). +$$ + +This indicates that $\bar{\lambda}$ is peaked sharply around $\lambda_{i}$ with a variance + +$$ +\sigma_ {i} ^ {2} = \frac {1}{- F ^ {\prime \prime} (\lambda_ {i}) P}. +$$ + +In summary, most probable $i$ -th largest eigenvalue $\lambda_{(i)}$ is determined by + +$$ +\int_ {\lambda_ {i}} ^ {\infty} \rho (\lambda) d \lambda = \frac {i}{P}, +$$ + +with fluctuation $O(1 / \sqrt{P})$ + +# A.6 Statistics of sample eigenvalues and its concentration properties + +Note that unlike eigenvalue density converges in law, eigenvector statistics Eq. (S21) is noisy even when $P \to \infty$ [43]. In this case, we define the $Q$ matrix as the expectation over different trials as in Eq. (S39). Equivalently, this could be obtained by averaging over a small eigenvalue interval, which could be done by plugging in a small $\eta = 1 / \sqrt{P}$ to extract the pole. Note that this $1 / \sqrt{P}$ is also obtained by analyzing fluctuation around the most probable $i$ -th biggest eigenvalue as above. This is essentially averaging over a Cauchy distribution centered at $\lambda$ with width $\eta$ . Thus, for practical usage of Eq. (S40), we simply plug this most likely $i$ -th eigenvalue [8], with $\eta = 1 / \sqrt{P}$ . + +# B Relation to regression-based similarity measures + +Regression Score is not a representational similarity measure but is commonly used for scoring model closeness to the brain [46, 12]. Here, we discuss how our theoretical analysis for the overlap matrix $\mathbf{M}$ can also be applied to the regression setting. Regression score measures how well a model's activations $\mathbf{X}$ predict neural responses $\mathbf{Y}$ via a linear probe. Concretely, one performs ridge regression on a training subset $(\mathbf{X}_{1:p}, \mathbf{Y}_{1:p})$ of size $p < P$ , obtaining: + +$$ +\hat {\mathbf {X}} (p) = \mathbf {Y} \hat {\boldsymbol {\beta}} (p). \tag {S42} +$$ + +$$ +\hat {\beta} (p) = \arg \min _ {\beta} \| \mathbf {Y} _ {1: p} \beta - \mathbf {X} _ {1: p} \| _ {F} ^ {2} + \alpha_ {\mathrm {r e g}} \| \beta \| _ {F} ^ {2}, \tag {S43} +$$ + +Then the regression score gives the neural prediction error, + +$$ +E _ {g} (p) = \frac {\| \hat {\mathbf {X}} (p) - \mathbf {X} \| _ {F} ^ {2}}{\| \mathbf {X} \| _ {F} ^ {2}}, \tag {S44} +$$ + +Note that this error can be decomposed to each error mode, where $E_{g}(p) = \sum_{i}\widetilde{W}_{i}(p)$ where $\widetilde{W}_i(p)\coloneqq \frac{\kappa^2}{1 - \gamma}\frac{W_i}{(p\lambda_i + \kappa)^2}$ . + +The quantity $W_{i}$ denotes the projection of target labels on the $i^{\mathrm{th}}$ -model eigenvalue and hence can be expressed in terms of eigencomponents, $W_{i} = \sum_{j}\frac{\lambda_{j}}{\sum_{k}\lambda_{k}} M_{ij}$ . However, calculating $W_{i}$ assumes that there is access to population-level eigenvalues and poses a problem with limited data. In future work, we would like to test whether our analyses help improve the reliability of regression-based similarity methods. + +# C Theory of Power-Law Spectrum + +Here, we consider the case where the population spectrum obeys a power-law: + +$$ +\tilde {\lambda} _ {k} = \left(\frac {k}{P}\right) ^ {- s}, \quad k = 1, \dots , P, \quad s > 1 \tag {S45} +$$ + +where we normalized eigenvalue indices explicitly by $P$ . For large $P$ , the population density becomes: + +$$ +\tilde {\rho} (\tilde {\lambda}) = \frac {1}{P} \sum_ {k = 1} ^ {P} \delta (\tilde {\lambda} - \tilde {\lambda} _ {k}) \sim \frac {1}{P} \int_ {1} ^ {P} \delta (\tilde {\lambda} - \tilde {\lambda} _ {k}) d k, \tag {S46} +$$ + +We change the variables to $\mu \coloneqq \tilde{\lambda}_k$ for which we get: + +$$ +d \mu = - s P ^ {s} k ^ {- s - 1} d k = - \frac {s}{P} \mu^ {1 + 1 / s} d k. \tag {S47} +$$ + +In the limit $P\to \infty$ , the density becomes + +$$ +\tilde {\rho} (\tilde {\lambda}) = \frac {1}{s} \int_ {1} ^ {\infty} \mu^ {- 1 - 1 / s} \delta (\tilde {\lambda} - \mu) d \mu = \gamma \tilde {\lambda} ^ {- 1 - \gamma}, \quad \tilde {\lambda} \in [ 1, \infty ], \quad \gamma = s ^ {- 1}, \tag {S48} +$$ + +where we defined $\gamma \in [0,1]$ for notational convenience. Note that, in this definition, the expectation value of $\tilde{\lambda}$ diverges. + +# C.1 Solving the Stieltjes transform + +Next, we need to solve the self-consistent equation for the Stieltjes transform Eq. (S11) which reads: + +$$ +\mathfrak {g} (z) = \int_ {1} ^ {\infty} \frac {\tilde {\rho} (\tilde {\lambda})}{z - \tilde {\lambda} (1 - q + q z \mathfrak {g} (z))} d \tilde {\lambda}, \quad \tilde {\rho} (\tilde {\lambda}) = \gamma \tilde {\lambda} ^ {- 1 - \gamma}. \tag {S49} +$$ + +This integral has a closed-form solution expressed in terms of hypergeometric functions [3]. To evaluate this integral, it is convenient to work in terms of the following quantities: + +$$ +w := \frac {z}{1 - q}, \quad \beta := \frac {q}{1 - q}, \quad \mathfrak {g} ^ {\prime} := z \mathfrak {g} (z), \quad \kappa := \frac {w}{1 + \beta \mathfrak {g} ^ {\prime}}. \tag {S50} +$$ + +Then the integral equation becomes + +$$ +\begin{array}{l} \mathfrak {g} ^ {\prime} = \gamma \kappa \int_ {1} ^ {\infty} \frac {\tilde {\lambda} ^ {- 1 - \gamma}}{\kappa - \tilde {\lambda}} d \tilde {\lambda} = - \gamma \kappa^ {- \gamma} \int_ {0} ^ {\kappa} \frac {u ^ {\gamma}}{1 - u} d u \\ = - \gamma \kappa^ {- \gamma} \mathrm {B} (\kappa ; 1 + \gamma , 0), \tag {S51} \\ \end{array} +$$ + +where we made a change of variables $u \coloneqq \kappa / \tilde{\lambda}$ in the first line and used the integral definition of the incomplete Beta function + +$$ +\mathrm {B} (z; a, b) = \int_ {0} ^ {z} u ^ {a - 1} (1 - u) ^ {b - 1} d u. \tag {S52} +$$ + +The integral solution reported in [3] can be obtained from Eq. (S51) by using the following identity [17] in terms of hypergeometric functions: + +$$ +\mathrm {B} (z; a, b) = \frac {z ^ {a} (1 - z) ^ {b}}{a} _ {2} F _ {1} (1, a + b; a + 1; z). \tag {S53} +$$ + +We rewrite the self-consistent equation by replacing the definition of $\kappa$ + +$$ +\mathfrak {g} ^ {\prime} = - \gamma \left(\frac {w}{1 + \beta \mathfrak {g} ^ {\prime}}\right) ^ {- \gamma} \mathsf {B} \left(\frac {w}{1 + \beta \mathfrak {g} ^ {\prime}}; 1 + \gamma , 0\right). \tag {S54} +$$ + +While this equation is exact, it is not possible to solve for $\mathfrak{g}$ . To obtain an analytical solution for the self-consistent equation, we need to expand the r.h.s. to leading order in $\mathfrak{g}'$ : + +$$ +\mathfrak {g} ^ {\prime} = - \gamma w ^ {- \gamma} \mathrm {B} (w; 1 + \gamma , 0) + \mathfrak {g} ^ {\prime} \beta \gamma \left(\frac {w}{1 - w} - \gamma w ^ {- \gamma} \mathrm {B} (w; 1 + \gamma , 0)\right) + \mathcal {O} \left(\left(\beta \mathfrak {g} ^ {\prime}\right) ^ {2}\right), \tag {S55} +$$ + +which can be truncated to the linear order provided that $\beta \mathfrak{g}' \ll 1$ . Solving for $\mathfrak{g}'$ , we get + +$$ +\mathfrak {g} ^ {\prime} = \frac {- \gamma w ^ {- \gamma} \mathrm {B} (w ; 1 + \gamma , 0)}{1 - \beta \gamma \left(\frac {w}{1 - w} - \gamma w ^ {- \gamma} \mathrm {B} (w ; 1 + \gamma , 0)\right)}. \tag {S56} +$$ + +We also provide a power series expansion of the incomplete beta function: + +$$ +B (w; 1 + \gamma , 0) = \left\{ \begin{array}{l l} \sum_ {n = 0} ^ {\infty} \frac {1}{n - \gamma} w ^ {\gamma - n} + \pi (\cot (\pi \gamma) - i), & \text {w h e n} w \gg 1 \\ \sum_ {n = 1} ^ {\infty} \frac {1}{n + \gamma} w ^ {\gamma + n}, & \text {w h e n} w \ll 1, \end{array} \right. \tag {S57} +$$ + +which will be helpful when we implement these functions numerically. In terms of the power series, the solution becomes: + +$$ +\begin{array}{l} \mathfrak {g} ^ {\prime} = - \gamma \frac {\pi (\cot (\pi \gamma) - i) w ^ {- \gamma} + \sum_ {n = 0} ^ {\infty} \frac {1}{n - \gamma} w ^ {- n}}{1 + \beta \gamma \left(\pi \gamma (\cot (\pi \gamma) - i) w ^ {- \gamma} + \sum_ {n = 0} ^ {\infty} \frac {\gamma}{n - \gamma} w ^ {- n} - \frac {w}{1 - w}\right)} \\ = - \gamma \frac {\pi (\cot (\pi \gamma) - i) w ^ {- \gamma} + \sum_ {n = 0} ^ {\infty} \frac {1}{n - \gamma} w ^ {- n}}{1 + \beta \gamma \left(\pi \gamma (\cot (\pi \gamma) - i) w ^ {- \gamma} + \sum_ {n = 0} ^ {\infty} \frac {n}{n - \gamma} w ^ {- n}\right)}, \tag {S58} \\ \end{array} +$$ + +where we simplified the denominator in the last line using $\frac{w}{1 - w} = -\sum_{n=0}^{\infty} w^{-n}$ . + +Next, we compute the sample eigenvalue density $\rho (\lambda)$ and its Hilbert transform $\mathfrak{h}(\lambda)$ by computing + +$$ +\lim _ {\eta \rightarrow 0 ^ {+}} \mathfrak {g} (\lambda - i \eta) = \lim _ {\eta \rightarrow 0 ^ {+}} \frac {\mathfrak {g} ^ {\prime} (\lambda - i \eta)}{\lambda - i \eta} = \mathfrak {h} (\lambda) + i \pi \rho (\lambda). \tag {S59} +$$ + +This is an extremely tedious calculation that we perform using Mathematica. Furthermore, we expand the results in $q$ and, assuming $q \ll 1$ , keep only the linear term. In this regime, the leading order behavior of $\rho(\lambda)$ and $\mathfrak{h}(\lambda)$ looks like: + +$$ +\rho (\lambda) = \gamma \lambda^ {- 1 - \gamma} \left(1 - q \gamma \left(2 \pi \gamma \cot (\pi \gamma) \lambda^ {- \gamma} + \sum_ {n = 1} ^ {\infty} \frac {n + \gamma}{n - \gamma} \lambda^ {- n}\right)\right) + \mathcal {O} (q ^ {2}) +$$ + +$$ +\begin{array}{l} \mathfrak {h} (\lambda) = \lambda^ {- 1} \left(1 - \lambda^ {- \gamma} \pi \gamma \cot (\pi \gamma) - \lambda^ {- 1} \frac {\gamma}{1 - \gamma}\right) \\ + \pi \gamma^ {2} q \left(\pi \gamma \lambda^ {- 2 \gamma - 1} \left(\cot^ {2} (\pi \gamma) - 1\right) + \lambda^ {- \gamma - 2} \frac {(\gamma + 1) \cot (\pi \gamma)}{1 - \gamma}\right) + \mathcal {O} \left(q ^ {2}, \lambda^ {3}\right). \tag {S60} \\ \end{array} +$$ + +Here, we did not include higher-order terms for $\mathfrak{h}(\lambda)$ to avoid clutter. + +Finally, we use the formula for estimating sample eigenvalues Eq. (S41) for which we obtain an explicit formula: + +$$ +\mathfrak {F} (\lambda , q; \gamma) := \int_ {\lambda} ^ {\infty} \rho (\lambda) d \lambda = \lambda^ {- \gamma} \left(1 - q \gamma^ {2} \left(\lambda^ {- \gamma} \pi \cot \pi \gamma + \sum_ {n = 1} ^ {\infty} \frac {1}{n - \gamma} \lambda^ {- n}\right)\right). \tag {S61} +$$ + +Here, the semi-colon separates sample-related arguments that we have empirical access to $(\lambda_i,q)$ from the only population-related quantity, $\gamma$ . Hence, using the following relation [34, 8] + +$$ +\mathfrak {F} (\lambda_ {i}, q; \gamma) = \frac {i}{P} \tag {S62} +$$ + +we can either predict the shape of empirical eigenvalues given the decay rate of population spectrum (forward), or infer the population decay rate given the empirical observations of eigenvalues (backward). Finally, we numerically test our theory and obtain perfect agreement with empirical data in Fig. S1. + +![](images/a6244653033ee23f470a399edc77fb401efd92b46ea3786d2acf067352941dce.jpg) +Figure S1: For a population spectrum with $\tilde{\lambda}_k = k^{-1.1}$ (Left) and $\tilde{\lambda}_k = k^{-2.4}$ (Right), we show the spectra of the empirical eigenvalues for different $N$ . Black solid line indicates the true eigenvalue decay. The numbers in parentheses in the legend indicate the inferred true decay rate from a population of $N$ . In the regime $s < 2$ ( $\gamma > 0.5$ ), the empirical eigenvalues are always overestimated (Left), and in the regime $s \geq 2$ ( $\gamma < 0.5$ ) they are always underestimated (Right). + +![](images/a4e16eee0b898334391c0d1412b3340abf8c42d7794e1e66970b0bbb46920688.jpg) + +# D Experimental Details + +Code for all experiments is publicly available in this Github repository. All experiments were done using a single A100 GPU. + +# D.1 Synthetic Data + +For the synthetic experiments, we generate a population activation matrix in $\mathbb{R}^{P\times \tilde{N}}$ whose Gram matrix follows a chosen spectral distribution (e.g., a power-law). We then form the sample activation matrix by projecting onto a random subset (or random linear subspace) of size $N$ , yielding $\mathbb{R}^{P\times N}$ . This procedure enables us to directly control the underlying population eigenvalues and eigenvectors, facilitating clean comparisons between sample-level and population-level similarity measures. + +# D.2 Brain Data + +We employ a set of publicly available neural recordings from primate visual cortex (e.g., V2) and compare these against the representations of various vision models, similarly to the methodology in [12]. In total, we evaluate 32 models spanning supervised, self-supervised, and adversarially trained architectures, including well-known families such as ResNet, DenseNet, MobileNet, EfficientNet, and Vision Transformers. We extract intermediate-layer activations for each model on the same set of visual stimuli used in the neural recordings, applying the standard preprocessing routines (e.g., image resizing, ImageNet normalization). + +Within each model, we select one or more representative layers (e.g., post-ReLU or transformer blocks). We then compute Gram matrices from those activations, matching the dimensionality of the neural dataset. In scenarios where the dataset contains more neurons than we wish to analyze, we project the data into a lower-dimensional subspace of size $N$ . Finally, we compute representational similarity (e.g., CKA or (SV)CCA) between these model-derived Gram matrices and the neural Gram matrices, both in their raw (sample) forms and using our denoising procedure for backward inference. + +# E Another denoising method: truncated inverse + +We utilize a truncated Singular Value Decomposition (SVD) to obtain a regularized estimate of $\hat{\mathbf{M}}$ : + +$$ +\tilde {\mathbf {M}} = \mathbf {V} \boldsymbol {\Sigma} _ {\text {t r u n c}} ^ {- 1} \mathbf {U} ^ {\top} \mathbf {M}, \tag {S63} +$$ + +where $\mathbf{Q}^{(x)} = \mathbf{U}\pmb{\Sigma}\mathbf{V}^{\top}$ is the SVD of $\mathbf{Q}^{(x)}$ , and $\pmb{\Sigma}_{\mathrm{trunc}}^{-1}$ is the truncated inverse of the singular values, defined as: + +$$ +\left(\boldsymbol {\Sigma} _ {\text {t r u n c}} ^ {- 1}\right) _ {i i} = \left\{ \begin{array}{l l} \frac {1}{\sigma_ {i}} & \text {i f} i \leq \tau , \\ 0 & \text {o t h e r w i s e}, \end{array} \right. \tag {S64} +$$ + +# F Sample CKA with population eigenvalue term + +We demonstrate that eigenvector delocalization is the dominant factor causing the decrease in sample CKA for rapidly decaying spectra such as power-law or exponential distributions. + +In Fig. S2, we consider a population with eigenvalues following $\tilde{\lambda}_i = i^{-1.2}$ and in Fig. S3, we examine a population with eigenvalues following $\tilde{\lambda}_i = e^{-i}$ . These experiments confirm that while sampling affects both eigenvalues and eigenvectors, the systematic underestimation of CKA is predominantly caused by eigenvector delocalization rather than changes to the eigenvalue distribution. This holds true for both power-law and exponential eigenvalue spectra, which are common in neural data. + +# G Confidence Interval under Maximum Likelihood Estimation + +Although it is hard to calculate the uncertainty of the estimator from constrained optimization, we can compute it using maximum likelihood estimation for $\tilde{M}_{ja}$ . + +![](images/89142be761b1c501af8cc479325771ff521aa8421cfb9d8f0169132175c0e63b.jpg) +Figure S2: Left: Normalized sample eigenvalues for $P = 100$ and $\tilde{\lambda}_i = i^{-1.2}$ with varying $N$ . The first few terms of normalized eigenvalues remain relatively stable despite neuron sampling. Right: Sample CKA between brain and model with identical representations (true CKA = 1) as neurons are sampled. The deviation is primarily due to eigenvector delocalization, as shown by the close match between observed sample CKA and CKA calculated with population eigenvalues but sample eigenvectors. + +![](images/b8a0e4d645277a4729f6f663b52863a031a2cb7d6770d2c9ec226123db47829c.jpg) + +![](images/299e74729f301d9deec8bd64dbee0546fcfb137eff36b2fbfb3b14520430af15.jpg) +Figure S3: Left: Normalized sample eigenvalues for $P = 100$ and $\tilde{\lambda}_i = e^{-i}$ with varying $N$ . As with the power-law case, the dominant normalized eigenvalue components remain relatively stable under sampling. Right: Sample CKA behavior with exponential eigenvalue decay. The observed sample CKA closely tracks the hybrid CKA (population eigenvalues with sample eigenvectors), confirming that eigenvector delocalization primarily drives the CKA reduction. + +![](images/a7118bc4865bd761edb4eeb7efd5b1e86051ca861402c91c3571b11aa977f6f9.jpg) + +# G.1 Problem Setup + +For two sets of eigenvectors $\{|u_j\rangle \}_{j = 1}^P$ (unobserved) and $\{|v_{a}\rangle \}_{a = 1}^{P}$ (observed), we estimate confidence intervals for $\tilde{M}_{ja} = \left|\left\langle u_j\big|v_a\right\rangle \right|^2$ based on empirical eigenvectors $\{| \hat { u } _ { i } ^ { ( t ) } \rangle \} _ { i = 1 } ^ { P }$ across trials $t$ + +Our neuron-wise sampling model assumes: + +$$ +\boxed {| \hat {u} _ {i} ^ {(t)} \rangle = \sum_ {j = 1} ^ {P} \epsilon_ {i j} ^ {(t)} \sqrt {Q _ {i j}} | u _ {j} \rangle , \quad \epsilon_ {i j} ^ {(t)} \sim \mathcal {N} (0, 1) i. i. d.} \tag {S65} +$$ + +where $Q\in \mathbb{R}^{P\times P}$ is fixed across trials. + +Projecting onto $|v_{a}\rangle$ gives: + +$$ +\left\langle v _ {a} \mid \hat {u} _ {i} ^ {(t)} \right\rangle \sim \mathcal {N} \left(0, \sigma_ {i a} ^ {2}\right), \quad \sigma_ {i a} ^ {2} = \sum_ {j = 1} ^ {P} Q _ {i j} \tilde {M} _ {j a} \tag {S66} +$$ + +The squared overlaps follow: + +$$ +M _ {i a} ^ {(t)} \equiv \left| \left\langle v _ {a} \right| \hat {u} _ {i} ^ {(t)} \right\rangle \Bigg | ^ {2} \sim \sigma_ {i a} ^ {2} \chi_ {1} ^ {2} \tag {S67} +$$ + +# G.2 Confidence Intervals (CI) + +# G.2.1 Single-Trial Case + +The negative log-likelihood for a single trial is: + +$$ +\ell_ {a} \left(\tilde {M}. a\right) = \frac {1}{2} \sum_ {i = 1} ^ {P} \left[ \ln \sigma_ {i a} ^ {2} + \frac {M _ {i a}}{\sigma_ {i a} ^ {2}} \right] \tag {S68} +$$ + +For a confidence interval on $\tilde{M}_{ka}$ , we use profile likelihood: + +$$ +L _ {\text {p r o f i l e}} (m) = \max _ {\left\{\tilde {M} _ {j a}: j \neq k, 0 \leq \tilde {M} _ {j a} \leq 1 \right\}} \exp \left(- \ell_ {a} (\tilde {M}. _ {a})\right) \quad \text {w i t h} \tilde {M} _ {k a} = m \tag {S69} +$$ + +The $(1 - \alpha)$ CI is: + +$$ +\mathrm {C I} _ {1 - \alpha} (\tilde {M} _ {k a}) = \left\{m \in [ 0, 1 ]: - 2 \log \left(L _ {\text {p r o f i l e}} (m) / L _ {\max }\right) \leq \chi_ {1, 1 - \alpha} ^ {2} \right\} \tag {S70} +$$ + +Algorithm 2 Profile CI for $\tilde{M}_{ka}$ (Single Trial) + +1: Input: $M_{ia}$ , $Q$ , index $k$ , level $1 - \alpha$ +2: Output: $[\tilde{M}_{ka}^{\mathrm{lower}},\tilde{M}_{ka}^{\mathrm{upper}}]$ +3: Set $\tau = \chi_{1,1 - \alpha}^2$ . Compute $L_{\mathrm{max}}$ by minimizing $\ell_{a}$ in Eq. (S68). +4: for $m$ on a grid in $[0,1]$ do +5: Minimize $\ell_{a}(\tilde{M}_{-a})$ subject to $\tilde{M}_{ka} = m$ and $0\leq \tilde{M}_{ja}\leq 1(j\neq k)$ +6: Set $\Lambda(m) = -2\log\left(L_{\text{profile}}(m) / L_{\max}\right)$ . +7: end for +8: Return the smallest and largest $m$ with $\Lambda(m) \leq \tau$ . + +# G.2.2 Multiple Trials + +For $T$ trials with the same $Q$ , the joint negative log-likelihood is: + +$$ +\ell_ {a} ^ {\text {j o i n t}} (\tilde {M}. _ {a}) = \sum_ {t = 1} ^ {T} \frac {1}{2} \sum_ {i = 1} ^ {P} \left[ \ln \sigma_ {i a} ^ {2} + \frac {M _ {i a} ^ {(t)}}{\sigma_ {i a} ^ {2}} \right] \tag {S71} +$$ + +The joint profile likelihood and CI follow analogously: + +$$ +\mathrm {C I} _ {1 - \alpha} ^ {(T)} (\tilde {M} _ {k a}) = \{m \in [ 0, 1 ]: \Lambda_ {T} (m) \leq \chi_ {1, 1 - \alpha} ^ {2} \} \tag {S72} +$$ + +Algorithm 3 Profile CI for $\tilde{M}_{ka}$ (Multiple Trials) + +1: Input: $\{M_{ia}^{(t)}\}_{t=1}^T$ , $Q$ , index $k$ , level $1-\alpha$ , optional weights $w_t$ +2: Output: $[\tilde{M}_{ka}^{\mathrm{lower}},\tilde{M}_{ka}^{\mathrm{upper}}]$ +3: Set $\tau = \chi_{1,1 - \alpha}^{2}$ . Compute $L_{\mathrm{max}}^{(T)}$ by minimizing $\ell_{a}^{\mathrm{joint}}$ . +4: for $m$ on a grid in $[0,1]$ do +5: Minimize $\ell_{a}^{\mathrm{joint}}(\tilde{M}_{a})$ subject to $\tilde{M}_{ka} = m$ and $0\leq \tilde{M}_{ja}\leq 1(j\neq k)$ +6: Set $\Lambda_T(m) = -2\log \left(L_{\mathrm{profile}}^{(T)}(m) / L_{\mathrm{max}}^{(T)}\right)$ . +7: end for +8: Return the smallest and largest $m$ with $\Lambda_T(m) \leq \tau$ . + +# G.3 CIs for CKA and CCA + +For weighted functionals like CKA or CCA: + +$$ +f (\tilde {M}) = \sum_ {j = 1} ^ {P} \sum_ {a = 1} ^ {P} w _ {j} w _ {a} ^ {\prime} \tilde {M} _ {j a} \tag {S73} +$$ + +Apply the delta method: + +1. Compute gradient: $\frac{\partial f}{\partial\tilde{M}_{j a}} = w_{j}w_{a}^{\prime}$ +2. Estimate Fisher information: $\mathcal{I}_{\hat{\tilde{M}}} = \nabla^{2}\ell^{\mathrm{joint}}(\hat{\tilde{M}})$ +3. Calculate variance: $\operatorname{Var}(f) = \nabla f^{\top} \mathcal{I}_{\hat{\hat{M}}}^{-1} \nabla f$ +4. CI: $f(\hat{\dot{M}}) \pm z_{1 - \alpha / 2} \cdot \sqrt{\operatorname{Var}(f)}$ + +![](images/9b0444d65e3d5696c17982cce10872d57df6109a2d4356b5b6d3ae1cebb39724.jpg) +Figure S4: MLE estimation and confidence interval of 5 trials. Here $P = 100$ , and the two population Gram matrices had power-law eigenvalues with exponent $-1.2$ . Left: For CKA. Blue dots are the empirical sample CKA, where the orange dotted line is the theoretical line for sample CKA. The green solid line is the estimated population CKA using MLE, and the blue shades are the $95\%$ confidence interval. Right: Same analysis but for CCA. + +![](images/3de8d1d6ee54ce9b5961e75f4bc95cf1fba99baba60200d6bc331ff3f1dc63bb.jpg) + +# H Eigenvector Delocalization via Support Mergers + +# H.1 Setup and diagnostic + +Let $\Sigma \in \mathbf{R}^{P\times P}$ have eigenvalues $\{\tilde{\lambda}_i\}_{i = 1}^P$ . From $N$ i.i.d. samples $X = \Sigma^{1 / 2}Z$ with $Z_{ij}\sim \mathcal{N}(0,1)$ , the sample covariance is $S = \frac{1}{N} XX^{\top}$ . A convenient diagnostic for support components is + +$$ +\mathcal {B} (x) := \frac {1}{x} + \frac {1}{N} \sum_ {i = 1} ^ {P} \frac {1}{\bar {\lambda} _ {i} - x}, \tag {S74} +$$ + +whose monotonicity between its poles $x_{i} = 1 / \tilde{\lambda}_{i}$ tracks gaps vs. support. Intervals with $\mathcal{B}'(x) < 0$ correspond to gaps; support edges occur where the local monotonicity changes (tangency/inflection). See [8] for related transform pictures. + +# H.2 From support mergers to eigenvector delocalization + +Decreasing $N$ is akin to increasing an effective noise level: nearby spikes mix first (Dyson Brownian motion intuition). Thus leading (well-separated) spikes remain isolated, while deeper ones merge into a common bulk. Empirically, the diagonal overlaps $Q_{ii}$ stay near 1 up to an index $i'$ and then drop sharply; the drop aligns with the point where a local neighborhood can no longer sustain two separated support components. + +![](images/c0c25e9c440f289feb122b29b794882682e30850221055602b86a55c52520e76.jpg) +Figure S5: Counting components via $\mathcal{B}$ . Blue segments on the vertical axis correspond to support intervals (adapted from Fig. 6 of [8]). +Figure S6: Heuristic. Additive Dyson Brownian motion shown; in our multiplicative setting, the same local-mixing picture applies: smaller inter-spike spacing $\lambda_{i} - \lambda_{i + 1}\Rightarrow$ earlier merger. + +$$ +\mathrm {d} \lambda_ {i} = \sqrt {\frac {2}{\beta N}} \mathrm {d} B _ {i} + \frac {1}{N} \sum_ {j = 1} ^ {N} \frac {\mathrm {d} t}{\lambda_ {i} - \lambda_ {j}} +$$ + +$$ +\mathrm{d}\mathbf{v}_{i} = \frac{1}{\sqrt{N}}\sum_{\substack{j = 1\\ j\neq i}}^{N}\frac{\mathrm{d}B_{ij}}{\lambda_{i} - \lambda_{j}}\mathbf{v}_{j} - \frac{1}{2N}\sum_{\substack{j = 1\\ j\neq i}}^{N}\frac{\mathrm{d}t}{(\lambda_{i} - \lambda_{j})^{2}}\mathbf{v}_{i} +$$ + +# H.3 Two-peak approximation and the $\sqrt{N}$ law + +Assume a power-law population spectrum + +$$ +\tilde {\lambda} _ {i} = i ^ {- 1 - \gamma}, \quad \gamma > 0, \tag {S75} +$$ + +so the poles of $\mathcal{B}$ are $x_{i} = 1 / \tilde{\lambda}_{i} = i^{1 + \gamma}$ . Between two consecutive poles $b = x_{i}$ and $a = x_{i + 1}$ $(a > b)$ , approximate locally + +$$ +\mathcal {B} _ {\mathrm {l o c}} (x) \approx \frac {1}{x} + \frac {1}{N} \left[ \frac {1}{a - x} + \frac {1}{b - x} \right]. \tag {S76} +$$ + +A local merger occurs when $x \mapsto \frac{1}{x}$ is tangent to the rational term (equivalently, equal value and slope at $x^{\star} \in (b, a)$ ). Solving the tangency system gives the critical sample size + +$$ +N _ {i} ^ {\star} = \frac {\left[ (i + 1) ^ {\frac {2}{3} (1 + \gamma)} + i ^ {\frac {2}{3} (1 + \gamma)} \right] ^ {3}}{\left[ (i + 1) ^ {1 + \gamma} - i ^ {1 + \gamma} \right] ^ {2}}. \tag {S77} +$$ + +For $N > N_{i}^{\star}$ , the two local components near $x_{i}$ and $x_{i + 1}$ remain separated; for $N < N_{i}^{\star}$ , they merge. + +Remark H.1 (Key: $\sqrt{N}$ scaling of eigenvector delocalization). Using $(i + 1)^{1 + \gamma} - i^{1 + \gamma} = (1 + \gamma)i^{\gamma} + o(i^{\gamma})$ and $(i + 1)^{\frac{2}{3}(1 + \gamma)} + i^{\frac{2}{3}(1 + \gamma)} = 2i^{\frac{2}{3}(1 + \gamma)} + o(i^{\frac{2}{3}(1 + \gamma)})$ , Eq. (S77) yields + +$$ +N _ {i} ^ {\star} = \frac {8}{(1 + \gamma) ^ {2}} i ^ {2} [ 1 + o (1) ]. +$$ + +Hence the delocalization threshold (where $Q_{ii}$ drops) occurs at + +$$ +i ^ {\star} (N) \asymp \frac {1 + \gamma}{\sqrt {8}} \sqrt {N}. +$$ + +Interpretation: the number of population-aligned eigenvectors grows only like $\sqrt{N}$ ; beyond $i^{\star}$ , local support components have merged and the corresponding sample eigenvectors are mixed with the bulk. This $\sqrt{N}$ law is the main practical takeaway. See Fig. S7. + +![](images/05b653e151e705122a565cfbb6843d76f8ba70dc4724cf2019fcefef8dc5f882.jpg) +Figure S7: Left: PR of sample eigenvector when population eigenvalue followed power-law with exponent $-1.2$ . Vertical lines with colors are theoretical predictions for the critical index from the two-peak approximation. The black horizontal dashed line corresponds to $\mathrm{PR} = 2$ , which roughly marks the index at which the eigenvector becomes delocalized. Right: Critical index $i'$ , where population eigenvalue followed power-law with exponent $-1 - \gamma$ . Solid lines are theoretical predictions from the two-peak approximation, $i' = \frac{1 + \gamma}{\sqrt{8}} \sqrt{N}$ . 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Moreover, the captured radiance inherently depends on scene illumination, making it difficult to recover the intrinsic spectral reflectance that remains invariant to lighting conditions. To address these challenges, we propose a chromaticity-intensity decomposition framework, which disentangles an HSI into a spatially smooth intensity map and a spectrally variant chromaticity cube. The chromaticity encodes lighting-invariant reflectance, enriched with high-frequency spatial details and local spectral sparsity. Building on this decomposition, we develop CIDNet—a Chromaticity-Intensity Decomposition unfolding network within a dual-camera CASSI system. CIDNet integrates a hybrid spatial-spectral Transformer tailored to reconstruct fine-grained and sparse spectral chromaticity and a degradation-aware, spatially-adaptive noise estimation module that captures anisotropic noise across iterative stages. Extensive experiments on both synthetic and real-world CASSI datasets demonstrate that our method achieves superior performance in both spectral and chromaticity fidelity. Code is released at: https://github.com/xiaodongwo/CIDNet. + +# 1 Introduction + +Coded aperture snapshot spectral imaging (CASSI) has emerged as a promising architecture for capturing hyperspectral images (HSIs) in a single shot [1, 25, 34]. By jointly modulating the spectral cube with a coded aperture and dispersing it spatially through a prism, CASSI produces a 2D compressed measurement that encodes both spatial and spectral information. This compressive measurement fuses (shears) the spectral bands, making each pixel of the 2D sensor a mixture of many wavelengths. As a result, recovering the full 3D spectral image becomes a severely under-determined, ill-posed inverse problem. + +![](images/a9a90e3eb3ffd9ab8c0873603a253088ee10667221180aa75c6354972d5e8e57.jpg) +Figure 1: (a) Chromaticity-Intensity decomposition of HSI images (b) Chromaticity exhibits highlight removal, lowlight enhancement and high-frequency textures. + +The difficulty comes mainly from two aspects. One is that spatial and spectral signals are highly overlapped and entangled in the compressed measurement. Many works attempt to address this through various priors or deep models, broadly categorized into four paradigms. Optimization-based methods [17, 33] introduce hand-crafted priors such as total variation or low-rank constraints. + +However, their performance is often limited in recovering spatial structures, especially under complex textures or noise. Plug-and-Play (PnP) approaches [24, 39] integrate powerful pre-trained denoisers into iterative solvers, yet these methods typically denoise each or few spectral band independently, neglecting spectral correlation and structure. Deep unfolding methods [4, 7, 14, 19, 37, 38] bridge model-based and data-driven paradigms by learning iterative modules guided by the CASSI physics. End-to-end networks [3, 10, 20, 21, 26] leverage CNN or Transformer to directly infer the spectral cube from measurements. These deep learning-based frameworks implicitly exploit spatial-spectral dependencies and have shown promising performance. Recently, diffusion model [22, 30, 35] and Mamba[23] have been used for spectral reconstruction. Nonetheless, all these approaches rely on network backbones to learn spatial-spectral features in an implicit manner. There lacks a clear and interpretable decomposition or quantitative structure that explicitly characterizes the physical roles of spatial and spectral components during reconstruction. + +The second challenge is that existing methods often overlook the impact of illumination. Since the captured spectral measurement is radiance-based, it inherently entangles the intrinsic surface reflectance with scene illumination. This coupling makes the reconstruction sensitive to lighting variations across time and environments, thereby limiting spectral accuracy. To address similar issues in the RGB image, prior works have explored intrinsic image decomposition [2, 8, 15, 16] and Retinex-based models [28, 29] to explicitly separate reflectance from illumination, enabling applications such as shadow removal and low-light enhancement. In the hyperspectral remote sensing community, several studies have also extended intrinsic decomposition to spectral reflectance and illumination separation [11, 12, 32], offering better invariance to lighting conditions. However, to the best of our knowledge, such decomposition has not yet been incorporated into CASSI reconstruction. + +In this paper, we propose a novel chromaticity learning framework for compressive spectral imaging, which leverages a chromaticity-intensity decomposition prior under the CASSI sensing mechanism. Our motivation is illustrated in Fig. 1, where the spectral image cube $\mathbf{X}$ is factorized as: + +$$ +\mathbf {X} = \mathbf {C} \odot \mathbf {I}, \tag {1} +$$ + +where $\mathbf{C}$ denotes the chromaticity cube and $\mathbf{I}$ represents the intensity image. Notably, $\mathbf{C}$ exhibits several desirable properties: (i) spatially invariant to illumination, suppressing highlights and enhancing details in low-light regions; (ii) spectrally sparse with localized support (as illustrated latter); (iii) enriched with high-frequency texture, essential for fine-detail recovery. In contrast, the intensity component $\mathbf{I}$ captures the global illumination structure in the scene. It is interesting to note that the chromaticity exhibits more intrinsic characteristics of the sample compared to hyperspectral images. Hence, learning the chromaticity instead of HSIs seems to benefit the field more. + +Building upon the above observations, we propose a physically interpretable chromaticity-intensity decomposition model tailored for CASSI systems. By leveraging a dual-camera CASSI setup, we validate this decomposition paradigm within both traditional optimization-based solvers and deep unfolding frameworks. To further explore its potential, we design a novel Chromaticity-Intensity Decomposition Network (CIDNet), which incorporates the spectral sparsity of chromaticity through a sparse TopK spectral Transformer, and models spatially anisotropic noise via a degradation-aware, spatially-adaptive variance estimator. + +In summary, our main contributions are summarized as follows: + +i) We propose a novel chromaticity-intensity decomposition model for spectral compressive imaging, which explicitly separates hyperspectral images into lighting-invariant chromaticity and smooth intensity. We further validate its effectiveness on optimization-based and unfolding algorithms in a dual-camera CASSI setting. +ii) We develop an intensity-guided deep unfolding network that incorporates the chromaticity decomposition into unfolding algorithm. The network features a hybrid spatial-spectral Transformer (HSST) architecture, where the encoder leverages window-based local spatial attention (Spa-LWSA) and the decoder employs sparse TopK spectral attention (Spec-TKSA) to capture localized spectral structures. +iii) We introduce a degradation-aware, spatially-adaptive dual noise estimation module (DNEM) to model anisotropic noise across different reconstruction stages. This module enables each iteration to adaptively handle varying noise levels across spatial locations. +iv) Extensive experiments on both synthetic and real datasets demonstrate that our method achieves state-of-the-art performance in terms of spectral reconstruction and chromaticity fidelity. + +![](images/08e224f4bfcdbbe2d158626c10cac3cd598b433d073517111bdef166344118c3.jpg) +Figure 2: (a) The architecture of our CIDNet with $K$ stages (iterations). (b) The CASSI system uses an intensity-guided mask to modulate the chromaticity. (c) Diagram of asymmetric backbone for our Hybrid Spatial-Spectral Transformer (HSST), with a local window spatial attention (Spa-LWSA) in Encoder and Sparse TopK spectral self-attention module (Spec-TKSA) in Decoder. (d) Details of Spec-TKSA. + +# 2 Proposed Method + +# 2.1 Degradation Model of CASSI + +Inspired by chromaticity-intensity decomposition in RGB intrinsic image analysis, we extend this concept to the hyperspectral domain. Given a hyperspectral image cube $\mathbf{X} \in \mathbb{R}^{H \times W \times N_{\lambda}}$ , we decompose it into a spatially smooth intensity image $\mathbf{I} \in \mathbb{R}^{H \times W}$ and a chromaticity cube $\mathbf{C} \in \mathbb{R}^{H \times W \times N_{\lambda}}$ as: + +$$ +\mathbf {X} (u, v, \lambda) = \mathbf {C} (u, v, \lambda) \odot \mathbf {I} (u, v), \tag {2} +$$ + +where $(u,v)$ denotes spatial location, $\lambda$ is the spectral band index, and $\odot$ denotes pixel-wise multiplication. Specifically, the intensity image is defined as the average spectral energy per pixel: + +$$ +\mathbf {I} (u, v) = \frac {1}{N _ {\lambda}} \sum_ {\lambda = 1} ^ {N _ {\lambda}} \mathbf {X} (u, v, \lambda), \tag {3} +$$ + +We found that intensity image can be approximated as a PAN image in dual-camera CASSI. A detailed proof in this PAN-Intensity Equivalence is provided in supplement materials. Hence, the chromaticity is computed as the normalized spectral signature: + +$$ +\mathbf {C} (u, v, \lambda) = \frac {\mathbf {X} (u , v , \lambda)}{\mathbf {I} (u , v) + \epsilon}, \tag {4} +$$ + +where $\epsilon$ is a small constant to avoid division by zero. This decomposition separates the multiplicative effect of illumination $\mathbf{I}(u,v)$ from the spectral reflectance $\mathbf{C}(u,v,\lambda)$ , which captures intrinsic scene properties. Importantly, $\mathbf{C}$ is invariant to changes in illumination intensity and direction, enabling more robust modeling of reflectance and spectral reconstruction under varying lighting conditions. After decomposition, the CASSI measurement process can be modeled as follows. The hyperspectral cube $\mathbf{X}$ is modulated by a coded aperture $\mathbf{M}^0\in \mathbb{R}^{H\times W}$ , resulting in a spatially coded cube: + +$$ +\mathbf {X} ^ {\prime} (u, v, \lambda) = \mathbf {C} (u, v, \lambda) \odot \mathbf {I} (u, v) \odot \mathbf {M} ^ {0} (u, v). \tag {5} +$$ + +Now we can treat $\mathbf{I}(u,v)\odot \mathbf{M}^0 (u,v)$ as a new formation of coded mask $\mathbf{M}'(u,v) = \mathbf{I}(u,v)\odot$ $\mathbf{M}^0 (u,v)$ incorporating the spatial intensity. This leaves the chromaticity an unknown variable when the intensity is obtained beforehand. Following a typical CASSI formulation, the modulated cube $\mathbf{X}'$ is then passed through a dispersive element that shifts each spectral band $\lambda_{n_{\lambda}}$ by a wavelength-dependent displacement $d(\lambda_{n_{\lambda}} - \lambda_{c})$ along the spatial axis (e.g., the $x$ -axis). The sheared datacube can be expressed as: + +$$ +\mathbf {X} ^ {\prime \prime} (u, v, n _ {\lambda}) = \mathbf {X} ^ {\prime} (u, v + d \left(\lambda_ {n _ {\lambda}} - \lambda_ {c}\right), \lambda_ {n _ {\lambda}}), \tag {6} +$$ + +![](images/0efb7623c2201bb712b70783808397fd0ce603c141e1005dfc8951ac4959d074.jpg) +Figure 3: Simulation HSIs reconstruction comparisons of Scene 7 with 4 (out of 28) spectral channels. The left shows the spectral curves corresponding to the two red boxes of the RGB image. The top-right depicts the enlarged patches corresponding to the yellow boxes in the bottom HSIs. Zoom in for a better view. + +![](images/cb5dfad2f700bfdc18402579dda411eddcf86a448ca32c1267458f684a9189e9.jpg) + +where $\lambda_{c}$ is the reference wavelength that remains unshifted. Finally, the 2D measurement $\mathbf{Y} \in \mathbb{R}^{H \times (W + d \cdot (N_{\lambda} - 1))}$ acquired by the camera is a summation of all dispersed bands: + +$$ +\mathbf {Y} (u, v) = \sum_ {n _ {\lambda} = 1} ^ {N _ {\lambda}} \mathbf {X} ^ {\prime \prime} (u, v, n _ {\lambda}) + \mathbf {N} (u, v), \tag {7} +$$ + +where $\mathbf{N}$ is additive measurement noise. This can be written compactly in vectorized form as: + +$$ +\mathbf {y} = \boldsymbol {\Phi} (\mathbf {c} \odot \mathbf {i}) + \mathbf {n}, \tag {8} +$$ + +where $\mathbf{c} = \mathrm{vec}(\mathbf{C})$ , $\mathbf{i} = \mathrm{vec}(\mathbf{I})$ , $\Phi$ is the sensing matrix determined by the modulation and dispersion process, and $\mathbf{n}$ is the vectorized noise term. Given that the intensity map $\mathbf{I}$ is known (PAN or RGB image in dual-camera CASSI scheme), we formulate the chromaticity-based measurement model as a standard linear inverse problem: + +$$ +\mathbf {y} = \mathbf {H} \mathbf {c} + \mathbf {n}, \tag {9} +$$ + +where $\mathbf{c}$ denotes the vectorized chromaticity, $\mathbf{H}$ is the effective sensing matrix that incorporates both the CASSI modulation-dispersion process and the known intensity modulation, and $\mathbf{n}$ is the vectorized noise. + +# 2.2 Optimization Framework of CASSI + +To characterize realistic imaging noise, we assume an anisotropic Gaussian noise model $\mathbf{n} \sim \mathcal{N}(\mathbf{0}, \boldsymbol{\Sigma})$ , where $\boldsymbol{\Sigma} = \mathrm{diag}(\sigma_1^2, \ldots, \sigma_M^2)$ represents the spatially varying noise covariance. Under a Bayesian framework, the posterior probability of the chromaticity $\mathbf{c}$ is given by $p(\mathbf{c} \mid \mathbf{y}) \propto p(\mathbf{y} \mid \mathbf{c}) \cdot p(\mathbf{c})$ , and the likelihood is: + +Table 1: Data-consistency projection comparison. + +
MethodGradient projection updating
ISTA [36]c = z + H^T(y - Hz)
GAP [19]c = z + H^T(HH^T)^{-1}(y - Hz)
HQS [4]c = z + H^T(HH^T + μI)^{-1}(y - Hz)
Oursc = z + H^T(HH^T + μΣ)^{-1}(y - Hz)
+ +$$ +p (\mathbf {y} \mid \mathbf {c}, \boldsymbol {\Sigma}) \propto \exp \left(- \frac {1}{2} (\mathbf {y} - \mathbf {H c}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} (\mathbf {y} - \mathbf {H c})\right), \tag {10} +$$ + +and $p(\mathbf{c}) \propto \exp(-\tau R(\mathbf{c}))$ is a generic prior over chromaticity with regularization function $R(\cdot)$ and weight $\tau > 0$ . Maximizing the posterior leads to the following Maximum A Posteriori (MAP) estimation problem: + +$$ +\hat {\mathbf {c}} = \operatorname {a r g m i n} _ {\mathbf {c}} \left(- \frac {1}{2} (\mathbf {y} - \mathbf {H c}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} (\mathbf {y} - \mathbf {H c})\right) + \tau R (\mathbf {c}). \tag {11} +$$ + +When the noise is homoscedastic (i.e., $\pmb{\Sigma} = \sigma^{2}\mathbf{I}$ ), the problem reduces to the common quadratic form: $\hat{\mathbf{c}} = \mathrm{argmin}_{\mathbf{c}}\frac{1}{2}\| \mathbf{y} - \mathbf{H}\mathbf{c}\|_2^2 +\tau R(\mathbf{c})$ . We rewrite Eq. (11) using an auxiliary variable $\mathbf{z}$ : + +$$ +\hat {\mathbf {c}}, \hat {\mathbf {z}} = \operatorname {a r g m i n} _ {\mathbf {c}, \mathbf {z}} \left(- \frac {1}{2} (\mathbf {y} - \mathbf {H c}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} (\mathbf {y} - \mathbf {H c})\right) + \tau R (\mathbf {z}), \quad \text {s . t .} \quad \mathbf {c} = \mathbf {z}. \tag {12} +$$ + +Using the half-quadratic splitting (HQS) framework, Eq. (12) is minimized by solving the following subproblems iteratively: + +$$ +\mathbf {c} ^ {(k + 1)} = \operatorname {a r g m i n} _ {\mathbf {c}} \left(- \frac {1}{2} (\mathbf {y} - \mathbf {H c}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} (\mathbf {y} - \mathbf {H c})\right) + \frac {\mu}{2} \| \mathbf {c} - \mathbf {z} ^ {(k)} \| _ {2} ^ {2}, \tag {13} +$$ + +$$ +\mathbf {z} ^ {(k + 1)} = \operatorname {a r g m i n} _ {\mathbf {z}} \frac {\mu}{2} \| \mathbf {z} - \mathbf {c} ^ {(k + 1)} \| _ {2} ^ {2} + \tau R (\mathbf {z}), \tag {14} +$$ + +where $\mu$ is a penalty parameter. The c-subproblem in Eq. (13) is quadratic and has a closed-form solution: + +$$ +\mathbf {c} ^ {(k + 1)} = \left(\mathbf {H} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {H} + \mu \mathbf {I}\right) ^ {- 1} \left(\mathbf {H} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {y} + \mu \mathbf {z} ^ {(k)}\right). \tag {15} +$$ + +Note that $\mathbf{H}^{\top}\pmb{\Sigma}^{-1}\mathbf{H}$ is a fat matrix and $(\mathbf{H}^{\top}\pmb{\Sigma}^{-1}\mathbf{H} + \mu \mathbf{I})^{-1}$ will be difficult to compute and thus we simplify it based on the Sherman-Morrison-Woodbury formula, + +$$ +\left(\mathbf {H} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {H} + \mu \mathbf {I}\right) ^ {- 1} = \mu^ {- 1} \mathbf {I} - \mu^ {- 2} \mathbf {H} ^ {\top} \left(\boldsymbol {\Sigma} + \mu^ {- 1} \mathbf {H} \mathbf {H} ^ {\top}\right) ^ {- 1} \mathbf {H}. \tag {16} +$$ + +In CASSI systems, $\mathbf{H}\mathbf{H}^{\top}$ is a diagonal matrix defined as $\mathbf{H}\mathbf{H}^{\top} \triangleq \mathrm{diag}\{h_1, \ldots, h_n\}$ . With $\boldsymbol{\Sigma} \triangleq \mathrm{diag}(\sigma_1^2, \ldots, \sigma_M^2)$ and detailed derivation in supplement materials, we obtain a generalized form of gradient descent, which is expressed as, + +$$ +\mathbf {c} ^ {(k + 1)} = \mathbf {z} ^ {(k)} + \mathbf {H} ^ {\top} \left(\mathbf {H} \mathbf {H} ^ {\top} + \mu \boldsymbol {\Sigma}\right) ^ {- 1} \left(\mathbf {y} - \mathbf {H} \mathbf {z} ^ {(k)}\right). \tag {17} +$$ + +Define this gradient projection as $\mathbf{c}^{(k + 1)} = \mathrm{proj}_{\Sigma}(\cdot)$ . Interestingly, we observed that this gradient projection resembles previous optimization-based methods but introduces a key change relates to the noise modeling. As summarized in Tab. 1, while traditional ISTA[36], GAP [19], and HQS [4] methods employ static data-consistency steps with fixed regularization, our method proposes a dynamic, spatially-adaptive correction mechanism. Finally, we update $\mathbf{z}^{(k + 1)}$ using any proximal operator depending on the prior $R(\cdot)$ , + +$$ +\mathbf {z} ^ {(k + 1)} = \operatorname {p r o x} _ {\tau / \mu \cdot R} \left(\mathbf {c} ^ {(k + 1)}\right). \tag {18} +$$ + +If the noise variance $\pmb{\Sigma}$ is known a priori, with Eq. (17) and Eq. (18), this concludes the efficient HQS derivation with anisotropic Gaussian noise. However, in practice, the noise map is unavailable and may vary dynamically across iterations (e.g., in PnP or unfolding methods). To effectively account for the degradation-varying characteristics in the CASSI system, we parameterize the anisotropic noise covariance $\pmb{\Sigma}^{(k)}$ and the denoising strength $\tau_{k} / \mu_{k}$ in a stage-specific manner. Both parameters are learned by a degradation-aware estimator $\mathcal{E}$ that takes as input the current iterate $\mathbf{z}^{(k)}$ and the measurement $\mathbf{y}$ : + +$$ +\left\{\boldsymbol {\Sigma} ^ {(k)}, \tau_ {k} / \mu_ {k} \right\} = \mathcal {E} \left(\mathbf {z} ^ {(k)}, \mathbf {y}\right). \tag {19} +$$ + +The estimator $\mathcal{E}$ is implemented as a lightweight CNN that jointly captures spatial structure in $\mathbf{z}^{(k)}$ and the encoded degradation in $\mathbf{y}$ . A detailed network module is found in supplement materials. We denote $\boldsymbol{\Sigma}^{(k)}$ and $\omega^{(k)} = \tau^{(k)} / \mu^{(k)}$ as the noise map for gradient projection and proximal mapping (denoiser) respectively (Dual Noise-Estimation Module (DNEM), as we referred to). The estimated $\boldsymbol{\Sigma}^{(k)}$ reflects the anisotropic uncertainty in the current iterate, and modulates the linear update of $\mathbf{c}^{(k + 1)}$ via Eq. (17), while $\omega^{(k)}$ controls the noise level fed into the proximal denoiser for $\mathbf{z}^{(k + 1)}$ via Eq. (18). The final iterative process can be expressed as: + +$$ +\left\{\boldsymbol {\Sigma} ^ {(k)}, \omega^ {(k)} \right\} = \mathcal {E} (\mathbf {z} ^ {(k)}, \mathbf {y}), +$$ + +$$ +\mathbf {c} ^ {(k + 1)} = \operatorname {p r o j} _ {\boldsymbol {\Sigma} ^ {(k)}} (\mathbf {z} ^ {(k)}) = \mathbf {z} ^ {(k)} + \mathbf {H} ^ {\top} \left(\mathbf {H} \mathbf {H} ^ {\top} + \boldsymbol {\Sigma} ^ {(k)}\right) ^ {- 1} (\mathbf {y} - \mathbf {H} \mathbf {z} ^ {(k)}), \tag {20} +$$ + +$$ +\mathbf {z} ^ {(k + 1)} = \operatorname {p r o x} _ {\omega^ {(k)} \cdot R} (\mathbf {c} ^ {(k + 1)}). +$$ + +Here, $\mu^{(k)}$ is omitted due to the usage of the network, $\mathbf{H}$ denotes the CASSI forward operator and $R(\cdot)$ is a regularization prior, which could be total variation, or a learned denoiser as in our experiments. This stage-adaptive formulation enables flexible and efficient recovery under spatially variant degradation patterns. + +To validate the effectiveness of our chromaticity-intensity decomposition strategy, we explore two integration paradigms: a traditional model-based iterative scheme and a deep unfolding network. The classical iterative algorithm is described in supplement materials, leveraging analytical priors and explicit update rules based on the degradation model. In contrast, our primary design adopts a learnable unfolding structure, as illustrated in Fig. 2. Each stage is composed of a learnable noise estimation, an analytical reconstruction step shown in Eq. (17) and a learned proximal denoiser. This framework offers the interpretability of traditional optimization while benefiting from the expressiveness and efficiency of deep networks. + +![](images/d3e255fa16ece3f4e74480cb3210a9363d7d3317037a1e8590b11ebac6a4662e.jpg) +Figure 4: Simulation: chromaticity reconstruction of Scene 8 with 4 (out of 28) spectral channels. The spectral curves correspond to the two red boxes in the RGB image (top-middle). The top-right depicts the zoomed patches corresponding to the yellow boxes in the bottom chromaticity. + +# 2.3 Hybrid Spatial-Spectral Transformer + +To better reconstruct the chromaticity component C, which inherently contains rich spatial textures and locally correlated spectral patterns (illustrated in Fig. 5), we propose an asymmetric UNet backbone using Hybrid Spatial-Spectral Transformer (HSST). This module is specifically designed to simultaneously learn the high-frequency details in spatial dimensions and sparse-local dependencies in the spectral domain, as shown in Fig. 2. + +We adopt a dual-branch design: the spatial attention branch captures intra-image textures through a Swin Transformer in Encoder, while the spectral attention branch is tailored to exploit sparse and locally correlated spectral features using a TopK spectral attention mechanism in Decoder. This asymmetric design is inspired by [37] and motivated by experimental verification, where the asymmetric design has better reconstruction results. + +Spectral Attention. Unlike the spectral correlation in HSIs, chromaticity spectra features exhibit structured sparsity and localized correlation, see Fig. 5. Motivated by this, we introduce a window-based spectral TopK attention mechanism, where attention is applied across the spectral channels within each local spatial window. Specifically, each spectral token attends only to its $K$ most relevant spectral neighbors, enforcing both sparsity and locality. + +Given an input feature cube $\mathbf{X} \in \mathbb{R}^{H \times W \times C}$ , we first divide it into non-overlapping spatial windows of size $N \times N$ , resulting in a batch of local cubes $\{\mathbf{X}_w\} \subset \mathbb{R}^{N^2 \times C}$ . Within each window, we perform spectral self-attention across the $C$ channels for every spatial location. We begin by computing the query, key, and value embeddings using learned $1 \times 1$ convolutions: + +![](images/2a4d12341207e909b88dc23be1699e4de7badd685f2b8984a11b52492c20daff.jpg) +Figure 5: Demonstration of sparse and local spectral correlation of chromaticity. Top: RGB contents of the benchmark testing data. Middle: spectral correlation coefficient matrices of the HSIs $(28\times 28)$ . Bottom: Corresponding matrices by the chromaticity. + +$$ +\left\{\mathbf {Q} _ {\mathbf {i}}, \mathbf {K} _ {\mathbf {i}}, \mathbf {V} _ {\mathbf {i}} \right\} = \operatorname {C o n v} _ {1 \times 1} \left(\mathbf {X} _ {w}\right) \in \mathbb {R} ^ {N ^ {2} \times C \times d}, \tag {21} +$$ + +where $d$ is the embedding dimension per head. To model inter-channel dependencies, we transpose the last two dimensions and perform attention along the channel axis. For each position $i \in \{1, \dots, N^2\}$ , the attention is computed as: + +$$ +\mathbf {A} _ {i} = \operatorname {S o f t m a x} \left(\operatorname {T o p K} \left(\frac {\mathbf {Q} _ {i} \mathbf {K} _ {i} ^ {\top}}{\sqrt {d}}\right)\right) \in \mathbb {R} ^ {C \times C}, \tag {22} +$$ + +Table 2: Comparisons of HSIs between CIDNet and SOTA methods on KAIST simulation dataset. PSNR (upper entry in each cell), and SSIM (lower entry in each cell) are reported. The best result is highlighted in bold. + +
MethodParams(M)GFLOPs(G)S1S2S3S4S5S6S7S8S9S10Avg
DeSCI [17]--27.1323.0426.6234.9623.9422.3824.4522.0324.5623.5925.27
0.7480.6200.8180.8970.7060.0.6830.7430.6730.7320.5870.721
GAP-Net [19]4.2778.5833.7433.2634.2841.0331.4432.4032.2730.4633.5130.2433.26
0.9110.9000.9290.9670.9190.9250.9020.9050.9150.8950.917
MST-L [3]2.0328.1535.4035.8736.5142.2732.7734.8033.6632.6735.3932.5035.18
0.9410.9440.9530.9730.9470.9550.9250.9480.9490.9410.948
DAUHST-9stg [4]6.1579.5037.2539.0241.0546.1535.8037.0837.5735.1040.0234.5938.36
0.9580.9670.9710.9830.9690.9700.9630.9660.9700.9560.967
SSR-9stg [38]5.1878.9339.0742.0444.4948.8038.6438.5039.1636.9643.1236.0840.69
0.9700.9810.9800.9900.9800.9780.9710.9760.9800.9680.978
PIDS-RGB [5]--42.0940.0841.5048.5540.0539.0036.6337.0238.8238.6440.24
0.9830.9490.9680.9890.9820.9740.9400.9480.9530.9800.967
In2SET-9stg [27]9.6959.4042.5646.4244.5550.6342.0142.4941.5940.5343.8342.3343.69
0.9890.9940.9860.9960.9920.9910.9830.9890.9900.9940.990
CIDNet-3stg1.4024.8040.8845.3943.5547.5440.3741.9440.9841.1142.5240.7942.51
0.9860.9930.9830.9930.9900.9010.9810.9920.9870.9920.989
CIDNet-5stg2.3341.2641.5646.3643.9847.9241.4742.2741.2741.3643.9040.5643.07
0.9870.9940.9840.9930.9920.9920.9820.9920.9900.9920.990
CIDNet-7stg3.2657.7141.6646.7944.5248.5141.4442.5641.4641.9344.3641.4943.47
0.9880.9950.9850.9940.9920.9930.9830.9930.9900.9930.990
CIDNet-9stg4.1974.1642.7247.8844.8748.8342.5943.0142.2842.2644.6842.0544.12
0.9900.9960.9860.9940.9930.9930.9850.9940.9910.9940.991
+ +where $\mathrm{TopK}(\cdot)$ retains only the TopK values per row and masks out the rest with $-\infty$ before applying softmax. This yields a sparse attention map across spectral channels for each spatial location $i$ in the window $\mathbf{Z}_i = \mathbf{A}_i\mathbf{V}_i \in \mathbb{R}^{C \times d}$ . To improve the robustness and expressiveness of spectral modeling, we further adopt a multi-ratio strategy. Instead of selecting a single sparsity level, we generate multiple attention maps using different TopK ratios (e.g., $\{1/2, 2/3, 3/4, 4/5\}$ of $C$ ), and then aggregate them adaptively. Specifically, let $\mathbf{A}^{(r)}$ denote the sparse attention computed under ratio $r$ , and $\alpha^{(r)}$ be a learnable scalar weight. The final attention output is then formulated as: + +$$ +\mathbf {Z} _ {i} = \sum_ {r \in \mathcal {R}} \alpha^ {(r)} \cdot \operatorname {S o f t m a x} \left(\mathbf {A} _ {i} ^ {(r)}\right) \mathbf {V} _ {i}, \tag {23} +$$ + +where $\mathcal{R}$ denotes the set of TopK ratios. This fusion across multiple sparsity levels enables the network to capture both dominant and complementary spectral correlations, further improving spectral detail preservation. and reshaped back to the original window structure. After processing all windows, the outputs are stitched to reconstruct the full feature map. This spectral TopK attention not only reduces the computational burden from dense $O(C^2)$ to $O(KC)$ , but also explicitly captures the structured sparsity observed in reflectance spectra—where only a few wavelengths contribute significantly. Empirically, we find that this strategy enhances spectral sharpness and suppresses irrelevant crossband mixing. Following the standard Transformer architecture, we apply a conventional feedforward network after the sparse TopK attention, which is not the focus of our work. + +Spatial Attention. Inspired by recent advances in Swin Transformers [18], we employ the Swin Transformer as our spatial modeling backbone. The input feature map $\mathbf{X} \in \mathbb{R}^{H \times W \times C}$ is divided into non-overlapping windows of size $M \times M$ . Within each window, we perform multi-head self-attention (MSA) by computing + +$$ +\mathbf {Z} _ {\mathrm {s p a}} = \operatorname {M S A} (\mathrm {L N} (\mathbf {X})) + \mathbf {X}, \quad \mathbf {Z} _ {\text {o u t}} = \operatorname {F F N} (\mathrm {L N} (\mathbf {Z} _ {\mathrm {s p a}})) + \mathbf {Z} _ {\mathrm {s p a}}, \tag {24} +$$ + +where $\mathrm{LN}(\cdot)$ denotes layer normalization, and FFN is a standard feedforward network. The relative position bias and shifted-window mechanism in Swin Transformer enhance local texture modeling while maintaining global continuity across windows. + +# 3 Experiments + +We conduct comprehensive experiments on both simulated and real-world CASSI systems. The datasets, training settings, and implementation details are introduced as follows. + +Table 3: Comparisons of intensity (left) and chromaticity (right) between CIDNet and SOTA methods on KAIST simulation dataset. PSNR (upper entry in each cell), and SSIM (lower entry in each cell) are reported. + +
MethodS1S2S3S4S5S6S7S8S9S10Avg
DeSCI [17]29.42/14.7627.48/23.0231.01/26.7541.71/19.3126.62/20.1425.12/17.6627.08/20.8724.55/19.4829.19/24.9025.61/10.9528.78/19.79
0.84/0.490.72/0.670.92/0.830.97/0.750.81/0.720.79/0.500.82/0.600.77/0.540.86/0.790.69/0.320.82/0.62
GAP-Net [19]36.78/21.8535.84/20.8839.31/25.6645.90/18.6333.94/23.3034.37/19.4335.65/24.4332.57/20.3137.69/17.3432.05/13.1936.41/20.50
0.95/0.620.93/0.610.98/0.790.98/0.660.94/0.760.95/0.570.95/0.640.94/0.540.95/0.520.92/0.390.95/0.61
MST-L [3]38.05/23.9538.42/30.2641.07/32.3647.73/19.5935.14/28.8936.44/29.6837.04/27.1834.89/25.9439.26/20.6534.42/17.5638.25/25.61
0.96/0.720.96/0.800.98/0.890.99/0.750.96/0.840.97/0.780.96/0.750.96/0.750.97/0.740.95/0.600.97/0.76
DAUHST-9stg [4]39.68/25.7240.84/29.5445.08/34.9849.24/31.5437.84/34.0838.79/32.6340.33/31.9536.71/31.3944.55/34.3035.91/26.8840.90/31.30
0.97/0.770.97/0.850.99/0.940.99/0.860.98/0.910.98/0.840.98/0.830.97/0.800.98/0.900.96/0.660.98/0.84
SSR-9stg [38]41.19/24.4543.35/31.2648.50/40.5150.01/37.6540.85/33.4640.13/36.0541.86/32.7238.44/33.0345.98/37.7637.09/27.8642.74/33.47
0.98/0.830.98/0.930.99/0.970.99/0.950.98/0.950.98/0.920.98/0.860.98/0.910.99/0.950.97/0.860.98/0.91
PIDS [5]37.44/18.2038.81/17.2734.81/22.7442.83/17.9233.71/19.8936.66/15.9035.95/19.1036.81/17.5137.06/16.2435.43/9.3736.95/17.41
0.99/0.720.98/0.330.98/0.800.98/0.680.98/0.780.98/0.320.97/0.610.98/0.360.98/0.370.98/0.150.98/0.51
In2SET-9stg [27]58.06/29.4259.61/31.1559.60/36.0262.63/25.5657.55/33.6258.55/27.6857.77/32.0557.82/26.5659.44/25.2656.52/12.9358.75/28.03
1.00/0.771.00/0.831.00/0.931.00/0.811.00/0.901.00/0.791.00/0.831.00/0.761.00/0.841.00/0.491.00/0.80
CIDNet-3stg63.78/28.0264.70/31.1063.18/39.7767.12/36.1461.77/37.5464.73/35.8862.72/32.7165.24/33.9763.65/37.6064.93/33.1564.18/34.59
1.00/0.831.00/0.931.00/0.971.00/0.951.00/0.961.00/0.931.00/0.851.00/0.941.00/0.951.00/0.881.00/0.92
CIDNet-5stg64.67/29.7065.40/33.4163.41/39.5466.83/38.0264.03/38.4665.85/36.3663.18/32.7065.84/36.7063.65/38.4564.98/32.5964.78/35.59
1.00/0.851.00/0.931.00/0.971.00/0.961.00/0.961.00/0.931.00/0.861.00/0.941.00/0.961.00/0.891.00/0.92
CIDNet-7stg61.89/30.8162.96/32.0660.99/41.1765.42/38.0661.26/36.9163.87/36.4960.32/33.2363.88/35.9861.14/38.8062.52/33.4362.43/35.69
1.00/0.851.00/0.931.00/0.971.00/0.961.00/0.961.00/0.931.00/0.861.00/0.941.00/0.961.00 / 0.891.00/0.93
CIDNet-9stg62.28/25.8963.54/34.2961.42/41.4861.26/38.2264.00/39.7164.00/37.1060.80/33.3164.13/36.6962.02/38.0062.71/33.4362.80/35.81
1.00/0.861.00/0.951.00/0.971.00/0.971.00/0.941.00/0.941.00/0.871.00/0.951.00/0.961.00/0.901.00/0.93
+ +Simulation Dataset. We adopt two widely used hyperspectral datasets: CAVE [31] and KAIST [6]. The CAVE dataset contains 32 hyperspectral images with a spatial resolution of $512 \times 512$ . The KAIST dataset provides 30 high-resolution hyperspectral scenes of size $2704 \times 3376$ . We obtain ground-truth multi-spectral chromaticity and intensity using chromaticity-intensity decomposition Eq. (28). Following prior works [20, 21, 3], we use all CAVE images for training and select 10 scenes from KAIST for evaluation. + +Implementation Details. Our model is implemented in PyTorch and trained using the Adam optimizer [13] for 300 epochs. The initial learning rate is set to $4 \times 10^{-4}$ and updated using a cosine annealing schedule. We employ the $\ell_2$ loss between the reconstructed chromaticity and ground-truth chromaticity as the objective function. For training, we randomly extract 3D hyperspectral patches from each scene. For simulated data, the patch size is $256 \times 256 \times 28$ , for real-world data, we use patches of size $350 \times 260 \times 26$ . In simulation experiments, the forward imaging model is configured with a dispersion shift step $d = 2$ , directing dispersion along the horizontal axis (rightward). In real-world scenarios, we assume a vertical dispersion direction and set $d = 1$ , consistent with the dual-camera hardware setup. All experiments are conducted in Nvidia A40 GPU. + +Comparing Methods. We compared the HSIs and chromaticity reconstruction performance of our CIDNet with other 7 SOTA methods, including DeSCI[17], GAP-Net[19], MST-L[3], DAUHST-9stg[4], SSR-9stg[38] and two dual-camera CASSI algorithm: PIDS[5] and In2SET-9stg[27]. PIDS is compared with RGB image as guidance (in original paper). We evaluate the reconstruction performance from two perspectives: hyperspectral images (HSIs) and chromaticity. Unlike the baseline methods that directly reconstruct HSIs, our method assumes that the intensity is known and focuses on reconstructing the chromaticity. For a fair comparison in the HSI domain, we + +![](images/ddded3d827e9a6caef4530e0ab0c24e3b15e6f8529f76de494d3d4049cb76ac4.jpg) +PIDS + +![](images/b14d94f02bc1339a4fec39bc318625931880e0ad7338aa3e707d1ecea5af62ba.jpg) + +![](images/504daf9e011a30021daf68b283606875d7143058aba7a3020470f1f48eb86b23.jpg) + +![](images/d3df83b8220e8233abdd33696dcb33473e4736db8f795536f6498985c56bd49c.jpg) + +![](images/adb2155138d069e5fa7839be9010d80151553d1f46917ba7c91c2cca1a483e3b.jpg) +CID-TV + +![](images/ac78839bf094e3877650ecbf4da73c0ebdd007cd79e7c90554dda9ee3b9785d9.jpg) + +![](images/555eaa896a08b18ed9d2ce343cdc961f231aed1f13ad2ee0b170a9736efb65cb.jpg) + +![](images/28959d6c0f5bebd5c4f892dceb1e51ae8ad311e93c6e582b1f027fc68698e716.jpg) + +![](images/beb56b9916d20dbc2081f049fe406c0053f7bad219d9871f6d2c3177ddef03e2.jpg) +CIDNet + +![](images/bc813fb27bad1269bdef4071162daa235ba592388e4f025b89e9cfc0f03fcc85.jpg) +Figure 6: Real-data reconstruction in dual-camera CASSI. + +![](images/9f04a0f7fbcd0a6035dcb2a86d0380044a30285d9d88a2fbc3e2e123079b9248.jpg) + +![](images/b5f484a784624e559297d890039b3a456b18a93b81734cca14952547691938fc.jpg) + +obtain our reconstructed HSIs by multiplying the recovered chromaticity with the known intensity. Conversely, for chromaticity-level comparison, we perform chromaticity-intensity decomposition on the HSIs reconstructed by the baseline methods to extract their chromaticity components. The reconstruction quality of HSIs is evaluated using peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM). + +Table 4: Break-down ablation study on individual components of the proposed method. + +
Base-1Int.HSSTDNEMPSNRSSIMParamsFLOPs
✓35.770.9491.1116.13
✓✓40.830.9841.1116.13
✓✓✓42.300.9881.4425.04
✓✓✓✓42.510.9891.4024.80
+ +Table 5: Break-down ablation study on spectral self-attention mechanism. + +
MethodBase-2WSSA-WSSATKSA-TKSALWSA+TKSALWSA-TKSA
PSNR40.9541.9842.0542.3642.51
SSIM0.9850.9880.9880.9880.989
Params1.121.331.331.271.40
FLOPs18.0323.2925.1623.4724.80
+ +Table 6: Break-down ablation study on intensity-guided mask. + +
MethodADMMADMM-IntMSTMST-IntDAUHSTDAUHST-Int
PSNR24.8037.0934.2637.2837.2142.64
SSIM0.7120.9650.9350.9710.9590.989
+ +# 3.1 Quantitative Results + +As shown in Tab. 2, we compare the PSNR and SSIM of HSIs with SOTA methods. our proposed CIDNet excels in 8 out of 10 scenes, particularly in CIDNet-9stg, achieving an average PSNR of 44.12dB and SSIM of 0.991. This significantly surpasses previous unfolding and end-to-end networks, and also in dual-camera CASSI systems, such as In2SET-9stg. A visual comparison is shown in Fig. 3. We provide the visual comparison of simulation Scene7 with 4 out of 28 spectral channels. In addition, we plot the spectral density curves of two regions in the top left RGB image. Our CIDNet-9stg achieves relatively higher spectral accuracy with reference spectra, demonstrating the effectiveness of our method. To verify the reconstructed quality of chromaticity and intensity, we compare the PSNR and SSIM of reconstructed chromaticity and intensity (with decomposition of HSIs), which is shown in Table 3. Note that our method assumes a known intensity. Therefore, to ensure a fair comparison of intensity, we decompose the reconstructed HSIs to extract their intensity component for evaluation. Our CIDNet achieves significant improvements in metrics of chromaticity and intensity. For dual-camera CASSI algorithm PIDS and In2SET, we achieve the best chromaticity metrics with a PSNR of 35.81 and a SSIM of 0.93. A visual comparison of reconstructed chromaticity is shown in Fig. 4. In this research, we used a real-world DCCHI measurement Ninja, taken from publicly available data as detailed in [9]. Fig. 6 illustrates the reconstruction results for four spectral bands in this scene, using two dual-camera CASSI reconstruction algorithms, PIDS and CID-TV, where CID-TV is the iterative CID algorithm using Total Variationa as Regularizer, details can be found in supplement materials. The comparison highlights the superior image restoration quality of our model over other methods, validating its effectiveness and reliability in real-world applications. + +# 3.2 Ablation Study + +Effectiveness of Intensity, HSST and DNEM. We verify the effectiveness of our proposed intensity-guided mask and two network module, HSST and DNEM. We adopt Base-1, derived by retaining binary mask and removing spatial-spectral attention and noise estimation from CIDNet-3stg to conduct the ablation study, where we used ground-truth HSIs instead of chromaticity for supervision. Tab. 4 shows the results of PSNR and SSIM of different settings, and our method achieves a significant 6.74dB PSNR improvements compared with Base-1. + +Robustness of intensity-guided mask. We test the robustness of intensity-guided mask by employing this intensity mask to ADMM, DAUHST and MST, the result is shown in Tab. 6. We compare the reconstruction quality with/without intensity-guided mask and find that it is significantly improving the base (without intensity) in iterative, end-to-end and unfolding framework. Note that DAUHST achieves slightly higher metric than ours. However its flops and parameters are also greater. + +Self-Attention Scheme Comparison. We compare Window-based Spectral Self-Attention (WSSA) [38] and spectral TKSA and its variants within the encoder and decoder. We use $^{\prime} + ^{\prime}$ to signify a + +parallel implementation of both attentions (each processing half of the feature channels), and $^{\prime}-$ to represent an encoder-decoder implementation. Base-2 is CIDNet-3stg that removes the attention module. Tab. 5 shows the ablation results and LWSA-TKSA yields the most prominent improvement of 1.56dB PSNR compared with Base-2, which shows the effectiveness of our method. + +# 4 Conclusion + +We present CIDNet, a novel reconstruction framework for CASSI, which leverages a physically motivated chromaticity-intensity decomposition. By disentangling the hyperspectral image into a spatially smooth intensity map and a spectrally informative chromaticity cube, our method enables lighting-invariant reflectance modeling and better preserves spatial-spectral details. We have designed a hybrid spatial-spectral Transformer to recover the sparse and high-frequency chromaticity components and introduce a degradation-aware unfolding strategy with spatially adaptive noise modeling to handle anisotropic noise inherent in the dual-camera CASSI system. Extensive experiments on both simulated and real-world CASSI datasets validate the effectiveness of our approach, achieving state-of-the-art performance in spectral reconstruction and chromaticity fidelity. This work highlights the benefit of physics-aware decomposition and hybrid attention mechanisms in addressing the ill-posed inverse problem of CASSI reconstruction. + +# Acknowledgments + +This work was supported by National Key R&D Program of China (2024YFF0505603), the National Natural Science Foundation of China (grant number 62271414), Zhejiang Provincial Distinguished Young Scientist Foundation (grant number LR23F010001), Zhejiang 'Pioneer' and 'Leading Goose' R&D Program (grant number 2024SDXHDX0006, 2024C03182), the Key Project of Westlake Institute for Optoelectronics (grant number 2023GD007), the 2023 International Sci-tech Cooperation Projects under the purview of the 'Innovation Yongjiang 2035' Key R&D Program (grant number 2024Z126). 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix + +In the supplementary material, we provide more details that are not in our main paper: + +(a) Multispectral Image Formation in Sec. A.1. +(b) Proof of Illumination-Invariance of Chromaticity in Sec. A.2. +(c) Proof of PAN-Intensity Equivalence in Sec. A.3. We demonstrate the PAN image in dual-camera CASSI is equivalent to the Intensity image. +(d) Closed-form Solution of Data-fidelity Term in Sec. A.4. +(e) Traditional Optimization-based Methods in Sec. A.5. Some results are visualized. +(f) Visual Comparison of HSIs and Chromaticity in Sec. A.6. +(g) Noise Map Visualization of DNEM in Sec. A.7. +(h) Limitation and Broader Impact of Our Work in Sec. A.8 + +# A.1 Multispectral Image Formation + +Let $(u,v)$ be spatial coordinates and $\lambda \in [\lambda_{\mathrm{min}},\lambda_{\mathrm{max}}]$ the wavelength. Assume the scene reflectance is $\mathbf{R}(u,v,\lambda)$ (this is not the same as data-prior term), and the illumination spectral power is $L(\lambda)$ , spatially uniform. Let $s(\lambda)$ be the spectral response function of the grayscale camera, which converts the spectral radiance to a scalar intensity. Then the intensity recorded by the grayscale camera at $(u,v)$ is: + +$$ +\mathbf {I} (u, v) = \int_ {\lambda_ {\min }} ^ {\lambda_ {\max }} s (\lambda) \cdot L (\lambda) \cdot \mathbf {R} (u, v, \lambda) d \lambda . \tag {25} +$$ + +This is the measurement we observe from the grayscale camera. We define a multispectral distribution: + +$$ +\mathbf {X} (u, v, \lambda) = s (\lambda) \cdot L (\lambda) \cdot \mathbf {R} (u, v, \lambda) \tag {26} +$$ + +Then the multispectral chromaticity function is defined as the normalized form of $X(u,v,\lambda)$ over $\lambda$ : + +$$ +\mathbf {C} (u, v, \lambda) = \frac {\mathbf {X} (u , v , \lambda)}{\mathbf {I} (u , v)} = \frac {s (\lambda) \cdot L (\lambda) \cdot \mathbf {R} (u , v , \lambda)}{\int s (\lambda) \cdot L (\lambda) \cdot \mathbf {R} (u , v , \lambda) d \lambda}, \tag {27} +$$ + +which leads to the final formulation of spectral product of intensity and chromaticity: + +$$ +\mathbf {X} (u, v, \lambda) = \mathbf {C} (u, v, \lambda) \odot \mathbf {I} (u, v). \tag {28} +$$ + +Next we demonstrate that the chromaticity is illumination invariant, and the intensity can be obtained via a dual-camera setting. + +# A.2 Proof of Illumination-Invariance of Chromaticity + +Now suppose the illumination changes globally: + +$$ +L (\lambda) \to \alpha \cdot L (\lambda), \quad \alpha > 0 +$$ + +Then: + +$$ +\begin{array}{l} \mathbf {C} ^ {\prime} (u, v, \lambda) = \frac {s (\lambda) \cdot \alpha L (\lambda) \cdot \mathbf {R} (u , v , \lambda)}{\int s \left(\lambda^ {\prime}\right) \cdot \alpha L \left(\lambda^ {\prime}\right) \cdot \mathbf {R} (u , v , \lambda^ {\prime}) d \lambda^ {\prime}} (29) \\ = \frac {s (\lambda) \cdot L (\lambda) \cdot \mathbf {R} (u , v , \lambda)}{\int s \left(\lambda^ {\prime}\right) \cdot L \left(\lambda^ {\prime}\right) \cdot \mathbf {R} (u , v , \lambda^ {\prime}) d \lambda^ {\prime}} = \mathbf {C} (u, v, \lambda) (30) \\ \end{array} +$$ + +Thus, the chromaticity $\mathbf{C}(u,v,\lambda)$ is invariant to uniform intensity changes in illumination, even when considering the grayscale camera's spectral sensitivity. + +# A.3 Proof of PAN-Intensity Equivalence + +Proposition A.1 (PAN-Intensity Equivalence Under Uniform Illumination). In a dual-camera system comprising a CASSI sensor and a grayscale PAN camera exposed under the same illumination $L(\lambda)$ , let $s(\lambda)$ denote the spectral response of the camera. The PAN image $\mathbf{I}_{\mathrm{PAN}}(u,v)$ provides a relative estimate of the scene intensity $\mathbf{I}(u,v)$ defined by the chromaticity-intensity decomposition of the hyperspectral image $\mathbf{X}(u,v,\lambda)$ . That is, + +$$ +\mathbf {I} _ {\text {P A N}} (u, v) \approx k \cdot \mathbf {I} (u, v), \tag {31} +$$ + +where $k$ is a scalar constant that is approximately invariant across spatial coordinates $(u, v)$ . + +Proof. Our derivation begins with the Retinex theory, which decomposes an image into chromaticity and intensity components. For hyperspectral images, this decomposition is generalized as: + +$$ +\mathbf {X} (u, v, \lambda) = \mathbf {C} (u, v, \lambda) \cdot \mathbf {I} (u, v), \tag {32} +$$ + +where the chromaticity and intensity are defined as: + +$$ +\mathbf {C} (u, v, \lambda) = \frac {\mathbf {X} (u , v , \lambda)}{\int \mathbf {X} (u , v , \lambda^ {\prime}) d \lambda^ {\prime}}, \tag {33} +$$ + +$$ +\mathbf {I} (u, v) = \int \mathbf {X} (u, v, \lambda) d \lambda . \tag {34} +$$ + +In our dual-camera setup with CASSI and PAN sensors under identical illumination $L(\lambda)$ , the PAN image formation is modeled as: + +$$ +\mathbf {I} _ {\text {P A N}} (u, v) = \int s (\lambda) \cdot L (\lambda) \cdot \mathbf {X} (u, v, \lambda) d \lambda . \tag {35} +$$ + +Substituting the $\mathbf{X}$ into the product of chromaticity and intensity yields: + +$$ +\mathbf {I} _ {\mathrm {P A N}} (u, v) = \int s (\lambda) L (\lambda) [ \mathbf {C} (u, v, \lambda) \cdot \mathbf {I} (u, v) ] d \lambda . \tag {36} +$$ + +Since $\mathbf{I}(u,v)$ is independent of wavelength $\lambda$ , it can be factored out of the integral: + +$$ +\mathbf {I} _ {\mathrm {P A N}} (u, v) = \mathbf {I} (u, v) \cdot \int s (\lambda) L (\lambda) \mathbf {C} (u, v, \lambda) d \lambda . \tag {37} +$$ + +The key insight is that $\mathbf{C}(u,v,\lambda)$ is normalized $(\int \mathbf{C}(u,v,\lambda')d\lambda' = 1)$ and exhibits smooth spectral variation, while $s(\lambda)L(\lambda)$ acts as a broadband low-pass filter. This justifies the approximation: + +$$ +\int s (\lambda) L (\lambda) \mathbf {C} (u, v, \lambda) d \lambda \approx k, \tag {38} +$$ + +where $k$ is a spatial-invariant scalar constant. Therefore, we obtain: + +$$ +\mathbf {I} _ {\text {P A N}} (u, v) \approx k \cdot \mathbf {I} (u, v). \tag {39} +$$ + +To address scale ambiguity, we normalize the PAN image to $[0,1]$ during training and inference, which justifies using PAN as a relative intensity estimate in our chromaticity-intensity framework. + +# A.4 Closed-form Solution of Data-fidelity Term + +The c-subproblem in Eq. (13) is quadratic and has a closed-form solution: + +$$ +\mathbf {c} ^ {(k + 1)} = \left(\mathbf {H} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {H} + \mu \mathbf {I}\right) ^ {- 1} \left(\mathbf {H} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {y} + \mu \mathbf {z} ^ {(k)}\right). \tag {40} +$$ + +Note that $\mathbf{H}^{\top}\pmb{\Sigma}^{-1}\mathbf{H}$ is a fat matrix and $(\mathbf{H}^{\top}\pmb{\Sigma}^{-1}\mathbf{H} + \mu \mathbf{I})^{-1}$ will be difficult to compute and thus we simplify it based on the Sherman-Morrison-Woodbury formula, + +$$ +\left(\mathbf {H} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {H} + \mu \mathbf {I}\right) ^ {- 1} = \mu^ {- 1} \mathbf {I} - \mu^ {- 2} \mathbf {H} ^ {\top} \left(\boldsymbol {\Sigma} + \mu^ {- 1} \mathbf {H} \mathbf {H} ^ {\top}\right) ^ {- 1} \mathbf {H}. \tag {41} +$$ + +By plugging Eq. (41) into Eq. (40), we formulate it as + +$$ +\mathbf {c} ^ {(k + 1)} = \frac {\mathbf {H} ^ {\top} \mathbf {y} + \mu \mathbf {z} ^ {(k)}}{\mu} - \frac {\mathbf {H} ^ {\top} \left(\boldsymbol {\Sigma} + \mu^ {- 1} \mathbf {H} \mathbf {H} ^ {\top}\right) ^ {- 1} \mathbf {H} \mathbf {H} ^ {\top} \mathbf {y}}{\mu^ {2}} - \frac {\mathbf {H} ^ {\top} \left(\boldsymbol {\Sigma} + \mu^ {- 1} \mathbf {H} \mathbf {H} ^ {\top}\right) ^ {- 1} \mathbf {H} \mathbf {z} ^ {(k)}}{\mu}. \tag {42} +$$ + +In CASSI systems, $\mathbf{HH}^{\top}$ is a diagonal matrix defined as $\mathbf{HH}^{\top} \triangleq \mathrm{diag}\{h_1, \ldots, h_n\}$ . With $\boldsymbol{\Sigma} \triangleq \mathrm{diag}(\sigma_1^2, \ldots, \sigma_M^2)$ , we obtain: + +$$ +\left(\boldsymbol {\Sigma} + \mu^ {- 1} \mathbf {H} \mathbf {H} ^ {\top}\right) ^ {- 1} = \operatorname {d i a g} \left\{\frac {\mu}{\mu \sigma_ {1} ^ {2} + h _ {1}}, \dots , \frac {\mu}{\mu \sigma_ {n} ^ {2} + h _ {n}} \right\}, \tag {43} +$$ + +$$ +\left(\boldsymbol {\Sigma} + \mu^ {- 1} \mathbf {H} \mathbf {H} ^ {\top}\right) ^ {- 1} \mathbf {H} \mathbf {H} ^ {\top} = \operatorname {d i a g} \left\{\frac {\mu h _ {1}}{\mu \sigma_ {1} ^ {2} + h _ {1}}, \dots , \frac {\mu h _ {n}}{\mu \sigma_ {n} ^ {2} + h _ {n}} \right\}. \tag {44} +$$ + +Let $\mathbf{y} \triangleq [y_1, \ldots, y_n]^\top$ and $[\mathbf{H}\mathbf{c}^{(k)}]_i$ denote the $i$ -th element of $\mathbf{H}\mathbf{c}^{(k)}$ . We plug Eq. (43) and Eq. (44) into Eq. (42) as + +$$ +\begin{array}{l} \mathbf {c} ^ {(k + 1)} = \mu^ {- 1} \mathbf {H} ^ {\top} \mathbf {y} + \mathbf {c} ^ {(k)} - \mu^ {- 1} \mathbf {H} ^ {\top} \left[ \frac {y _ {1} h _ {1} + \mu [ \mathbf {H c} ^ {(k)} ] _ {1}}{\mu \sigma_ {1} ^ {2} + h _ {1}}, \dots , \frac {y _ {n} h _ {n} + \mu [ \mathbf {H c} ^ {(k)} ] _ {n}}{\mu \sigma_ {n} ^ {2} + h _ {n}} \right] ^ {\top} (45) \\ = \mathbf {c} ^ {(k)} + \mathbf {H} ^ {\top} \left[ \frac {y _ {1} - \left[ \mathbf {H c} ^ {(k)} \right] _ {1}}{\mu \sigma_ {1} ^ {2} + h _ {1}}, \dots , \frac {y _ {n} - \left[ \mathbf {H c} ^ {(k)} \right] _ {n}}{\mu \sigma_ {n} ^ {2} + h _ {n}} \right] ^ {\top}. (46) \\ \end{array} +$$ + +Generally this is a generalized form of gradient descent, which is expressed as, + +$$ +\mathbf {c} ^ {(k + 1)} = \mathbf {z} ^ {(k)} + \mathbf {H} ^ {\top} \left(\mathbf {H} \mathbf {H} ^ {\top} + \mu \boldsymbol {\Sigma}\right) ^ {- 1} (y - \mathbf {H} \mathbf {z} ^ {(k)}) \tag {47} +$$ + +# A.5 Traditional Optimization-based Methods + +We explore two traditional optimization-based paradigms considering a PAN-guided and RGB-guided intensity respectively, where we formulate the optimization problem using TV prior as, + +$$ +\hat {\mathbf {c}} = \operatorname {a r g m i n} _ {\mathbf {c}} \frac {1}{2} \| \mathbf {y} - \boldsymbol {\Phi} (\mathbf {c} \odot \mathbf {i}) \| _ {2} ^ {2} + \tau \mathbf {T V} (\mathbf {c}), \tag {48} +$$ + +where $\Phi$ is the sensing matrix determined by the modulation and dispersion process, $\mathbf{c}$ and $\mathbf{i}$ are chromaticity and intensity respectively, the noise estimation term is omitted since it is hard to be estimated in iterative methods. We consider a dual-camera setting where the second camera could be a grayscale or RGB camera, which satisfy $\mathbf{i} = \mathbf{i}^{\mathrm{PAN}}$ or $\mathbf{i}^{\mathrm{RGB}}$ . In RGB scenarios, the RGB-guided intensity is a three-channel image, which cannot be multiplied directly with $\Phi$ due to the channel mismatch. Hence, we interpolate the RGB image to the same spectral channels with corresponding HSIs. Using the HQS framework, Eq. (48) is minimized by solving the following subproblems iteratively by introducing $\mathbf{c} = \mathbf{z}$ : + +$$ +\mathbf {c} ^ {(k + 1)} = \operatorname {a r g m i n} _ {\mathbf {c}} \frac {1}{2} \| \mathbf {y} - \mathbf {H c} \| _ {2} ^ {2} + \frac {\mu}{2} \| \mathbf {c} - \mathbf {z} ^ {(k)} \| _ {2} ^ {2}, \tag {49} +$$ + +$$ +\mathbf {z} ^ {(k + 1)} = \operatorname {a r g m i n} _ {\mathbf {z}} \frac {\mu}{2} \| \mathbf {z} - \mathbf {c} ^ {(k + 1)} \| _ {2} ^ {2} + \tau \mathbf {T V} (\mathbf {z}), \tag {50} +$$ + +Following previous derivation on Eq.49 and traditional TV denoising term Eq.50, we iterate these two steps to approach its finest solution. The reconstruction is compared with iterative methods such as GAP-TV, DeSCI and PIDS and presented in Fig.7. Scene5 is selected for better visualization purpose. + +![](images/6917053a131566d4ef0811fa326533d2761bc0a639aa5776f8fd1c2ccbe6ffa7.jpg) +Figure 7: Reconstructed HSIs using traditional optimization-based methods. + +# A.6 Visual Comparison of HSIs and Chromaticity + +We compare the HSIs and chromaticity using KAIST datasets. As shown in Fig. 9, we select 4 out of 10 test datasets, with the top row being the RGB reference and intensity image. Four spectral images are selected for visualization. It can be seen from the figure that chromaticity enhances the low-light regions while preserving more spatial textures as compared with regular spectral images, demonstrating that chromaticity contains more features. + +# A.7 Noise Map + +In this paper we propose a dual noise estimation module for data-fidelity term and denoising network. This network module is designed to estimate a spatially adaptive noise map $\mathcal{E}$ , two noise map are output corresponding to the gradient projection noise map and the proximal mapping noise map respectively, where we use convolution and channel attention (CA) to enhances informative channels by modeling inter-channel relationships, as shown in Fig. 8. This structure guarantees positivity and adaptiveness of the output noise + +map, suitable for uncertainty modeling or variance-aware image restoration tasks. We conduct the experiment and find in CIDNet-3stg, the gradient projection noise map is prominent in the 1st stage and fades away in the latter 2 stages. While the proximal mapping noise map exhibits finer structure in the 2nd and 3rd stages. This is reasonable since more uncertainty is present in the 1st stage introduced by the noisy measurement while the denosing network focus more on the residual learning. + +![](images/9a025d8418b6f3a5d8cc772a801691782f30fb60050a136bc03f824ebdc9017a.jpg) +Figure 8: The network structure of step map estimation. + +# A.8 Limitation and Broader Impact of Our Work + +Our work assumes a pre-measured intensity and utilizes a grayscale or RGB image obtained by a dual-camera CASSI system to serve as an intensity image. This is how we obtain intensity image and also our limitation. However, we expect that this decomposition framework is applicable to regular CASSI system. By obtaining the intensity image though a regular CASSI training and then freeze this intensity network, continue training the CIDNet for chromaticity reconstruction. This could be further explored in our future work. Moreover, our chromaticity-intensity decomposition framework opens a new paradigm for low-light or shadow-removal hyper-spectral reconstruction since the chromaticity represents more abundant scene/sample information. + +![](images/a9a201f4d14760dc7a7b39a818937e75860f6baa5d5b0863ce5e33ea8b572549.jpg) +Figure 9: Comparison between HSIs and chromaticity on KAIST test dataset. + +![](images/6a14a93726762d983f8257d62af689a93a4a56e11ce7bdf90444962a8ec7250d.jpg) +Figure 10: Visualization of dual noise estimation module in CIDNet-3stg. + +![](images/e3979bedab98857c05921bd05d176232c1ddbd9a9f7ce52a652e278c69b9b983.jpg) \ No newline at end of file diff --git a/NeurIPS/2025/Spectral Compressive Imaging via Chromaticity-Intensity Decomposition/images.zip b/NeurIPS/2025/Spectral Compressive Imaging via Chromaticity-Intensity Decomposition/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..edf88a12d08b5764115c8619276e6b267119d79c --- /dev/null +++ b/NeurIPS/2025/Spectral Compressive Imaging via Chromaticity-Intensity Decomposition/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d5ab4be5c6c840c523b0d06ad38df6d04e2bf152c9e6e85a13a0506d391a6ef0 +size 1335275 diff --git a/NeurIPS/2025/Spectral Compressive Imaging via Chromaticity-Intensity Decomposition/layout.json b/NeurIPS/2025/Spectral Compressive Imaging via Chromaticity-Intensity Decomposition/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..9a1843163acc174b8836e17e28e442199feb4583 --- /dev/null +++ b/NeurIPS/2025/Spectral Compressive Imaging via Chromaticity-Intensity Decomposition/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ad808206f9abb129147750e9f4bcb6a89abb9a8d32f545554b96b50136febde2 +size 816181 diff --git a/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_content_list.json b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2895b385f225eb2aa4da1313ed6cec87e8409aef --- /dev/null +++ b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eb88deedb8701d5a65b9c6c27ef1eaefd76d5fb19cb665de33a60a88881ee464 +size 193841 diff --git a/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_model.json b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_model.json new file mode 100644 index 0000000000000000000000000000000000000000..209ac8b5379ef33ebcda1dd38a675e5625829d40 --- /dev/null +++ b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:38e33da98183ff8601b326f2bf04440185f154334642275586c457b19905b4f4 +size 239653 diff --git a/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_origin.pdf b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f99834f2d1bfd7c5aea569aed4e255c0d0341b78 --- /dev/null +++ b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/d6f26d2e-400e-43db-b4c0-feea2d973fa0_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f9a6ca3d86e35037bfd932a5ff9590494dd11ab7c2d0deb83dcaee2e81352f5c +size 898688 diff --git a/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/full.md b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/full.md new file mode 100644 index 0000000000000000000000000000000000000000..0d406c99fe9319a54a48af41aaa877a02472198c --- /dev/null +++ b/NeurIPS/2025/Spectral Conditioning of Attention Improves Transformer Performance/full.md @@ -0,0 +1,1020 @@ +# Spectral Conditioning of Attention Improves Transformer Performance + +Hemanth Saratchandran + +Australian Institute for Machine Learning Adelaide University + +hemanth.saratchandran@adelaide.edu.au + +Simon Lucey + +Australian Institute for Machine Learning Adelaide University + +simon.lucey@adelaide.edu.au + +# Abstract + +We present a theoretical analysis of the Jacobian of a attention block within a transformer, showing that it is governed by the query, key, and value projections that define the attention mechanism. Leveraging this insight, we introduce a method that systematically alters the spectral properties of each attention layer to reduce the Jacobian's condition number, thereby improving the overall conditioning of the attention layers within a transformer network. We empirically show that this improved Jacobian conditioning translates to enhanced performance in practice. Our approach is simple, broadly applicable, and can be easily integrated as a drop-in replacement for a wide range of existing attention mechanisms. We validate its effectiveness across diverse transformer architectures and tasks, demonstrating consistent improvements in performance. + +# 1 Introduction + +Since its introduction in [34], the modern transformer has emerged as one of the most influential architectures in modern machine learning, yielding advances across a wide range of domains, including natural language processing (NLP) [34, 42, 41], computer vision [7, 20, 33, 5], and robotics [27, 23]. At the heart of its success lies the attention mechanism, which enables the model to capture complex relationships by computing pairwise interactions between all input tokens. By dynamically assigning importance to different tokens, attention allows transformers to model global dependencies, making them a foundational architecture in deep learning. + +In this work, we investigate the conditioning of the Jacobian associated with the attention layer in transformer models. The conditioning of a matrix is measured by its condition number, the ratio of its largest to smallest singular value, with high values indicating ill-conditioning. Poorly conditioned Jacobians can hinder performance for gradient-based optimizers [25]. Recent advances in feedforward neural networks have shown that improving Jacobian conditioning can lead to better optimization and generalization performance. For instance, [30] proposed a weight normalization method to address ill-conditioning, while [19] demonstrated that reducing the condition number of the neural tangent kernel (NTK) leads to better convergence. Despite these insights, the conditioning of Jacobians in transformer models, particularly within attention layers remains largely unexamined. Addressing this gap is the central focus of our work. + +We present a theoretical framework showing that the conditioning of the Jacobian in a attention layer is influenced by the conditioning of the queries, keys, and values matrices that constitute the attention matrix within that layer. Building on this insight, we introduce a method that modifies the spectrum of these matrices by adding a carefully designed correction term, yielding a new attention block that we call spectral conditioned attention, resulting in significantly improved Jacobian conditioning. Our analysis shows that the optimal correction can be derived using the singular value decomposition (SVD) of the query, key, and value matrices. However, computing the SVD directly can + +be prohibitively expensive in large-scale models. To address this, we also propose a computationally efficient approximation that achieves comparable improvements in conditioning with substantially lower overhead. + +While a full theoretical proof linking our methodology to improved convergence in gradient-based optimization would require analyzing the NTK of a transformer, an extremely difficult task, we provide strong empirical evidence that our approach improves transformer performance across a wide range of applications, including image classification, object detection, instance segmentation, and natural language processing. A key advantage of our approach is its compatibility with a broad class of different attention mechanisms [4, 20, 33, 39, 6, 38], which we confirm empirically. Our main contributions include: + +1. A theoretical framework that analyses the condition number of the Jacobian of attention showing its dependence on the condition number of the query, key and value matrices comprising an attention layer. +2. The introduction of spectral conditioned attention that adds a correction term to each of the query, key and value matrices resulting in a better conditioned attention Jacobian. + +We validate the above contributions across a range of transformer architectures and applications, including image classification, object detection, instance segmentation, and natural language processing, consistently demonstrating improved performance in each setting. + +# 2 Related Work + +Conditioning. Several recent works have highlighted the critical role of conditioning in the optimization and generalization of neural networks. In [30], the authors focus on the condition numbers of weight matrices in feedforward architectures, showing that well-conditioned weights correlate with improved performance across tasks. They propose a preconditioning strategy that multiplies each weight matrix by a carefully constructed transformation, effectively reducing its condition number and facilitating more stable training. In a complementary line of work, [19] investigates neural network optimization from the perspective of the neural tangent kernel (NTK). The authors demonstrate that improving the conditioning of the NTK leads to better convergence, especially in the infinite-width regime where the NTK governs training dynamics [14]. In [21], it is shown that normalization improves the conditioning of various neural network architectures. Similarly, [40] demonstrate that normalizing attention weights enhances training convergence, which can be interpreted as an alternative means of improving the conditioning of attention layers. In a different approach, [1] shows that simply increasing the depth of feedforward networks can enhance their conditioning, thereby improving the effectiveness of gradient-based optimization methods. Recent work on transformers has examined conditioning from multiple perspectives, including tokenization, activation functions, and optimization [28, 17, 31, 15, 29], as well as within the context of fine-tuning [16, 3, 2, 12]. + +Attention. A range of methods have been proposed to improve the efficiency and scalability of Transformer architectures, often by rethinking the design of attention mechanisms or reducing computational complexity. For example, the Data-Efficient Image Transformer (DeiT) [33] introduces distillation tokens to achieve competitive performance in vision tasks without relying on large-scale datasets. The Cross-Covariance Image Transformer (XCiT) [4] reformulates attention by leveraging cross-covariances of spatial features, enabling efficient spatial interactions with reduced overhead. Swin Transformer [20] introduces a hierarchical architecture and a novel shifted window-based attention mechanism, significantly improving efficiency and effectiveness in vision tasks. DaViT [6] extends traditional vision transformers by incorporating both spatial and channel attention mechanisms, enhancing feature extraction across multiple dimensions. The Nyströmformer [38] takes a different approach by approximating full self-attention using the Nyström method, significantly lowering the quadratic complexity to near-linear while maintaining the core properties of attention, making it especially effective for long-sequence modeling. + +# 3 Methodology + +# 3.1 Preliminaries + +In this section, we define the transformer architecture by describing the structure of a transformer layer, along with establishing notation for various mathematical quantities. + +A transformer layer can be represented as a mapping + +$$ +\mathbf {T}: \mathbb {R} ^ {N \times D} \rightarrow \mathbb {R} ^ {N \times D} \tag {1} +$$ + +which is formally defined by + +$$ +\mathbf {T} (X) = \mathbf {F} (\mathbf {A} (X) + X), \tag {2} +$$ + +where $\mathbf{F}$ denotes a feed forward neural network with a residual connection, and $\mathbf{A}$ represents an attention mechanism. In general, layer normalization is added however for simplicity we omit layer normalization for this discussion. + +For the theoretical framework we will primarily focus on self-attention, which is one of the most common forms of attention. Self-attention is composed of three learnable matrices, query $W_{Q} \in \mathbb{R}^{D \times d}$ , key $W_{K} \in \mathbb{R}^{D \times d}$ , and value $W_{V} \in \mathbb{R}^{D \times d}$ , defined for an input sequence $X \in \mathbb{R}^{N \times D}$ by + +$$ +\mathbf {A} (X) = \operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X W _ {V} \tag {3} +$$ + +where $\text{softmax}$ is the softmax activation that acts row-wise on a matrix [26]. Note that then $A(X) \in \mathbb{R}^{N \times d}$ . When training a transformer the self-attention parameters are given by the weight matrices $W_{Q}, W_{K}$ and $W_{V}$ . For further details on transformers readers may consult [26]. + +The self-attention map of a layer in a transformer $\mathbf{A}(X)$ has parameters given by those parameters in $X$ from the previous layer and those given by $W_{Q}$ , $W_{K}$ and $W_{V}$ that define $\mathbf{A}(X)$ . Our work will consider the Jacobian of $\mathbf{A}(X)$ with respect to the parameters within the layer of $\mathbf{A}(X)$ , namely $W_{Q}$ , $W_{K}$ and $W_{V}$ . Therefore, when we speak of the Jacobian of $\mathbf{A}(X)$ it will be with respect to $W_{Q}$ , $W_{K}$ , $W_{V}$ . We will denote this Jacobian by $J(\mathbf{A}(X))$ and note that it is defined by + +$$ +J (\mathbf {A} (X)) = \left[ \frac {\partial \mathbf {A} (X)}{\partial W _ {Q}}, \frac {\partial \mathbf {A} (X)}{\partial W _ {K}}, \frac {\partial \mathbf {A} (X)}{\partial W _ {V}} \right] ^ {T} \tag {4} +$$ + +Given a matrix $W \in \mathbb{R}^{m \times n}$ we denote the vectorization of $W$ by $\mathbf{vec}(W) \in \mathbb{R}^{mn \times 1}$ [22]. Note that for such a matrix there is a transformation $T_{mn} \in \mathbb{R}^{mn \times mn}$ such that $T_{mn}\mathbf{vec}(W) = \mathbf{vec}(W^T)$ where $W^T$ denotes the transpose of $W$ . The matrix $T_{mn}$ is known as a commutation matrix and is a permutation matrix [22]. The maximum singular value of a matrix $W$ will be denoted by $\sigma_{\max}(W)$ and the minimum singular value by $\sigma_{\min}(W)$ . We will use the standard terminology SVD to denote the singular value decomposition of a matrix. Given a vector $z \in \mathbb{R}^n$ the notation $||z||_2$ will denote the vector 2-norm of $z$ . + +# 3.2 Main Theorems + +In this section, we analyze the conditioning of the Jacobian of the self-attention matrix in a transformer. We will show that the condition number of the Jacobian depends on the condition number of the queries, keys and values matrices. This then motivates our key insight: reducing the condition number of the queries, keys and values matrices can lead to a lower condition number for the Jacobian of the self-attention matrix in each layer of a transformer, which we empirically verify in Section 4. We have chosen to focus our theory on self-attention so as to yield concrete formulas. However, our theory goes through for more general attention mechanisms, see Section A.1 for a discussion. Proofs of all lemmas and theorems in this section can be found in Section A.1. + +Definition 3.1. Let $A$ be an $N \times d$ matrix of full rank. The condition number of $A$ , denoted by $\kappa$ , is defined as + +$$ +\kappa (A) = \frac {\sigma_ {\max } (A)}{\sigma_ {\min } (A)} \tag {5} +$$ + +where $\sigma_{\mathrm{max}}(A)$ denotes the maximum singular value of $A$ and $\sigma_{\mathrm{min}}(A)$ the minimum singular value of $A$ , which we know is non-zero as $A$ is full rank. + +Our objective is to analyze the condition number of the self-attention block within a transformer. We demonstrate that the condition number of its Jacobian is influenced by the condition numbers of the query, key, and value weight matrices. By appropriately conditioning these matrices, we show that the Jacobian's condition number can be significantly reduced, leading to a more stable and well-conditioned attention mechanism suitable for applications. + +To establish this, we first examine the derivatives of the self-attention block with respect to the parameters $W_{Q}$ , $W_{K}$ , and $W_{V}$ , yielding Theorem 3.3 which is then used to establish a condition number bound on the Jacobian given in Theorem 3.4. As a starting point, Theorem 3.2 presents the derivative of the softmax function, which plays a central role in the analysis. + +Lemma 3.2. Let $\Lambda : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ denote the function $\Lambda(z) = \operatorname{Diag}(z) - z \cdot z^T$ . We then have that + +$$ +\frac {\partial \operatorname {s o f t m a x}}{\partial x} (z) = \Lambda (\operatorname {s o f t m a x} (z)). \tag {6} +$$ + +Theorem 3.3. Let $\mathbf{A}(X)$ denote a self-attention matrix with input $X$ as defined by Equation (3). Then + +$$ +\frac {\partial \mathbf {A} (X)}{\partial W _ {Q}} = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \left(\Lambda \left(\mathbf {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)\right)\right) \left(X W _ {K} \otimes X\right) \tag {7} +$$ + +$$ +\frac {\partial \mathbf {A} (X)}{\partial W _ {K}} = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \left(\Lambda \left(\mathbf {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)\right)\right) \left(X \otimes X W _ {Q}\right) \cdot T _ {D d} \tag {8} +$$ + +$$ +\frac {\partial \mathbf {A} (X)}{\partial W _ {V}} = I _ {d} \otimes \operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X \tag {9} +$$ + +where $\Lambda$ is defined in Theorem 3.2 and $T_{Dd}$ is the commutation matrix satisfying $T_{Dd}\mathbf{vec}(W_K) = \mathbf{vec}(W_K^T)$ (see Section 3.1). + +Theorem 3.4. Let $\mathbf{A}(X)$ denote a self-attention matrix, as defined in Equation (3), with input $X$ and let $J(\mathbf{A}(X))$ denote its Jacobian with respect to the parameter matrices $W_{Q}$ , $W_{K}$ and $W_{V}$ as defined in Equation (4). Assume that $J(\mathbf{A}(X))$ has full rank so that $\kappa(J(\mathbf{A}(X)))$ is finite. Then + +$$ +\begin{array}{l} \kappa (J (\mathbf {A} (X))) \leq \kappa (X) ^ {3} \kappa \left(\Lambda \left(\operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)\right)\right) \kappa \left(W _ {V}\right) \left(\kappa \left(W _ {Q}\right) + \kappa \left(W _ {K}\right)\right) \tag {10} \\ + \kappa (X) \kappa (\operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)) \\ \end{array} +$$ + +where $\Lambda$ is defined in Theorem 3.2. + +Theorem 3.4 shows that $\kappa(J(\mathbf{A}(X)))$ is bounded above by a sum of two terms: + +$$ +\kappa (X) ^ {3} \cdot \kappa (\Lambda (\operatorname {s o f t m a x} (X W _ {Q} W _ {K} ^ {T} X ^ {T}))) \kappa (W _ {V}) (\kappa (W _ {Q}) + \kappa (W _ {K})) \tag {11} +$$ + +$$ +\kappa \left(\operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {\top} X ^ {\top}\right)\right) \tag {12} +$$ + +Observation: Directly conditioning the Jacobian of the self-attention map with respect to the parameters $W_{Q}, W_{K}$ , and $W_{V}$ during training would require computing the Jacobian at every iteration, which is prohibitively expensive. However, Theorem 3.4 offers a more efficient alternative: reducing the upper bound in Equation (10). Since we have access to $W_{Q}, W_{K}$ , and $W_{V}$ , lowering their condition numbers reduces the quantity in Equation (11), thereby tightening the bound in Equation (10). The following theorem shows how to do this. + +Theorem 3.5. Let $W_{Q}, W_{K}$ and $W_{V}$ denote the parameters of a self-attention matrix $\mathbf{A}(X)$ . Then there exist matrices $C_{Q}, C_{K}$ and $C_V$ such that + +$$ +\kappa \left(W _ {Q} + C _ {Q}\right), \kappa \left(W _ {K} + C _ {K}\right), \kappa \left(W _ {V} + C _ {V}\right) \leq 2 \tag {13} +$$ + +![](images/a09dad5fc2c4570c5c71d14c6eeb7b60407719a1fa4d10ef5b5aaea6d46fc9af.jpg) +Figure 1: An illustration of spectrally conditioned self-attention within a transformer layer. At each layer, the self-attention weights $W_{Q}, W_{K}$ , and $W_{V}$ are modified by adding correction terms $C_{Q}, C_{K}$ , and $C_{V}$ , respectively. The correction terms $C_{Q}, C_{K}$ , and $C_{V}$ are initialized before training using Theorem 3.8 and remain fixed throughout training. + +Overview of proof of Theorem 3.5: The same proof works for $W_{Q}$ , $W_{K}$ and $W_{V}$ . Thus we will give the main steps for $W_{Q}$ . As in Section 3.1 let $W_{Q} \in \mathbb{R}^{D \times d}$ . + +1. Using the SVD write $W_{Q} = USV^{T}$ where $U \in \mathbb{R}^{D \times D}$ is the left singular vectors, $S \in \mathbb{R}^{D \times d}$ is the matrix of singular values on the main diagonal and $V \in \mathbb{R}^{d \times d}$ are the right singular vectors. +2. Define a new matrix $C_Q = U\overline{S} V^T$ where $\overline{S}$ has $\sigma_{\max}(W_Q)$ on its main diagonal and zeros everywhere else. +3. The matrix $W_{Q} + C_{Q}$ then satisfies $\kappa (W_{Q} + C_{Q})\leq 2$ . + +# 3.3 Spectral Conditioned Attention + +In Theorem 3.5, we showed that adding correction terms $C_{Q}, C_{K}$ , and $C_V$ to the weight matrices $W_{Q}, W_{K}$ , and $W_{V}$ , respectively, significantly reduces their condition numbers $\kappa(W_{Q}), \kappa(W_{K})$ , and $\kappa(W_{V})$ . This motivates the following definition. + +Definition 3.6. We define spectral conditioned attention by + +$$ +\operatorname {S p e c} \mathbf {A} (X) := \operatorname {s o f t m a x} \left(X \left(W _ {Q} + C _ {Q}\right) \left(W _ {K} + C _ {K}\right) ^ {T} X ^ {T}\right) X \left(W _ {V} + C _ {V}\right) \tag {14} +$$ + +where $C_Q, C_K$ and $C_V$ are the correction terms given in Theorem 3.5. + +We call the process of adding correction matrices $C_{Q}, C_{K}$ and $C_V$ to $W_{Q}, W_{K}$ and $W_{V}$ respectively to lower their condition number as spectral conditioning. + +Remark 3.7. Theorem 3.5, together with Theorem 3.4, shows that spectrally conditioned attention reduces the upper bound in Equation (10), thereby yielding a tighter bound on the condition number of the Jacobian of the self-attention layer. + +The proof overview for Theorem 3.5 showed the correction term $C_{Q}$ (and $C_K, C_V$ ) is derived by taking an SVD. Computing the SVD of each of $W_{Q}, W_{K}, W_{V}$ at each iteration of training would lead to a significant memory overhead. We therefore provide an implementation friendly form of spectral conditioned attention, motivated by the following theorem (proof given in Section A.1). + +Theorem 3.8. Let $A \in \mathbb{R}^{m \times n}$ and let $I_k \in \mathbb{R}^{m \times n}$ be the matrix that has 1 on its main $k \times k$ diagonal, where $k = \min\{m, n\}$ , and zeros elsewhere. Let $\lambda \geq 2$ be a fixed constant and assume that $\frac{\sigma_{\max}(A) + \lambda}{\lambda - \sigma_{\min}(A)} \leq \frac{\sigma_{\max}(A)}{\sigma_{\min}(A)}$ . Then $\kappa(A + \lambda I_k) < \kappa(A)$ . + +The key difference between Theorem 3.5 and Theorem 3.8 when applied to $W_{Q}$ , $W_{K}$ , and $W_{V}$ is that Theorem 3.5 guarantees a condition number strictly less than 2, whereas Theorem 3.8 only ensures a reduction relative to the original i.e. $\kappa(W_{Q} + \lambda I_{k}) < \kappa(W_{Q})$ , and similarly for $W_{K}$ and $W_{V}$ . However, the advantage of Theorem 3.8 over Theorem 3.5 is that the correction term $\lambda I_{k}$ does not require computing the SVD of $W_{Q}$ , $W_{K}$ or $W_{V}$ and hence is much more memory efficient. + +Implementation: We initialized correction matrices $C_{Q}, C_{K}$ , and $C_{V}$ as $\lambda I_{k}$ with $\lambda = 10$ (see Section A.2.1 for an ablation study on $\lambda$ ), and used spectral conditioned attention where the modified weights were $W_{Q} + C_{Q}, W_{K} + C_{K}$ , and $W_{V} + C_{V}$ . These correction matrices are fixed during training and are not updated, incurring no additional memory overhead during backpropagation. An overview of the resulting attention architecture is shown in Figure 1. + +For a comparison of spectral conditioned attention using Theorem 3.5 and Theorem 3.8 on vision transformers see Section A.2.1. + +# 4 Experiments + +In this section, we evaluate our insights from Section 3 on a variety of transformer applications. For each application we will consider an original transformer architecture used within the literature and compare it with one employing spectral conditioning on its attention layer. + +Implementation. In all cases, we used the implementation described in Section 3.3, where fixed correction terms $C_Q$ , $C_K$ , and $C_V$ are added to the query, key and value matrices $W_Q$ , $W_K$ and $W_V$ respectively within each attention layer, with $\lambda = 10$ (see Section A.2.1 for an ablation on $\lambda$ ). These terms are not updated during training and therefore do not introduce any additional trainable parameters. For more details see Section A.2. + +# 4.1 Image Classification + +Vision transformers. We applied spectral conditioned attention to a variety of modern vision transformers: the original vision transformer (ViT-B) [7], Swin transformer (Swin-B) [20], Cross-Covariance image transformer (XCiT-M) [4], Data efficient image transformer (DeiT-B) [33], Dual attention vision transformer (DaViT-B) [6] for image classification on ImageNet-1k. Each of these vision transformers uses a different attention block to the standard self-attention used in the ViT-B architecture. However, spectral conditioned attention (see Theorem 3.6) works by adding a correction term to each query, key and value matrix and hence can easily be plugged into these modern variants of self-attention. We include these latter models to illustrate that, while our theoretical analysis in Section 3 centers on self-attention, spectral conditioned attention can be readily incorporated into modern transformer architectures, including those with more complex designs. + +**Validating the theory.** We validate the theoretical results from Section 3 on a ViT-B model trained on the ImageNet-1k dataset. For each experiments we ran five trials and took the mean and standard deviations. For each attention head in every transformer layer, we compute the minimum singular value of the query $(W_{Q})$ , key $(W_{K})$ , and value $(W_{V})$ weight matrices, and then average these values across all heads and layers. To improve the conditioning, we add correction terms $C_{Q}$ , $C_{K}$ , and $C_{V}$ as described after Theorem 3.8, resulting in spectral conditioned attention matrices $W_{Q} + C_{Q}$ , $W_{K} + C_{K}$ , and $W_{V} + C_{V}$ . The left figure in Figure 2 shows the average minimum singular value during training and in each case we see that the minimum singular value of the corrected weights is higher. Similar plots for the maximum singular value is shown in Section A.2.1. The middle plot in Figure 2 compares the average condition numbers of the original and spectral conditioned query, key and value matrices, demonstrating that the latter are substantially better conditioned, thus empirically verifying the claim of Theorem 3.8. Finally, the right plot reports the average condition number of the + +Jacobian of the self-attention matrices, both for the standard ViT-B and for the spectral conditioned version. It also includes the theoretical upper bound from Theorem 3.4. While the spectral correction was motivated by this theoretical bound, the results clearly show that it leads to better-conditioned Jacobians in practice, supporting its use in designing more stable attention mechanisms. Note that each plot shows the mean over five trials. Standard deviations cannot be seen due to the log plot of the y-axis but can be found in separate plots in Section A.2.1. We repeated the experiment using the XCiT-M transformer [4]. As shown in Figure 3, the results exhibit the same trend observed with ViT-B, further supporting the theoretical findings in Section 3. + +![](images/8950555c42ff8494a702262bd2f7a58606eeeeeb1a759d5e1ab78c3e459c8c21.jpg) +Figure 2: Analysis for ViT-B. Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ and their spectrally conditioned counterparts $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ throughout training. Middle: Condition numbers of $W_{Q}$ , $W_{K}$ , and $W_{V}$ , and their spectrally conditioned forms during training. Right: Average condition number of the self-attention Jacobian over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). + +![](images/e106249032c6d6112d743aec1195aa5482791c855f9be24f33e71fde4704ffa5.jpg) + +![](images/863047f27ffb2c6ac6c37eb34da6efaf4f837bf2a4d393f70053fd8216d69fd3.jpg) + +![](images/4bae58881288a68a1e626cb5b37af302dab30ec1d3a70042658827b535ba18ea.jpg) +Figure 3: Analysis for XCiT-M. Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ and their spectrally conditioned counterparts $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ throughout training. Middle: Condition numbers of $W_{Q}$ , $W_{K}$ , and $W_{V}$ , and their spectrally conditioned forms during training. Right: Average condition number of the self-attention Jacobian over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). + +![](images/09e7016885a441bce7929eca2e778009997e03bbd349b4dc23580d63b11e6fb3.jpg) + +![](images/a3df516528219051ffa2144f178f0029113c88c97f1a319dfc5542461e53d693.jpg) + +Rank assumption. In the statement of Theorem 3.4, we assumed that the Jacobian is full rank, ensuring the condition number in Equation (10) remains finite. Empirically, we observed this assumption holds across all vision transformer architectures considered. For both ViT-B and XCiT-M, the final plots in Figure 2 and Figure 3 confirm that the average condition number remains finite and stable throughout training, indicating no rank deficiency in the Jacobian of each layer. + +Results on ImageNet-1k. We trained two versions of each transformer: a baseline model and one incorporating spectral conditioned attention. For each base architecture, we observed that the minimum singular values of the query, key, and value matrices remained below 1 throughout training (see Figure 2 for ViT-B and Figure 3 for XCiT-M), allowing us to apply Theorem 3.8 and implement spectral conditioning as described in Section 3.3. All models were trained using the AdamW optimizer, see Section A.2.1. We trained each model five times with five different random seeds. The final results, summarized in Table 1, show the mean test accuracy with the standard deviation in brackets. We observe that in every case, the spectral conditioned variant achieves higher test accuracy than its unmodified counterpart. Hardware, memory cost and FLOPS analysis is shown + +in Section A.2.1. An ablation on $\lambda$ is given in Section A.3. Furthermore, spectral conditioning was applied together with layer normalization as shown in Figure 1. A discussion comparing the two is given in Section A.3. + +Table 1: Comparison of Vision Transformers with their original attention architecture versus one with spectrally conditioned attention (Spec. cond.), pre-trained on the ImageNet-1k dataset. We report Top-1% classification accuracy. In each case, spectral conditioning improves performance over the original attention mechanism. + +
ViT-BDeiT-BSwin-BXCiT-MDaViT-B
Original80.7 (±0.41)81.6 (±0.30)83.4 (±0.28)82.6 (±0.39)84.3 (±0.26)
Spec. cond.81.7 (±0.38)82.6 (±0.32)84.1 (±0.25)83.5 (±0.35)84.9 (±0.21)
+ +# 4.2 Object Detection and Instance Segmentation + +In this section, we evaluate our methodology on two downstream tasks: object detection and instance segmentation. The goal is to assess the effectiveness of spectral conditioned attention in a fine-tuning setting. We begin by pretraining an XCiT architecture [4] on the ImageNet-1k dataset, followed by fine-tuning on the COCO 2017 dataset [18]. The XCiT models are used as backbones within the Mask R-CNN framework [13], enhanced with a Feature Pyramid Network (FPN) to improve multi-scale feature representation. To integrate XCiT with FPN, we modify its column structure to extract intermediate features from multiple layers. Specifically, we use the 12-layer XCiT-Small (XCiT-S) adapting their its strides from the original fixed stride of 16 to [4, 8, 16, 32] to match the FPN hierarchy. Downsampling is performed via max pooling, and upsampling is achieved using a single transposed convolution layer. We conduct this procedure for both the original XCiT-S and one incorporating spectral conditioned attention blocks. + +Downstream results. The results for both object detection and instance segmentation are shown in Table 2. We ran five trials each with a different seed and plotted the mean metric with standard deviation in brackets. The reported metrics include $AP^b$ (Average Precision for bounding box predictions), $AP_{50/75}^b$ (Average Precision at IoU thresholds of 0.50 and 0.75 for bounding boxes), $AP^m$ (Average Precision for mask predictions), and $AP_{50/75}^m$ (Average Precision at IoU thresholds of 0.50 and 0.75 for mask predictions). In each case we see the XCiT employing spectral conditioned (spec. cond.) out performs the original architecture. Hardware and memory overheads are discussed in Section A.2.2. + +Table 2: Performance evaluation of object detection and instance segmentation on the COCO dataset. For each metric, our spectrally conditioned architecture (Spec. cond.) outperforms the original. + +
ModelAPbAPb50APb75APmAPm50APm75
original44.9 (±0.33)66.1 (±0.31)48.9 (±0.35)40.1 (±0.29)63.1 (±0.31)42.8 (±0.34)
Spec. cond.45.6 (±0.31)66.7 (±0.36)49.6 (±0.32)40.5 (±0.33)63.4 (±0.28)43.3 (±0.32)
+ +# 4.3 Nyströmformer on LRA Benchmark + +To evaluate the effectiveness of our method, we applied it to the Nyströmformer architecture [38], which is specifically designed to handle long-range dependencies efficiently. Experiments were conducted on the Long-Range Arena (LRA) benchmark [32], a suite of tasks targeting the ability of models to process extended input sequences. We trained two variants of the Nyströmformer: one baseline and one with spectral conditioned attention. + +**Validating the theory.** We validate the theoretical results from Section 3 on a Nyströmformer trained on the LRA text classification task. Each experiment was repeated five times, reporting the mean and standard deviation. We computed the minimum singular values and condition numbers of $W_{Q}$ , $W_{K}$ , $W_{V}$ , and their spectral-conditioned forms $W_{Q} + C_{Q}$ , $W_{K} + C_{K}$ , and $W_{V} + C_{V}$ . We + +also evaluated the average condition number of the attention Jacobian, its conditioned variant, and the theoretical bound from Equation (10). The observed trends closely follow those for ViT-B and XCiT-M in Section 4.1, supporting our theoretical findings. Results are shown in Figure 4; standard deviations and maximum singular value plots are provided in Section A.2.1. + +![](images/245a2efb441d9d4b31565cc9f0232e5de2fabdb4989f0ce98516157ab1350607.jpg) +Figure 4: Analysis for Nyströmformer on text classification task. Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ and their spectrally conditioned counterparts $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ throughout training. Middle: Condition numbers of $W_{Q}$ , $W_{K}$ , and $W_{V}$ , and their spectrally conditioned forms during training. Right: Average condition number of the attention Jacobian over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). + +![](images/435fbff294d7a0146bbcbf45d14fc7121beb943f0f382b2ac6100dfaea294656.jpg) + +![](images/e71f1a005f3b26049ef98afd301849745e23cb1806957b0d4a2f7b0228ed3a26.jpg) + +Results. Table 3 shows the results of the experiment. As can be seen from the table the Nyströmformer that employed spectral conditioned attention outperformed the original Nyströmformer from [38] in every task in the LRA benchmark. We used the exact training setup and hyperparameters from [38]. Hardware and memory overheads are discussed in Section A.3 and an ablation on $\lambda$ is given in Section A.3. A discussion comparing spectral conditioning and layer normalization is given in Section A.3. + +Table 3: Comparison of a Nyströmformer using its original architecture (original) with a Nyströmformer using spectral conditioned attention (Spec. cond.) on the LRA benchmark. We report evaluation accuracy (\%). As shown, our methodology consistently improves performance across all tasks. + +
ModelListOpsTextRetrievalImagePathfinder
Original37.1 (±0.21)63.8 (±0.24)79.8 (±0.26)39.9 (±0.20)72.9 (±0.25)
Spec. cond.37.8 (±0.23)64.8 (±0.20)80.6 (±0.27)40.2 (±0.22)73.7 (±0.23)
+ +# 4.4 Language Modeling with Crammed BERT + +We applied our insights from Section 3 to a Crammed BERT language model, trained entirely from scratch using masked language modeling following the approach in [11]. The model consists of 110 million parameters, with 12 transformer layers and 12 attention heads per layer. For this experiment, we train two versions from scratch: the original Crammed BERT baseline model from [11] and a variant incorporating spectral conditioned attention in each layer, following the implementation from Section 3.3. Both models are trained on The Pile dataset [9], a large-scale corpus designed for language model training, following the pretraining regime carried out in [11]. After pretraining, we evaluate the performance on the GLUE benchmark [35] following the evaluation methodology outlined in [11]. Each model was trained five times on the Pile dataset [9] with five different random seeds. Table 4 shows the mean results with standard deviations shown in brackets. As can be seen from the table, spectral conditioned attention outperforms the baseline model in every downstream task in the GLUE benchmark. For the Hardware and memory overheads we refer the reader to Section A.4. + +Table 4: Evaluation of a pre-trained Crammed BERT on the GLUE benchmark using the original architecture (Original) and our spectrally conditioned variant (Spec. cond.). Spectral conditioning improves performance across all tasks. + +
MNLISST-2STSBRTEQNLIQQPMRPCCoLAAvg.
Original83.8 (±0.21)92.3 (±0.18)86.3 (±0.21)55.1 (±0.20)90.1 (±0.23)87.3 (±0.15)85.0 (±0.25)48.9 (±0.26)78.6
Spec. cond.84.3 (±0.20)92.7 (±0.19)86.6 (±0.22)55.5 (±0.24)91.0 (±0.22)87.5 (±0.23)86.1 (±0.23)51.7 (±0.24)79.4
+ +# 5 Limitations + +Our spectral conditioning approach is derived by optimizing an upper bound on the condition number of the self-attention Jacobian, rather than directly minimizing the Jacobian's condition number itself. While this bound provides useful theoretical guidance and aligns with empirical improvements, it remains an indirect proxy. Developing methods to efficiently estimate and control the exact Jacobian conditioning during training would be a valuable direction for future work. Additionally, due to computational resource constraints, our experiments were limited to models with up to 100 million parameters. Whether spectral conditioning offers similar benefits for large-scale transformer models with billions of parameters remains an open question. + +# 6 Conclusion + +We presented a theoretical framework linking the conditioning of self-attention Jacobians to the spectral properties of the query, key, and value matrices in transformer architectures. Building on this insight, we introduced spectral conditioned self-attention, a simple and broadly applicable method that modifies these matrices with carefully designed correction terms to improve Jacobian conditioning. Our empirical results demonstrate that this approach consistently yields good performance across diverse transformer models and tasks, including image classification, object detection, language modeling, and long-range sequence learning. + +# References + +[1] Naman Agarwal, Pranjal Awasthi, and Satyen Kale. A deep conditioning treatment of neural networks. In Algorithmic Learning Theory, pages 249-305. PMLR, 2021. +[2] Paul Albert, Frederic Z Zhang, Hemanth Saratchandran, Anton van den Hengel, and Ehsan Abbasnejad. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix + +# A.1 Theoretical Framework + +In this section we will give the proofs of the lemma and theorems in Section 3.2 and Section 3.3. + +To begin with we will need the following standard facts on derivatives of matrices. + +Lemma A.1. Let $A \in R^{n \times m}$ , $B \in \mathbb{R}^{k \times l}$ and $C \in \mathbb{R}^{m \times k}$ . Then + +$$ +\frac {\partial A C B}{\partial C} = B ^ {T} \otimes A. \tag {15} +$$ + +Proof. We start by using a well known vectorization identity [22] + +$$ +\operatorname {v e c} (A C B) = \left(B ^ {T} \otimes A\right) \operatorname {v e c} (C) \tag {16} +$$ + +where vec denotes the vectorization operator which takes a matrix and maps it to a vector by stacking its columns on top of each other, see [22]. We then differentiate Equation (16) to obtain + +$$ +\frac {\partial \operatorname {v e c} (A C B)}{\partial \operatorname {v e c} (C)} = B ^ {T} \otimes A. \tag {17} +$$ + +The result of the lemma follows. + +![](images/c988fb6dbe965383c32a9c0e90655d0bcb0434fedf41ee1b6317acf7893d0bc6.jpg) + +Lemma A.2. Let $A \in \mathbb{R}^{n \times m}$ so that $A^T \in \mathbb{R}^{m \times n}$ . Then + +$$ +\mathbf {v e c} \left(A ^ {T}\right) = T _ {m n} \mathbf {v e c} (A) \tag {18} +$$ + +$$ +\frac {\partial \operatorname {v e c} \left(A ^ {T}\right)}{\partial \operatorname {v e c} (A)} = T _ {m n} \tag {19} +$$ + +where $T_{mn}$ is a commutation matrix. + +Proof. The first equation follows from the definition of the transpose of a matrix [22]. The second equation then follows from the first. $\square$ + +We now give the proof of Theorem 3.2. + +Proof of Theorem 3.2. Let $[z_1, \ldots, z_n]$ denote a row vector in $\mathbb{R}^n$ . Then by definition + +$$ +\operatorname {s o f t m a x} \left(\left[ z _ {1}, \dots , z _ {n} \right]\right) = \left[ \frac {e ^ {z _ {1}}}{\sum_ {i = 1} ^ {n} e ^ {z _ {i}}}, \dots , \frac {e ^ {z _ {n}}}{\sum_ {i = 1} ^ {n} e ^ {z _ {i}}} \right]. \tag {20} +$$ + +From the above equation we can compute the partial derivative and find + +$$ +\frac {\partial \operatorname {s o f t m a x} (z) _ {i}}{\partial z _ {j}} = \operatorname {s o f t m a x} (z) _ {i} \left(\delta_ {i j} - \operatorname {s o f t m a x} (z) _ {j}\right). \tag {21} +$$ + +The term $\frac{\partial\mathrm{softmax}(z)_i}{\partial z_j}$ is precisely the $ij$ component of the matrix $\frac{\partial\mathrm{softmax}(z)}{\partial z}$ . Putting each of these $ij$ terms into an $n\times n$ matrix we find + +$$ +\frac {\partial \mathbf {s o f t m a x} (z)}{\partial z} = D i a g (z) - z \cdot z ^ {T} \tag {22} +$$ + +which proves the lemma. + +![](images/2cd172eb574c728523210d919a4004c50876128b73003229f83540e8a8dfa013.jpg) + +We can use the above lemmas to give the proof of Theorem 3.3. + +Proof of Theorem 3.3. We start by establishing the derivative formula for the term $\frac{\partial\mathbf{A}(X)}{\partial W_Q}$ . We will use the notation used in Section 3.1. Note that by definition $\frac{\partial\mathbf{A}(X)}{\partial W_Q} \in \mathbb{R}^{dN \times dD}$ . Using Equation (3) + +$$ +\mathbf {A} (X) = I _ {N} \operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X W _ {V} \tag {23} +$$ + +where $I_N$ is the $N \times N$ identity matrix. This is done so that we can apply Theorem A.1. We then compute + +$$ +\begin{array}{l} \frac {\partial \mathbf {A} (X)}{\partial W _ {Q}} = \frac {\partial \left(I _ {N} \operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X W _ {V}\right)}{\partial W _ {Q}} (24) \\ = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \frac {\partial \text {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)}{\partial W _ {Q}} \text {u s i n g T h e o r e m A . 1} (25) \\ = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \Lambda \left(\mathbf {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)\right) \frac {\partial \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)}{\partial W _ {Q}} \text {u s i n g T h e o r e m 3 . 2} (26) \\ = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \Lambda \left(\operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)\right) \left(X W _ {K} \otimes X\right) (27) \\ \end{array} +$$ + +which proves the first equality in Theorem 3.3. + +To compute $\frac{\partial\mathbf{A}(X)}{\partial W_K}\in \mathbb{R}^{dN\times dD}$ we proceed in a similar way. + +$$ +\begin{array}{l} \frac {\partial \mathbf {A} (X)}{\partial W _ {K}} = \frac {\partial \left(I _ {N} \mathbf {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X W _ {V}\right)}{\partial W _ {K}} (28) \\ = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \frac {\partial \mathbf {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)}{\partial W _ {K}} \text {u s i n g T h e o r e m A . 1} (29) \\ = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \Lambda \left(\operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)\right) \frac {\partial \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)}{\partial W _ {K}} \text {u s i n g T h e o r e m 3 . 2} (30) \\ = \left(W _ {V} ^ {T} X ^ {T} \otimes I _ {N}\right) \Lambda \left(\operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right)\right) \left(X \otimes X W _ {Q}\right) T _ {D d} (31) \\ \end{array} +$$ + +where the last equality follows from Theorems A.1 and A.2 This establishes the second equality in Theorem 3.3. + +To prove the identity for $\frac{\partial\mathbf{A}(X)}{\partial W_V}\in \mathbb{R}^{dN\times dD}$ we write + +$$ +\mathbf {A} (X) = \operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X W _ {V} I _ {d} \tag {32} +$$ + +where $I_{d}$ is the $d \times d$ identity matrix. Then we simply apply Theorem A.1 to obtain + +$$ +\begin{array}{l} \frac {\partial \mathbf {A} (X)}{\partial W _ {V}} = \frac {\partial (\operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X W _ {V} I _ {d})}{\partial W _ {V}} (33) \\ = I _ {d} \otimes \operatorname {s o f t m a x} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T}\right) X (34) \\ \end{array} +$$ + +which proves the final equality in Theorem 3.3. + +Proof of Theorem 3.4. The proof of Theorem 3.4 follows from using Theorem 3.3 and the definition of the Jacobian of $\mathbf{A}(X)$ with respect to $W_{Q}$ , $W_{K}$ and $W_{V}$ given by + +$$ +J (\mathbf {A} (X)) = \left[ \frac {\partial (\mathbf {A} (X))}{\partial W _ {Q}}, \frac {\partial (\mathbf {A} (X))}{\partial W _ {K}}, \frac {\partial (\mathbf {A} (X))}{\partial W _ {V}} \right] ^ {T}. \tag {35} +$$ + +We recall that the condition number is defined as $\kappa(J(\mathbf{A}(X))) = \frac{\sigma_{\max}(J(\mathbf{A}(X)))}{\sigma_{\min}(J(\mathbf{A}(X)))}$ where $\sigma_{\max}(J(\mathbf{A}(X)))$ is the maximum singular value of $J(\mathbf{A}(X))$ and $\sigma_{\min}(J(\mathbf{A}(X)))$ the minimum singular value which we know is non-zero because of the assumption that $J(\mathbf{A}(X))$ has full rank. Note that, using the notation in Section 3.1, we have that $J(\mathbf{A}(X)) \in \mathbb{R}^{3dN \times dD}$ as $\frac{\partial \mathbf{A}(X)}{\partial W_Q}$ , $\frac{\partial \mathbf{A}(X)}{\partial W_K}$ , $\frac{\partial \mathbf{A}(X)}{\partial W_V} \in \mathbb{R}^{dN \times dD}$ . For each of our notation we will write $\mathbf{A}_Q := \frac{\partial \mathbf{A}(X)}{\partial W_Q}$ , $\mathbf{A}_K := \frac{\partial \mathbf{A}(X)}{\partial W_K}$ , $\mathbf{A}_V := \frac{\partial \mathbf{A}(X)}{\partial W_V}$ . + +We will start by computing a bound for the maximum singular value. We have for any vector $z \in \mathbb{R}^{dD}$ we have + +$$ +\begin{array}{l} \left\| J (\mathbf {A} (X)) z \right\| _ {2} ^ {2} = \left\| \left[ \frac {\partial (\mathbf {A} (X))}{\partial W _ {Q}} (z), \frac {\partial (\mathbf {A} (X))}{\partial W _ {K}} (z), \frac {\partial (\mathbf {A} (X))}{\partial W _ {V}} (z) \right] ^ {T} \right\| _ {2} ^ {2} (36) \\ = \left\| \left[ \mathbf {A} _ {Q} (z), \mathbf {A} _ {K} (z), \mathbf {A} _ {V} (z) \right] \right\| _ {2} ^ {2} (37) \\ = \| \mathbf {A} _ {Q} (z) \| _ {2} ^ {2} + \| \mathbf {A} _ {K} (z) \| _ {2} ^ {2} + \| \mathbf {A} _ {V} (z) \| _ {2} ^ {2} (38) \\ \leq \left(\sigma_ {\max } \left(\mathbf {A} _ {Q}\right) ^ {2} + \sigma_ {\max } \left(\mathbf {A} _ {K}\right) ^ {2} + \sigma_ {\max } \left(\mathbf {A} _ {V}\right) ^ {2}\right) \| z \| _ {2} ^ {2}. (39) \\ \end{array} +$$ + +This implies that + +$$ +\begin{array}{l} \sigma_ {\max } (J (\mathbf {A} (X))) := \max _ {z \neq 0} \frac {\left\| J (\mathbf {A} (X)) z \right\| _ {2} ^ {2}}{\left\| z \right\| _ {2} ^ {2}} (40) \\ \leq \sqrt {\sigma_ {\max } (\mathbf {A} _ {Q}) ^ {2} + \sigma_ {\max } (\mathbf {A} _ {K}) ^ {2} + \sigma_ {\max } (\mathbf {A} _ {V}) ^ {2}} (41) \\ \leq \sigma_ {\max } (\mathbf {A} _ {Q}) + \sigma_ {\max } (\mathbf {A} _ {K}) + \sigma_ {\max } (\mathbf {A} _ {V}). (42) \\ \end{array} +$$ + +The next step is to compute a lower bound for the minimum singular value $\sigma_{\mathrm{min}}(J(\mathbf{A}(X)))$ . The approach is similar to the above, using the fact that $\sigma_{\mathrm{min}}(J(\mathbf{A}(X))) = \min_{||z||_2 = 1}J(\mathbf{A}(X)(z))$ . We can then use the inequality + +$$ +\begin{array}{l} \| J (\mathbf {A} (X) (z) \| _ {2} ^ {2} = \left\| \left[ \frac {\partial (\mathbf {A} (X))}{\partial W _ {Q}} (z), \frac {\partial (\mathbf {A} (X))}{\partial W _ {K}} (z), \frac {\partial (\mathbf {A} (X))}{\partial W _ {V}} (z) \right] ^ {T} \right\| _ {2} ^ {2} (43) \\ \geq \max \left\{\left\| \frac {\partial (\mathbf {A} (X))}{\partial W _ {Q}} (z) \right\|, \left\| \frac {\partial (\mathbf {A} (X))}{\partial W _ {K}} (z) \right\|, \left\| \frac {\partial (\mathbf {A} (X))}{\partial W _ {V}} (z) \right\| \right\} (44) \\ \end{array} +$$ + +Then minimizing the above over the constraint $z \in \mathbb{R}^{dD}$ such that $||z|| = 1$ we obtain + +$$ +\sigma_ {\min } (J (\mathbf {A} (X))) \geq \max \left\{\sigma_ {\min } \left(\frac {\partial (\mathbf {A} (X))}{\partial W _ {Q}}\right), \sigma_ {\min } \left(\frac {\partial (\mathbf {A} (X))}{\partial W _ {K}}\right), \sigma_ {\min } \left(\frac {\partial (\mathbf {A} (X))}{\partial W _ {V}}\right) \right\}. \tag {45} +$$ + +Combining the bounds on $\sigma_{\mathrm{max}}(J(\mathbf{A}(X)))$ and $\sigma_{\mathrm{min}}(J(\mathbf{A}(X)))$ we obtain + +$$ +\kappa (J (\mathbf {A} (X))) \leq \kappa \left(\frac {\partial (\mathbf {A} (X))}{\partial W _ {Q}}\right) + \kappa \left(\frac {\partial (\mathbf {A} (X))}{\partial W _ {K}}\right) + \kappa \left(\frac {\partial (\mathbf {A} (X))}{\partial W _ {V}}\right). \tag {46} +$$ + +The final step is to get a bound on the condition numbers for each term on the right hand side in the above inequality. This is done using two facts: Firstly, given two matrices $C$ and $D$ such that the product $CD$ is full rank then $\kappa(CD) \leq \kappa(C)\kappa(D)$ and the second that $\kappa(C \otimes D) = \kappa(C)\kappa(D)$ . Using these two facts we can then use Theorem 3.3 to obtain the bound of Theorem 3.4 and the proof is finished. + +We will now give the proof of Theorem 3.5. + +Proof of Theorem 3.5. We will give the proof for $W_{Q}$ as the proof for $W_{K}$ and $W_{V}$ are exactly analogous. + +Take the SVD of $W_{Q}$ to obtain $W_{Q} = USV^{T}$ where $U \in \mathbb{R}^{D \times D}$ is the left singular vectors, $S \in \mathbb{R}^{D \times d}$ is the matrix of singular values having the singular values of $W_{Q}$ on its main diagonal, $V \in \mathbb{R}^{d \times d}$ is the right singular vectors. We then define a new matrix $C_{Q}$ by the formula + +$$ +C _ {Q} := U \bar {S} V ^ {T} \tag {47} +$$ + +where $\overline{S} \in \mathbb{R}^{D \times d}$ has $\sigma_{\max}(W_Q)$ on its main diagonal and zeros elsewhere. We then have that + +$$ +W _ {Q} + C _ {Q} = U (S + \bar {S}) V ^ {T}. \tag {48} +$$ + +We can then compute $\sigma_{\mathrm{max}}(W_Q + C_Q) = 2\sigma_{\mathrm{max}}(W_Q)$ by definition of $C_Q$ and $\sigma_{\mathrm{min}}(W_Q + C_Q) = \sigma_{\mathrm{min}}(W_Q) + \sigma_{\mathrm{max}}(W_Q)$ . This implies + +$$ +\begin{array}{l} \kappa \left(W _ {Q} + C _ {Q}\right) = \frac {\sigma_ {\max } \left(W _ {Q} + C _ {Q}\right)}{\sigma_ {\min } \left(W _ {Q} + C _ {Q}\right)} (49) \\ = \frac {2 \sigma_ {\max} \left(W _ {Q}\right)}{\sigma_ {\min} \left(W _ {Q}\right) + \sigma_ {\max} \left(W _ {Q}\right)} (50) \\ < 2 (51) \\ \end{array} +$$ + +which finishes the proof of the theorem. + +![](images/a5052e9ac5306db97d810d050a4a18a890fcbace71cbfa7cda953b24155a89d6.jpg) + +A key issue for implementing the approach in Theorem 3.5 into the attention layer is that it requires the computation of the SVD of each of the matrices $W_{Q}$ , $W_{K}$ and $W_{V}$ . Computing the SVD each time for a large matrix is extremely memory intensive. This is exactly the premise of Theorem 3.8 whose proof we now give. + +Proof of Theorem 3.8. The proof starts by making use of the Weyl inequalities for singular values [8] to obtain + +$$ +\sigma_ {\max } (A + \lambda \cdot I _ {k}) \leq \sigma_ {\max } (A) + \sigma_ {\max } (\lambda \cdot I _ {k}). \tag {52} +$$ + +$$ +\sigma_ {\min } (A + \lambda \cdot I _ {k}) \geq \sigma_ {\min } (\lambda \cdot I _ {k}) - \sigma_ {\max } (A). \tag {53} +$$ + +We then compute + +$$ +\begin{array}{l} \kappa (A + \lambda \cdot I _ {k}) := \frac {\sigma_ {\max } (A + \lambda \cdot I _ {k})}{\sigma_ {\min } (A + \lambda \cdot I _ {k})} (54) \\ \leq \frac {\sigma_ {\max } (A) + \sigma_ {\max } (\lambda \cdot I _ {k})}{\sigma_ {\min } (\lambda \cdot I _ {k}) - \sigma_ {\min } (A)} (55) \\ = \frac {\sigma_ {\max } (A) + \lambda}{\lambda - \sigma_ {\min } (A)} (56) \\ \leq \frac {\sigma_ {\max } (A)}{\sigma_ {\min } (A)} \text {b y} (57) \\ = \kappa (A) (58) \\ \end{array} +$$ + +which completes the proof. + +![](images/690c2b34992b3ff8fc723bf300f05c837b4e881688909f7b7dd0eb6cc0d3360a.jpg) + +Other forms of Attention. The theory developed in Section 3 was presented for the self-attention mechanism defined in Section 3.1, to allow for concrete analytical expressions. However, as evident from the derivations, the proposed corrections to the weight matrices $W_{Q}$ , $W_{K}$ , and $W_{V}$ apply to any attention layer. Since the results depend only on the derivatives of the attention function with respect to these matrices, analogous theorems can be established for more general mechanisms such as cross-attention. Moreover, in Section 4, we empirically demonstrate that our theoretical results extend to a broad class of attention architectures. + +# A.2 Experimental Analysis + +Details of implementation of spectral conditioned attention. For all of the experiments in Section 4 we implemented spectral conditioned attention following the strategy outlined in Section 3.3 via Theorem 3.8. We describe this implementation in detail. For each attention layer in a transformer fix the query, key and value weights as $W_{Q}, W_{K}, W_{V}$ . We then form three matrices denoted by $C_{Q}$ , $C_{K}$ and $C_{V}$ with each being defined by the equation + +$$ +C _ {Q} = \lambda I _ {Q} \tag {59} +$$ + +$$ +C _ {K} = \lambda I _ {K} \tag {60} +$$ + +$$ +C _ {V} = \lambda I _ {V} \tag {61} +$$ + +with $\lambda = 10$ and where $C_Q, C_K$ and $C_V$ have the same shape as $W_Q, W_K$ and $W_V$ and $I_Q, I_K$ and $I_V$ has the same shape as $C_Q, C_K$ and $C_V$ with 1's on their main diagonal and zeros elsewhere. + +The matrices $C_Q, C_K$ and $C_V$ , which we call the correction matrices, are fixed throughout training the transformer and never change. They are not updated and hence do not add any memory to the backpropagation algorithm and the only memory comes from storing them. During the training of the transformer, we add each of these correction matrices to the weights $W_Q, W_K$ and $W_V$ during each forward pass forming. Thus after initializing the weights $W_Q^0, W_K^0$ and $W_V^0$ we add the correction terms $C_Q, C_K$ and $C_V$ as follows + +$$ +1. W _ {Q} ^ {0} \rightarrow W _ {Q} ^ {0} + C _ {Q} \tag {62} +$$ + +$$ +2. W _ {K} ^ {0} \rightarrow W _ {K} ^ {0} + C _ {K} \tag {63} +$$ + +$$ +3. W _ {V} ^ {0} \rightarrow W _ {V} ^ {0} + C _ {V} \tag {64} +$$ + +perform a forward pass using $W_{Q}^{0} + C_{Q}$ , $W_{K}^{0} + C_{K}$ and $W_{V}^{0} + C_{V}$ , then perform a backward pass and during the backward pass only the weights $W_{Q}^{0}$ , $W_{K}^{0}$ and $W_{V}^{0}$ are updated forming $W_{Q}^{1}$ , $W_{K}^{1}$ and $W_{V}^{1}$ . This process then repeats, we add the fixed correction terms to $W_{Q}^{1}$ , $W_{K}^{1}$ and $W_{V}^{1}$ forming + +$$ +1. W _ {Q} ^ {1} \rightarrow W _ {Q} ^ {1} + C _ {Q} \tag {65} +$$ + +$$ +2. W _ {K} ^ {1} \rightarrow W _ {K} ^ {1} + C _ {K} \tag {66} +$$ + +$$ +3. W _ {V} ^ {1} \rightarrow W _ {V} ^ {1} + C _ {V} \tag {67} +$$ + +then a backward pass updates $W_{Q}^{1}$ , $W_{K}^{1}$ and $W_{V}^{1}$ and this process continues. Note that Theorem 3.8 allows any $\lambda \geq 2$ when forming Through ablations we found the best value is $\lambda = 10$ . + +# A.2.1 Vision transformers on Imagenet-1k + +**Validating the theory.** In Section 4.1 we validated the theory given in Section 3 on a ViT-B and XCiT-M architecture. For each experiment we ran five trials with five different random seeds and plotted the mean. The plots in Section 4.1 were shown in a log scale where the standard deviations were not visible. We have plotted each plot in a new scale that clearly shows the standard deviations. The plots for ViT-B can be seen in Figures 5 to 8 and the plots for XCiT-M can be seen from Figures 9 to 12. Furthermore, Figure 6 validates the assumption in Theorem 3.8 on the maximum singular value for ViT-B and Figure 10 for the maximum singular for XCiT justifying the use of Theorem 3.8. + +Ablation on $\lambda$ from Theorem 3.8. In Section 4 we used the implementation for the correction terms given in Section 3. This required choosing a factor $\lambda \geq 2$ . For the experiments in Section 4 we chose $\lambda$ through a grid search treating it as a hyperparameter. An ablation for $\lambda$ on the ViT-B architecture trained on ImageNet-1k is shown in Table 5. What we found was that $\lambda \geq 10$ gives the best result. + +Table 5: Ablation on $\lambda$ for ViT-B + +
λ246810121416
ViT-B Acc.80.881.181.481.681.781.781.681.5
+ +Relation to Normalization. Our conditioning methodology can be seen as being related to normalization. By adding correction terms to the weights $W_{Q}$ , $W_{K}$ and $W_{V}$ to improve their spectrum we are in effect producing a normalization for their spectrum. We therefore decided to test our spectral conditioning when we remove layer normalization in the vision transformers to understand whether spectral conditioning can replace layer normalization. Table 6 shows the results on the ViTs on ImageNet-1k. We ran each experiments 5 times and plot the mean and standard deviation in brackets. We found that if we remove layer normalization and keep spectral conditioning (with $\lambda = 10$ ) the performance drops. We noticed that this was due to the fact that the weight values in the corrected terms $W_{Q} + C_{Q}$ , $W_{K} + C_{K}$ and $W_{V} + C_{V}$ were much larger than without layer normalization which can lead to issues with backpropagation and suggesting that layer normalization was still crucial to maintain weights from getting too large. + +Table 6: Performance comparison of spectral conditioning with and without layer normalization. + +
ViT-BDeiT-BSwin-BXCiT-MDaViT-B
Original (with only layer norm.)80.7 (±0.41)81.6 (±0.30)83.4 (±0.28)82.6 (±0.39)84.3 (±0.26)
Spec. cond. + layer norm.81.7 (±0.38)82.6 (±0.32)84.1 (±0.25)83.5 (±0.35)84.9 (±0.21)
Spec. cond. - layer norm.79.8 (±0.31)80.4 (±0.33)80.5 (±0.23)80.0 (±0.35)80.3 (±0.21)
+ +Comparing Theorem 3.5 and Theorem 3.8. In Section 3, we saw that Theorem 3.5 gave a methodology that could reduce the condition number of the query weights, key weights and value weights to less than 2. However, the reason we did not use this theorem as the basis of our experiments is that it requires an SVD computation which would significantly increase training times when compared to the implementation friendly theorem Theorem 3.8. We carried out a comparison on the vision transformer for both methods. As can be seen from Table 7 the vision transformer employing the methodology from Theorem 3.5 take much longer to train. + +Table 7: Comparison of accuracy and training time across transformer variants with spectral conditioning of attention following Theorem 3.5 and Theorem 3.8. + +
ModelAcc.Training time (hrs:mins)
ViT-B (original)80.729:29
ViT-B spec. cond. (Thm. 3.5)82.041:38
ViT-B spec. cond. (Thm. 3.8)81.729:33
DeiT-B (original)81.626:16
DeiT-B spec. cond. (Thm. 3.5)82.937:54
DeiT-B spec. cond. (Thm. 3.8)82.626:24
Swin-B (original)83.453:12
Swin-B spec. cond. (Thm. 3.5)84.368:32
Swin-B spec. cond. (Thm. 3.8)84.153:26
XCiT-M (original)82.691:03
XCiT-M spec. cond. (Thm. 3.5)83.9109:12
XCiT-M spec. cond. (Thm. 3.8)83.591:18
DaViT-M (original)84.375:09
DaViT-M spec. cond. (Thm. 3.5)85.293:12
DaViT-M spec. cond. (Thm. 3.8)84.975:28
+ +Hardware and implementation. The image classification experiments in Section 4.1 of the paper were done on Nvidia A100 GPUs. The implementation of the ViTs were all done using the Timm code base [36]. The architectures were all trained from scratch on the ImageNet-1k dataset using the AdamW optimizer following the hyperparameters used in the original papers [7, 20, 4, 33, 6]. + +Memory Cost. We analyze the memory overhead introduced by using spectrally conditioned weight matrices $W_{Q} + C_{Q}$ , $W_{K} + C_{K}$ , and $W_{V} + C_{V}$ in the forward pass for the ViT-B architecture. A similar analysis gives a similar profile for all the other ViT architectures we consider in Section 4.1. The correction matrices $C_{Q}, C_{K}$ , and $C_V$ are fixed diagonal matrices with constant entries (value 10) on the main diagonal, are non-trainable, and have gradients disabled (requires_grad = False). + +Computing $X(W_{Q} + C_{Q})$ expands as: + +$$ +X \left(W _ {Q} + C _ {Q}\right) = X W _ {Q} + X C _ {Q}, \tag {68} +$$ + +where $XW_{Q}$ is a standard matrix multiplication and $XC_{Q}$ corresponds to scaling each column of $X$ by 10. This operation is efficiently implemented as a column-wise scaling, requiring no explicit storage of $C_{Q}$ beyond a scalar or a small constant vector. + +- Parameters: No new trainable parameters are introduced; $C_Q$ , $C_K$ , and $C_V$ are fixed and non-trainable. +- Gradient Storage: Since the correction matrices do not participate in backpropagation, no additional gradient memory is used. + +- Activation Memory: The computation of $XC_{Q}$ involves element-wise scaling and summation, which reuses existing output tensors without requiring new memory allocations. + +In practice, the additional memory overhead from spectral conditioning is negligible, as it involves no new trainable parameters, no gradient storage, and no extra activation tensors. + +Table 8: Comparison of memory overhead for spectral conditioning relative to standard attention projections. + +
Memory AspectOriginal AttentionWith Spectral Conditioning
Trainable Parameters3 × Dd3 × Dd
Gradient Storage3 × Dd3 × Dd
Correction Matrices (CQ, CK, CV)-negligible (constant scalar)
Activation Memory3 × Nd3 × Nd (no extra tensors)
+ +FLOPS Cost. We analyze the additional FLOPS overhead introduced by using spectrally conditioned weight matrices $W_{Q} + C_{Q}$ , $W_{K} + C_{K}$ , and $W_{V} + C_{V}$ in the forward pass of the attention mechanism in the forward pass for the ViT-B architecture. A similar analysis gives a similar profile for all the other ViT architectures we consider in Section 4.1. Consider the self-attention formulation: + +$$ +\operatorname {s o f t m a x} \left(X \left(W _ {Q} + C _ {Q}\right) \left(W _ {K} + C _ {K}\right) ^ {\top} X ^ {\top}\right) \cdot X \left(W _ {V} + C _ {V}\right), \tag {69} +$$ + +where $X \in \mathbb{R}^{N \times D}$ is the input, and $W_{Q}, W_{K}, W_{V} \in \mathbb{R}^{D \times d}$ are the learnable projection matrices. The correction matrices $C_{Q}, C_{K}$ , and $C_{V}$ are fixed diagonal matrices with constant entries 10 along their main diagonal. + +Computing the term $X(W_{Q} + C_{Q})$ expands as: + +$$ +X \left(W _ {Q} + C _ {Q}\right) = X W _ {Q} + X C _ {Q}. \tag {70} +$$ + +The first term, $XW_{Q}$ , corresponds to a standard matrix multiplication with computational complexity $\mathcal{O}(NDd)$ , requiring $2NDd$ FLOPS. The second term, $XC_{Q}$ , corresponds to scaling each column of $X$ by 10, which requires only $Nd$ FLOPS (one multiplication per entry). + +Thus, the total FLOPS for computing $X(W_{Q} + C_{Q})$ is: + +$$ +2 N D d + N d. \tag {71} +$$ + +An identical cost arises for $X(W_{K} + C_{K})$ and $X(W_{V} + C_{V})$ . Therefore, the additional FLOPS introduced by the correction terms across all three projections is: + +$$ +3 N d. \tag {72} +$$ + +Compared to the original cost of $6NDd$ FLOPS for the query, key, and value projections without spectral conditioning, the overhead introduced by spectral conditioning is negligible: + +$$ +\frac {3 N d}{6 N D d} = \frac {1}{2 D}, \tag {73} +$$ + +which is insignificant for typical values of $D$ used in practice (e.g., $D = 512, 768, 1024$ are common dimension for ViTs). + +Table 9: FLOPS comparison for the forward pass of the attention projection layers with and without spectral conditioning. $N$ denotes the sequence length, $D$ the embedding dimension,and $d$ the projection dimension per head. + +
OperationFLOPS (Original)FLOPS (With Spectral Conditioning)
Query projection (XWQ)2NDd2NDd + Nd
Key projection (XWK)2NDd2NDd + Nd
Value projection (XWV)2NDd2NDd + Nd
Total Projection FLOPS6NDd6NDd + 3Nd
+ +![](images/ca1543fe18f9b7f404b74fd3f290b7de968a0542713cd8b628b96eec803cce5b.jpg) +Figure 5: Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a ViT-B during training. We plot the mean over five trials and the standard deviation. Right: Average minimum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a ViT-B during training. We plot the mean over five trials and the standard deviation. + +![](images/f86e1cc3c69ad3c89069cdb9268ad2ea2b49d3bc17bf021292e33dfcdd3ee9ed.jpg) + +![](images/ceffcc998752335d815eb19e79c33cfa38b69079104fc8d26644ea9bcfd9d622.jpg) +Figure 6: Left: Average maximum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a ViT-B during training. We plot the mean over five trials and the standard deviation. Right: Average maximum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a ViT-B during training. We plot the mean over five trials and the standard deviation. + +![](images/4020956b2379f7c50ba84780c2162d5083d1defb3531c16171cd32c2c0ebfa67.jpg) + +![](images/fc4ef10be9932a71ba5939e8dc79861453f1c33d1fa81729176d86430299348a.jpg) +Figure 7: Left: Average condition number of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a ViT-B during training. We plot the mean over five trials and the standard deviation. Right: Average condition number of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a ViT-B during training. We plot the mean over five trials and the standard deviation. + +![](images/99524eb297727bda178929c79abaaff7f7aa3af0aa0b8eaeadcf86fe8abe144b.jpg) + +# A.2.2 Object detection and instance segmentation + +Hardware and Implementation: The experiments for Section 4.2 of the paper on object detection and instance segmentation were carried out on Nvidia A100 GPUs. The implementation followed [13]. We used the code base given by the GitHub [24] following their exact training regime. + +![](images/0796d3df45a76f7b48101fb2175642240d7af035fe63ae28b98d28c9cd906365.jpg) +Figure 8: Average condition number of the self-attention Jacobian of a ViT-B over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). We ran five trials with five different random seeds with plots showing the mean and standard deviations. + +![](images/4c762495f3dcab05dcc8e9d0721ac5024145bf564b4e4438151dfa47f57f7f5d.jpg) +Figure 9: Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a XCiT-M during training. We plot the mean over five trials and the standard deviation. Right: Average minimum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a XCiT-M during training. We plot the mean over five trials and the standard deviation. + +![](images/f900e50f5e06a731d6241baf03d58535fd75999e26b7df6cba9bb147f8739cdd.jpg) + +![](images/fd62d8d17b351082ba23d65c49766dc0408b6f06158517029997b95e20f248ab.jpg) +Figure 10: Left: Average maximum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a XCiT-M during training. We plot the mean over five trials and the standard deviation. Right: Average maximum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a XCiT-M during training. We plot the mean over five trials and the standard deviation. + +![](images/7131cd8b7363ec9099c5a7ae5ac69f478e05f44fd22f019a6c76ff7c8d16a8e4.jpg) + +Memory and Overheads: For this experiment we used a pre-trained XCiT-S and since, as explained in Section A.2.1, the correction terms are not trained and fixed throughout training there was no real memory overhead. + +![](images/fc3feb3db7ea72b4b989f81826f8099bb2a67de8ebce0b84735acb12da3c5028.jpg) +Figure 11: Left: Average condition number of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a XCiT-M during training. We plot the mean over five trials and the standard deviation. Right: Average condition number of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a XCiT-M during training. We plot the mean over five trials and the standard deviation. + +![](images/2512badc6c464c3dc31b632477e8b06e8a192cc7a18f1bab8846b05877286c3e.jpg) + +![](images/24c3157e0b42db04de926158a5873cab3accc3ce2be827883830bbbff72717be.jpg) +Figure 12: Average condition number of the self-attention Jacobian of a XCiT-M over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). We ran five trials with five different random seeds with plots showing the mean and standard deviations. + +Further results: We also ran the object detection and instance segmentation results using a Swin-base vision transformer as the backbone. The result are shown in Table 10. + +Table 10: Comparison of Swin-base and spectrally conditioned (spec. cond.) Swin-base on COCO benchmarks. + +
ModelAPbAPb50APb75APmAPm50APm75
Swin-base45.967.150.141.163.743.8
Spec. cond. Swin-base46.868.150.741.864.244.6
+ +# A.3 Nyströmformer on LRA benchmark + +**Validating the theory.** In Section 4.3 we validated the theory given in Section 3 on a Nyströmformer on the text classification task in the LRA benchmark. For each experiment we ran five trials with five different random seeds and plotted the mean. The plots in Section 4.3 were shown in a log scale where the standard deviations were not visible. We have plotted each plot in a new scale that clearly shows the standard deviations. The plots are shown from Figures 13 to 16. We also ran a similar empirical analysis for the listops task in the LRA benchmark. The results of which can be seen in + +Figures 17 to 21. Furthermore, Figure 14 and Figure 19 validates the assumption in Theorem 3.8 on the maximum singular value justifying the use of Theorem 3.8. + +Ablation on $\lambda$ from Theorem 3.8. For the experiments in Section 4.3 we chose $\lambda$ through a grid search treating it as a hyperparameter. An ablation for $\lambda$ on the Nyströmformer trained on the text classification task on the LRA benchmark is shown in Table 11. Once again we found that $\lambda \geq 10$ gives the best result. + +Table 11: Ablation on $\lambda$ for Nyströmformer on text classification task. + +
λ246810121416
Acc.63.764.164.364.664.864.864.764.7
+ +Relation to Normalization. We compared spectral conditioning on a Nyströmformer with and without layer normalization. As can be seen in Table 12 we see that layer normalization is important for spectral conditioning. Once again when we removed layer normalization and just applied spectral conditioning we found that the weights were large resulting in poorer performance. + +Table 12: Comparing spectral conditioning with and without layer normalization for the Nyströmformer on text classification. + +
Acc.(%)
Original (with only layer norm.)63.8 (±0.24)
Spec. cond. + layer norm.64.8 (±0.20)
Spec. cond. - layer norm62.3 (±0.23)
+ +Hardware. All the experiments for the Nyströmformer on LRA benchmark results in Section 4.3 were carried out on Nvidia A100 GPUs following the implementation and hyperparameter settings given in [37]. + +Memory and Overheads: The analysis carried out on FLOPS and memory in Section A.2.1 apply in this case as once again the correction matrices $C_{Q}$ , $C_{K}$ and $C_{V}$ are fixed throughout training and only added for forward passes. + +![](images/197b38eb842ab269fc37d3aaaa16d55580e414cf19edc734c692f1dedc7b5a1e.jpg) +Figure 13: Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a Nyströmformer on the text classification task during training. We plot the mean over five trials and the standard deviation. Right: Average minimum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a Nyströmformer on the text classification during training. We plot the mean over five trials and the standard deviation. + +![](images/b60d3dde74d68b31c01d64668a54f86c0272d67ae1d086868ad00a5bd2ca29df.jpg) + +# A.4 Language Modeling with Crammed BERT + +Hardware. The language modeling experiment in Section 4.4 were all carried out on a Nvidia A6000 GPU. The Crammed-Bert was implemented following the original paper [11] and the original GitHub [10]. The training regime follows [10]. + +![](images/2f3283cecc2380fec800f713da4cea1fc5bf979ff152b56a26d3c7125af7e3c7.jpg) +Figure 14: Left: Average maximum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a Nyströmformer on the text classification task during training. We plot the mean over five trials and the standard deviation. Right: Average maximum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a Nyströmformer on the text classification during training. We plot the mean over five trials and the standard deviation. + +![](images/6708811f610be3f520bcd8af5b491d777ee1cee1453da30051ca3400f85dff15.jpg) + +![](images/b78aaa9c16867f6e68f9945169c5331d1818a1a21cae8fcd5616e311d56612e8.jpg) +Figure 15: Left: Average condition number of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a Nyströmformer on the text classification task during training. We plot the mean over five trials and the standard deviation. Right: Average condition number of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a Nyströmformer on the text classification task during training. We plot the mean over five trials and the standard deviation. + +![](images/e59c41af713a701e69bce291fdc52fee1695cf022cd1f1ee3585f78bc10fb79c.jpg) + +![](images/264e7ea34cbe88c395c35e9d53289e991f8cf1d7e8729817d81fa7be187413c4.jpg) +Figure 16: Average condition number of the self-attention Jacobian of a Nyströmformer on the text classification task over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). We ran five trials with five different random seeds with plots showing the mean and standard deviations. + +![](images/0c0decc995fe4b6861d76a671b7fc9adb2b85aea08e307029debc0a7c7655c3e.jpg) +Figure 17: Analysis for Nyströmformer on listops task. Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ and their spectrally conditioned counterparts $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ throughout training. Middle: Condition numbers of $W_{Q}, W_{K}$ , and $W_{V}$ , and their spectrally conditioned forms during training. Right: Average condition number of the attention Jacobian over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). + +![](images/2c0fbcc2701342fbb6a6925662afd2c25b3c09055a5738b00f3096053a17fa27.jpg) + +![](images/7c1da54e18cb8df7b87936dbb2eb7cb9bb587521b1c013d7dc799162551fb82a.jpg) + +![](images/45624eba0e2e889ea7aeec66d36a95b27ba896a2a178f862483899240925bd5c.jpg) +Figure 18: Left: Average minimum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a Nyströmformer on the listops task during training. We plot the mean over five trials and the standard deviation. Right: Average minimum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a Nyströmformer on the listops task during training. We plot the mean over five trials and the standard deviation. + +![](images/3a9e9a30c26c3d0ab15266ef240b380ec34e8494287a29ce65723734f86b06a4.jpg) + +![](images/4f00bb8254cd98239b270d66ab185f0bad6770d14857f7d0427f527499104ab2.jpg) +Figure 19: Left: Average maximum singular value of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a Nyströmformer on the listops task during training. We plot the mean over five trials and the standard deviation. Right: Average maximum singular value of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a Nyströmformer on the listops task during training. We plot the mean over five trials and the standard deviation. + +![](images/c48dc391e051fc7b263098f34a9faaeef088e640f4d3ccf7026fd623a5d4b88b.jpg) + +Memory and Overheads: The analysis carried out on FLOPS and memory in Section A.2.1 apply in this case as once again the correction matrices $C_{Q}$ , $C_{K}$ and $C_{V}$ are fixed throughout training and only added for forward passes. + +![](images/60770ff120fa9fb441403214aafb648eee509b38e8e56cc1c9bf3aae45c00ec7.jpg) +Figure 20: Left: Average condition number of the query, key, and value projection matrices $(W_{Q}, W_{K}, W_{V})$ for a Nyströmformer on the listops task during training. We plot the mean over five trials and the standard deviation. Right: Average condition number of the corrected query, key, and value projection matrices $(W_{Q} + C_{Q}, W_{K} + C_{K}, W_{V} + C_{V})$ for a Nyströmformer on the listops task during training. We plot the mean over five trials and the standard deviation. + +![](images/5572324ed3140ef25ff852ac04f4e69a22edfa1ff60bf66830604a4e81e93e7a.jpg) + +![](images/03c4a1db1502da16c87c7b12478871fc19fd6efc1c3cd9c435221afe143897c8.jpg) +Figure 21: Average condition number of the self-attention Jacobian of a Nyströmformer on the listops task over the course of training, before and after spectral conditioning, along with the theoretical bound from Equation (10). 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Their ability to handle variable-sized collections of unstructured observations, combined with simple maximum-likelihood training and uncertainty-aware predictions, makes them well-suited for modeling data over continuous domains. Since their introduction, several variants have been proposed. Early approaches typically represented observed data using finite-dimensional summary embeddings obtained through aggregation schemes such as mean pooling. However, this strategy fundamentally mismatches the infinite-dimensional nature of the generative processes that NPs aim to capture. Convolutional conditional neural processes (ConvCNPs) address this limitation by constructing infinite-dimensional functional embeddings processed through convolutional neural networks (CNNs) to enforce translation equivariance. Yet CNNs with local spatial kernels struggle to capture long-range dependencies without resorting to large kernels, which impose significant computational costs. To overcome this limitation, we propose the Spectral ConvCNP (SCNNP), which performs global convolution in the frequency domain. Inspired by Fourier neural operators (FNOs) for learning solution operators of partial differential equations (PDEs), our approach directly parameterizes convolution kernels in the frequency domain, leveraging the relatively compact yet global Fourier representation of many natural signals. We validate the effectiveness of SConvCNP on both synthetic and real-world datasets, demonstrating how ideas from operator learning can advance the capabilities of NPs. + +# 1 Introduction + +Stochastic processes offer a mathematical framework for modeling systems that evolve with inherent randomness over continuous domains such as time and space. They underpin a wide range of scientific applications—from spatio-temporal climate dynamics to biological and physical systems—thereby motivating the development of machine learning methods that can learn from data generated by such phenomena [Mathieu et al., 2021, Vaughan et al., 2021, Allen et al., 2025, Ashman et al., 2025, Dupont et al., 2021]. Among classical approaches, Gaussian processes (GPs; Rasmussen et al., 2006) provide a Bayesian framework with closed-form inference and uncertainty quantification. However, their cubic computational cost from matrix inversion and the difficulty of specifying suitable kernels—especially in high-dimensional settings—limit their scalability. + +Motivated by the success of deep neural networks in large-scale function approximation, neural network-based alternatives have emerged. Neural processes (NPs; Garnelo et al., 2018a,b) exemplify this paradigm, combining ideas from GPs and deep learning within a meta-learning framework. By exposing the model to multiple realizations of an underlying stochastic process, each treated as a distinct task, NPs learn shared structure across tasks to parameterize a neural mapping that directly gives an approximation of the corresponding posterior predictive distribution [Bruinsma, 2024]. Once trained, the model enables efficient probabilistic predictions on new tasks without further training. + +Since the introduction of conditional neural processes (CNPs, Garnelo et al. [2018a]) as the first class within the NPs family [Jha et al., 2022], a wide range of extensions and variants have been proposed to enhance their effectiveness. One active line of research introduces explicit inductive biases into CNPs to encode the symmetries commonly present in scientific domains [Gordon et al., 2019, Kawano et al., 2021, Holderrieth et al., 2021, Huang et al., 2023, Ashman et al., 2024a,b]. Another major direction seeks to move beyond the mean-field factorized Gaussian predictive distributions to which CNPs are limited. A popular approach augments CNPs with stochastic latent variables, giving rise to the family of latent neural processes (LNPs; Garnelo et al. [2018b], Louizos et al. [2019], Wang and Van Hoof [2020], Foong et al. [2020], Lee et al. [2020], Volpp et al. [2021], Wang et al. [2022], Wang and van Hoof [2022], Kim et al. [2022], Jung and Park [2023], Lee et al. [2023], Xu et al. [2023]). Alternative approaches explore autoregressive prediction schemes [Bruinsma et al., 2023, Nguyen and Grover, 2022], Gaussian predictive distributions with non-diagonal covariances [Bruinsma et al., 2021, Markou et al., 2022], and quantile-based parameterizations of the predictive distribution [Mohseni et al., 2023]. + +This work focuses on CNPs, particularly convolutional CNPs (ConvCNPs; Gordon et al. [2019]), which were the first to endow NPs with translation equivariance. ConvCNPs introduce the convolutional deep set construction which characterizes a broad class of translation-equivariant mappings over finite, potentially unstructured sets of observations as a composition of a functional embedding with a translation-equivariant operator, typically realized through convolutional neural networks (CNNs; Fukushima [1980], LeCun et al. [1989, 1998]). + +Despite their effectiveness, ConvCNPs can struggle to aggregate information from observations spread across large spatial domains—a challenge that becomes particularly pronounced in sparse data regimes. This limitation stems from their reliance on local convolutional kernels with small receptive fields, which hampers their ability to model long-range dependencies. A natural remedy is to enlarge the kernel size to extend the receptive field; however, this approach rapidly increases the number of model parameters and computational cost [Romero et al., 2021]. Alternatively, transformer-based architectures can capture long-range interactions but incur quadratic, rather than linear, computational complexity in the number of observations [Vaswani et al., 2017, Nguyen and Grover, 2022]. + +In this work, we propose an alternative approach that represents functions in the frequency domain, inspired by the well-established observation that many natural processes exhibit energy concentration in low-frequency bands [Field, 1987, Ruderman and Bialek, 1993, Wainwright and Simoncelli, 1999]. This characteristic allows for efficient approximation using only a subset of dominant spectral coefficients, enabling tractable computations while preserving the signal's global structure. By parameterizing convolution kernels directly in the Fourier domain over a finite set of frequencies and leveraging the convolution theorem, our method attains large effective receptive fields without incurring prohibitive computational costs. + +While spectral methods have been extensively studied in neural operator learning for partial differential equations (PDEs) (e.g., [Li et al., 2020a, Gupta et al., 2021, Kovachki et al., 2021, 2023, Helwig et al., 2023]), their application within NPs framework remains relatively unexplored. To bridge this gap, we propose the Spectral Convolutional Conditional Neural Process (SCConvCNP)—a model that adopts Fourier neural operators (FNOs; Li et al. [2020a]) to realize global convolution while maintaining computational efficiency. Across a suite of synthetic and real-world benchmarks, SConvCNP performs competitively with state-of-the-art baselines, illustrating how ideas from neural operators can enhance the flexibility and performance of NPs. + +# 2 Preliminaries + +# 2.1 Fourier Neural Operators + +Neural operators [Chen and Chen, 1995, Li et al., 2020a, Kovachki et al., 2023] are neural network architectures designed to learn mappings between function spaces rather than finite-dimensional vectors. Much like conventional feed-forward networks, they consist of stacked layers that alternate between operator-based transformations and pointwise nonlinearities. The common transformation choice is an integral operator with kernel $\kappa : \mathcal{X} \times \mathcal{X} \to \mathcal{Y}$ , acting on an input function $v: \mathcal{X} \to \mathcal{Y}$ as + +$$ +\mathcal {K} [ v ] (x) = \int \kappa (x, s) v (s) d s, +$$ + +where $\mathcal{X} = \mathbb{R}^{d_x}$ and $\mathcal{Y} = \mathbb{R}^{d_y}$ are Euclidean spaces with dimensions $d_{x}$ and $d_{y}$ , respectively. While this work focuses on linear operators, nonlinear formulations have also been explored, including continuous formulations of softmax attention [Ashman et al., 2024a, Calvello et al., 2024]. When the kernel is stationary ( $\kappa(x,s) = \kappa(x-s)$ ), the operator simplifies to a convolution, $\mathcal{K}[v] = \kappa * v$ , connecting neural operators to CNNs, where $\kappa$ is parameterized by learnable weights. + +Typically, convolutional kernels are spatially local and possess a limited receptive field [Luo et al., 2016, Peng et al., 2017, Wang et al., 2018]. Consequently, modeling long-range dependencies requires increasingly large kernels, which substantially inflates the parameter count. The Fourier Neural Operator (FNO, Li et al. [2020a]) overcomes this limitation by exploiting the convolution theorem [Bracewell and Kahn, 1966, Oppenheim, 1999], which re-expresses convolution in the frequency domain as: + +$$ +\mathcal {K} [ v ] (x) = \mathcal {F} ^ {- 1} \left[ \mathcal {F} [ \kappa ] (\xi) \cdot \mathcal {F} [ v ] (\xi) \right] (x), \tag {1} +$$ + +where $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier and inverse Fourier transforms, respectively. Rather than parameterizing the kernel $\kappa$ in the spatial domain, the FNO learns its representation directly in the Fourier domain. When the input function $v$ is approximately band-limited—i.e., its spectrum $\mathcal{F}[v](\xi)$ carries negligible energy for frequencies $|\xi| > \xi_0$ —the high-frequency components can be truncated with minimal loss of information. This property, observed in many natural signals [Field, 1987, Ruderman and Bialek, 1993, Wainwright and Simoncelli, 1999], allows setting $\mathcal{F}[\kappa](\xi) = 0$ outside the retained frequency band. + +In practice, functions are accessible only through discrete samples, so transitions between spatial and frequency domains rely on the discrete Fourier transform (DFT). Given samples of $v$ on a uniform grid $\mathcal{G} \subset \mathcal{X}$ , the FNO applies the DFT, $\hat{\mathcal{F}}\{(v(x_g))_{x_g \in \mathcal{G}}\}$ , computed efficiently via the fast Fourier transform (FFT; Cooley and Tukey [1965], Frigo and Johnson [2005]). The resulting spectrum is then truncated to a finite set of frequency modes $\hat{\Xi} \subset \mathbb{R}^{d_x}$ , which are assumed to capture the dominant energy of the signal $v$ . For each retained frequency $\hat{\xi} \in \hat{\Xi}$ , the Fourier representation of the kernel, $\mathcal{F}[\kappa](\hat{\xi})$ , is parameterized by learnable complex weights. In the simplest case, this takes the form $\mathcal{F}[\kappa](\hat{\xi}) = w_{\hat{\xi}} \in \mathbb{C}^{d_y}$ , with $\mathbb{C}$ denoting the complex field1. Note that this formulation implicitly imposes periodicity in the physical domain, since the kernel is represented using a discrete set of harmonics (equivalently, a Dirac comb in frequency space). After pointwise multiplication in the frequency domain, an inverse FFT maps the result back to the physical domain, producing the operator output. + +# 2.2 Neural Processes + +Let $\mathcal{P}(\mathcal{Y}^{\mathcal{X}})$ denote the set of probability measures over functions $f:\mathcal{X}\to \mathcal{Y}$ . We assume a data-generating process $\mu \in \mathcal{P}(\mathcal{Y}^{\mathcal{X}})$ , from which functions are sampled. A task $\mathcal{D}$ consists of a finite collection of noisy input-output observations drawn from a realization $f\sim \mu$ , partitioned into a context set and a query $^2$ set [Bruinsma, 2024, Ashman et al., 2024a,b]: + +$$ +\mathcal {D} = \left(\mathcal {D} _ {c}, \mathcal {D} _ {q}\right) = \left(\left\{\left(x _ {c, k}, y _ {c, k}\right) \right\} _ {k \in \mathcal {I} _ {c}}, \left\{\left(x _ {q, k}, y _ {q, k}\right) \right\} _ {k \in \mathcal {I} _ {q}}\right), \tag {2} +$$ + +where + +$$ +y _ {c, k} = f (x _ {c, k}) + \epsilon_ {c, k}, \quad y _ {q, k} = f (x _ {q, k}) + \epsilon_ {q, k}, \quad \epsilon_ {c, k}, \epsilon_ {q, k} \stackrel {\text {i . i . d .}} {\sim} \mathcal {N} (0, \sigma_ {0} ^ {2}). +$$ + +Here, $\sigma_0 > 0$ denotes the observation noise scale, and $\mathcal{I}_c,\mathcal{I}_q\subset \mathbb{N}$ index the context and query sets, respectively. Neural processes (NPs; [Garnelo et al., 2018a,b]) constitute a class of models that employ neural networks to learn a mapping $\eta :\bigcup_{k = 0}^{\infty}(\mathcal{X}\times \mathcal{Y})^{k}\to \mathcal{P}(\mathcal{Y}^{\mathcal{X}})$ , which takes a finite (possibly empty) context set $\mathcal{D}_c$ and gives a stochastic process intended to approximate the Bayesian posterior induced by $\mu$ . This approximation is constructed implicitly through its finite-dimensional marginals[Garnelo et al., 2018b, Bruinsma et al., 2021, Bruinsma, 2024, Mathieu et al., 2023]. Denote $\pmb{x}_q = (x_{q,k})_{k\in \mathcal{I}_q}$ and $\pmb{y}_q = (y_{q,k})_{k\in \mathcal{I}_q}$ . The NP approximation can then be expressed as + +$$ +q \left(\boldsymbol {y} _ {q} \mid \boldsymbol {x} _ {q}, \mathcal {D} _ {c}\right) \approx p \left(\boldsymbol {y} _ {q} \mid \boldsymbol {x} _ {q},, \mathcal {D} _ {c}\right), +$$ + +where $q(\pmb{y}_q \mid \pmb{x}_q, \mathcal{D}_c)$ and $p(\pmb{y}_q \mid \pmb{x}_q, \mathcal{D}_c)$ are the finite-dimensional distributions—evaluated at the query inputs $\pmb{x}_q$ —of the learned process $\eta[\mathcal{D}_c]$ and of the posterior process induced by the true data-generating measure $\mu$ , respectively. + +In this work, we focus on conditional NPs (CNPs, Garnelo et al. [2018a]), which restrict $q$ to the family of mean-field Gaussians, i.e. $q(\boldsymbol{y}_q \mid \boldsymbol{x}_q, \mathcal{D}_c) = \prod_{k \in \mathcal{I}_q} q(y_{q,k} \mid x_{q,k}, \mathcal{D}_c)$ . In general, CNPs parameterization of $q(y_{q,k} \mid x_{q,k}, \mathcal{D}_c)$ can be abstracted as a two-stage encoder-decoder pipeline [Bruinsma, 2024, Ashman et al., 2024a, 2025]. The encoder $\varphi_e: \mathcal{X} \times \bigcup_{k=0}^{\infty} (\mathcal{X} \times \mathcal{Y})^k \to \mathcal{H}$ maps the query location and context set into a latent representation, while the decoder $\varphi_d: \mathcal{X} \times \mathcal{H} \to \Theta$ transforms the query location and latent code into parameters $\theta \in \Theta$ of the predictive distribution $q(y_{q,k} \mid x_{q,k}, \mathcal{D}_c)$ . + +The vanilla CNP encoder summarizes the context set $\mathcal{D}_c$ using a permutation-invariant architecture [Qi et al., 2017, Zaheer et al., 2017]. Each context pair $(x_{c,k},y_{c,k})\in \mathcal{D}_c$ is independently mapped by a shared feedforward network to a finite-dimensional embedding $\varepsilon_{c,k}$ , which are subsequently aggregated—typically via mean pooling—into a single representation $\varepsilon_c$ . Notably, this encoding is independent of the query location (i.e., $\varphi_e$ does not condition on the query). The decoder then fuses $\varepsilon_c$ with the query input through another feedforward network to produce the parameters of the Gaussian predictive distribution. + +Although sum-pooling aggregation provides universal approximation guarantees [Zaheer et al., 2017, Bloem-Reddy and Teh, 2020], NPs employing such mechanisms often exhibit underfitting in practice [Kim et al., 2019]. Prior works have partly attributed this phenomena to two primary factors[Xu et al., 2020]: (1) the limitation of summaries with prespecified finite dimensionality in representing context sets of arbitrary size [Wagstaff et al., 2019], and (2) the shortcomings of simple sum or mean pooling operations to effectively capture rich interactions between context and query points[Xu et al., 2020, Nguyen and Grover, 2022]. + +Given that the problems tackled by NPs are inherently functional in nature, it is natural to seek embeddings that also exhibit a functional character. In this spirit, Gordon et al. [2019] formalize a general framework for constructing translation-equivariant prediction maps over sets, satisfying + +$$ +\eta \Big [ \big \{(x _ {c} + \tau , y _ {c}) \mid (x _ {c}, y _ {c}) \in \mathcal {D} _ {c} \big \} \Big ] \big ((x _ {q, k}) _ {k \in \mathcal {I} _ {q}} \big) = \eta \Big [ \mathcal {D} _ {c} \Big ] \big ((x _ {q, k} - \tau) _ {k \in \mathcal {I} _ {q}} \big), +$$ + +for every $\tau \in \mathcal{X}$ . They show that a broad family of such maps is expressed as $\eta[\mathcal{D}_c] = \varphi_d[\varphi_e[\mathcal{D}_c]]$ , where the functional embedding is defined by + +$$ +\varphi_ {e} \left[ \mathcal {D} _ {c} \right] (\cdot) = \sum_ {\left(x _ {c}, y _ {c}\right) \in \mathcal {D} _ {c}} \phi \left(y _ {c}\right) \psi_ {e} (\cdot - x _ {c}), \tag {3} +$$ + +and $\varphi_d:\mathcal{H}\to C_b(\mathcal{X},\mathcal{Y})$ is a translation-equivariant decoder operating on a function space $\mathcal{H}$ . Here, $C_b(\mathcal{X},\mathcal{Y})$ denotes the space of bounded continuous functions from $\mathcal{X}$ to $\mathcal{Y}$ ; $\phi (y) = (1,y)^3$ and $\psi_e:\mathcal{X}\rightarrow \mathbb{R}$ is a continuous, strictly positive-definite kernel typically chosen to be a Gaussian. + +In implementing the ConvCNP, $\varphi_{e}[\mathcal{D}_{c}]$ is first evaluated on a uniform grid $\mathcal{G}\subset \mathcal{X}$ that spans the joint support of both the context and query locations. This yields a discretized representation $(\varphi_{e}[\mathcal{D}_{c}](x_{g}))_{x_{g}\in \mathcal{G}}$ , which is then passed through $\varphi_{d}$ to produce $\left(\varphi_{d}\bigl [\varphi_{e}[\mathcal{D}_{c}]\bigr ](x_{g})\right)_{x_{g}\in \mathcal{G}}$ . Because the query inputs $(x_{q,k})_{k\in \mathcal{T}_q}$ may not lie exactly on the grid $\mathcal{G}$ , ConvCNP employs an additional interpolation step to obtain predictions at query locations. Specifically + +$$ +\left(\theta_ {q, k}\right) _ {k \in \mathcal {I} _ {q}} = \left(\sum_ {x _ {g} \in \mathcal {G}} \varphi_ {d} \left[ \varphi_ {e} \left[ \mathcal {D} _ {c} \right] \right] \left(x _ {g}\right) \psi_ {d} \left(x _ {q, k} - x _ {g}\right)\right) _ {k \in \mathcal {I} _ {q}}, \tag {4} +$$ + +where $\psi_d$ is another strictly positive kernel. Note that this step can be viewed as part of the decoder itself, thereby preserving the overall encoder-decoder abstraction discussed earlier. + +# 3 Spectral Convolutional Conditional Neural Process + +The decoder $\varphi_d$ in ConvCNP is typically parameterized using standard CNNs such as U-Net [Ronneberger et al., 2015] or ResNet [He et al., 2016]. These architectures employ discrete convolutional + +kernels—finite sets of learnable parameters that define localized filters operating over neighboring grid points. The kernel size, fixed a priori, determines the receptive field of each convolution [Ding et al., 2022] and is generally much smaller than the overall spatial extent of the input signals [Romero et al., 2021, Knigge et al., 2023]. + +This locality constraint limits the model's capacity to capture long-range dependencies and to integrate information from observations distributed across large domains [Peng et al., 2017, Wang et al., 2018, Ramachandran et al., 2019, Wang et al., 2020]. The issue becomes particularly pronounced when handling sparse or irregularly sampled data, where effective global reasoning cannot emerge solely from local convolution operations. While increasing the kernel size could enlarge the receptive field, it leads to a rapid growth in both parameter count and computational cost. Transformer-based architectures offer an alternative by enabling global interactions [Vaswani et al., 2017]; however, they typically incur quadratic rather than linear complexity in the number of inputs [Nguyen and Grover, 2022, Feng et al., 2022, Ashman et al., 2024a, 2025]. + +To overcome this limitation without relying on prohibitively large filters or expensive transformers, we exploit the Fourier representation of signals. This is motivated by the well-established observation that many natural signals are approximately band-limited (see Section 2.1), implying that their Fourier representation offers a more compact encoding relative to its physical-domain counterpart. Concretely, we instantiate $\varphi_d$ via spectral convolution modules based on equation 1. This substitution effectively enlarges the receptive field, enabling the model to capture global structures from sparse or irregularly sampled data without incurring a parameter count explosion. We refer to the resulting model as the Spectral Convolutional Conditional Neural Process (SConvCNP). + +Computational Complexity. The computational cost of the SConvCNP comprises three parts: (i) $\mathcal{O}(|\mathcal{D}_c||\mathcal{G}|)$ to compute the discretized functional embedding on the grid $\mathcal{G}$ (equation 3); (ii) $\mathcal{O}(|\mathcal{G}|\log |\mathcal{G}|)$ for the FFT-based spectral convolution (equation 1); and (iii) $\mathcal{O}(|\mathcal{D}_q||\mathcal{G}|)$ to interpolate grid embeddings at the query locations (equation 4). Overall, the total complexity is $\mathcal{O}\big(|\mathcal{G}||\mathcal{D}_c| + \log |\mathcal{G}| + |\mathcal{D}_q|)$ , matching the ConvCNP complexity of $\mathcal{O}\big(|\mathcal{G}||\mathcal{D}_c| + 1 + |\mathcal{D}_q|)\big)$ up to logarithmic factors. Both architectures therefore scale linearly with task size. A key limitation, however, is that $|\mathcal{G}|$ grows exponentially with input dimension, restricting these methods to low-dimensional domains. In contrast, transformer-based NPs (TNPs; Kim et al. [2019], Nguyen and Grover [2022], Feng et al. [2022], Ashman et al. [2024a]) avoid gridding entirely and thus scale more gracefully with input dimension. Instead, they scale quadratically in task size $\mathcal{O}(|\mathcal{D}_c|^2 +|\mathcal{D}_c||\mathcal{D}_q|)$ . + +Positional Encodings The convolution operator preserves translation equivariance under the Fourier transform (see equation 1). However, practical FNOs implementations often include explicit positional information to improve predictive accuracy [Li et al., 2020a, Tran et al., 2021, Gupta et al., 2021, Rahman et al., 2022b, Helwig et al., 2023, Tripura and Chakraborty, 2023, Liu et al., 2023a, Li et al., 2024]. Accordingly, we augment the functional embedding $\varphi_{e}[\mathcal{D}_{c}](x)$ with positional information: + +$$ +\tilde {\varphi} _ {e} [ \mathcal {D} _ {c} ] (x) = (\varphi_ {e} [ \mathcal {D} _ {c} ] (x), x). +$$ + +Although this addition breaks translation equivariance, it helps improving the performance (see Section B.5 for ablation results). Future work may explore relative positional encodings [Shaw et al., 2018, Su et al., 2024], which preserve translation equivariance while providing spatial context. + +Discretization Sensitivity of DFT. Unlike the continuous Fourier transform, the DFT—and therefore the FFT—depends inherently on the grid $\mathcal{G}$ on which $\varphi_{e}[\mathcal{D}_{c}]$ (or $\widetilde{\varphi}_e[\mathcal{D}_c]$ ) is represented. This dependence arises from both the grid resolution and its physical extent: changing either alters the resulting Fourier coefficients and can lead to inconsistent behavior in the outputs (see Section A). Sensitivity to resolution is not unique to DFT; CNNs exhibit analogous issues [Raonic et al., 2023, Bartolucci et al., 2023]. For example, ConvCNP mitigates this effect by fixing the grid resolution. While spatial CNNs become stable once the resolution is fixed, DFT-based methods remain sensitive unless both the resolution and the physical range are controlled. Accordingly, we fix both, choosing a domain sufficiently large to cover all context and query inputs across tasks. When this is impractical, the domain can instead be divided into (possibly overlapping) fixed-size patches, with spectral convolutions applied independently to each patch and the outputs aggregated—mirroring the mechanism of standard convolution layers. Related ideas in efficient transformer architectures suggest that this is a promising direction for future work [Beltagy et al., 2020, Zaheer et al., 2020, Liu et al., 2021, Ding et al., 2023]. + +![](images/6aee6a8aa80a25c2ca10a30396b31aab21d6acf767773e9e082426808c2ca34d.jpg) +Figure 1: Example predictions for synthetic data. For each model, the blue curve shows its predictive mean and the shaded region indicates $\pm 2$ standard deviations under the corresponding Gaussian predictive distribution. In the first two rows, where the data are generated from a Gaussian process, the true joint density is shown in purple: the dash-dotted line marks the true mean, and the shaded band spans $\pm 2$ standard deviations around that mean. Red points denote query samples drawn from the ground-truth process, and black points indicate the context observations. + +# 4 Experiments + +We evaluate our framework on four regression benchmarks and compare its performance against several members of CNPs family. Specifically, we include the original CNP [Garnelo et al., 2018a], the Attentive CNP (AttCNP; Kim et al. [2019]), the Convolutional CNP (ConvCNP; Gordon et al. [2019]), the Transformer Neural Process (TNP; Nguyen and Grover [2022]), and the Translation-Equivariant Transformer Neural Process (TETNP; Ashman et al. [2024a]). To evaluate model performance, we report both the log-likelihood and root-mean-squared error (RMSE) metrics. Additional details on datasets, architectures, and training procedures are provided in Appendix B. The implementation and experimental code are available at https://github.com/peiman-m/SConvCNP. + +# 4.1 Synthetic 1-D Regression + +We begin by evaluating models on four synthetic benchmarks generated from distinct stochastic processes: a GP with a Matérn-5/2 kernel, a GP with a periodic kernel, a sawtooth-wave generator, and a square-wave generator. For each benchmark, the parameters of the generative process—kernel hyperparameters for the GPs, frequency and direction for the sawtooth generator, and frequency and duty cycle for the square-wave generator—are sampled randomly. Complete experimental details are provided in Appendix B.1. Table 1 reports the average evaluation metrics over 1,000 test batches, each containing 16 tasks (each task defined in equation 2). For every task—independent of batch—both $x_{c}$ and $x_{q}$ are sampled independently from $\mathcal{U}[-3,3)$ . For each test batch, the number of context + +Table 1: Comparison of predictive performance across methods on synthetically generated tasks (4 seeds). Lower RMSE and higher log-likelihood indicate better performance. For each metric and experimental setting, boldface highlights the models ranking in the top two. + +
MetricDataModel
CNPAttCNPTNPTETNPConvCNPSConvCNP
Log-likelihood ↑Matérn 5/2-0.54±0.01-0.32±0.00-0.29±0.00-0.28±0.00-0.30±0.01-0.29±0.01
Periodic-1.20±0.00-0.87±0.05-0.76±0.03-0.57±0.02-0.73±0.01-0.66±0.00
Sawtooth-0.90±0.00-0.90±0.00-0.90±0.00-0.90±0.000.10±0.300.82±0.03
Square-1.39±0.00-1.41±0.01-1.33±0.02-1.17±0.06-1.35±0.04-1.13±0.03
RMSE ↓Matérn 5/20.50±0.000.45±0.000.44±0.000.44±0.000.45±0.000.45±0.00
Periodic0.81±0.000.65±0.020.60±0.020.50±0.010.57±0.000.53±0.00
Sawtooth0.58±0.000.58±0.000.58±0.000.58±0.000.40±0.060.24±0.00
Square0.98±0.001.00±0.010.93±0.020.81±0.010.89±0.020.79±0.01
+ +points, shared by all tasks within that batch, is drawn independently according to $|\mathcal{I}_c| \sim \mathcal{U}[5,25)$ . The number of query points, however, is fixed at $|\mathcal{I}_q| = 256$ for all the test tasks. + +As shown, SConvCNP consistently outperforms or closely matches strong baselines, including TETNP and ConvCNP, which represent the current state of the art. Figure 1 provides qualitative comparisons of predictive maps produced by SConvCNP, ConvCNP, and TETNP (other models are omitted due to space constraints). + +For the sawtooth-wave benchmark, we were unable to successfully train either TNP or TETNP. Across the configurations we attempted—including increased model capacity and multiple randomized initializations—their predictions consistently collapsed to zero. We hypothesize that this failure mode stems from spectral bias, the tendency of neural networks to favor low-frequency structure over high-frequency components [Rahaman et al., 2019, Ronen et al., 2019, Basri et al., 2020, Tancik et al., 2020, Fridovich-Keil et al., 2022]. Recent findings by Vasudeva et al. [2025] further suggest that this bias is exacerbated in transformer architectures compared with convolutional networks. This interpretation aligns with observations by Nguyen and Grover [2022], who originally introduced TNPs. They report degraded performance on GP samples drawn from a periodic kernel—reflected in poor log-likelihood scores—despite strong results on Matérn-kernel tasks. Interestingly, our experiments on periodic-kernel GP tasks (Table 1) do not replicate this limitation. A key distinction is that we employ the Efficient Query TNP (EQTNP) architecture of Feng et al. [2022], rather than the original TNP design [Nguyen and Grover, 2022] (see Appendix B.1.1 for architectural details). Finally, although sawtooth and square-wave functions share similar discontinuity and high-frequency characteristics, Table 1 shows that TNP and TETNP do not collapse when trained on square-wave signals. Investigating the source of this discrepancy falls beyond the scope of this work but represents an intriguing direction for future study. + +# 4.2 Predator-Prey Dynamics + +We next assess performance on simulated trajectories from a stochastic variant of the Lotka-Volterra predator-prey system [Lotka, 1910, Volterra, 1926], following the formulation in Bruinsma et al. [2023]. Let $U_{t}$ and $V_{t}$ denote the prey and predator populations at time $t$ , respectively. Their dynamics evolve according to the stochastic Lotka-Volterra system + +$$ +d U _ {t} = \alpha U _ {t} d t - \beta U _ {t} V _ {t} d t + \sigma U _ {t} ^ {\nu} d B _ {t} ^ {(1)}, \qquad d V _ {t} = - \gamma V _ {t} d t + \delta U _ {t} V _ {t} d t + \sigma V _ {t} ^ {\nu} d B _ {t} ^ {(2)}, +$$ + +where $B_{t}^{(1)}$ and $B_{t}^{(2)}$ are independent Brownian motions. In the deterministic part of the model, the prey population grows exponentially at rate $\alpha$ in the absence of predation, while the predator population declines at rate $\gamma$ without access to prey. The interaction terms $\beta U_{t}V_{t}$ and $\delta U_{t}V_{t}$ describe predation and the resulting reproductive gain for predators. Stochasticity is introduced through the multiplicative noise terms $\sigma U_{t}^{\nu}$ and $\sigma V_{t}^{\nu}$ , where $\sigma$ controls the overall noise intensity and $\nu$ determines how fluctuations scale with population size. + +Table 2: Comparison of predictive performance across methods on test tasks constructed from the Pred-Prey population dynamics simulation (4 seeds). Lower RMSE and higher log-likelihood indicate better performance. Boldface highlights the models ranking in the top two for each metric. + +
CNPAttCNPTNPTETNPConvCNPSConvCNP
Log-likelihood ↑-0.27±0.010.08±0.000.12±0.000.16±0.000.14±0.000.14±0.00
RMSE ↓0.37±0.000.31±0.000.31±0.000.30±0.000.31±0.000.30±0.00
+ +![](images/70625bd563adc6756f8150e1c7cecf856986350f1c89169217cbc0c6885b528a.jpg) + +![](images/13961ac5afc4162b0194fc489892ee9c92427d27b5211b344dbda9e3ec160f00.jpg) +Figure 2: Illustrative model predictions based on simulated trajectories from Lotka-Volterra predator-prey dynamics. Black points represent the context observations. For each model, the blue curve depicts the predictive mean, and the shaded region marks the $\pm 2$ -standard-deviation band of the associated Gaussian predictive distribution. Ground-truth traffic measurements, shown in red, are included as a reference. + +![](images/31ab8c82adf0a9cfae4567dcd3c02f8a0bc1e4265652e2be2575bb9cccc8f84e.jpg) + +![](images/725970e7d41af5cd5fea0dbf462c361562062575a284141b172176fb929c5fbe.jpg) + +We simulate trajectories over a dense uniform grid on $t \in [-10, 100)$ , discarding the initial 10 years as burn-in. For each task, context points $x_{c}$ and query points $x_{q}$ are drawn independently and uniformly from the retained interval [0, 100). Population values (corresponding to $y = [U_{t}, V_{t}]$ ) and time inputs (corresponding to $x = t$ ) are rescaled by factors of 0.01 and 0.1, respectively, before being fed to the models. A complete description of the experimental protocol is provided in Appendix B.2. + +Table 2 reports average evaluation metrics over 1,000 test batches, each containing 16 tasks. For each batch, the number of context points—shared across all tasks—is sampled as $|\mathcal{I}_c| \sim \mathcal{U}[5,25)$ , and the number of query points is fixed at $|\mathcal{I}_q| = 256$ . SConvCNP achieves log-likelihood performance comparable to ConvCNP, and both are competitive with TETNP. In terms of RMSE, SConvCNP matches TETNP, indicating that its predictive uncertainty intervals are slightly wider. Qualitative comparisons of predictive maps for the top three models appear in Figure 2. + +# 4.3 Traffic Flow + +For our third experiment, we evaluate on the California traffic-flow dataset from LargeST [Liu et al., 2023b]. This dataset comprises five years (2017-2021) of traffic measurements recorded every 5 minutes by approximately 8,600 loop-detector sensors deployed across California's highway network. We focus on the year 2020, where traffic patterns are expected to exhibit heightened variability due to the abrupt onset of the COVID-19 pandemic. For each sensor, we segment its year-long time series into non-overlapping 14-day windows and downsample each window by a factor of 6 (from 5-minute to 30-minute resolution). Each window is treated as a dense trajectory from which individual tasks are constructed. Additional details of the experimental setup are provided in Appendix B.3. + +Table 3: Comparison of predictive performance across methods on test tasks constructed from California traffic flow measurements (4 seeds). Lower RMSE and higher log-likelihood indicate better performance. Boldface highlights the models ranking in the top two for each metric. + +
CNPAttCNPTNPTETNPConvCNPSConvCNP
Log-likelihood ↑1.73±0.101.80±0.011.93±0.061.72±0.021.98±0.022.05±0.02
RMSE ↓0.05±0.000.05±0.000.04±0.000.05±0.000.04±0.000.04±0.00
+ +![](images/488b1207362b8e7c22ddb5a07c0e20039ab44ef5f104291273f5640f9f926199.jpg) +Figure 3: Illustrative model predictions on the California traffic-flow dataset. Black points represent the context observations. For each model, the blue curve depicts the predictive mean, and the shaded region marks the $\pm 2$ -standard-deviation band of the associated Gaussian predictive distribution. Ground-truth traffic measurements, shown in red, are included as a reference. + +Table 3 reports evaluation metrics averaged over 66,586 test tasks, constructed from 2,561 held-out test sensors and partitioned into batches of size 32. For each batch, the number of context points—shared across all tasks—is drawn as $|\mathcal{I}_c| \sim \mathcal{U}[5,25)$ , and the number of query points is fixed at $|\mathcal{I}_q| = 50$ . Across all metrics, SConvCNP achieves the best performance, attaining the highest log-likelihoods and lowest RMSE among the baselines. Figure 3 provides qualitative comparisons of the predictive maps produced by the three models. + +# 4.4 Image Completion + +For our final experiment, we evaluate model performance on an image-completion task formulated as a spatial regression problem, where the model maps 2D pixel coordinates to their corresponding intensity values. We use images from the Describable Textures Dataset (DTD; Cimpoi et al. [2014]), and construct each task from a processed $64 \times 64$ subsampled crop of an original image. Because each task contains a large number of context and query pixels, we were unable to fit TETNP within our computational budget—even after reducing its size—so we exclude it from this experiment. Additional experimental details are provided in Appendix B.4. Table 4 reports results averaged over 1,880 test tasks, evaluated in batches of 16. For each batch, the number of context points—shared across all tasks—is sampled as $|\mathcal{I}_c| \sim \mathcal{U}[5,1024)$ ; all remaining pixels serve as query points, so that $|\mathcal{I}_q| = 64 \times 64 - |\mathcal{I}_c|$ . Among the baselines, SConvCNP achieves the highest log-likelihoods and the second-lowest RMSE. Figure 4 presents qualitative comparisons of the predictive maps produced by the top three models. + +![](images/4793e1a1ceb3a2bea0bd2f339de5a07e632b378a27d434fe7d3d2a9ce8490d19.jpg) +Figure 4: Illustrative model outputs for image-completion tasks using samples from the DTD dataset. Grey pixels indicate query regions, and the remaining pixels serve as context observations. For each model, query pixels are filled with the mean of the predictive distribution. + +Table 4: Comparison of predictive performance across methods on image-completion test tasks constructed from images of the DTD dataset (4 seeds). Lower RMSE and higher log-likelihood indicate better performance. Boldface highlights the models ranking in the top two for each metric. + +
CNPAttCNPTNPConvCNPSConvCNP
Log-likelihood ↑0.67±0.011.39±.021.39±0.051.48±0.001.50±0.02
RMSE ↓0.14±0.000.08±0.000.06±0.000.08±0.000.08±0.00
+ +# 5 Related Works + +Neural PDE Solvers The high computational cost of traditional numerical PDE solvers has motivated the development of more efficient alternatives based on deep learningGupta and Brandstetter [2022]. Among these, neural operators [Li et al., 2020b, Kovachki et al., 2023], and in particular Fourier neural operators (FNOs, Li et al., 2020a), have become popular. Numerous extensions have since been proposed: Helwig et al. [2023] introduced rotation- and reflection-equivariant FNOs; Gupta et al. [2021] developed multiwavelet neural operators by projecting kernels onto predefined polynomial bases; and Tran et al. [2021] reduced model complexity through separable Fourier representations. Xiao et al. [2024] proposed the Amortized FNO (AM-FNO), which substantially cuts parameter count by amortizing the Fourier kernel parameterization, while Qin et al. [2024] analyzed FNOs through the lens of spectral bias. Koshizuka et al. [2024] provided a mean-field theoretical perspective, and Zheng et al. [2024] introduced the Mamba Neural Operator, which couples global context via a state-space model to achieve linear complexity and representation equivalence. Bartolucci et al. [2023] examined continuous-discrete consistency, showing that it holds for convolutional neural operators [Raonic et al., 2023]. Additional works include orthogonal-attention eigenfunction methods for operator learning [Xiao et al., 2023] and hierarchical transformers with frequency-aware priors for resolution-invariant super-resolution [Luo et al., 2024]. + +Function Space Inference In non-parametric Bayesian modeling, GPs and deep GPs [Damianou and Lawrence, 2013] provide flexible function-space priors with well-calibrated uncertainty estimates, but their computational cost becomes prohibitive on large datasets. This challenge has motivated the development of Bayesian neural networks (BNNs; MacKay [1992], Hinton and Van Camp [1993], Neal [2012]), which combine neural network expressiveness with Bayesian uncertainty quantification. However, specifying meaningful priors over network weights remains notoriously difficult. Recent work therefore reframes Bayesian inference in neural networks as learning a posteriori over the functions induced by stochastic weights [Wolpert, 1993, Qiu et al., 2023]. Variational implicit processes (VIPs; Ma et al. [2019], Santana et al. [2021], Ortega et al. [2022]) generalize GPs by defining implicit stochastic processes through latent variables, while functional variational BNNs (fBNNs; Sun et al. [2019]) enforce alignment between BNN-induced priors and target priors by minimizing a functional KL divergence—though this objective can be difficult to compute and, in some cases, ill-posed [Burt et al., 2020]. Subsequent work has aimed to address these limitations [Ma and Hernandez-Lobato, 2021, Rudner et al., 2022, Wild et al., 2022, Rudner et al., 2023, Wu et al., 2024]. Orthogonally, energy-based models have been explored for representing stochastic processes in function space [Yang et al., 2020, Lim et al., 2022]. More recently, several approaches have extended diffusion and flow models to function spaces [Phillips et al., 2022, Dutordoir et al., 2023, Mathieu et al., 2023, Franzese et al., 2023, Lim et al., 2023, Kerrigan et al., 2024, Zhang and Scott, 2025]. + +# 6 Conclusion + +In this work, we introduced the Spectral Convolutional Conditional Neural Process (SConvCNP), a new member of the CNP family that leverages advances in operator learning—particularly the Fourier Neural Operator—to model probabilistic prediction maps within the neural processes framework, with a special emphasis on the Convolutional Conditional Neural Process (ConvCNP). Through experiments on both synthetic and real datasets, we show that SConvCNP delivers highly competitive performance relative to recent state-of-the-art methods, highlighting the promise of bridging research on neural processes with developments in operator learning. + +# Acknowledgments + +We thank Arman Hasanzadeh and Jonathan W. Siegel for their insightful feedback and constructive suggestions. We also acknowledge the Texas A&M High Performance Research Computing facility for providing the computational resources used in this study. Finally, we thank the anonymous reviewers for their valuable comments and suggestions, which helped improve the quality of this work. + +# References + +Anna Allen, Stratis Markou, Will Tebbutt, James Requeima, Wessel P Bruinsma, Tom R Andersson, Michael Herzog, Nicholas D Lane, Matthew Chantry, J Scott Hosking, et al. End-to-end data-driven weather prediction. 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Flow straight and fast in hilbert space: Functional rectified flow. arXiv preprint arXiv:2509.10384, 2025. +Jianwei Zheng, Wei Li, Ni Xu, Junwei Zhu, and Xiaoqin Zhang. Alias-free mamba neural operator. Advances in Neural Information Processing Systems, 37:52962-52995, 2024. + +# A Discretization Sensitivity of the DFT + +To illustrate the sensitivity of the discrete Fourier transform (DFT) to grid shape—and how changes in discretization can disturb predictive performance—we consider a simple one-dimensional example (although the analysis extends to higher dimensions). Let $0 < \Delta \ll 1$ denote a fixed discretization resolution, interpreted as the spacing between successive points. Define the grid $\mathcal{G}_1 = (x_0,\ldots ,x_{m_1 - 1})$ as a uniform discretization of the interval [0, 1] with $m_{1} = \lfloor \frac{1}{\Delta}\rfloor$ and ordered such that $x_{k - 1} < x_k$ for all $k\in \{1,\dots,m_1 - 1\}$ . Let the sampled values of $\varphi_e[\mathcal{D}_c]$ at $\mathcal{G}_1$ be denoted by $(\varphi_{e}[\mathcal{D}_{c}](x))_{x\in \mathcal{G}_{1}} = (\varphi_{e}[\mathcal{D}_{c}](x_{0}),\ldots ,\varphi_{e}[\mathcal{D}_{c}](x_{m_{1} - 1}))$ . The DFT of $\varphi_{e}[\mathcal{D}_{c}]$ over this grid is given by: + +$$ +\hat {\mathcal {F}} \left\{(\varphi_ {e} [ \mathcal {D} _ {c} ] (x)) _ {x \in \mathcal {G} _ {1}} \right\} (k) = \sum_ {n = 0} ^ {m _ {1} - 1} \varphi_ {e} [ \mathcal {D} _ {c} ] (x _ {n}) e ^ {- i 2 \pi \frac {k}{m _ {1}} n}, \quad \forall k \in \{0, \ldots , m _ {1} - 1 \}. +$$ + +Thus, the DFT yields Fourier coefficients at the normalized frequencies $\hat{\Xi}_1 = \left(\frac{k}{m_1}\right)_{k=0}^{m_1-1}$ . As described in Section 2.1, the FNO implicitly ties its kernel spectrum parameterization to the set $\hat{\Xi}_1$ encountered during training. Now suppose we wish to evaluate the model on a larger domain, say [0, 2], using the same resolution $\Delta$ . Then the corresponding grid $\mathcal{G}_2$ has size $m_2 = \lfloor \frac{2}{\Delta} \rfloor$ , and the DFT computes coefficients at normalized frequencies $\hat{\Xi}_2 = \left(\frac{k}{m_2}\right)_{k=0}^{m_2-1}$ . Since $m_2 \geq 2m_1$ , we have $|\hat{\Xi}_2| \geq 2|\hat{\Xi}_1|$ . Let $1 \leq k_{\max} \leq m_1$ denote the number of frequency modes retained during training. Clearly, the first $k_{\max}$ frequencies of $\hat{\Xi}_2$ do not align with those of $\hat{\Xi}_1$ , resulting in a mismatch. + +# B Experimental Details + +All implementations are written in PyTorch [Paszke et al., 2019] and publicly available at https://github.com/peiman-m/SConvCNP. We used a single NVIDIA A100 GPU with 40 GB of memory for all the computations. Our code is based on the implementations of Ashman et al. [2024a] and Bruinsma et al. [2023]. + +# B.1 Synthetic 1-D Regression + +# B.1.1 Model Architectures + +This section details the architectures of all CNP variants used in our experiments. Each model outputs a Gaussian predictive distribution parameterized by a mean and a pre-softplus [Dugas et al., 2000] scale. The pre-softplus value is passed through a softplus transformation, and a minimum noise term of $10^{-6}$ is then added to the resulting scale. The Gaussian factorizes across query points—as is standard for CNPs (see Section 2.2)—and, for multidimensional outputs, across output dimensions as well. Thus, predictions assume independence both across query locations and across components of each output vector. Unless otherwise noted, all nonlinearities use ReLU activations [Nair and Hinton, 2010]. + +Conditional Neural Process (CNP): The CNP encodes each context pair $(x_{c,k},y_{c,k})\in \mathcal{D}_c$ using separate input and output pathways. The input $x_{c,k}$ and output $y_{c,k}$ are first passed through distinct MLPs with two hidden layers each of dimension 256, producing 256-dimensional representations $\varepsilon_{c,k}^{(x)}$ and $\varepsilon_{c,k}^{(y)}$ . Their concatenation $\varepsilon_{c,k} = [\varepsilon_{c,k}^{(x)},\varepsilon_{c,k}^{(y)}]$ is then processed by a deeper MLP with six hidden layers of width 256 and a final 256-dimensional output layer, yielding an embedding for each context pair. Context embeddings $\varepsilon_{c,k}$ are averaged to form a single aggregated representation $\varepsilon_{c}$ of dimension 256. For prediction, each query input $x_{q,k}$ is concatenated with the context representation $\varepsilon_{c}$ and passed through a decoder MLP consisting of six hidden layers of dimension 256. The decoder's final output layer has dimensionality $2d_{y}$ , parameterizing the mean and the log-scale parameters of the Gaussian predictive distribution. + +Attentive Conditional Neural Process (AttCNP). Our AttCNP implementation follows the deterministic architecture introduced by Kim et al. [2019]. The initial encoding stage mirrors that of the CNP: each context pair $(x_{c,k},y_{c,k})\in \mathcal{D}_c$ is processed by two separate MLPs (each with + +two hidden layers of width 256), producing embeddings $\varepsilon_{c,k}^{(x)}$ and $\varepsilon_{c,k}^{(y)}$ . These are concatenated to form $\varepsilon_{c,k} = [\varepsilon_{c,k}^{(x)}, \varepsilon_{c,k}^{(y)}]$ , which is subsequently passed through an additional two-layer MLP with hidden width 256. Departing from the CNP, the AttCNP applies self-attention to the set of context embeddings. We use two layers of multi-head self-attention, each with 8 heads (head dimension 32), a feedforward subnetwork with hidden width 256, residual connections, and pre-layer normalization for both the attention and feedforward blocks. The input and output dimensionality of each attention layer is fixed at 256. Query representations $\varepsilon_{q,k}$ are computed using a single layer of multi-head cross-attention between the (attended) context embeddings and the query embeddings, with the same configuration as above (8 heads of dimension 32, feedforward width 256, residual connections, and pre-normalization). The decoder is an MLP with six hidden layers of width 256. Its final layer outputs $2d_{y}$ units, corresponding to the mean and log-scale parameters of a Gaussian predictive distribution. + +Transformer Neural Process (TNP). To apply the TNP, we first construct token representations for both context and query points. For each context pair $(x_{c,k},y_{c,k})\in \mathcal{D}_c$ , we form the token + +$$ +[ x _ {c, k}, y _ {c, k}, 1 ], +$$ + +where the final singleton "1" serves as a density flag indicating that the observation $y_{c,k}$ at $x_{c,k}$ is available. For each query location $x_{q,k}$ , we instead construct + +$$ +[ x _ {q, k}, \mathbf {0}, 0 ], +$$ + +where the zero-vector matches the shape of $y_{c,k}$ and the final "0" indicates the absence of an observation [Nguyen and Grover, 2022, Ashman et al., 2024a]. All tokens are then passed through a shared two-layer MLP (width 256), producing initial embeddings $\varepsilon_{c,k}$ and $\varepsilon_{q,k}$ . + +In the original TNP architecture [Nguyen and Grover, 2022], context and query embeddings are processed jointly by a transformer encoder. An attention mask prevents (i) interactions among queries and (ii) context $\rightarrow$ query attention, ensuring that context representations remain query-independent, while queries may attend to contexts. This design incurs quadratic complexity in the total sequence length, + +$$ +\mathcal {O} \left(\left(\left| \mathcal {D} _ {c} \right| + \left| \mathcal {D} _ {q} \right|\right) ^ {2}\right). +$$ + +Because queries never attend to one another, Feng et al. [2022] introduced the Efficient Query TNP, which we adopt here. This variant first applies self-attention to context tokens to produce updated context embeddings, then processes each query using cross-attention over these updated contexts. The resulting two-branch structure (contexts processed twice, queries once) reduces the overall complexity to + +$$ +\mathcal {O} \big (| \mathcal {D} _ {c} | ^ {2} + | \mathcal {D} _ {q} | | \mathcal {D} _ {c} | \big). +$$ + +Our model uses six transformer layers with eight attention heads (head dimension 32), each followed by a feedforward subnetwork of width 256. We use the standard pre-norm architecture, applying layer normalization before both the attention and feedforward modules. All attention operations share a 256-dimensional embedding dimension. At each layer, we first apply context self-attention and then query-context cross-attention. Notably, the same multi-head attention parameters are used for both operations; the distinction arises only from which embeddings serve as queries, keys, and values (yielding self-attention when they coincide, and cross-attention otherwise). This is not a modeling requirement—separate parameter sets could be used, as demonstrated in Appendix B.4.1. The final-layer query embeddings are passed to a decoder MLP with two hidden layers (width 256), whose output parameterizes a Gaussian predictive distribution via $2d_{y}$ units corresponding to the mean and log-scale. + +Translation Equivariant Transformer Neural Process (TETNP). The token construction for TETNP closely mirrors that of the TNP, with the key difference that token representations omit the input locations in order to preserve translation equivariance. For each context pair $(x_{c,k},y_{c,k})\in \mathcal{D}_c$ we therefore construct the token + +$$ +[ y _ {c, k}, 1 ], +$$ + +and for each query location $x_{q,k}$ we use + +$$ +[ \mathbf {0}, 0 ]. +$$ + +All tokens are processed by a shared two-layer MLP (width 256), yielding initial embeddings $\varepsilon_{c,k}$ and $\varepsilon_{q,k}$ . TETNP replaces the standard multi-head attention mechanism with the translation-equivariant + +variant introduced by Ashman et al. [2024a]. To compute attention scores, the pairwise scaled dot-product between token embeddings for each head is concatenated with the corresponding pairwise differences in token locations. This concatenated feature is passed through an MLP with two hidden layers (width 256) and an output dimension equal to the number of attention heads, producing translation-equivariant attention logits. Except for this modified attention module, the remaining architecture and computation flow follow the implementation of the TNP. + +Convolutional Conditional Neural Process (ConvCNP): For ConvCNP, we begin by determining, for each input dimension, the minimum and maximum coordinates observed across both the context and query sets. These extrema are expanded by a small margin of 0.1 in each dimension. The interval between the expanded minima and maxima is then uniformly discretized at a resolution of 64 points per unit. When necessary—for example, to satisfy the CNN's minimum grid-size requirements—the discretization range is further enlarged while maintaining the same resolution. The resulting one-dimensional grids are combined via a Cartesian product to yield a uniform $d_x$ -dimensional grid $\mathcal{G}$ . + +The functional embedding in equation 3 is evaluated on this grid. The Gaussian kernels used in this embedding are initialized with per-dimension length scales set to twice the grid resolution, i.e., $2/64$ . For multi-dimensional inputs and outputs, the model uses separate length scales for each dimension. The embedding values are further normalized using a density channel, defined as + +$$ +\operatorname {D e n s i t y} (x) = \sum_ {\left(x _ {c}, y _ {c}\right) \in \mathcal {D} _ {c}} \psi_ {e} \left(x - x _ {c}\right). +$$ + +The functional and density channels are concatenated to form the representation + +$$ +\left(\operatorname {D e n s i t y} \left(x _ {g}\right), \frac {\sum_ {\left(x _ {c} , y _ {c}\right) \in \mathcal {D} _ {c}} \phi \left(y _ {c}\right) \psi_ {e} \left(x _ {g} - x _ {c}\right)}{\sum_ {\left(x _ {c} , y _ {c}\right) \in \mathcal {D} _ {c}} \psi_ {e} \left(x _ {g} - x _ {c}\right)}\right) _ {x _ {g} \in \mathcal {G}}. \tag {5} +$$ + +Each grid location is processed by an MLP with two hidden layers of width 128. The resulting features are passed to a CNN based on a U-Net architecture [Ronneberger et al., 2015], consisting of six residual convolutional blocks (kernel size 11, stride 2, 128 channels) in the encoder, followed by a symmetric sequence of six transposed convolutional blocks in the decoder, with skip connections following the design of Bruinsma et al. [2023]. Because the U-Net downsamples the spatial resolution by a factor of 64, we ensure that the constructed grid size is divisible by 64. For the one-dimensional benchmarks considered here, we therefore enlarge the discretization interval when necessary so that the CNN's output aligns exactly with the input grid size. + +To obtain predictive distribution parameters at query locations—which may not lie on the grid $\mathcal{G}$ —an interpolation scheme analogous to equation 5 is applied, except that the weighted sum over the feature-map values is not normalized. For this interpolation, we employ a separate Gaussian kernel with learnable length-scale parameters, distinct from the kernel used in the functional embedding. The resulting query-specific embeddings are then processed by a decoder MLP with two hidden layers (width 128), whose output consists of $2d_{y}$ units parameterizing the mean and log-scale of a Gaussian predictive distribution. + +Spectral Convolutional Conditional Neural Process (SCovCNP) The SConvCNP replaces the ConvCNP's U-Net backbone with a U-shaped Fourier Neural Operator (FNO) architecture [Li et al., 2020a, Rahman et al., 2022a]. A residual Fourier block is denoted + +$$ +\mathrm {F} (c _ {\mathrm {i n}}, c _ {\mathrm {o u t}}, s _ {\mathrm {i n}}, s _ {\mathrm {o u t}}, m _ {\mathrm {f}}), +$$ + +where $c_{\mathrm{in}}$ and $c_{\mathrm{out}}$ are channel dimensions, $s_{\mathrm{in}}$ and $s_{\mathrm{out}}$ are spatial sizes, and $m_{\mathrm{f}}$ is the number of retained Fourier modes. + +The representation from equation 5, augmented with positional encodings (grid coordinates), is first processed by an MLP with one hidden layer of width 64 and GELU activations [Hendrycks and Gimpel, 2016]. Its output is then passed through the following sequence of residual Fourier blocks: + +$\mathrm{L}_1 = \mathrm{F}(64,128,|\mathcal{G}|,\lfloor |\mathcal{G}| / 2\rfloor ,32)$ +$\mathrm{L}_2 = \mathrm{F}(128,128,\lfloor |\mathcal{G}| / 2\rfloor ,\lfloor |\mathcal{G}| / 4\rfloor ,32)$ + +$\mathrm{L}_3 = \mathrm{F}(128,256,\lfloor |\mathcal{G}| / 4\rfloor ,32)$ +$\mathrm{L}_4 = \mathrm{F}(256,128,\lfloor |\mathcal{G}| / 2\rfloor ,32)$ +$\mathrm{L}_5 = \mathrm{F}(256,128,|\mathcal{G}|,32)$ + +Layers $\mathrm{L}_1\to \mathrm{L}_2\to \mathrm{L}_3\to \mathrm{L}_4$ form the contractive-expansive path. The final block $\mathrm{L}_5$ receives the channel-wise concatenation of the outputs of $\mathrm{L}_4$ and $\mathrm{L}_1$ , yielding the U-shaped skip connection. Its output is concatenated with the initial MLP features and processed by a final MLP (one hidden layer, width 128, GELU) to produce the SConvCNP representation for the decoder. + +Parameter Count. Table 5 summarizes the number of learnable parameters for all models evaluated in our 1D synthetic regression experiments. + +Table 5: Learnable parameter counts for all models evaluated in the 1D synthetic regression experiments. + +
CNPAttCNPTNPTETNPConvCNPSConvCNP
Number of parameters (million)1.32.12.63.13.83.7
+ +Forward run time. Table 6 reports the forward-pass runtime (in seconds) for a batch of 16 tasks during both training and validation across all models evaluated in our 1D synthetic regression experiments. As described in Appendix B.1.2, the number of query points is fixed to 256 during validation, whereas during training it is sampled uniformly from $\mathcal{U}[5,25)$ . + +Table 6: Forward-pass runtime (in seconds) for a batch of 16 tasks during training and validation across all models in the 1D synthetic regression setting. + +
CNPAttCNPTNPTETNPConvCNPSConvCNP
Train0.0030.0040.0070.0140.0090.009
Validation0.0060.0090.0130.0230.0330.030
+ +# B.1.2 Data and Experimental Setup + +We evaluate our methods on four families of synthetic 1D regression tasks, each defined by a distinct stochastic generative process: a Gaussian process (GP) with a Matérn-5/2 kernel, a GP with a periodic kernel, a sawtooth-wave generator, and a square-wave generator. GP tasks are sampled using GPyTorch [Gardner et al., 2018]. All processes include independent Gaussian observation noise. + +Process Parameters. For each task family, hyperparameters are sampled independently at the task level: + +- GP with Matérn-5/2 kernel. The scaled distance is + +$$ +d = \sqrt {(x - x ^ {\prime}) ^ {\top} \Lambda^ {- 2} (x - x ^ {\prime})}, \qquad \Lambda = \lambda I, \quad \lambda \sim \mathcal {U} [ 0. 2 5, 1). +$$ + +The kernel is + +$$ +k _ {m 2. 5} (x, x ^ {\prime}) = \frac {2 ^ {- 1 . 5}}{\Gamma (2 . 5)} (\sqrt {5} d) ^ {2. 5} K _ {2. 5} (\sqrt {5} d), +$$ + +where $K_{2.5}$ is a modified Bessel function of the second kind. Observation noise has variance $\sigma_0^2 = 0.01$ . + +- GP with periodic kernel. The kernel is parameterized by period $\rho \sim \mathcal{U}[0.5,2)$ and lengthscale $\lambda \sim \mathcal{U}[0.25,1)$ : + +$$ +k _ {p} (x, x ^ {\prime}) = \exp \left(- 2 \sum_ {i = 1} ^ {d} \frac {\sin^ {2} \left(\frac {\pi}{\rho} \left(x _ {i} - x _ {i} ^ {\prime}\right)\right)}{\lambda}\right), +$$ + +with observation noise variance $\sigma_0^2 = 0.01$ + +- Sawtooth wave. Piecewise-linear periodic functions with frequency $\xi \sim \mathcal{U}(0.5, 5)$ , direction $u \in \{+1, -1\}$ , and offset $c \sim \mathcal{U}[0, 1)$ . Noise follows $\mathcal{N}(0, 0.0025)$ . +- Square wave. Discontinuous periodic functions with frequency $\xi \sim \mathcal{U}(0.5, 5)$ , duty cycle $D \sim \mathcal{U}[0.25, 0.75)$ , and offset $c \sim \mathcal{U}[0, 1)$ . Noise follows $\mathcal{N}(0, 0.0025)$ . + +Table 7: Synthetic generative processes used in 1D regression experiments. + +
Processf(x)
GP (Matérn 5/2)f ~ GP(0, km2.5 + σ02I)
GP (Periodic)f ~ GP(0, kp + σ02I)
Sawtoothf(x) = 2((ξ(uT x) - c) mod 1) + σ0(x), σ0(x) ~ N(0, σ02)
Square Wavef(x) = 21{((ξx-c) mod 1) < D} - 1 + σ0(x), σ0(x) ~ N(0, σ02)
+ +Each model is trained for 500 epochs using AdamW [Loshchilov and Hutter, 2017] with a learning rate of $5 \times 10^{-4}$ . We apply gradient clipping with a maximum norm of 0.5 [Pascanu et al., 2013]. Each epoch consists of 1000 iterations, and every iteration processes a batch of 16 tasks, yielding a total of 8 million on-the-fly sampled training tasks. For each batch, the numbers of context and query points are independently drawn as $|\mathcal{I}_c| \sim \mathcal{U}[5,25)$ and $|\mathcal{I}_q| \sim \mathcal{U}[5,25)$ . These values are shared across all tasks in the batch. Input locations are sampled uniformly and independently from $[-3,3)$ for each task. + +For validation, we use a fixed set of 4,096 tasks, organized into 256 batches of 16 tasks. In these tasks, the number of query points is fixed at 256, while the number of context points is sampled using the same procedure as during training. Testing follows the same configuration as validation, except that we evaluate on 16,000 test tasks. Unlike the dynamically generated training tasks, the validation and test sets remain fixed across all experiments. + +# B.2 Predator-Prey Model + +# B.2.1 Model Architectures + +The model architectures and parameter counts closely follow those described in B.1.1. The only modification concerns the discretization of the functional embedding in ConvCNP and SConvCNP. Specifically, we first expand the range defined by the minima and maxima of all context and query coordinates by a margin of 0.5 in each dimension. The resulting interval is then uniformly discretized at a resolution of 48 points per unit length. + +# B.2.2 Data and Experimental Setup + +We generate data from a stochastic variant of the Lotka-Volterra predator-prey system [Lotka, 1910, Volterra, 1926], following the formulation of Bruinsma et al. [2023]. Let $U_{t}$ and $V_{t}$ denote the prey and predator populations at time $t$ . Their dynamics evolve according to + +$$ +d U _ {t} = \alpha U _ {t} d t - \beta U _ {t} V _ {t} d t + \sigma U _ {t} ^ {\nu} d B _ {t} ^ {(1)}, +$$ + +$$ +d V _ {t} = - \gamma V _ {t} d t + \delta U _ {t} V _ {t} d t + \sigma V _ {t} ^ {\nu} d B _ {t} ^ {(2)}, +$$ + +where $B_{t}^{(1)}$ and $B_{t}^{(2)}$ are independent Brownian motions. The deterministic drift recovers classical predator-prey behavior: prey grow exponentially at rate $\alpha$ in the absence of predators, while predators decline at rate $\gamma$ without prey. The interaction terms $\beta U_{t}V_{t}$ and $\delta U_{t}V_{t}$ model predation and predator reproduction. Stochasticity enters through the multiplicative noise terms $\sigma U_{t}^{\nu}$ and $\sigma V_{t}^{\nu}$ , where $\sigma$ controls noise intensity and $\nu$ determines how fluctuations scale with population size. Setting $\nu = 1$ yields noise proportional to population level, while $\nu < 1$ and $\nu > 1$ induce sublinear and superlinear scaling, respectively. + +Table 8: Parameter distributions for the stochastic Lotka-Volterra equations. + +
ParameterDistribution
Initial condition U-10U([5, 100])
Initial condition V-10U([5, 100])
αU([1.0, 5.0])
βU([0.04, 0.08])
γU([1.0, 2.0])
δU([0.04, 0.08])
σU([0.5, 10])
ηU([1, 5])
νFixed at 1/6
+ +We simulate trajectories using the parameter distributions in Table 8 over a total of 110 years, discarding the first 10 years as burn-in to reduce sensitivity to initial conditions. Integration is performed using the Euler-Maruyama method with time step $\Delta t = 0.022$ , producing 5000 solver steps over the interval $t \in [-10, 100)$ . Although the solver runs at this finer resolution, we record states only on a uniform grid with spacing 0.05, yielding approximately 2200 selected time points spanning $[-10, 100)$ . + +Models are trained for 500 epochs using AdamW with a learning rate of $10^{-4}$ . We apply gradient clipping with a maximum norm of 0.5. Each epoch consists of 1,000 iterations, each processing a batch of 16 tasks, yielding a total of 8 million on-the-fly sampled training tasks. For every batch, the numbers of context and query points are drawn independently as $|\mathcal{I}_c| \sim \mathcal{U}[5,25)$ and $|\mathcal{I}_q| \sim \mathcal{U}[5,25)$ ; the sampled values are shared across all tasks within the batch. + +For validation, we use a fixed set of 4,096 tasks, arranged into 256 batches of 16 tasks. In these tasks, the number of query points is fixed at 256, while the number of context points is sampled using the same procedure as in training. Testing follows the same protocol as validation, except that we evaluate on 16,000 test tasks. + +To construct a task from a simulated trajectory, input locations $t$ are sampled uniformly and independently from the recorded time points in [0, 100) and paired with the corresponding values $(U_{t}, V_{t})$ . We rescale time by a factor of 0.1, mapping the 100-year interval to [0, 10), and rescale population sizes by multiplying the values by 0.01. Unlike the dynamically generated training tasks, the randomization used to construct the validation and test tasks is fixed across all experiments. + +# B.3 Traffic Flow + +# B.3.1 Model Architectures + +Consistent with the Predator-Prey experiments described in Section B.2.1, the model architectures follow those introduced in Section B.1.1. For the discretization of the functional embedding in both ConvCNP and SConvCNP, we use a margin of 0.5 and a resolution of 48 grid points per unit length. + +# B.3.2 Data and Experimental Setup + +We use the California traffic-flow dataset from LARGEST [Liu et al., 2023b], a large-scale benchmark for traffic forecasting available at https://www.kaggle.com/datasets/liusx77/largest. The dataset provides 5-minute traffic-flow measurements from 8,600 loop-detector sensors across California's highway system, collected from 2017-2021. Our experiments focus on the year 2020. + +We discard sensors with more than $50\%$ missing values. For the remaining sensors, missing entries are linearly interpolated between observed measurements, with leading or trailing gaps filled using forward/backward propagation. Sensors are randomly partitioned into training, validation, and test splits using a 6:1:3 ratio. + +Each sensor's time series is segmented into non-overlapping 14-day windows (4,032 time steps), yielding 26 windows per sensor. Within each window, timestamps are reset to start at 0 and increment + +in 5-minute steps. We then downsample each window by a factor of 6, producing a 30-minute resolution. Time indices are rescaled to days by dividing by $(60 \times 24)$ . Traffic-flow values are normalized to the $[0,1]$ range using min-max statistics computed from the training set. + +The resulting dataset contains: + +- Training: 5,119 sensors $\rightarrow$ 133,094 windows +- Validation: 853 sensors $\rightarrow$ 22,178 windows +- Test: 2,561 sensors $\rightarrow$ 66,586 windows + +Models are trained for 100 epochs using AdamW with a learning rate of $10^{-4}$ . We apply gradient clipping with a maximum norm of 0.5. Each epoch consists of approximately 4,160 iterations, each processing a batch of 32 tasks. For every batch, the numbers of context and query points are drawn independently as $|\mathcal{I}_c| \sim \mathcal{U}[5,25)$ and $|\mathcal{I}_q| \sim \mathcal{U}[5,25)$ ; the sampled values are shared across all tasks within the batch. A task is formed by independently sampling $|\mathcal{I}_c| + |\mathcal{I}_q|$ time steps from a window without replacement. Training tasks are sampled on-the-fly from training windows. For validation and testing, we pre-generate fixed sets of tasks: + +- Validation: 22,178 tasks (batched in 32), with $|\mathcal{I}_q| = 256$ and $|\mathcal{I}_c|$ sampled as in training. +- Test: 66,586 tasks (batched in 32), with $|\mathcal{I}_q| = 50$ and $|\mathcal{I}_c|$ sampled as in training. + +# B.4 Image Completion + +# B.4.1 Model Architectures + +All models output Gaussian predictive distributions parameterized by a mean and a pre-softplus standard deviation. Because pixel values are normalized to [0, 1], the predicted mean is passed through a sigmoid. The predicted standard deviation is obtained by applying a softplus to the network output, scaling the result by 0.99, and adding a minimum noise of 0.01. + +Conditional Neural Process (CNP). The CNP architecture follows the design used in previous experiments (see Section B.1.1), with the only modification being an increase in the MLP hidden size from 256 to 512. + +Attentive Conditional Neural Process (AttCNP). The AttCNP architecture is likewise based on the design used in earlier experiments (see Section B.1.1). The MLP hidden size is increased from 256 to 512. For the self-attention applied to the set of context embeddings, we increase the per-head dimension from 32 to 64, set the input/output dimension of each attention layer to 512, and increase the feedforward subnetwork width from 256 to 512. The same modifications apply to the multi-head cross-attention used to update query embeddings (eight heads of dimension 64 and feedforward width 512). + +Transformer Neural Process (TNP). As with the other models, all MLP hidden sizes are increased from 256 to 512. We adopt the Efficient Query TNP design described in Section B.1.1, which employs a two-branch structure in which context embeddings are processed twice and query embeddings once. In contrast to previous sections, we allocate separate parameters for the two branches: at each layer, the multi-head self-attention over contexts and the multi-head cross-attention between queries and contexts no longer share parameters. For each multi-head attention module, the feedforward width is increased from 256 to 512, and all input/output embedding dimensions are set to 512. All other configurations follow Section B.1.1. + +Convolutional Conditional Neural Process (ConvCNP). Because the data lie on a regular grid, we use the on-the-grid implementation of ConvCNP [Gordon et al., 2019], eliminating the discretization step and improving computational efficiency. Let I denote an incomplete image, with unobserved pixels filled with dummy values, and let $\mathbf{M}_c$ be the binary mask indicating observed (context) pixels. For multi-channel images, the mask is broadcast along the channel dimension. + +As in Section B.1.1, the output of the convolutional deep set module has two components: (i) a density channel capturing the spatial distribution of context pixels, and (ii) a kernel-smoothed representation. + +The kernel is implemented via a 2D convolutional layer (kernel size 11, $d_y$ input channels, 128 output channels, no bias) with a nonnegativity constraint enforced by taking absolute values of the learned weights during the forward pass [Gordon et al., 2019]. + +The density channel is obtained by convolving this modified filter with the mask $\mathrm{M}_c$ . The kernel-smoothed component is computed by first multiplying I elementwise with $\mathrm{M}_c$ (setting non-context pixels to zero) and then applying the same modified convolution. Following Gordon et al. [2019], we omit normalization by the density channel, as it did not provide empirical benefits and occasionally introduced instability; however, we retain the positivity constraint on the kernel. + +The resulting representation is processed pointwise by an MLP with two hidden layers of width 128. These features are then fed into a ResNet-style CNN [He et al., 2016] consisting of six residual convolutional blocks (kernel size 11, 128 channels), following the implementation of Bruinsma et al. [2023]. The embeddings corresponding to query pixels are finally passed to a decoder MLP with two hidden layers of width 128. + +Spectral Convolutional Conditional Neural Process (SCNNP). The SConvCNP mirrors the on-grid ConvCNP architecture described above, with the distinction that the ResNet backbone is replaced by a U-shaped FNO. To construct positional information, we uniformly discretize the interval $[-1, 1]$ along each spatial axis into grids whose sizes match the corresponding image dimensions. Their Cartesian product yields the 2D positional encoding, which is concatenated with the output of the convolutional deep set. This concatenated tensor is then passed pointwise through an MLP with one hidden layer of width 128 and GELU activations. + +Let $s_h$ and $s_w$ denote the image height and width, respectively. Using the notation introduced in Section B.1.1, the operator consists of the following sequence of residual Fourier blocks: + +- $\mathrm{L}_1 = \mathrm{F}(512, 512, (s_h, s_w), (s_h, s_w), 32)$ +- $\mathrm{L}_2 = \mathrm{F}(512, 512, (s_h, s_w), (\lfloor s_h / 4 \rfloor, \lfloor s_w / 4 \rfloor)$ , 8) +- $\mathrm{L}_3 = \mathrm{F}(512, 512, (\lfloor s_h / 4 \rfloor, \lfloor s_w / 4 \rfloor), (\lfloor s_h / 16 \rfloor, \lfloor s_w / 16 \rfloor)$ , 2) +- $\mathrm{L}_4 = \mathrm{F}(512, 512, (\lfloor s_h / 16 \rfloor, \lfloor s_w / 16 \rfloor), (\lfloor s_h / 16 \rfloor, \lfloor s_w / 16 \rfloor), 2)$ +- $\mathrm{L}_5 = \mathrm{F}(512, 512, (\lfloor s_h / 16 \rfloor, \lfloor s_w / 16 \rfloor), (\lfloor s_h / 4 \rfloor, \lfloor s_w / 4 \rfloor), 8)$ +- $\mathrm{L}_6 = \mathrm{F}(512, 512, (\lfloor s_h / 4 \rfloor, \lfloor s_w / 4 \rfloor), (s_h, s_w), 16)$ +$\mathrm{L}_7 = \mathrm{F}(512,512,(s_h,s_w),(s_h,s_w),32)$ + +The forward pass follows a standard U-shaped pattern. Layers $\mathrm{L}_1\rightarrow \mathrm{L}_2\rightarrow \mathrm{L}_3\rightarrow \mathrm{L}_4$ are applied sequentially. On the upward path, $\mathrm{L}_5$ receives the channel-wise concatenation of the outputs of $\mathrm{L}_4$ and $\mathrm{L}_3$ . Similarly, $\mathrm{L}_6$ receives the concatenation of the outputs of $\mathrm{L}_5$ and $\mathrm{L}_2$ , and $\mathrm{L}_7$ receives the concatenation of the outputs of $\mathrm{L}_6$ and $\mathrm{L}_1$ . The output of $\mathrm{L}_7$ is then concatenated with the features from the initial MLP and fed into a final MLP (one hidden layer, width 128, GELU) to yield the SConvCNP representation consumed by the decoder. + +Parameter Count. Table 9 summarizes the number of learnable parameters for all models evaluated in our image completion experiments. + +Table 9: Learnable parameter counts for all models evaluated in the image completion experiments. + +
CNPAttCNPTNPConvCNPSConvCNP
Number of parameters (million)58.413.712.214.2
+ +# B.4.2 Data and Experimental Setup + +For our image-completion experiments, we use the Describable Textures Dataset (DTD; Cimpoi et al. [2014]). We adopt the standard DTD split, which contains 1,880 images each for training, validation, and testing. All images are RGB, with heights ranging from 231 to 778 pixels and widths from 271 to 900 pixels. During training, validation, and testing, we sample from each image a random $192 \times 192$ crop, which we then resize to $64 \times 64$ . Pixel coordinates of the resulting $64 \times 64$ grid along each + +axis are linearly mapped to $[-1, 1]$ , and pixel intensities are normalized to $[0, 1]$ independently across channels. + +Models are trained for 500 epochs using AdamW with a learning rate of $10^{-4}$ . We apply gradient clipping with a maximum norm of 0.5. At each epoch, the model processes batches of 16 tasks. For each batch, we draw the number of context pixels as $|\mathcal{I}_c| \sim \mathcal{U}[5, 1024)$ , and treat all remaining pixels as queries, i.e. $|\mathcal{I}_q| = 64 \times 64 - |\mathcal{I}_c|$ . A single sampled value of $|\mathcal{I}_c|$ is shared across all tasks within the batch. + +# B.5 Ablation Studies + +We performed a series of ablation studies to quantify the contribution of three key design choices in SCONVCNP: (i) the number of retained Fourier modes, (ii) the discretization resolution of the functional embedding, and (iii) the inclusion of positional encodings. Experiments were conducted on two 1D function families—samples from a Gaussian process with a Matérn-5/2 kernel and a sawtooth wave. To isolate the effect of each component, we used a slightly reduced-capacity SCONVCNP model (3.1M parameters, 32 Fourier modes). Training followed the same protocol as in Section B.1, and evaluation was performed on 8,096 test tasks. A summary of our findings is provided below. + +Number of Fourier modes. Increasing the number of Fourier modes substantially improves performance on the sawtooth signal but yields negligible gains for Matérn-5/2 functions (Table 10). This difference is spectral: the sawtooth's Fourier amplitudes decay only as $1 / \xi$ , leaving substantial energy at high frequencies and requiring many modes for accurate representation. In contrast, Matérn-5/2 samples exhibit spectra decaying as $1 / \xi^6$ , with extremely weak high-frequency content; thus, a small number of low-frequency modes already captures nearly all signal energy. + +Table 10: Predictive log-likelihood of SConvCNP for different numbers of Fourier modes $m$ (4 seeds). + +
m=8m=16m=32
Matérn 5/2-0.29±0.00-0.29±0.00-0.29±0.00
Sawtooth wave-0.14±0.030.20±0.020.80±0.03
+ +Discretization resolution. Finer discretization improves performance in the sawtooth case but has minimal effect for Matérn-5/2 samples (Table 11). The sawtooth's slowly decaying spectrum implies a high Nyquist frequency, so denser sampling is required to resolve its high-frequency components. In contrast, the Matérn-5/2 spectrum concentrates almost all energy at low frequencies; coarser discretizations already capture the relevant structure. + +Table 11: Predictive log-likelihood of SConvCNP for different discretization resolutions (number of points per unit; 4 seeds). + +
163264
Matérn 5/2-0.30±0.00-0.29±0.00-0.29±0.00
Sawtooth wave0.03±0.020.26±0.080.20±0.02
+ +Positional encoding. Adding positional encodings yields consistent improvements, particularly for the sawtooth wave (Table 12). + +Table 12: Predictive log-likelihood of SConvCNP with and without positional encoding (4 seeds). + +
with positional encodingwithout positional encoding
Matérn 5/2-0.29±0.00-0.31±0.00
Sawtooth wave0.80±0.030.67±0.02
+ +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction clearly state that we introduce Spectral Convolutional Conditional Neural Processes (SCovCNPs), explain the motivation for using spectral methods to address limitations in ConvCNPs, and accurately describe the scope of our experimental validation. The claims match the actual content and results presented in the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: Section 3.1 discusses the limitations of our approach, including the trade-off between positional encodings and translation equivariance, and the discretization sensitivity of the FFT. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [NA] + +Justification: This paper primarily presents a methodological contribution with empirical validation rather than theoretical results requiring formal proofs. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: The paper provides detailed descriptions of the model architecture, data generation process, and experimental setup, ensuring that the main results can be reproduced. In addition, all implementations are publicly available in our code repository. + +# 5. 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Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [Yes] + +Justification: All experimental results in Tables 1-3 report standard deviations across 4 seeds, providing a clear measure of the statistical variability in our results. + +# 8. Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +Answer: [Yes] + +Justification: The paper explicitly states that all computations were performed on a single NVIDIA A100 GPU with 40 GB of memory. Additionally, forward-pass runtimes for all models on one of the benchmarks are reported, providing further clarity on the execution time requirements. + +# 9. 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Mahoney + +ICSI, LBNL, and Department of Statistics + +University of California, Berkeley + +mmahoney@stat.berkeley.edu + +# Abstract + +Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in scale, the corresponding covariance and kernel matrices become increasingly large, often reaching magnitudes that make their direct formation impractical or impossible. Existing techniques typically rely on matrix-vector products, which can provide efficient approximations, if the matrix spectrum behaves well. However, in settings like distributed learning, or when the matrix is defined only indirectly, access to the full data set can be restricted to only very small sub-matrices of the original matrix. In these cases, the matrix of nominal interest is not even available as an implicit operator, meaning that even matrix-vector products may not be available. In such settings, the matrix is "impalpable," in the sense that we have access to only masked snapshots of it. We draw on principles from free probability theory to introduce a novel method of "free decompression" to estimate the spectrum of such matrices. Our method can be used to extrapolate from the empirical spectral densities of small submatrices to infer the eigenspectrum of extremely large (impalpable) matrices (that we cannot form or even evaluate with full matrix-vector products). We demonstrate the effectiveness of this approach through a series of examples, comparing its performance against known limiting distributions from random matrix theory in synthetic settings, as well as applying it to submatrices of real-world datasets, matching them with their full empirical eigenspectra. + +# 1 Introduction + +Eigenvalues encode essential information about matrices and are central to the evaluation of spectral invariants in computational linear algebra. Two such invariants, the log-determinant and the trace of the inverse, appear frequently in statistical modeling and Bayesian inference (Ubaru & Saad, 2017). However, these quantities require access to the full range of eigenvalues, posing substantial challenges in modern large-scale settings. As datasets continue to scale up, it is becoming increasingly common to encounter matrices that cannot be practically formed in their entirety. This may occur, for example, if the matrices are so large that they exceed the storage capacity in local memory. Popular methods for operating with large matrices often rely on randomized low-rank approximations from Randomized Numerical Linear Algebra (RandNLA) (Drineas & Mahoney, 2016; Murray et al., 2023; Dereziński & Mahoney, 2021), and can broadly be classified as either iterative subspace or sampling methods, although hybrid methods have become prominent (Ubaru et al., 2017; Saibaba et al., 2017; Chen + +et al., 2021). Implicit methods (Adams et al., 2018) that rely on Krylov subspaces using Arnoldi- or Lanczos-based techniques (Ubaru et al., 2017) have been successful for estimating arbitrary spectral invariants (Gardner et al., 2018; Potapczynski et al., 2023), and they require access only to matrix-vector products. These techniques avoid explicit matrix storage and can often converge quickly. However, their performance degrades significantly when the matrix is ill-conditioned. + +The situation is particularly dire when most matrix entries are inaccessible and matrix-vector products are unavailable or too costly. This can occur when the data used to generate the matrix is spread across a distributed system, or when the matrix itself is enormous (see e.g., Ameli et al. (2025)). Such matrices can be considered impalpable, because neither the matrix nor its matrix-vector products can be handled in their entirety. Despite being inaccessible, the information these impalpable matrices contain can be crucial, making methods to summarize or access them highly valuable. Approaches which subsample a few rows and columns are feasible, assuming that any observed quantities can be directly extended to the full matrix (Drineas et al., 2008). This is the strategy behind the Nyström method, which accesses a matrix via column/row sampling, and then reconstructs the matrix under linearity assumptions (Gittens & Mahoney, 2016; Gittens et al., 2016). Unfortunately, such techniques incur significant bias, as they target behavior dictated by large eigenvalues by design, giving less prominence to the near-singular dimensions that play a crucial role in determinants and inverses. + +In the setting of impalpable matrices, or when otherwise fine-grained behavior of ill-conditioned matrices becomes important and implicit methods fail, there are very few tools currently available (Couillet & Liao, 2022; Ameli et al., 2025). To address this gap, we provide a novel method for approximating the empirical spectral density of extremely large (impalpable) matrices, requiring only access to appropriate submatrices. Our technique is based on tools from random matrix theory and free probability, and it applies to virtually any class of matrices. Assuming only that the underlying matrix is sufficiently large, the empirical spectral density of a (sampled) submatrix can be evolved by a partial differential equation (PDE) in terms of the dimension of the matrix to access the spectral density of the full (impalpable) matrix. Due to connections with free probability theory, which we explain below, we call this process free decompression. + +# To summarize, we: + +(I) introduce and derive free decompression, the first method for spectral density estimation of impalpable matrices, by evolving the empirical spectral density of submatrices, and provide rigorous polynomial error bounds under mild assumptions (Proposition 4); +(II) demonstrate the utility of our free decomposition method for a range of synthetic examples, social media graph data, and empirical neural tangent kernels; and +(III) provide a user-friendly Python package, freealg, implementing all proposed algorithms, that reproduces the numerical results of this paper. + +The remainder of this paper is structured as follows: Section 2 provides background and outlines the different classes of matrices of interest; Section 3 provides the necessary tools in order to construct our free decompression procedure; Section 4 describes its implementation and issues faced in operationalization; Section 5 provides numerical examples. We conclude with discussion in Section 6. A summary of the notation used throughout the paper is provided in Appendix A. An overview of free decompression and its application to well-studied random matrix ensembles is presented in Appendix B.1. Discussions of the significant numerical considerations required to perform free decompression are provided in Appendix C and Appendix D. Proofs of our theoretical contributions can be found in Appendices E and F. Background on the neural tangent kernel example is provided in Appendix G. Finally, Appendix H presents a guide to our implementation of free decompression. + +# 2 Impalpable Matrices + +We are interested in estimating the spectral density of a wide class of matrices for which operating on the full matrix is infeasible. In the most extreme case, these matrices cannot be formed in their entirety, either implicitly or explicitly. We call these matrices impalpable. We will consider the case where the impalpable matrix of interest is Hermitian. Since we are interested in the setting where we + +cannot access the full matrix, we will work with submatrices indexed by randomly sampled columns of the impalpable matrix of interest, where the submatrix consists of the intersection of the impalpable matrix's rows and columns that appear in this index set. This corresponds to an assumption that the matrices are part of a sequence that is asymptotically free (Mäïda, 2023). We will then see that it is convenient to work with tools from random matrix theory and free probability, namely the familiar Stieltjes transform (Couillet & Liao, 2022), whose favorable properties we exploit in our free decompression procedure. + +# 2.1 Categories of Matrix Difficulty + +We find it useful to classify matrices by four modes of computational difficulty: explicit, implicit, out-of-core, and impalpable, the latter of which is our focus. + +Explicit Matrices. This is the most straightforward class of matrices. They can be formed explicitly as contiguous arrays and operated on in memory. Existing implementations on such matrices are highly optimized, with their runtime limited only by their computational complexity. + +Implicit Matrices. Difficulties begin to arise once matrices become too large to be stored in memory. Implicit matrices are those that can be computed or estimated only in terms of matrix-vector products $\pmb{v} \mapsto \mathbf{A}\pmb{v}$ (Adams et al., 2018). Treating a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ as implicit often has significant computational benefits, even before the memory wall (Gholami et al., 2024); for iterative methods, computation time can often be reduced, e.g., from $\mathcal{O}(n^3)$ operations to $\mathcal{O}(n^2)$ . If the formation of the matrix-vector product is efficient, this can be further reduced to $\mathcal{O}(n)$ . Memory requirements may follow similar reductions. Implicit methods exist for most standard linear algebra operations, often based on Arnoldi- or Lanczos-based iterations, and appear in mature software packages (Potapczynski et al., 2023). One of the most popular implicit matrix algorithms for spectral function estimation is stochastic Lanczos quadrature (SLQ) (Ubaru et al., 2017); however, the performance of such methods may deteriorate for highly ill-conditioned matrices (Ameli et al., 2025). + +Out-of-Core Matrices. Implicit methods typically operate on Krylov iteration schemes, with error rates depending on condition numbers of the matrix—see Bhattacharjee et al. (2025), e.g., in the case of SLQ. This can be disastrous for large and ill-conditioned dense matrices, with accurate estimates requiring a large number of matrix-vector products. Exact methods become necessary in this setting if a high level of accuracy is required, but the memory bottleneck still remains. Out-of-core matrices have the property that for some integer $m \ll n$ , any subblock of size $\mathbb{R}^{m \times m}$ can be formed explicitly and operated on in memory, and the full matrix can be stored locally by iterating through these blocks (Aggarwal & Vitter, 1988). Decompositions (Rabani & Toledo, 2001; Mu et al., 2014), inverses (Caron & Utard, 2002), and products of out-of-core matrices can be (slowly) computed to high precision, assuming adequate (typically very large) available external storage. + +Impalpable Matrices. The most extreme category of matrices includes cases that do not fit into the other categories. Here, the matrix cannot be formed in memory, as it is either too large for out-of-core operations, or has missing portions. Matrix-vector products are assumed too expensive to compute accurately, or are rendered useless due to strong ill-conditioning. In principle, such a matrix can be considered "impalpable," as the vast proportion of either its elements or properties cannot be operated on. Comprising the worst-case + +
TypeAccess in Memory
MatrixMatrix-Vector ProductAny Subblock
Explicit✓✓✓
ImplicitX✓X
Out-of-coreX~✓
ImpalpableXXX
+ +Table 1: Comparison of matrix classes, according to attributes that can be formed and operated on in memory. + +scenario, impalpable matrices offer unique challenges for linear algebra practitioners. Determining accuracy of any proposed solution is an underdetermined problem in general. The only feasible strategy is extrapolation, using available information. While this is inherently risky, the class of impalpable matrices simply offer no alternative. To our knowledge, only the FLODANCE algorithm of Ameli et al. (2025) offers a valid approach for estimating log-determinants of impalpable matrices. + +# 2.2 Examples of Impalpable Matrices + +**Enormous Gram Matrices.** Our motivating example of an impalpable matrix is an ill-conditioned Gram matrix $\mathbf{A} = (\kappa(\boldsymbol{x}_i, \boldsymbol{x}_j))_{i,j=1}^n$ comprised of a massive number $n$ of points $\boldsymbol{x}_i$ and a kernel $\kappa$ that is expensive to compute. In this case, computing a few subblocks of $\mathbf{A}$ is viable, but out-of-core methods are too expensive. Ameli et al. (2025) demonstrates the challenges of computing the log-determinant of a neural tangent kernel matrix for a large neural network model. Their most cost-effective strategy extrapolated the behavior of the log-determinant from smaller submatrices in accordance with a scaling law. This is an example of an impalpable matrix strategy, relying on smaller portions of the matrix without matrix-vector products. The potential gains are best demonstrated with a synthetic example: the covariance matrix of $\boldsymbol{x}_1, \ldots, \boldsymbol{x}_n$ , where each $\boldsymbol{x}_i \sim \mathcal{N}(\mathbf{0}_p, \mathbf{I}_{p \times p})$ is $p$ -dimensional. Once $p$ and $n$ become extremely large, forming this matrix is prohibitive. However, its spectral density is well-approximated by the Marchenko-Pastur law (Marčenko & Pastur, 1967), and its log-determinant over submatrices of increasing size is predictable (Nguyen & Vu, 2014; Cai et al., 2015; Ameli et al., 2025; Hodgkinson et al., 2023a). + +Catastrophic Cancellation. Impalpable matrices typically arise due to issues of scale, but they need not be large themselves. Indeed, such matrices can arise due to catastrophic cancellation. Some matrices can be comprised of a non-trivial product of rectangular matrices; for example, $\mathrm{cond}(\mathbf{A}^{\mathsf{T}}\mathbf{A}) = \mathrm{cond}(\mathbf{A})^{2}$ , so representing the outer product $\mathbf{A}^{\mathsf{T}}\mathbf{A}$ (e.g. in the normal equations) in floating-point precision can incur greater numerical error (Ameli et al., 2025). Higham (2022) refers to the formation of $\mathbf{A}^{\mathsf{T}}\mathbf{A}$ as a "cardinal sin" of numerical linear algebra. More products yield even larger errors. General Gram matrices exhibit similar behavior: the condition number can become so large that the matrix cannot be accurately represented in double precision, rendering even exact methods hopeless (Lowe et al., 2025). Fortunately, submatrices have smaller condition numbers, so there may be a submatrix that can be represented in double precision. Ideally, spectral behavior of these submatrices can then be extrapolated to obtain insight into the behavior of the full matrix in a meaningful way. + +# 3 Free Decompression + +Our objective is to approximate the spectral distribution of an impalpable matrix $\mathbf{A}$ given only explicit access to a randomly sampled $n\times n$ submatrix. This seemingly intractable task can be accomplished by working with the Stieltjes transform of the submatrix, and observing that for large $n$ this Stieltjes transform varies with $n$ in a manner that is well approximated by a PDE. + +Free decompression of a random submatrix $\mathbf{A}_n$ to a larger matrix $\mathbf{A}$ requires: + +1. estimation of its Stieltjes transform $m_{\mathbf{A}_n}$ +2. evolution of $m_{\mathbf{A}_n}$ in $n$ via equation (1); +3. evaluation of the spectral distribution of $\mathbf{A}$ + +This enables an approximation of the Stieltjes transform of the full matrix $\mathbf{A}$ , from which a corresponding spectral distribution is easily computed. We emphasize that the matrix $\mathbf{A}$ itself may be arbitrary and deterministic. Asymptotic freeness enters our model through the method of sumbatrix selection, making it amenable to random matrix theoretic and free probabilistic tools. + +# 3.1 Tools from Free Probability + +In order to make free decompression concrete, we require tools from free probability. Let $\mathbf{A}$ be a fixed matrix of large size. If we can approximate the spectral density of a small randomly sampled submatrix, and extrapolate its behavior to the full matrix, it should also be possible to extrapolate further to larger matrices than the one we are given. To this end, we treat $\mathbf{A}$ as an element in an infinite sequence of matrices $\{\mathbf{A}_1,\mathbf{A}_2,\ldots \}$ of increasing size, where $\mathbf{A}_n\in \mathbb{R}^{n\times n}$ . For consistency, we assume that for each $n$ , the matrix $\mathbf{A}_n$ is the top-left $n\times n$ submatrix of $\mathbf{A}_{n + 1}$ . This ensures that $[\mathbf{A}_n]_{ij} = [\mathbf{A}]_{ij}$ is constant in $n$ for fixed $i,j\leq n$ . We will also assume that for any integer $k\geq 1$ , the quantity $\frac{1}{n}\mathrm{tr}(\mathbf{A}_n^k)$ converges as $n\to \infty$ . This is equivalent to assuming that the empirical spectral distributions of $\mathbf{A}_n$ converge weakly as $n\to \infty$ (Voiculescu, 1991). Note that this imposes no requirements on the matrix to be approximated. + +![](images/c4263773f39a18ef5cd52dd2b22934a07e20f04c63204f1dfab4d26029cc04ad.jpg) +Figure 1: (a) Estimation of the Marchenko-Pastur law with ratio $\lambda = \frac{1}{50}$ from eigenvalues of Wishart matrices with $d = 50,000$ degrees of freedom and size $n \times n = 1000 \times 1000$ ; (b) approximation of densities for various values of $n$ via free decompression (Algorithm 1); and (c) comparison of our decompressed approximation of $n = 32,000$ with the known limiting analytic density for $\lambda = \frac{32}{50}$ . + +![](images/3dac46692140f1a2711f92e0499297475e404106afa8fac6de66f0b2b917abb3.jpg) + +![](images/08d912628d4f6fc8920288e4c7ed12ce6d8c63c27a099ce65e946f30ef2f9d48.jpg) + +Still, we cannot predict the upcoming row and column in $\mathbf{A}_{n + 1}$ from $\mathbf{A}_n$ alone, and we will need to appeal to random transformations to connect each element in the sequence together. One way to transform $\mathbf{A}$ into a random matrix, while preserving its eigenspectrum, is to perform a similarity transform with a random matrix $\mathbf{P}$ , recovering $\mathbf{PAP}^{-1}$ . The most inexpensive choice of random matrix $\mathbf{P}$ is a (Haar) random permutation matrix, so that the (random) similarity transform is equivalent to the following operation: let $S_{n}$ be the uniform distribution on the space of permutations on $\{1,\ldots ,n\}$ , and consider the matrix $\mathbf{A}_n^\sigma = (\mathbf{A}_{\sigma (i)\sigma (j)})_{i,j = 1}^n$ , where $\sigma \sim S_n$ . + +Free Compression. Due to a result of Nica (1993), the sequence of matrices $\mathbf{A}_n^\sigma$ is asymptotically free as $n\to \infty$ , in the sense of free probability theory. We shall not further discuss free random matrices here, but we refer the interested reader to Maida (2023) for a detailed introduction. Instead, we rely on one useful property. + +Let $m_{\mathbf{A}}(z)$ denote the Stieltjes transform of $\mathbf{A}$ of size $n \times n$ , given by $m_{\mathbf{A}}(z) = n^{-1}\mathbb{E}[\mathrm{tr}(\mathbf{A} - z\mathbf{I})^{-1}]$ and let $\omega (z)$ denote the functional inverse of $m(z)$ , satisfying $m(\omega (z)) = z$ . The corresponding $R$ -transform is then defined as $R(z) = \omega (-z) - z^{-1}$ . Our method depends on the following fundamental theorem of Nica & Speicher (1996); see also Olver & Nadakuditi (2012, Section 7). + +Theorem 1 (NICA-SPEICHER). For a free random matrix $\mathbf{A}$ of size $n\times n$ with $R$ -transform $R(z)$ , the $R$ -transform of the top-left $n_s\times n_s$ submatrix of $\mathbf{A}$ is given by $R_{n_s}(z) = R(z\frac{n_s}{n})$ . + +In other words, to arrive at the $R$ -transform for a submatrix comprised of only the top-left proportion $\alpha$ of rows and columns, one need only scale the argument by $\alpha$ . In free probability theory, this procedure is called free compression (Olver & Nadakuditi, 2012). + +Free Decompression. Our main approach is based on the observation that nothing obstructs this operation from being conducted in reverse, to glean information about a larger matrix from a smaller one. For this reason, we call our procedure free decompression: starting from the $R$ -transform of a smaller $n_s \times n_s$ submatrix $R_{n_s}(z)$ , the $R$ -transform of the larger $n \times n$ matrix is given by $R_n(z) = R_{n_s}\left(\frac{n}{n_s} z\right)$ . + +Example (COVARIANCE MATRICES). Let $\mathbf{X} \in \mathbb{R}^{n \times d}$ have elements $X_{ij} \sim \mathcal{N}(0,1)$ and let $\mathbf{A} \coloneqq \frac{1}{d}\mathbf{X}\mathbf{X}^{\intercal} \in \mathbb{R}^{n \times n}$ . Assuming $d \geq n$ , then $\mathbf{A}$ is almost surely full rank, and it is $n \times n$ Wishart-distributed with $d$ degrees of freedom. The top-left $n_s \times n_s$ corner of $\mathbf{A}^\sigma$ is also Wishart-distributed with $d$ degrees of freedom; and so, for large $n_s$ and $d$ , its eigenvalues approximately follow the Marchenko-Pastur law with ratio $\lambda = n_s / d$ . The $R$ -transform is given by $R_{n_s}(z) = (1 - zn_s / d)^{-1}$ . Applying free decompression, we estimate the $R$ -transform of the full matrix to be + +$$ +R _ {n} (z) = \left(1 - z \cdot \frac {n}{n _ {s}} \cdot \frac {n _ {s}}{d}\right) ^ {- 1} = \left(1 - z \cdot \frac {n}{d}\right) ^ {- 1}. +$$ + +This is the $R$ -transform of a Marchenko-Pastur law with ratio $\lambda = n / d$ , and it approximates the spectral density of a Wishart-distributed matrix with $d$ degrees of freedom and size $n \times n$ . This + +is precisely what we expected for the full matrix $\mathbf{A}$ . In effect, this allows us to infer spectral properties of the $\mathbf{A} = \frac{1}{d}\mathbf{X}\mathbf{X}^{\intercal} \in \mathbb{R}^{n \times n}$ from the submatrix $\mathbf{A}_{n_s} = \frac{1}{d}\mathbf{X}_{n_s}\mathbf{X}_{n_s}^{\intercal} \in \mathbb{R}^{n_s \times n_s}$ , where the $\mathbf{X}_{n_s} \in \mathbb{R}^{n_s \times d}$ are (much) shorter matrices with (much) smaller aspect ratio $\lambda$ (see Figure 1). + +However, for more general classes of matrices, inverting the Stieltjes transform in order to compute its $R$ -transform is not analytically tractable. Instead, we will work with the Stieltjes transform directly. + +# 3.2 Evolution of the Stieltjes Transform + +Our approach can be made precise by appealing to Theorem 1 and operating directly on the $R$ transform. We do this in Proposition 2, which shows that free decompression corresponds to the evolution of a PDE in the Stieltjes transform, given by (1). To do so, for a fixed $R$ -transform, we let $R(t,z) = R(ze^t)$ , so that $\alpha = n / n_s = e^t$ , and we consider the corresponding Stieltjes transform $m(t,z)$ and its inverse $\omega(t,z)$ . Then, the following proposition, proved in Appendix E, holds. + +Proposition 2. The Stieltjes transform $m(t,z)$ corresponding to $R$ -transforms $R(ze^t)$ satisfies + +$$ +\frac {\partial m}{\partial t} = - m + m ^ {- 1} \frac {\partial m}{\partial z}. \tag {1} +$$ + +Since (1) is a first-order quasilinear PDE, it can be readily solved using the method of characteristics. This is outlined in Proposition 3, which is proved in Appendix E. A key consequence of this proposition is that we obtain explicit solutions to (1) in terms of the initial data and desired "time" $t$ . + +Proposition 3. Suppose $m_0$ is analytic in an open domain $\Omega \subset \mathbb{C}$ , and $m_0(z) \neq 0$ for all $z \in \Omega$ . Consider the initial-value problem (1) with $m(0,z) = m_0(z)$ . For $z_0 \in \Omega$ , there is a unique solution $m(t,z)$ whose graph can be parametrized by curves $\tau \mapsto (t(\tau),z(\tau),m(\tau))$ , $\tau \in \mathbb{R}_{\geq 0}$ , where + +$$ +t (\tau) = \tau , \qquad z (\tau) = z _ {0} - m _ {0} (z _ {0}) ^ {- 1} (e ^ {\tau} - 1), \qquad m (\tau) = m _ {0} (z _ {0}) e ^ {- \tau}, \qquad (2) +$$ + +valid for $\tau$ such that $z(\tau) \in \Omega$ . Furthermore, letting $\Phi(t,z,z_0) \coloneqq z - z_0 + m(z_0)^{-1}(e^\tau - 1)$ , so $\Phi(t(\tau),z(\tau),z_0) = 0$ , there exists an inverse function $z_0 = \phi(t,z)$ solving $\Phi(t,z,\phi(t,z)) = 0$ which allows the explicit solution of (1) by $m(t,z) = m_0(\phi(t,z))e^{-t}$ . + +Finally, Proposition 4, proved in Appendix F, shows that errors in free decompression can grow, but no more than polynomially fast in the decompression ratio $\frac{n}{n_s}$ . + +Proposition 4. Let $m$ be a Stieltjes transform of a density $\rho$ such that $\| \rho \|_{L^{\infty}} < +\infty$ , and let $\delta m_{n_s}$ denote the error in an approximation to $m$ starting with a matrix size of $n_s$ . For $\delta m_n$ , the error under free decompression (1) to a ratio $n / n_s$ , there is a constant $\nu > 0$ such that for any $t > 0$ , + +$$ +\left\| \delta m _ {n} \right\| _ {L ^ {2}} \leq \left(n / n _ {s}\right) ^ {\nu} \left\| \delta m _ {n _ {s}} \right\| _ {L ^ {2}}. +$$ + +# 3.3 Heuristic Motivation + +While our method is ultimately constructed via Theorem 1, it is illustrative to consider the behavior of the Stieltjes transforms of successive matrices $\mathbf{A}_n$ in our sequence. In what follows, we will proceed informally, roughly following the reasoning outlined in Tao (2017). This heuristic approach is to aid the reader's intuition by viewing the evolution equations in terms of the Schur complement, to motivate the full proof in Appendix E. We let $e_i$ denote the $i$ -th column basis vector. Taking two successive matrices in our sequence $\mathbf{A}_n$ , we exploit properties of the Schur complement and the Woodbury matrix identity, to obtain the relationship between the trace of their inverses + +$$ +\operatorname {t r} \left(\mathbf {A} _ {n + 1} - z \mathbf {I}\right) ^ {- 1} = \operatorname {t r} \left(\mathbf {A} _ {n} - z \mathbf {I}\right) ^ {- 1} + \frac {\boldsymbol {e} _ {n + 1} ^ {\intercal} \left(\mathbf {A} _ {n + 1} - z \mathbf {I}\right) ^ {- 2} \boldsymbol {e} _ {n + 1}}{\boldsymbol {e} _ {n + 1} ^ {\intercal} \left(\mathbf {A} _ {n + 1} - z \mathbf {I}\right) ^ {- 1} \boldsymbol {e} _ {n + 1}}. \tag {3} +$$ + +Assuming the rows and columns of $\mathbf{A}$ have already been randomly permuted, then the distribution of each row and column of $(\mathbf{A}_{n + 1} - z\mathbf{I})^{-1}$ is the same. Therefore, in expectation, + +$$ +\mathbb {E} \left[ \boldsymbol {e} _ {n + 1} ^ {\intercal} (\mathbf {A} _ {n + 1} - z \mathbf {I}) ^ {- 1} \mathbf {e} _ {n + 1} \right] = \frac {1}{n + 1} \sum_ {i = 1} ^ {n + 1} \mathbb {E} \left[ \boldsymbol {e} _ {i} ^ {\intercal} (\mathbf {A} _ {n + 1} - z \mathbf {I}) ^ {- 1} \boldsymbol {e} _ {i} \right] = \frac {1}{n + 1} \mathbb {E} \left[ \operatorname {t r} (\mathbf {A} _ {n + 1} - z \mathbf {I}) ^ {- 1} \right]. +$$ + +![](images/97696a4c772c727772ce9056be8a13df0af5c73d4559a731a7a1bf806d5751e2.jpg) +Figure 2: Analytic continuation of the Stieltjes transform of a Marchenko-Pastur distribution (with ratio $\lambda = \frac{1}{50}$ ). (a) The principal branch contains a branch cut along the support of the spectral density, while (b) the secondary branch is continuous in this region. Curves (white) highlight the evolution of $t\mapsto \phi (t,z)$ over $t\in [0,4]$ , crossing the support of the spectral density. + +![](images/072f12eaeafa6c319e64c8bd460590a5fe95a6cabdd78b248df8672b3521a6a6.jpg) + +Algorithm 1: Pseudocode for Free Decompression +Input: random submatrix $\mathbf{A}_{n_s} \in \mathbb{R}^{n_s \times n_s}$ of Hermitian matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ ; spectral smoothing function EigApprox; gluing function Glue; Stieltjes transform approximation Stieltjes, perturbation $\delta$ +Output: Estimated spectral density values $\{\hat{\rho}(x)\}_{x \in \mathcal{X}}$ . +1 Compute the spectrum of $\mathbf{A}_{n_s}$ , $\{\lambda_i\}_{i=1}^n$ . // Step 1: Compute spectrum +2 $S = \text{EigApprox}(\{\lambda_i\}_{i=1}^n)$ // Step 2: Estimate smoothed spectral density +3 $G = \text{Glue}(S)$ // Step 3: Estimate glue function +4 $m(z) = \text{Stieltjes}(S, G)$ . // Step 4: Compute the Stieltjes transform +5 foreach $x \in \mathcal{X}$ do +6 Find $z_x \in \mathbb{C}^-$ such that $x + i\delta = z_x - \frac{n_s}{m(z_x)}$ . // Step 5: Solve characteristics +7 return $\left\{\frac{1}{\pi} \Im m(z_x) \frac{n_s}{n}\right\}_{x \in \mathcal{X}}$ . // Step 6: Return the estimated spectral density + +The same holds for $e_{n+1}^{\mathsf{T}}(\mathbf{A}_{n+1} - z\mathbf{I})^{-2}e_{n+1}$ . This is effectively a Hutchinson trace estimation procedure (Bekas et al., 2007), and we note that the variance of the Hutchinson estimators become small for large $n$ as well (Roosta-Khorasani & Ascher, 2015). Assuming this and noting that $\mathrm{tr}(\mathbf{A}_{n+1} - z\mathbf{I})^{-2} = \frac{d}{dz}\mathrm{tr}(\mathbf{A}_{n+1} - z\mathbf{I})^{-1}$ , normalizing by $n + 1$ we deduce the difference equation + +$$ +m _ {\mathbf {A} _ {n + 1}} (z) - m _ {\mathbf {A} _ {n}} (z) = n ^ {- 1} \left(- m _ {\mathbf {A} _ {n}} (z) + m _ {\mathbf {A} _ {n}} ^ {- 1} (z) m _ {\mathbf {A} _ {n}} ^ {\prime} (z)\right) + \mathcal {O} \left(n ^ {- 2}\right). \tag {4} +$$ + +For $n$ large, provided $e^t \approx n$ for $t > 0$ so that $\frac{\partial}{\partial t} m(e^t, z) \approx n[m_{\mathbf{A}_{n+1}}(z) - m_{\mathbf{A}_n}(z)]$ , (4) is well approximated by the differential equation for the Stieltjes transform $m_{\mathbf{A}}(t, z)$ given by (1). + +# 4 Approximating the Stieltjes Transform + +In principle, Propositions 2 and 3 suggest a straightforward approach to estimating the Stieltjes transform under free decompression, but these results disguise two main practical challenges: + +1. The Stieltjes transform is singular on the support of the spectral density, requiring analytic continuation in a way that does not disrupt the complex transport flow of (1). Without imposing additional properties, numerical analytic continuation is notoriously ill-posed (Trefethen, 2023). +2. The second challenge is accurately approximating the initial Stieltjes transform $m(z)$ from the submatrix eigenvalues. This is a well-studied problem, but as we will see, the nature of the PDE (1) necessitates accomplishing this task to a much higher degree of precision than is typically required. + +# 4.1 Analytic Continuation + +The Stieltjes transform $m$ is typically defined on the upper half-plane $\mathbb{C}^+ \coloneqq \{z : \Im(z) > 0\}$ and extended to the lower half-plane $\mathbb{C}^-$ via the Schwarz reflection principle, $m(\overline{z}) = \overline{m(z)}$ . This defines the principal branch, which exhibits a branch cut along the real axis on $I \coloneqq \operatorname{supp}(\rho)$ . However, the curves $\phi(t,z)$ traced by the PDE dynamics descend from $\mathbb{C}^+$ into $\mathbb{C}^-$ , crossing the real axis precisely through $I$ , where the principal branch becomes discontinuous. This is formalized in the following proposition. + +Proposition 5. Let the curve $t \mapsto \phi(t, z)$ be the solution of characteristic curve given in Proposition 3. Then, for each $z \in \mathbb{C}^+$ , there exists $t^* > 0$ such that $\Im \phi(t^*, z) = 0$ . In particular, $\Re \phi(t^*, z) \in I$ . + +Proposition 5 is proven in Appendix C.1, and this crossing necessitates defining a new branch of the transform—equal to the principal branch in $\mathbb{C}^+$ —that extends analytically into $\mathbb{C}^-$ and remains holomorphic across the interior $I^\circ$ . The existence of such a continuation is guaranteed by the principle of analytic continuation: since $m$ is holomorphic in $\mathbb{C}^+$ and has no singularities on $I$ , it can be extended across the cut into $\mathbb{C}^-$ . We refer to this extension as the secondary branch. + +Figure 2 illustrates this continuation for the Marchenko-Pastur law, showing both branches. Multiple approaches were considered to construct the secondary branch, and are discussed in Appendix D. The most successful approach, and the one considered in experiments to follow, introduces a gluing function $G(z)$ defined in (C.5), which compensates for the discontinuity across $I^{\circ}$ . In Appendix C, we show that this function can be accurately approximated by a low-degree rational function (a Padé approximant). When the analytic form of the Stieltjes transform is known, this construction is exact, as shown in Table B.3. In empirical settings—when the density is estimated from the eigenvalues of a submatrix—the same approximation still enables accurate continuation. + +In the sequel, all references to the Stieltjes transform $m$ implicitly refer to this secondary, analytically continued branch. + +# 4.2 Approximating the Empirical Spectral Density + +In light of Proposition 5, free decompression cannot be applied—even in a weak sense—to the empirical Stieltjes transform, $\hat{m}_{\mathbf{A}_n}(z) = \sum_{i=1}^{n} \frac{1}{\lambda_i - z}$ . Instead, we must work with the Stieltjes transform of a smoothed approximation to the empirical spectral distribution. For the synthetic examples we consider, the spectral densities of interest exhibit square-root behavior at the edges of their support, as shown in Table B.1, making Chebyshev polynomials a natural basis for approximation. + +However, for real datasets, which may exhibit more complex or irregular spectral behavior, it is often preferable to use the more flexible class of Jacobi polynomials. These introduce two tunable hyperparameters $(\alpha, \beta)$ and recover Chebyshev polynomials when $(\alpha, \beta) = \left(\frac{1}{2}, \frac{1}{2}\right)$ . To fit these polynomials, we found that Galerkin projections of kernel density estimates were often necessary, using Gaussian or Beta kernels. Once the spectral density is approximated in the Jacobi basis, the Stieltjes transform is computed directly using Gauss-Jacobi quadrature (Shen et al., 2011, Section 3.2.2). Details of these procedures and a full algorithmic description of our implementation can be found in Appendix D. Pseudocode for the full free decompression method is provided in Algorithm 1. + +# 5 Numerical Examples + +We now present numerical examples that demonstrate the utility of our free decompression method. To begin, we consider a number of synthetic examples for random matrices whose spectral densities and their corresponding Stieltjes transforms have known analytic expressions. We then consider two evaluations on real-world large-scale datasets: Facebook Page/Page network (Leskovec & Krevl, 2014; Rozemberczki et al., 2021); and the empirical Neural Tangent Kernel (NTK) corresponding to ResNet50 (He et al., 2016) trained on CIFAR10 (Krizhevsky, 2009). All experiments were conducted on a consumer-grade device with an AMD Ryzen®7 5800X processor, NVIDIA RTX 3080, and 64GB RAM. Hyperparameters used in each experiment are listed in Appendix H. + +![](images/f7928e63bc804bcb216880439b56f15f88fce0eab309ee416852135b17a4a426.jpg) +Figure 3: (a) Empirical spectral densities (dashed) of the symmetrically normalized Laplacian matrix of the SNAP Facebook dataset, compared against estimates obtained via free decompression from an initial submatrix of size $2^{12}$ (solid). (b) Empirical spectral densities (dashed) of submatrices of the log-NTK matrix of size 50,000, computed from the CIFAR-10 dataset using a ResNet-50 model, and estimates obtained via free decompression from an initial submatrix of size $2^{10}$ (solid). + +![](images/ab22c42d1d5f002614b11fb3ecc09bb3138b5c7e510d741e2476e41ba9e101d6.jpg) + +Synthetic Experiments: We consider five families of distributions that commonly arise as spectral distributions in random matrix theory: the Wigner (1955) semicircle law (free Gaussian); Marčenko & Pastur (1967) (free Poisson); Kesten (1959) and McKay (1981) (free binomial); Wachter (1978) (free Jacobi); and the general free-Meixner family (Saitoh & Yoshida, 2001; Anshelevich, 2008) (free analogs of the classical Meixner (1934) laws). In Table B.1, we list, for each law, its absolutely continuous density $\rho(x)$ supported on $[\lambda_{-},\lambda_{+}]$ , and any point masses. Immediately following, Table B.2 summarizes the simplest random-matrix or combinatorial constructions realizing these laws in practice. Figure 1 depicts our free decompression procedure for the Marchenko-Pastur case. We sampled $\mathbf{X}\in \mathbb{R}^{n\times d}$ , $n = 1000$ , $d = 50,000$ , with each $X_{ij}\sim \mathcal{N}(0,1)$ . The left panel shows the empirical density of $\mathbf{A} = \frac{1}{d}\mathbf{X}\mathbf{X}^{\intercal}$ as well as our density approximation using Beta kernels projected onto 50-degree Chebyshev polynomials. The center panel shows a range of densities estimated by free decompression. The right panel compares our freely decompressed approximation of the spectral distribution of $\mathbf{A}\in \mathbb{R}^{n\times n}$ , $n = 32,000$ , with the known exact Marchenko-Pastur law with ratio $\lambda = \frac{32}{50}$ . The principal and secondary branches are compared in Figure 2. The remaining models are treated in Appendix B.1, with comparison of the expected log-determinants and distributional distances. + +SNAP Facebook Dataset: We use the publicly available Facebook Page-Page network from SNAP (Leskovec & Krevl, 2014; Rozemberczki et al., 2021), where nodes represent verified Facebook pages (e.g., politicians, musicians) and edges indicate mutual likes. This graph is undirected with 22,470 nodes and 171,002 edges. We study the eigenvalue distribution of the symmetrically normalized Laplacian matrix of the graph (see e.g., (Mahoney, 2016, p. 16, Definition 4)), which yields a compact spectrum in [0, 2]. To avoid artificial spikes caused by degree-1 leaf nodes (which concentrate eigenvalues at $\lambda = 1$ ), we apply a small Erdős-Rényi perturbation graph $\mathcal{G}(n,p)$ at the connectivity regime where $p = \frac{1}{n}\log(n)$ (Huang & Landon, 2020) to the adjacency matrix of the graph before constructing the Laplacian. Comparison of the empirical spectral densities of randomly sampled submatrices with their free decompression approximations is shown in the left panel of Figure 3. + +Neural Tangent Kernel: Finally, we consider the spectrum of an empirical NTK derived from a ResNet50 network trained on CIFAR10. The NTK is a well-studied object in machine learning (Jacot et al., 2018; Novak et al., 2022); see Appendix G for background. This example is particularly challenging where naive approaches fail. The empirical NTK is known to be highly ill-conditioned for well-trained multi-class models (Ameli et al., 2025). When the NTK is constructed in blockwise fashion, classification models with $c$ classes often exhibit a low-rank structure, with approximately $1/c$ of the eigenvalues being extremely small. To address this, we project the NTK matrix onto its non-null eigenspace, removing the near-zero eigenvalues, and then reconstruct a Hermitian matrix in the reduced space. Subsamples are drawn from this reduced matrix. Specifically, we took $\hat{\mathbf{A}}$ to be the full $55,560 \times 55,560$ matrix (corresponding to 5556 images), and reduced this to a $50,000 \times 50,000$ matrix $\mathbf{A}$ after removing the null component. Submatrices of dimension 1024, 2048, 4096, 8192, 16,382, + +Table 2: Comparison of process time for direct computation of the spectral density versus free decompression (FD) on the NTK dataset. In the FD column, the first term is the time to compute eigenvalues of the initial $2^{10}$ -sized submatrix (common to all rows) and the second term is the FD step. Metrics (columns 4-8) compare the true spectral density with the FD approximation, and are reported as distributional distances (total variation, Jensen-Shannon, Kolmogorov-Smirnov) and relative errors of the first and second moments, all multiplied by 100 to give percentages. Values are the mean over 20 randomly sampled initial submatrices with standard deviations in parentheses. + +
Size nsProcess Time (sec)Distributional Distances (×100)Moments Rel. Error (×100)
DirectFD (ours)TVJSKSΔμ1/μ1Δμ2/μ2
2103.63.6 + 0.00.00% (±0.00)0.00% (±0.00)0.00% (±0.00)0.00% (±0.00)0.00% (±0.00)
21110.23.6 + 0.61.72% (±0.37)7.60% (±1.99)0.48% (±0.10)0.05% (±0.03)0.09% (±0.04)
21250.93.6 + 0.62.06% (±0.35)4.67% (±0.67)0.70% (±0.12)0.01% (±0.01)0.02% (±0.02)
213358.93.6 + 0.63.24% (±0.56)6.30% (±0.51)1.18% (±0.22)0.01% (±0.00)0.02% (±0.01)
2142820.23.6 + 0.74.33% (±0.87)7.55% (±0.80)1.76% (±0.48)0.01% (±0.01)0.03% (±0.01)
21520451.23.6 + 0.85.16% (±1.18)7.96% (±1.11)2.51% (±0.84)0.02% (±0.01)0.05% (±0.02)
50K67331.13.6 + 0.85.94% (±1.48)8.33% (±1.41)3.02% (±1.14)0.17% (±0.03)0.49% (±0.06)
+ +and 32,768 were then sampled, and their empirical spectral distributions are shown as dashed lines in the right panel of Figure 3. Free decompression approximations based on the Stieltjes transform of the $1024 \times 1024$ submatrix (see Figure G.1) are shown as solid lines. Table 2 reports the corresponding process times and errors in total variation (TV) distance, Jensen-Shannon (JS) distance, Kolmogorov-Smirnov (KS) distance, and the relative error in the first and second moments. + +# 6 Conclusion + +In modern settings, it is becoming increasingly common that the spectrum information of a matrix is required, even when the matrix is too large to fit into memory, or is otherwise unavailable. These matrices comprise the category of impalpable matrices, offering unique challenges to numerical linear algebra practitioners. We have outlined free decompression, a novel method of approximating the spectral density of a large impalpable Hermitian matrix from the spectrum of a randomly sampled submatrix. Our initial implementation of this method shows promise as a tool to extract accurate information about the spectrum of extremely large matrices that would otherwise be unavailable. We hope that this work motivates other researchers to further develop these and similar tools to further unlock information in impalpable matrices that is otherwise out of reach. + +Limitations. While our method imposes few restrictions on the large matrix of interest, the quality of approximation depends strongly on asymptotic estimates of the spectral density of $\mathbf{A}_n$ . While promising for future research, performing free decompression accurately remains highly challenging, with Proposition 4 highlighting the need for accurate spectral density estimation and analytic continuation. Without a general approach to estimate and operate on the Stieltjes transform on log-scale, that is, estimating $z \mapsto m(e^z)$ , highly ill-conditioned matrices still remain out of reach. At present, the implementation of our method is sensitive to the quality of approximation of the gluing function; and, while our approach works flawlessly in simple scenarios, it becomes more delicate with real data. Nevertheless, we stress that despite these limitations, no competing alternative exists for estimating the spectral density of impalpable matrices. + +# Acknowledgments and Disclosure of Funding + +LH is supported by the Australian Research Council through a Discovery Early Career Researcher Award (DE240100144). MWM acknowledges partial support from DARPA, DOE, NSF, and ONR. 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Uncertainty quantification with the empirical neural tangent kernel. arXiv preprint arXiv:2502.02870. + +# Appendices + +# Contents + +A Nomenclature 15 +B Benchmark Matrix Ensembles 15 + +B.1 Overview of Random Matrix Ensembles 15 +B.2 Free Decompression of Closed-Form Benchmark Distributions 18 + +C Considerations for Solving the PDE Using Characteristics 20 + +C.1 Challenge: Crossing the Cut 20 +C.2 Holomorphic Continuation of Stieltjes Transform 21 + +D Methods of Estimating Stieltjes Transform 23 + +D.1 Estimating Stieltjes Transform using Lanczos Method 24 +D.2 Estimating Stieltjes Transform using Jacobi Polynomials 25 +D.3 Estimating Stieltjes Transform using Chebyshev Polynomials 28 + +E Proofs of Propositions 1 and 2 30 +F Stability Analysis 31 +G Background Material for the Neural Tangent Kernel 33 +H Implementation and Reproducibility Guide 34 + +# Appendix A Nomenclature + +We use boldface lowercase letters for vectors, boldface upper case letters for matrices, and normal face letters for scalars, including the components of vectors and matrices. Table A.1 summarizes the main symbols and notations used throughout the paper, organized by context. + +# Appendix B Benchmark Matrix Ensembles + +We evaluate our method on a suite of benchmark matrix ensembles with known limiting spectral distributions. Appendix B.1 reviews the underlying matrix models and associated laws, while Appendix B.2 presents decompression results on these closed-form distributions. + +# B.1 Overview of Random Matrix Ensembles + +We review several canonical spectral laws arising from random matrix and combinatorial models, which serve as ground-truth distributions in our benchmarks. Concretely, we consider the Wigner (1955) semicircle law (free Gaussian), Marčenko & Pastur (1967) (free Poisson), Kesten (1959) and McKay (1981) (free binomial), Wachter (1978) (free Jacobi), and the general free-Meixner family (Saitoh & Yoshida, 2001; Anshelevich, 2008) (free analogs of the classical Meixner (1934) laws). + +Table A.1: Common notations used throughout the manuscript. + +
ContextSymbolDescription
Free ProbabilityρAbsolutely continuous empirical spectral density, R → R+
ICompact support interval of density ρ, I = [λ-, λ+] ⊆ R
mStieltjes transform of density, m(z) = ∫R ρ(y)/y-z dy, C \ I → C
H[ρ]Hilbert transform2 of density, H[ρ](x) = p.v. ∫R ρ(y)/y-x dy, R → R
RVoiculescu's R-transform, R(z) = ω(-z) - 1/z with ω(m(z)) = z
Free DecompressionnSize of the target (large) matrix
nsSize of the sampled (small) sub-matrix, ns < n
tDecompression scale, t = log(n/ns)
ρ0, m0Density and Stieltjes transform at t = 0
NotationC±Upper (lower) half complex plane
R, ζReal and imaginary part of complex variable
p.v.Cauchy principal value
+ +In Table B.1, we list, for each law, its absolutely continuous density $\rho(x)$ supported on $I = [\lambda_{-}, \lambda_{+}]$ , and any point masses. Immediately following, Table B.2 summarizes the simplest random-matrix or combinatorial constructions realizing these laws in practice. + +Table B.1: Spectral laws used in our analytic benchmarks, along with their free-probability analogs, absolutely continuous density functions $\rho(x)$ , compact support intervals $[\lambda_{-}, \lambda_{+}]$ , and any point-mass atoms. + +
DistributionFree Corresp.Abs. Cont. Density ρ(x)Support λ±Number of Atoms
Wigner semicircleFree Gaussian2√r2-x2/πr2±rNone
Marchenko-PasturFree Poisson√(λ+ - x)(x - λ-) / 2πλx(1 ± √λ)2(1 - 1/λ)δ(x) if λ > 1
Kesten-McKayFree Binomiald√4(d-1)-x2/2π(d2-x2)±2√d-1None
WachterFree Jacobi(a+b)√(λ+ - x)(x - λ-) / 2πx(1-x)(√b ± √a(a+b-1)/a+b)2x = 0, 1
MeixnerFree Meixnerc√4b-(x-a)2/2π((1-c)x2+acx+bc2)a ± 2√bAt most two
+ +All of the above distributions arise—up to affine rescaling—from the free Meixner class of measures (Anshelevich, 2007), whose orthogonal-polynomial recursion coefficients stabilize after the first level. Equivalently, the associated Jacobi matrix takes the bordered-Toeplitz form + +$$ +\mathbf {J} = \left[ \begin{array}{c c c c c} \alpha_ {0} & \beta_ {0} & & & \\ \hline \beta_ {0} & \alpha_ {1} & \beta_ {1} & & \\ & \beta_ {1} & \alpha_ {1} & \beta_ {1} & \\ & & \ddots & \ddots & \ddots \end{array} \right], \quad \alpha_ {n} = \alpha_ {1}, \beta_ {n} = \beta_ {1} \quad (n \geq 1). +$$ + +See Dubbs & Edelman (2015, Table 2) for the specific $(\alpha_{1},\beta_{1})$ giving each law. In particular, setting $\alpha_0 = 0$ , $\beta_0 = 1$ yields the standard zero-mean, unit-variance normalization. + +Table B.2: Empirical realizations of the spectral laws listed in Table B.1. The matrix $\mathbf{X}$ is assumed to have i.i.d. standard normal entries. + +
DistributionMatrix or Combinatorial ModelParameters
Wigner semicircleGaussian orthogonal ensemble √2(X + X*) +Adjacency matrix of Erdős–Rényi graph G(n, p)r = 2√n +pn = O(log(n))
Marchenko–PasturSample covariance (Wishart) 1/d XX*, X ∈ Rn×dλ = n/d
Kesten–McKayHaar-orthogonal Hermitian sum ∑i=1k(Oi + OiT) +Projection model d PODOT P (Longoria & Mingo, 2023) +Adjacency matrix of a random d-regular graphd = 2k +d ≥ 2 +d ≥ 2
WachterGeneralized eigenvalues of (S1, S1 + S2), Si = 1/di Xi Xiaa = d1/n, b = d2/n
Arises in MANOVA problems
MeixnerBordered Toeplitz tridiagonal with Jacobi coefficients α1, β1 +Block–Gaussian ensembles (Lenczewski, 2015)a = α1, b = β1 - 1
+ +Since the Jacobi parameters in the above become constant beyond the second level, the continued-fraction expansion of the Stieltjes transform + +$$ +m (z) = \frac {1}{z - \alpha_ {0} - \frac {\beta_ {0} ^ {2}}{z - \alpha_ {1} - \frac {\beta_ {1} ^ {2}}{z - \alpha_ {1} - \frac {\beta_ {1} ^ {2}}{\ddots}}}}, +$$ + +becomes eventually periodic, exhibiting a repeating tail that we encode by + +$$ +m (z) = \frac {1}{z - \alpha_ {0} - \beta_ {0} ^ {2} T}, \quad \text {w h e r e} \quad T = \frac {1}{z - \alpha_ {1} - \beta_ {1} ^ {2} T}. +$$ + +Eliminating $T$ yields the quadratic equation + +$$ +Q (z) m (z) ^ {2} - P (z) m (z) + 1 = 0, +$$ + +where + +$$ +P (z) = 2 (z - \alpha_ {0}) - \frac {\beta_ {0} ^ {2}}{\beta_ {1} ^ {2}} (z - \alpha_ {1}), +$$ + +$$ +Q (z) = \left(z - \alpha_ {0}\right) ^ {2} - \frac {\beta_ {0} ^ {2}}{\beta_ {1} ^ {2}} \left(z - \alpha_ {0}\right) \left(z - \alpha_ {1}\right) + \frac {\beta_ {0} ^ {4}}{\beta_ {1} ^ {2}}. +$$ + +Note that $Q(z)$ is at most quadratic (degenerating to linear when $\beta_0^2 = \beta_1^2$ ), and $P$ is at most linear. Solving for $m(z)$ yields + +$$ +m (z) = \frac {P (z) + \sqrt {P (z) ^ {2} - 4 Q (z)}}{2 Q (z)}, \tag {B.1} +$$ + +where the branch of the square root is chosen such that $m(z)$ defines a Herglotz map $\mathbb{C}^{+}\to \mathbb{C}^{+}$ . The corresponding Hilbert transform $\mathcal{H}[\rho ](x) = \Re m(x + i0)$ inside the support interval $I = [\lambda_{-},\lambda_{+}]$ is thus the rational function + +$$ +\mathcal {H} [ \rho ] (x) = \frac {P (x)}{2 Q (x)}, \quad x \in I. \tag {B.2} +$$ + +Table B.3 lists these polynomials $P, Q$ and the corresponding free cumulant $R$ -transforms for every distribution in Table B.1. + +Table B.3: Stieltjes, Hilbert, and $R$ -transforms for the spectral laws listed in Table B.1. The Stieltjes and Hilbert transforms are constructed from the polynomials $P(z)$ and $Q(z)$ according to (B.1) and (B.2). + +
DistributionStieltjes and Hilbert TransformsR-Transform
P(z)Q(z)
Wigner semicircle-zr2/4r2/4z
Marchenko-Pastur1 - λ - zλz1/1 - λz
Kesten-McKay(2 - d)z/d - 1d2 - z2/d - 1-d + d√1 + 4z2/2z
Wachtera - 1 - (a + b - 2)z/a + b - 1z(1 - z)/a + b - 1-(a + b) + z + √(a + b)2 + 2(a - b)z + z2/2z
Meixner(c - 2)z - ac(1 - c)z2 + acz + bc2(c/1 - c)1 - az + √(1 - az)2 - 4b(1 - c)z2/2z
+ +![](images/4b5389e39605eba2d5aec397e73fe889d06f39d6b79aece62270f93d8718c381.jpg) +Figure B.1: (a) Estimation of the Wigner semi-circle law for $\mathbf{A} = (\mathbf{X} + \mathbf{X}^{\mathsf{T}}) / \sqrt{2}$ , where $\mathbf{X}$ is a $n\times n$ matrix with standard normal entries for $n = 1000$ ; (b) approximation of densities for various values of $n$ via free decompression (Algorithm 1); and (c) comparison of our decompressed approximation of $n = 32,000$ with the known limiting analytic density. + +![](images/ef37aa51f0d37875bc389865f0e7ce910844b0f497b57c9912867913858f515b.jpg) + +![](images/cb9337f509d6e22f49ac6c9385d5a3c4423787fab6e439ad93c0569525777ec0.jpg) + +# B.2 Free Decompression of Closed-Form Benchmark Distributions + +In addition to the Marchenko-Pastur case discussed earlier (Figure 1), free decompression was tested on a matrices with Wigner, Kesten-McKay, Wachter, and Meixner limiting spectral distributions, as shown in Figures B.1 to B.4, respectively. To identify the Meixner distribution under free decompression, we make use of the $R$ -transform. Let $R_{a,b,c}(z)$ denote the $R$ -transform for the Meixner distribution with parameters $a, b, c$ as given in Table B.3. A few calculations show that for any $\alpha > 0$ , it holds $R_{a,b,c}(\alpha z) = R_{a(\alpha),b(\alpha),c(\alpha)}(z)$ where + +$$ +a (\alpha) = a \alpha , \qquad b (\alpha) = \frac {b \alpha^ {2} (1 - c (\alpha))}{1 - c}, \qquad c (\alpha) = \frac {c}{c + \alpha (1 - c)}. +$$ + +Consequently, performing free decompression on a Meixner distribution with parameters $a,b,c$ by a factor $\alpha = n / n_{s}$ will yield a Meixner distribution with parameters $a(n / n_s),b(n / n_s)$ and $c(n / n_{s})$ . + +For the Marchenko-Pastur density we also compared the spectral samples of our free decompression estimates with those taken from the analytic density, in order to measure sensitivity to the initial sample that was decompressed. We used three measures of accuracy: total variation, Jensen-Shannon distance, and log-determinants. To this end, free decompression was performed on ten different matrices of size $n = 1000$ , each with $d = 50,000$ degrees of freedom, to obtain approximations of the spectral density corresponding to 32,000 dimensional expansions. Eigenvalues were then sampled from the corresponding distributions via quasi-Monte Carlo, and compared to samples from the analytic density with aspect ratio $\lambda = \frac{32}{50}$ . The average distance in total variation was found to be $0.2\%$ , while the average Jensen-Shannon distance was $1.867\%$ . The mean log-determinant was + +![](images/9d3c21b5f841e84f8db4b5ac6a9b01252aec792502488b20a157408481074595.jpg) +Figure B.2: Estimation of the Kesten-McKay law from the model $\mathbf{A} = \sum_{i=1}^{k} (\mathbf{O}_i + \mathbf{O}_i^\intercal)$ (Longoria & Mingo, 2023, Section 5), with $d = 2k = 2$ and size $n = 32,000$ , where $\mathbf{O}_i$ are Haar-orthogonal matrices generated via Mezzadri (2007). (a) Spectral density from a subsample of size $n_s = 1000$ . (b) Densities for various $n$ via free decompression (Algorithm 1). (c) Comparison of decompressed and exact density at $n = 32,000$ . + +![](images/b38af28af493522c0c8e9d800459b2c2e55208c8d43faddb1158f59fb900553e.jpg) + +![](images/159f2577a40b00f9389c7cb8502949deb5c01fd769dc08cfcd05f4cfc214446d.jpg) + +![](images/0bc59f55fcc6ad401e975e9b2bf962a94caca718a093af0649828a03ef2e3d4f.jpg) +Figure B.3: (a) Estimation of the Wachter distribution with ratios $a = 80$ , $b = 50$ , from eigenvalues of Wishart matrices with $d_{1} = 80,000$ and $d_{2} = 50,000$ degrees of freedom, respectively, and size $n = 1000$ ; (b) approximation of densities for various values of $n$ via free decompression (Algorithm 1); and (c) comparison of our decompressed approximation of $n = 32,000$ with the known limiting analytic density for $a = \frac{80}{32}$ , $b = \frac{50}{32}$ . + +![](images/a6f5f6f0aaaf085c7e42af9055b6e1097419c84cee4d8d850437cf0952881322.jpg) + +![](images/a8696afc6d42b5953cb731fb980f95130bc0394f6a1a6f21fd22547a987f476d.jpg) + +![](images/6f11b13b44d6a909c5247161f46513f2ad9d602a3cc7e8b7cc571036fe25a379.jpg) +Figure B.4: (a) Estimation of the Meixner distribution with $a = 0.1$ , $b = 4$ , $c = 0.6$ from $n = 1000$ quasi-Monte Carlo samples; (b) approximation of densities for various values of $n$ via free decompression (Algorithm 1); and (c) comparison of our decompressed approximation of $n = 32,000$ with the known analytic density given by the Meixner distribution with $a \approx 0.566$ , $b \approx 253$ , $c \approx 0.210$ . + +![](images/08b791132019cd14bf34b728f8670f127979934e87a634ee1fb4ac57be5cf8aa.jpg) + +![](images/178de108584b8d0b710297824947afd1763142658b27c6f78940e600ff76bd2a.jpg) + +computed to be $-13514.88$ with a standard deviation of 1.503, versus the baseline of $-13538.94$ . This corresponds to an absolute relative error of $1.78\%$ with standard error of $0.01\%$ . + +# Appendix C Considerations for Solving the PDE Using Characteristics + +Our task is to compute $m(t,z)$ for a grid of points $z\in \mathbb{C}^{+}$ and times $t\in [0,\tau ]$ , given the initial condition $m(0,z) = m_0(z)$ . By Proposition 3, one has the explicit representation $m(t,z) = m_0(z_0)e^{-t}$ where the "initial label" $z_{0}$ satisfies the characteristic relation + +$$ +z = z _ {0} - \frac {e ^ {t} - 1}{m _ {0} (z _ {0})}. \tag {C.1} +$$ + +Equivalently, each characteristic is a straight line emanating from $z_0$ with direction vector $-m_0(z_0)^{-1}$ . + +To evaluate $m(t,z)$ at a given $(t,z)$ , one must solve the nonlinear equation (C.1) for the unknown initial point $z_0 = \phi(t,z)$ . Although this root-finding is numerically tractable, it raises two interlocking issues. First, the map $t \mapsto \phi(t,z)$ departs the upper half-plane and crosses the real axis through the spectral support $I \coloneqq \mathrm{supp}(\rho_0)$ , where $m_0$ is discontinuous. Second, evaluating $m_0$ on the lower half-plane requires a holomorphic extension of $m_0$ across the cut $I$ . + +In the remainder of this section, we address these points in two stages. In Appendix C.1, we prove that every trajectory $\phi(t,z)$ indeed intersects $\mathbb{R}$ precisely on $I$ . In Appendix C.2, we construct a second-sheet continuation of $m_0$ , holomorphic across $I$ , via an additive Riemann-Hilbert (gluing) ansatz. + +# C.1 Challenge: Crossing the Cut + +Consider a fixed target point $z \in \mathbb{C}^+$ . We seek to determine the initial points $z_0 = \phi(t, z)$ that, under the flow of the characteristic equation (C.1), pass through $z$ at time $t$ . Define the curve $\mathcal{C}: t \mapsto z_0$ by $z_0(t) = \phi(t, z)$ , given implicitly as the solution of (C.1) for fixed $z$ . This curve traces the locus of initial conditions $z_0$ whose corresponding characteristic trajectories intersect the point $z$ at time $t$ . + +Unlike the characteristic curves themselves—which are straight lines—this curve $z_0(t)$ is highly nonlinear, beginning at $z_0(0) = z$ . In the following propositions, we show that this curve descends from its initial point $z \in \mathbb{C}^+$ , eventually crossing into the lower half-plane $\mathbb{C}^-$ , and in fact intersects the real axis precisely on the support of $\rho$ . + +Proposition C.1 (Characteristic curve exiting $\mathbb{C}^+$ ). Let $\rho_0$ be an absolutely continuous probability density (with respect to Lebesgue measure) supported on a compact interval $I\subset \mathbb{R}$ . Assume there exist a compact subinterval $I^{\prime}\subset$ int $I$ and a constant $c_{0} > 0$ such that $\rho_0(x)\geq c_0$ for all $x\in I^{\prime}$ . Let $m_0(z)$ denote the Stieltjes transform of $\rho_0$ , which defines a Herglotz map $\mathbb{C}^{+}\to \mathbb{C}^{+}$ . For each $z\in \mathbb{C}^{+}$ and $t > 0$ , define $\phi (t)\coloneqq \phi (t,z)\in \mathbb{C}$ implicitly by + +$$ +z = \phi (t) - \frac {e ^ {t} - 1}{m _ {0} (\phi (t))}. \tag {C.2} +$$ + +Then there exists a finite time $t^* = t^*(z) > 0$ such that $\Im \phi(t) > 0$ for $0 \leq t < t^*$ and $\Im \phi(t^*) = 0$ . + +Proof. Let $a(t) = e^t - 1$ and define $w(z) = -1 / m_0(z)$ , so that $w: \mathbb{C}^+ \to \mathbb{C}^+$ . Fix $z = x + iy \in \mathbb{C}^+$ . Write $\phi(t) = u(t) + iv(t)$ and $w(\phi(t)) = \alpha(t) + i\beta(t)$ with $\beta(t) > 0$ . + +First, we show that when the density $\rho_0$ has no atoms, $\phi (t)$ does not asymptote to the edges of $I$ as $t\to \infty$ . Taking real parts in (C.2) gives + +$$ +x = u (t) + a (t) \alpha (t). +$$ + +If $u(t) \to u^{*} \in \partial I$ and $v(t) \to 0^{+}$ , then by Plemelj-Sokhotski $m_0(u^* + i0) \in \mathbb{R}$ is finite (no atom at $u^{*}$ ), hence $\alpha(t) \to \alpha^{*} := \Re w(u^{*} + i0) = -1 / m_{0}(u^{*} + i0) \neq 0$ , which forces $x = u(t) + a(t)\alpha(t) \to u^{*} + a(t)\alpha^{*}$ , a contradiction since $a(t) \to \infty$ . Thus the trajectory cannot converge to an endpoint while $v(t) > 0$ . This rules out the pathological possibility that the trajectory merely approaches the boundary without ever entering the support. Consequently, we may assume the curve eventually moves inside the interior of $I$ , where the following estimates apply. + +Next, we show $\beta$ is bounded below. By the Plemelj boundary values, for $u\in I$ we have + +$$ +\lim_{v\to 0^{+}}\mathfrak{S}w(u + iv) = \frac{\pi\rho_{0}(u)}{|m_{0}(u + i0)|^{2}}. +$$ + +Since $\rho_0 \geq c_0 > 0$ on $I'$ and $|m_0(u + i0)|$ is continuous on $I'$ , there exist $v_0 > 0$ and $c > 0$ such that $\mathfrak{S}w(u + iv) \geq c$ for all $u \in I'$ and $0 < v \leq v_0$ . Hence, once $u(t) \in I'$ with $v(t) \leq v_0$ , we have $\beta(t) \geq c$ . + +Finally, taking imaginary parts in (C.2) gives + +$$ +v (t) = y - a (t) \beta (t). +$$ + +As $t \to \infty$ , we have $a(t) \to \infty$ while $\beta(t)$ remains bounded below by $c$ whenever $u(t) \in I'$ and $0 < v(t) \leq v_0$ . Thus, $v(t) \to -\infty$ , so by continuity there exists a first time $t^*$ where $v(t^*) = 0$ . + +Proposition C.2 (Crossing on the support). Suppose the hypotheses of Proposition C.1 hold, and define the curve $t \mapsto \phi(t,z) \in \mathbb{C}$ implicitly by (C.2) for a fixed $z \in \mathbb{C}^+$ . Let $t^* > 0$ be the first time such that $\Im \phi(t^*) = 0$ , whose existence is guaranteed by Proposition C.1. Then the real part of the crossing point lies in the support of the density: $\Re \phi(t^*) \in I$ . + +Proof. Let $x^{*} \coloneqq \phi(t^{*})$ be the real-valued point where the curve $\phi$ intersects the real axis. Suppose, for contradiction, that $x^{*} \notin I$ . Since $I$ is compact, there exists an open interval $J \subset \mathbb{R} \setminus I$ containing $x^{*}$ . On such an interval $J$ , the Stieltjes transform $m_{0}(x)$ admits real boundary values. Indeed, by the Plemelj-Sokhotski formula, + +$$ +\lim _ {\epsilon \rightarrow 0 ^ {+}} m _ {0} (x + i \epsilon) = \mathcal {H} [ \rho_ {0} ] (x) + i \pi \rho_ {0} (x), \quad x \in \mathbb {R}, \tag {C.3} +$$ + +and since $\rho_0(x) = 0$ for all $x\in J$ , the imaginary part vanishes and $m_0(x)$ is real-valued on $J$ . In particular, $m_0(x^*)\in \mathbb{R}$ . Substituting $\phi (t^{*}) = x^{*}$ into (C.2), we find that the right-hand side of the equation is real, so $z\in \mathbb{R}$ , contradicting the assumption that $z\in \mathbb{C}^{+}$ . Therefore, the assumption $x^{*}\notin I$ is false, and we conclude $x^{*}\in I$ . + +Since the trajectory $\mathcal{C}$ enters the lower half-plane, we must evaluate the initial field $m_0$ both across the real line at points on $\mathrm{supp}(\rho_0)$ , and inside $\mathbb{C}^-$ . However, by definition, the Stieltjes transform + +$$ +m _ {0} (z) = \int_ {I} \frac {\rho_ {0} (x)}{x - z} \mathrm {d} x, +$$ + +is only defined for $\Im (z) > 0$ , and exhibits a jump discontinuity across the real axis precisely on $\mathrm{supp}(\rho_0)$ . This creates an analytic obstruction: to evaluate $m_0(\phi (t,z))$ for arbitrary $t$ , we require a holomorphic continuation of $m_0$ to the lower half-plane that is smooth across the cut $I$ . This continuation is constructed in the next section. + +# C.2 Holomorphic Continuation of Stieltjes Transform + +Let $m^{+}(z)$ denote the Stieltjes transform corresponding to a density $\rho$ supported on a compact interval $I\subset \mathbb{R}$ (such as the field $m_0(z)$ and the density $\rho_0$ in our case). By definition, this function is a Herglotz map $\mathbb{C}^+\to \mathbb{C}^+$ , holomorphic on $\mathbb{C}^+$ , and admits boundary values on $\mathbb{R}$ via the Sokhotski-Plemelj theorem: + +$$ +\lim _ {\epsilon \to 0 ^ {+}} m ^ {+} (x + i \epsilon) = \mathcal {H} [ \rho ] (x) + i \pi \rho (x). +$$ + +A natural extension of $m^{+}$ to $\mathbb{C}^{-}$ is the Schwarz reflection + +$$ +m ^ {+} (z) = \overline {{m ^ {+} (\bar {z})}}, \quad \Im (z) < 0. +$$ + +This extension is holomorphic on $\mathbb{R} \setminus I$ , but has a discontinuity across $I$ . Specifically, the discontinuity arises from a jump in the imaginary part: + +$$ +\lim _ {\epsilon \rightarrow 0 ^ {+}} \Im (m ^ {+} (x + i \epsilon)) - \Im (m ^ {+} (x - i \epsilon)) = 2 \pi \rho (x). +$$ + +Thus, $\mathfrak{S}(m^{+})$ is discontinuous on $I$ , while it vanishes and remains continuous on $\mathbb{R} \setminus I$ . We also note that the real part, $\Re(m^{+}) = \mathcal{H}[\rho](x)$ , is continuous across the entire real axis. + +Our goal is to construct a second analytic continuation of $m^+$ , denoted $m^-$ , that is holomorphic across the interior of the cut $I$ . That is, $m^-$ agrees with $m^+$ on $\mathbb{C}^+$ and continues analytically to $\mathbb{C}^-$ . + +through $I$ , at the expense of developing a discontinuity across $\mathbb{R} \setminus I$ . This setup forms an additive Riemann-Hilbert problem: find two branches $m^{+}$ and $m^{-}$ such that + +$$ +\lim _ {\epsilon \to 0 ^ {+}} m ^ {+} (x + i \epsilon) = \lim _ {\epsilon \to 0 ^ {+}} m ^ {-} (x - i \epsilon), \quad x \in I. +$$ + +We propose the following structure: + +$$ +m ^ {-} (z) := \left\{ \begin{array}{l l} m ^ {+} (z), & \Im (z) > 0, \\ - m ^ {+} (z) + G (z), & \Im (z) < 0, \end{array} \right. \tag {C.4} +$$ + +where $G(z)$ is a glue function chosen to cancel the jump across $I$ . In particular, we require that + +$$ +\begin{array}{l l} \Im (G (x + i 0)) = 0 & x \in I, \\ \Re (G (x + i 0)) = 2 \mathcal {H} [ \rho ] (x), & x \in I. \end{array} \tag {C.5} +$$ + +The first condition in the above ensures the continuity of the imaginary part on the whole of $\mathbb{R}$ : + +$$ +\lim _ {\epsilon \rightarrow 0 ^ {+}} \Im (m ^ {-} (x + i \epsilon)) - \Im (m ^ {-} (x - i \epsilon)) = \pi \rho (x) - \pi \rho (x) = 0, +$$ + +and the second condition guarantees continuity of the real part across $I$ : + +$$ +\lim _ {\epsilon \to 0 ^ {+}} \Re (m ^ {-} (x + i \epsilon)) - \Re (m ^ {-} (x - i \epsilon)) = 0, +$$ + +while allowing discontinuities on $\mathbb{R} \setminus I$ , which are harmless for our purposes. In this way, $m^{+}$ and $m^{-}$ form two complementary Riemann sheets: $m^{+}$ is holomorphic in $\mathbb{C}^{+} \cup (\mathbb{R} \setminus I)$ but discontinuous across $I$ , while $m^{-}$ is holomorphic in $\mathbb{C}^{-} \cup I$ , with a branch cut on $\mathbb{R} \setminus I$ . + +To characterize $G$ , note that $G(z) = m^{-}(z) + m^{+}(z)$ , and it must be analytic in $\mathbb{C}^{-}$ . On the real axis inside the support, this glue function satisfies $\Re G(x) = 2\mathcal{H}[\rho](x)$ , and thus inherits key structural properties from the Hilbert transform. In order to design an efficient functional form for $G$ , we first study the behavior of $\mathcal{H}[\rho]$ itself in Proposition C.3, which guides the minimal rational approximation ansatz in the next step. + +Proposition C.3 (Properties of the Hilbert transform). Let $\rho$ be a bounded, piecewise- $C^1$ density supported on a connected compact interval $I = [\lambda_{-},\lambda_{+}] \subset \mathbb{R}$ . Define the Hilbert transform + +$$ +\mathcal {H} [ \rho ] (x) = \mathrm {p . v .} \int_ {\lambda_ {-}} ^ {\lambda_ {+}} \frac {\rho (t)}{t - x} \mathrm {d} t, +$$ + +where p.v. denotes the Cauchy principal value. Then $\mathcal{H}[\rho]$ and its derivative with respect to $x$ satisfy: + +1. For $x\in (-\infty ,\lambda_{-})$ .. $\mathcal{H}[\rho ](x) > 0$ and $\mathcal{H}[\rho ]'(x) > 0$ +2. For $x \in [\lambda_{-}, \lambda_{+}]$ : the equation $\mathcal{H}[\rho](x) = 0$ has an odd (hence at least one) number of solutions $x^{*} \in (\lambda_{-}, \lambda_{+})$ . +3. For $x\in (\lambda_{+},\infty)$ .. $\mathcal{H}[\rho ](x) < 0$ and $\mathcal{H}[\rho ]^{\prime}(x) > 0$ +4. As $x \to \pm \infty$ , $\mathcal{H}[\rho](x) \sim -1/x$ . + +Proof. Step 1. Outside the support. If $x \notin [\lambda_{-}, \lambda_{+}]$ , differentiation under the integral sign is valid: + +$$ +\mathcal {H} [ \rho ] ^ {\prime} (x) = \int_ {\lambda_ {-}} ^ {\lambda_ {+}} \frac {\rho (t)}{(t - x) ^ {2}} \mathrm {d} t > 0. +$$ + +Hence, $\mathcal{H}[\rho]$ is strictly increasing on $(- \infty, \lambda_{-}) \cup (\lambda_{+}, \infty)$ . A direct sign check of the integrand confirms that $\mathcal{H}[\rho](x) > 0$ for $x < \lambda_{-}$ and $\mathcal{H}[\rho](x) < 0$ for $x > \lambda_{+}$ , establishing items (1) and (3). Furthermore, the asymptotic statement in item (4) follows directly from (King, 2009, Eq. 4.111), which gives $\mathcal{H}[\rho](x) \sim -\left(\int \rho \mathrm{d}x\right)/x$ as $|x| \to \infty$ ; since $\int \rho \mathrm{d}x = 1$ , this is $-1/x$ . + +Step 2. Existence of a zero. From steps (1), we have $\mathcal{H}[\rho] > 0$ as $x \downarrow \lambda_{-}$ , and $\mathcal{H}[\rho] < 0$ as $x \uparrow \lambda_{+}$ . Since $\mathcal{H}[\rho]$ is continuous—guaranteed by the boundedness of $\rho$ —the intermediate value + +Theorem ensures that there exists at least one root $x^{*} \in (\lambda_{-}, \lambda_{+})$ such that $\mathcal{H}[\rho](x^{*}) = 0$ . Moreover, every simple zero flips the sign of $\mathcal{H}[\rho]$ , so starting with positive sign on the left and finishing with negative sign on the right forces an odd number of sign-changes and hence an odd number of interior zeros. + +The properties in Proposition C.3 provide a clear picture of the behavior of the Hilbert transform of a probability density with compact support; see Figure C.1 for an example. + +![](images/98f912a72098598f5d2d02409ff660ff656224d241a47bcd82451a0af37894ba.jpg) +Figure C.1: A schematic example of the behavior of the Hilbert transform as prescribed by Proposition C.3. + +We now seek a practical approximation for the real part of the glue function $G(x)$ on the support interval $I$ . Motivated by the fact that many classical spectral laws—such as the free Meixner family of distributions in Appendix B.1—admit closed-form Hilbert transforms of rational form, we adopt a Padé-type ansatz: + +$$ +G (x) \approx \frac {P (x)}{Q (x)}, \tag {C.6} +$$ + +with $P$ and $Q$ polynomials of minimal degree. In the classical examples of Appendix B.1, this rational structure is not merely an approximation but exact, as shown in (B.2), with $P$ at most linear and $Q$ at most quadratic (see Table B.3). + +Even in more general settings—where the density $\rho$ is empirical or lacks a closed-form expression—the Padé approximant remains attractive as it provides a compact, flexible representation with interpretable parameters. Once we commit to this functional form, we can leverage structural properties of the Hilbert transform to constrain the numerator and denominator more effectively. Proposition C.3 provides key analytic features of $\mathcal{H}[\rho]$ that guide this design. Namely, $P$ should have at least a zero inside $I$ , while $Q$ should have no zeros in $\mathbb{C}^-$ or within $I$ , though it may have zeros in $\mathbb{R} \setminus I$ . Mirroring these features, we parametrize the Padé approximant as + +$$ +G (z) = \frac {P (z)}{Q (z)} = d + c z + \sum_ {j = 1} ^ {q} \frac {r _ {i}}{z - a _ {j}}, +$$ + +which is a Padé approximant of degree $[q + 1/q]$ , or $[q/q]$ if $c = 0$ or $[q - 1/q]$ if $c = d = 0$ . We constrain the parameters to ensure there is at least one zero in $I$ and no pole $a_j$ inside $I$ . + +# Appendix D Methods of Estimating Stieltjes Transform + +Given a probability density $\rho (x)$ supported on an interval $I\subset \mathbb{R}$ , its Stieltjes transform is defined by + +$$ +m (z) := \int_ {\mathbb {R}} \frac {\rho (x)}{x - z} d x, \qquad z \in \mathbb {C} \setminus I. +$$ + +The density $\rho$ can be recovered from the boundary behavior of $m(z)$ via the inversion formula: + +$$ +\rho (x) = \frac{1}{\pi}\lim_{\epsilon \to 0^{+}}\Im m(x + i\epsilon). +$$ + +Accurately estimating $m(z)$ is therefore essential for recovering the underlying spectral density. Given a Hermitian matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ , the Stieltjes transform of its empirical spectral distribution is + +defined by + +$$ +m (z) = \frac {1}{n} \mathrm {t r} \left(\mathbf {A} - z \mathbf {I}\right) ^ {- 1}, \quad z \in \mathbb {C} \setminus I. +$$ + +A direct way to compute $m(z)$ is + +$$ +m (z) = \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {1}{\lambda_ {i} - z}, +$$ + +where $\lambda_{i}$ are the eigenvalues of $\mathbf{A}$ . However, because the characteristic curves must pass through the roots of $m$ (Proposition C.2), applying free decompression to this empirical version of $m$ will not remove its poles. It is necessary to consider approximations of $m$ that correspond to Stieltjes transforms for smooth densities. + +In this section, we describe three numerical strategies for estimating the Stieltjes transform of a Hermitian matrix. The most widely used approach in the literature is based on the Lanczos algorithm (Appendix D.1), which constructs a continued fraction approximation to the resolvent. However, we find that this method often suffers from slow convergence and numerical instability, due in part to its reliance on a random initial vector and the large number of Lanczos steps required to achieve acceptable accuracy. As such, it is often ill-suited for tasks that demand high-precision estimates of $m(z)$ . + +To overcome these limitations, we present two alternative methods. The first, described in Appendix D.2, expands the spectral density in a Jacobi polynomial basis, offering flexibility in modeling various edge behaviors. The second, described in Appendix D.3, specializes this approach to Chebyshev polynomials, for which we derive a closed-form expression for the Stieltjes transform of each basis element. This enables direct and stable evaluation via power series. Together, these methods offer more accurate and robust alternatives to Lanczos-based approaches, especially in regimes where high-resolution spectral information is required. + +# D.1 Estimating Stieltjes Transform using Lanczos Method + +The key idea of the Lanczos method is to approximate $m(z)$ by constructing a rational function—specifically, a continued fraction—that reflects the spectral properties of $\mathbf{A}$ . This is achieved via a three-term recurrence relation satisfied by orthogonal polynomials associated with the spectral measure (Golub & Meurant, 2010; Meurant, 2006). The recurrence is obtained from the Lanczos algorithm, which builds a tridiagonal (Jacobi) matrix capturing the action of $\mathbf{A}$ on a low-dimensional Krylov subspace. In brief, the procedure is as follows. + +Given a normalized starting vector $\pmb{v}_0 \in \mathbb{R}^n$ , $\| \pmb{v} \|_2 = 1$ , the Lanczos algorithm constructs an orthonormal basis $\{\pmb{v}_0, \pmb{v}_1, \dots, \pmb{v}_{p-1}\}$ for the Krylov subspace + +$$ +\mathcal {K} _ {p} (\mathbf {A}, \boldsymbol {v} _ {0}) = \mathrm {s p a n} \left\{\boldsymbol {v} _ {0}, \mathbf {A} \boldsymbol {v} _ {0}, \mathbf {A} ^ {2} \boldsymbol {v} _ {0}, \ldots , \mathbf {A} ^ {p - 1} \boldsymbol {v} _ {0} \right\}, +$$ + +and produces the symmetric triadiagonal Jacobi matrix + +$$ +\mathbf {T} _ {p} := \left[ \begin{array}{c c c c} \alpha_ {0} & \beta_ {1} & & \\ \beta_ {1} & \alpha_ {1} & \beta_ {2} & \\ & \ddots & \ddots , & \ddots \\ & & \beta_ {p - 1} & \alpha_ {p - 1} \end{array} \right], +$$ + +which satisfies + +$$ +\mathbf {A} \mathbf {V} _ {p} \approx \mathbf {V} _ {p} \mathbf {T} _ {p}, +$$ + +where $\mathbf{V}_p\coloneqq [v_0,v_1,\ldots ,v_{p - 1}]\in \mathbb{R}^{n\times p}$ . This identity captures the three-term recurrence relation satisfied by the orthogonal polynomials associated with the spectral measure of $\mathbf{A}$ . The Stieltjes transform $m(z)$ can then be approximated by the first element of the resolvent of $\mathbf{T}_p$ : + +$$ +m _ {p} (z) = \pmb {e} _ {1} ^ {\intercal} (\mathbf {T} _ {p} - z \mathbf {I}) ^ {- 1} \pmb {e} _ {1}. +$$ + +where $e_1 = (1,0,\ldots,0) \in \mathbb{R}^n$ denotes the first standard basis vector. The above expression is equivalent to the continued fraction + +$$ +m _ {p} (z) = \frac {1}{z - \alpha_ {0} - \frac {\beta_ {0} ^ {2}}{z - \alpha_ {1} - \frac {\beta_ {1} ^ {2}}{\ddots - \frac {\beta_ {p - 1} ^ {2}}{z - \alpha_ {p}}}}}. +$$ + +This approximation is the $[0/p]$ Padé approximant of $m(z)$ , and typically exhibits convergence with increasing $p$ , especially when $m(z)$ is analytic away from the real axis (Golub & Meurant, 1997). However, the effectiveness of this approximation in practice depends on several implementation details, which we briefly summarize below. + +The choice of the starting vector $\boldsymbol{v}_0$ is not critical, as any unit-norm vector suffices. In practice, a randomly chosen unit vector often leads to a convergence, while in theory, the approximation $m_p(z)$ is independent of this choice when $p = n$ . However, for larger values of $p$ , numerical stability may require full or partial reorthogonalization during the Lanczos process to maintain orthogonality of the Krylov basis vectors (Meurant, 2006). + +A typical stopping criterion is based on monitoring the difference between successive approximants and terminating when + +$$ +\left| m _ {p} (z) - m _ {p - 1} (z) \right| < \varepsilon , +$$ + +for a fixed test point $z \in \mathbb{C}^+$ ; see (Frommer & Schweitzer, 2016; Ubaru et al., 2017) for detailed analyses of convergence rates and practical recommendations. + +The continued fraction $m_p(z)$ can be evaluated efficiently either through recursive computation or, equivalently, by solving the tridiagonal system $(\mathbf{T}_p - z\mathbf{I})\boldsymbol{x} = \boldsymbol{e}_1$ . The overall computational cost is $\mathcal{O}(pn^2)$ for dense matrices or $\mathcal{O}(p\mathrm{nnz}(\mathbf{A}))$ for sparse matrices (where $\mathrm{nnz}(\mathbf{A})$ is the number of non-zero elements of $\mathbf{A}$ ), plus $\mathcal{O}(p^2)$ for reorthogonalization. If the algorithm is run to completion with $p = n$ and numerical orthogonality is preserved, the approximation yields the exact spectrum of $\mathbf{A}$ , and thus fully recovers $m(z)$ . + +# D.2 Estimating Stieltjes Transform using Jacobi Polynomials + +Jacobi polynomials provide a highly flexible basis for approximating spectral densities, particularly when the distribution exhibits distinctive edge behavior—such as square-root singularities or vanishing derivatives at the boundaries of the support. This family of orthogonal polynomials can represent a wide range of distributions encountered in random matrix theory (e.g., distributions in Appendix B.1) and more general beta-like distributions. + +Let $\rho(x)$ be a probability density with support on the interval $[\lambda_{-}, \lambda_{+}]$ . Through an affine change of variables, we map the domain to the canonical interval $[-1, 1]$ via + +$$ +t (x) := \frac {2 x - (\lambda_ {+} + \lambda_ {-})}{\lambda_ {+} - \lambda_ {-}}. +$$ + +The density is then expanded in the Jacobi polynomial basis: + +$$ +\rho (x) \approx \sum_ {k = 0} ^ {K} \psi_ {k} w ^ {(\alpha , \beta)} (t (x)) P _ {k} ^ {(\alpha , \beta)} (t (x)), \tag {D.1} +$$ + +where $P_{k}^{(\alpha, \beta)}$ is the Jacobi polynomial of degree $k$ and parameters $\alpha, \beta > -1$ , orthogonal with respect to the weight + +$$ +w ^ {(\alpha , \beta)} (t) = (1 - t) ^ {\alpha} (1 + t) ^ {\beta}, \qquad \alpha , \beta > - 1. +$$ + +This expansion is especially effective when $\rho (x)\sim (\lambda_{+} - x)^{\alpha}(x - \lambda_{-})^{\beta}$ near the endpoints, aligning the basis with the singularity structure of $\rho$ , and accelerating the convergence. + +Projection. Given a set of eigenvalues $\{\lambda_i\}_{i=1}^N$ , we first construct a smoothed density estimate $\hat{\rho}(x)$ , e.g., via Gaussian or beta kernel density estimation. Mapping to the variable $t$ , we approximate the expansion coefficients using the orthogonality of Jacobi polynomials: + +$$ +\psi_ {k} = \frac {1}{\| P _ {k} ^ {(\alpha , \beta)} \| ^ {2}} \int_ {- 1} ^ {1} \hat {\rho} (t) P _ {k} ^ {(\alpha , \beta)} (t) d t \approx \frac {\sum_ {j} \hat {\rho} \left(t _ {j}\right) P _ {k} ^ {(\alpha , \beta)} \left(t _ {j}\right) \Delta t}{\| P _ {k} ^ {(\alpha , \beta)} \| ^ {2}}, \tag {D.2} +$$ + +where $\{t_j\}$ are grid points on $[-1, 1]$ and $\| P_k^{(\alpha, \beta)}\|^2$ denotes the squared norm, which admits a closed-form expression + +$$ +\| P _ {k} ^ {(\alpha , \beta)} \| ^ {2} = \frac {2 ^ {\alpha + \beta + 1}}{2 k + \alpha + \beta + 1} \frac {\Gamma (k + \alpha + 1) \Gamma (k + \beta + 1)}{\Gamma (k + 1) \Gamma (k + \alpha + \beta + 1)}. +$$ + +In the above, $\Gamma$ is the Gamma function. + +Stieltjes transform in Jacobi basis. Once the coefficients $\psi_{k}$ are obtained, the Stieltjes transform + +$$ +m (z) = \int_ {\lambda_ {-}} ^ {\lambda_ {+}} \frac {\rho (x)}{x - z} d x, +$$ + +can be approximated via the expansion (D.1). First, by mapping $x$ and $z$ as + +$$ +\begin{array}{l} x (t) := \frac {1}{2} (\lambda_ {+} + \lambda_ {-}) + \frac {1}{2} (\lambda_ {+} - \lambda_ {-}) t, \qquad t \in [ - 1, 1 ], \\ z (u) := \frac {1}{2} (\lambda_ {+} + \lambda_ {-}) + \frac {1}{2} (\lambda_ {+} - \lambda_ {-}) u, \qquad u \in \mathbb {C} \backslash [ - 1, 1 ], \\ \end{array} +$$ + +we arrive at the transformed expression + +$$ +m (u) \approx \frac {2}{\lambda_ {+} - \lambda_ {-}} \sum_ {k = 0} ^ {K} \psi_ {k} \int_ {- 1} ^ {1} \frac {w ^ {(\alpha , \beta)} (t) P _ {k} ^ {(\alpha , \beta)} (t)}{t - u} d t, \quad u \in \mathbb {C} ^ {+} \setminus [ - 1, 1 ]. +$$ + +The inner integral in the above defines the so-called Jacobi function of the second kind, + +$$ +Q _ {k} ^ {(\alpha , \beta)} (u) = \int_ {- 1} ^ {1} \frac {w ^ {(\alpha , \beta)} (t) P _ {k} ^ {(\alpha , \beta)} (t)}{t - u} d t, +$$ + +and the Stieltjes transform becomes + +$$ +m (z (u)) \approx \frac {2}{\lambda_ {+} - \lambda_ {-}} \sum_ {k = 0} ^ {K} \psi_ {k} Q _ {k} ^ {(\alpha , \beta)} (u). +$$ + +Gauss-Jacobi quadrature. In practice, we do not evaluate $Q_{k}^{(\alpha ,\beta)}$ directly via numerical integration. Instead, we compute each term using a dedicated Gauss-Jacobi quadrature rule (Shen et al., 2011, Section 3.2.2) tailored to the weight $w^{(\alpha ,\beta)}$ + +$$ +Q _ {k} ^ {(\alpha , \beta)} (u) \approx \sum_ {i = 1} ^ {n _ {k}} \frac {w _ {i} P _ {k} ^ {(\alpha , \beta)} (t _ {i})}{t _ {i} - u}, +$$ + +where $\{t_i, w_i\}$ are the quadrature nodes and weights for the given weight function. The number of nodes $n_k$ depends on the degree $k$ , typically chosen as $n_k \coloneqq \max(k + 1, n_0)$ . Evaluating each mode $Q_k^{(\alpha, \beta)}(u)$ independently using its own quadrature is critical for numerical stability, especially for large $k$ . This separation of summation and integration avoids artifacts that arise when attempting to quadrature-integrate the full series at once. + +Choosing parameters. Jacobi polynomials provide a flexible basis for modeling spectral densities with various edge behaviors, particularly those of the form $\rho(x) \sim (\lambda_{+} - x)^{\alpha}(x - \lambda_{-})^{\beta}$ . Different choices of $\alpha$ and $\beta$ yield classical orthogonal polynomials: + +![](images/150f49f2d8f4dd03032378f02e3c255cce1c888c23b64d0976c4bbf4017a28dd.jpg) +Figure D.1: Effect of Jackson damping on approximating the Marchenko-Pastur density from the Gram matrix $\frac{1}{d}\mathbf{X}\mathbf{X}^{\intercal}$ , where $\mathbf{X}$ is an $n\times d$ random matrix with $n = 1000$ and $n / d = 0.9$ . Parameters were intentionally selected to amplify Gibbs oscillations for illustrative purposes; typical cases exhibit much milder artifacts. + +- $\alpha = \beta = 0$ recovers Legendre polynomials. +- $\alpha = \beta = \frac{1}{2}$ gives Chebyshev polynomials of the second kind, well-suited for densities with square-root vanishing edges, such as $\rho(x) \sim \sqrt{(\lambda_{+} - x)(x - \lambda_{-})}$ , which are common in free probability models (see Table B.1). +- $\alpha = \beta = -\frac{1}{2}$ yields Chebyshev polynomials of the first kind, appropriate for densities with inverse square-root singularities at the edges. + +This parameterization aligns the orthogonal basis with the singular structure of $\rho(x)$ , often resulting in exponential convergence with respect to the number of terms $K$ . In practice, we find that the method is not highly sensitive to precise values of $\alpha$ and $\beta$ , as long as their qualitative effect on edge behavior is appropriate. + +Gibbs oscillations. Expansions in orthogonal polynomials can exhibit Gibbs oscillations, especially for large truncation orders $K$ . These oscillations can be suppressed by applying damping to the coefficients via a filter $g_{k}$ , resulting in the smoothed expansion: + +$$ +\rho (x) \approx \sum_ {k = 0} ^ {K} g _ {k} \psi_ {k} P _ {k} ^ {(\alpha , \beta)} (x). +$$ + +Several damping schemes exist, including Jackson (1912), Lanczos, Fejer, and Parzen filters. In this work, we use Jackson damping, defined as + +$$ +g _ {k} := \frac {(K - k + 1) \cos \left(\frac {\pi k}{K + 1}\right) + \sin \left(\frac {\pi k}{K + 1}\right) \cot \left(\frac {\pi}{K + 1}\right)}{K + 1}, \quad k = 0, 1, \ldots , K. +$$ + +Figure D.1 shows how Jackson damping improves the stability of the approximation. + +Regularization. To further stabilize the coefficients $\psi_{k}$ , we introduce Tikhonov regularization by modifying (D.2) as + +$$ +\psi_ {k} = \frac {1}{\gamma_ {k} + \| P _ {k} ^ {(\alpha , \beta)} \| ^ {2}} \int_ {- 1} ^ {1} \hat {\rho} (t) P _ {k} ^ {(\alpha , \beta)} (t) d t, +$$ + +with the penalty term $\gamma_{k}$ chosen as + +$$ +\gamma_ {k} := \gamma \left(\frac {k}{K + 1}\right) ^ {2}, +$$ + +where $\gamma > 0$ is a small regularization parameter. This form penalizes higher-order coefficients more strongly, resulting in smoother approximations without significantly biasing the shape or integral of the estimated density. + +Preserving positivity and mass. Finally, to enforce positivity of $\rho(x)$ over its support and ensure $\int \rho(x) \, \mathrm{d}x = 1$ , we perform a secondary constrained optimization over $\psi_k$ , initialized at the projected values, subject to constraints enforcing unit mass and pointwise non-negativity over a fine grid. + +# D.3 Estimating Stieltjes Transform using Chebyshev Polynomials + +Constructing the principal branch of the Stieltjes transform numerically is straightforward: one can estimate $m$ on the real axis by letting $m(x + i0^{+}) = \mathcal{H}[\rho](x) + i\pi \rho(x)$ and estimating the Hilbert transform using FFT techniques. Then, $m$ can be extended to $\mathbb{C}^{+}$ by convolution with the Poisson kernel. The lower-half plane is obtained by the Schwarz reflection principle. However, constructing the secondary branch of the Stieltjes transform numerically is ordinarily very challenging, as it is tantamount to numerical analytic continuation. + +Our primary mechanism for estimating the Stieltjes transform of a density with compact support is the following lemma. Here, we let $j(z) \coloneqq \frac{1}{2} (z + z^{-1})$ denote the Joukowski transform, and let + +$$ +J (z) := z - \sqrt {z ^ {2} - 1} +$$ + +denote the inverse Joukowski transform, where we take the branch of the square root to have positive imaginary part for $z \in \mathbb{C}^+$ . The Chebyshev polynomials of the second kind $U_k(x)$ are defined for $x \in [-1, 1]$ by the recurrence relation + +$$ +U _ {k + 1} (x) = 2 x U _ {k} (x) - U _ {k - 1} (x) +$$ + +from $U_{0}(x) = 1$ and $U_{1}(x) = 2x$ . + +Lemma D.1. The Stieltjes transform of $U_{k}(x)\sqrt{1 - x^{2}}$ on $[-1,1]$ is given by $\pi J(z)^{k + 1}$ for any $k\in \{0\} \cup \mathbb{N}$ . + +Proof. Let $m(z) = \pi J(z)^{k + 1}$ . Since $m:\mathbb{C}^{+}\to \mathbb{C}^{+}$ is analytic, by the Herglotz-Nevanlinna representation theorem, there exist constants $C,D > 0$ and a Borel measure $\mu$ such that + +$$ +m (z) = C + D z + \int \frac {1}{\lambda - z} \mathrm {d} \mu (\lambda). +$$ + +Since $J(i\eta) \to 0$ as $\eta \to \infty$ , $C = D = 0$ , and so $m$ is the Stieltjes transform of a measure $\mu$ . If $x \coloneqq \cos \theta$ for $\theta \in [0, \pi]$ , since $j(e^{i\theta}) = \cos \theta$ , $j(e^{i\theta}) = x$ and $J(x) = e^{i\theta}$ . Since $U_k(\cos \theta) \sin \theta = \sin((k + 1)\theta)$ , it follows that + +$$ +U _ {k} (x) \sqrt {1 - x ^ {2}} = \Im (e ^ {i (k + 1) \theta}) = \Im (J (x) ^ {k + 1}). +$$ + +Since $J$ is analytic on $\mathbb{C}^+$ , this, in turn, implies that + +$$ +\frac {\mathrm {d} \mu (x)}{\mathrm {d} x} = \lim _ {\delta \rightarrow 0 ^ {+}} \frac {1}{\pi} \Im [ m (x + i \delta) ] = U _ {k} (x) \sqrt {1 - x ^ {2}}, +$$ + +and the result follows. + +The significance of Lemma D.1 is that we can approximate a spectral density as a Chebyshev series, and immediately read off the corresponding Stieltjes transform. To see this, suppose that a spectral density $\rho$ on $[-1, 1]$ has the Chebyshev expansion + +$$ +\rho (x) = \sqrt {1 - x ^ {2}} \sum_ {k = 0} ^ {\infty} \psi_ {k} U _ {k} (x). +$$ + +By Fubini's Theorem, + +$$ +\mathbb {E} _ {X \sim \rho} [ U _ {k} (X) ] = \sum_ {l = 0} ^ {\infty} \psi_ {l} \int_ {- 1} ^ {1} U _ {k} (x) U _ {l} (x) \sqrt {1 - x ^ {2}} \mathrm {d} x = \frac {\pi}{2} \psi_ {k}, +$$ + +and so $\psi_{k}$ can be estimated by $\hat{\psi}_{n,k} = \frac{2}{\pi n}\sum_{i = 1}^{n}U_{k}(\lambda_{i})$ . The corresponding Stieltjes transform is then $m(z) = \pi \Lambda (J(z))$ where + +$$ +\Lambda (z) = \sum_ {k = 0} ^ {\infty} \psi_ {k} z ^ {k + 1}. +$$ + +Noting that $|J(z)| = \exp(-\Re(\operatorname{arccosh}(z)))$ , for $z \in \mathbb{C}^+$ , $|J(z)| \leq 1$ , and so $J(z)$ will lie in the domain of convergence of the power series for $\Lambda$ . However, this series is unlikely to converge for $z \in \mathbb{C}^-$ , and is therefore insufficient for estimating the secondary branch. The solution is to analytically continue $\Lambda$ outside the domain of convergence of its power series by appealing to Padé approximants. Computing Padé approximants algebraically is an expensive task, so we appeal to Wynn's $\epsilon$ -algorithm to rapidly evaluate the Padé approximant for $\Lambda$ , given estimates of its coefficients $\psi_k$ (Brezinski, 1996). For a sequence of partial sums $\Lambda_n(z) = \sum_{k=0}^{n} \psi_k z^{k+1}$ , $n = 0, 1, \ldots$ , the $\epsilon$ -algorithm constructs a tableau $\{\epsilon_k^{(n)}(z)\}_{n,k}$ by + +$$ +\epsilon_ {- 1} ^ {(n)} (z) = 0, \quad \epsilon_ {0} ^ {(n)} (z) = \Lambda_ {n} (z), \quad \epsilon_ {k + 1} ^ {(n)} (z) = \epsilon_ {k - 1} ^ {(n + 1)} (z) + (\epsilon_ {k} ^ {(n + 1)} (z) - \epsilon_ {k} ^ {(n)} (z)) ^ {- 1}. +$$ + +The coefficient $\epsilon_{2k}^{(n)}(z)$ provides the Padé approximant for $\Lambda$ with numerator and denominator degrees of $k + n$ and $k$ , respectively (Baker Jr. & Graves-Morris, 1996, Eq. (4.10)). In particular, Nuttal's Theorem (Baker Jr. & Graves-Morris, 1996, Theorem 6.5.3) shows that $\epsilon_{2n}^{(0)}(z)$ converges in measure to the analytic continuation of $\Lambda(z)$ regardless of whether the partial sums $\Lambda_n$ converge. + +Algorithm 2: Padé-Chebyshev Free Decompression +Input :Hermitian matrix $\mathbf{A}\in \mathbb{R}^{n\times n}$ Maximum Chebyshev polynomial order $K$ Evaluation set $\mathcal{X}\subset \mathbb{R}$ Subsampling parameter $n_{s}$ (with $1\leq n_{s}\leq n$ Perturbation $\delta$ where $0 < \delta \ll 1$ +Output:Estimated spectral density values $\{\hat{\rho} (x)\}_{x\in \mathcal{X}}$ + +// Step 1: Generate eigenvalues from a random submatrix + +1 Extract a random submatrix $\hat{\mathbf{A}}$ from $\mathbf{A}$ and compute its eigenvalues $\{\lambda_i\}_{i = 1}^n$ + +2 $\lambda_{-}\gets \min \{\lambda_i\}_{i = 1}^n -\delta n^{-1}$ + +// Estimate of the left endpoint of the support + +3 $\lambda_{+}\gets \max \{\lambda_i\}_{i = 1}^n +\delta n^{-1}$ + +// Estimate of the right endpoint of the support + +// Step 2: Estimate Chebyshev coefficients + +4 for $k = 0$ to $K$ do + +5 $\begin{array}{r}\big{\downarrow}\hat{\psi}_{n,k}\gets \frac{4}{\pi(\lambda_{+} - \lambda_{-})n}\sum_{i = 1}^{n}U_{k}(M_{\lambda_{-},\lambda_{+}}(\lambda_{i})) \end{array}$ + +// Step 3: Compute the Padé approximant + +6 Using Wynn's $\epsilon$ -algorithm, obtain the Padé approximant $\Lambda(z)$ for the power series + +$$ +\sum_ {k = 0} ^ {K} \hat {\psi} _ {n, k} z ^ {k + 1}. +$$ + +// Step 4: Form the Stieltjes transform + +7 $m(z)\gets \pi \Lambda (J(M_{a,b}(z)))$ + +// Step 5: Solve characteristic curves + +8 foreach $x\in \mathcal{X}$ do + +9 Using Newton's method, find $z_{x}\in \mathbb{C}$ such that + +$$ +x + i \delta = z _ {x} - \frac {\frac {n}{n _ {s}} - 1}{m (z _ {x})}. +$$ + +// Step 6: Return the estimated spectral density + +10 return $\left\{\frac{1}{\pi}\Im m(z_x)\frac{n_s}{n}\right\}_{x\in \mathcal{X}}$ + +To extend this procedure to an arbitrary support interval $[\lambda_{-},\lambda_{+}]$ , we can make use of a conformal map + +$$ +M _ {\lambda_ {-}, \lambda_ {+}} (z) = \frac {2 z - (\lambda_ {+} + \lambda_ {-})}{\lambda_ {+} - \lambda_ {-}}: [ \lambda_ {-}, \lambda_ {+} ] \mapsto [ - 1, 1 ]. +$$ + +In this case, the spectral density is given by + +$$ +\rho (x) = \frac {2 \sqrt {(\lambda_ {+} - x) (x - \lambda_ {-})}}{b - a} \sum_ {k = 0} ^ {\infty} \psi_ {k} U _ {k} (M _ {\lambda_ {-}, \lambda_ {+}} (x)), +$$ + +where the coefficients of the estimates are + +$$ +\hat {\psi} _ {n, k} = \frac {4}{\pi n (\lambda_ {+} - \lambda_ {-})} U _ {k} (\lambda_ {i}), +$$ + +and the Stieltjes transform becomes + +$$ +m (z) = \pi \Lambda (J (M _ {\lambda_ {-}, \lambda_ {+}} (z))). +$$ + +# Appendix E Proofs of Propositions 2 and 3 + +Proof of Proposition 2. By construction, $R(t,e^{-t}z) = R(0,z)$ . In terms of the inverse Stieltjes transform, this implies + +$$ +\omega (t, e ^ {- t} z) - \left(1 - e ^ {t}\right) \frac {1}{z} = \omega (0, z). +$$ + +Differentiating both sides in $t$ gives + +$$ +\frac {\partial \omega}{\partial t} (t, e ^ {- t} z) - z e ^ {- t} \frac {\partial \omega}{\partial z} (t, e ^ {- t} z) + \frac {e ^ {t}}{z} = 0. \tag {E.1} +$$ + +Since $m(t, \omega(t, z)) = z$ , $\frac{\partial m}{\partial z} \frac{\partial \omega}{\partial z} = 1$ and $\frac{\partial m}{\partial t} + \frac{\partial m}{\partial z} \frac{\partial \omega}{\partial t} = 0$ . Applying these relations to (E.1) gives + +$$ +- \frac {\partial m}{\partial t} (t, \omega (t, e ^ {- t} z)) - z e ^ {- t} + \frac {e ^ {t}}{z} \frac {\partial m}{\partial z} (t, \omega (t, e ^ {- t} z)) = 0. +$$ + +Now evaluate this expression at $z = \tilde{z}$ where $\tilde{z} = e^{t}m(t,z)$ so that $\omega (t,e^{-t}\tilde{z}) = z$ . Then + +$$ +- \frac {\partial m}{\partial t} (t, z) - m (t, z) + \frac {1}{m (t , z)} \frac {\partial m}{\partial z} (t, z) = 0. +$$ + +Rearranging this expression implies the result. + +Proof of Proposition 3. Let $\tau \mapsto (t(\tau), z(\tau))$ be a characteristic curve of the quasilinear PDE (1). By the chain rule, + +$$ +\frac {\mathrm {d} m (t (\tau) , z (\tau))}{\mathrm {d} \tau} = \frac {\partial m (t , z)}{\partial t} \frac {\mathrm {d} t (\tau)}{\mathrm {d} \tau} + \frac {\partial m (t , z)}{\partial z} \frac {\mathrm {d} z (\tau)}{\mathrm {d} \tau}. \tag {E.2} +$$ + +Matching coefficients in (E.2) with the PDE (1) yields the system of ODEs + +$$ +\frac {\mathrm {d} t (\tau)}{\mathrm {d} \tau} = 1, \quad t (0) = 0, \tag {E.3a} +$$ + +$$ +\frac {\mathrm {d} z (\tau)}{\mathrm {d} \tau} = - m (\tau) ^ {- 1}, \quad z (0) = z _ {0}, \tag {E.3b} +$$ + +$$ +\frac {\mathrm {d} m (\tau)}{\mathrm {d} \tau} = - m (\tau), \quad \quad \quad m (0) = m _ {0} \left(z _ {0}\right). \tag {E.3c} +$$ + +Solving (E.3) is straightforward, yielding (2). + +By standard ODE existence and uniqueness theorems, for each $z_0$ there is a unique characteristic curve $\tau \mapsto (t(\tau), z(\tau), m(\tau))$ near $\tau = 0$ . Since $\frac{dt}{d\tau} = 1$ and $\frac{dz}{d\tau} \neq 0$ if $m_0(z_0) \neq 0$ , we can (locally) invert $\tau \mapsto t$ and $(z_0) \mapsto z$ . This yields a unique $C^1$ function $m(t,z)$ solving (1) with $m(0,z) = m_0(z)$ . + +Finally, the condition $m_0(z)\neq 0$ for all $z\in \Omega$ ensures that $m(\tau)\neq 0$ in a neighborhood of $\tau = 0$ so the term $m^{-1}$ in (1) is well-defined. Uniqueness follows from the uniqueness of solutions to (E.3) with given initial data. We see that Proposition 3 gives explicit solutions to (1) in terms of the initial data and desired time $t$ . + +It remains to show the existence of $\phi$ . Define $\Phi(t, z, z_0) \coloneqq z_0 + m_0(z_0)^{-1}(1 - e^t) - z$ . If $\partial \Phi / \partial z_0 \neq 0$ , by the implicit function theorem (Hamilton, 1982), there exists an inverse function $z_0 = \phi(t, z)$ solving $\Phi(t, z, \phi(t, z)) = 0$ , which allows the explicit solution of (1) by $m(t, z) = m_0(\phi(t, z))e^{-t}$ . This completes the proof. + +# Appendix F Stability Analysis + +To analyze the stability of $m(t,z)$ over time $t$ , we perturb the PDE (1) by replacing $m$ with $m + \delta m$ and derive a PDE governing the evolution of the perturbation field $\delta m(t,z)$ . We then quantify the impact of this perturbation using an appropriate functional norm. + +A natural choice for this purpose is the $L^2$ -norm along horizontal slices in the upper half-plane, defined by + +$$ +\left\| m (t, \cdot + i y) \right\| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2} := \int_ {\mathbb {R}} | m (t, x + i y) | ^ {2} d x, +$$ + +where $\Gamma_y\coloneqq \{z = x + iy\mid x\in \mathbb{R}\}$ is the horizontal line at height $y > 0$ . This norm is particularly suitable for our analysis since we are ultimately interested in evaluating $m(t,x + i\delta)$ slightly above the real axis, in order to recover the spectral density from the Stieltjes transform. + +Moreover, this space includes the boundary values of Stieltjes transforms $m(t,x + i0^{+})$ whenever $\rho (t,\cdot)\in L^{2}(\mathbb{R})$ , since one can show that + +$$ +\| m (t, x + i 0 ^ {+}) \| _ {L ^ {2} \left(\Gamma_ {0} ^ {+}\right)} ^ {2} = \left(1 + \pi^ {2}\right) \| \rho (t, \cdot) \| _ {L ^ {2} (\mathbb {R})} ^ {2}. +$$ + +We now establish an upper bound on the growth of this norm over time, culminating in a Gronwall-type inequality. This is formalized in the following proposition. + +Proposition F.1. Suppose $\rho(t, \cdot) \in L^2(\mathbb{R})$ , and let $m(t, z)$ be the Stieltjes transform defined on the domain $\mathbb{C}^+$ . Let $\delta m$ be a small functional (Gâteaux) perturbation of $m$ (so $\|\delta m\| \ll 1$ ) satisfying the Bernstein inequality + +$$ +\left\| \partial_ {x} \delta m (t, z) \right\| _ {L ^ {2} \left(\Gamma_ {y}\right)} \leq M \| \delta m (t, z) \| _ {L ^ {2} \left(\Gamma_ {y}\right)}, \quad z \in \Gamma_ {y}, \forall y > 0 \tag {F.1} +$$ + +which means on each horizontal slice, $\delta m(t,\cdot +iy)$ is $M$ -band-limited (Paley-Wiener space), ensuring sufficient decay of the perturbations. Then, for every $y > 0$ , it holds that + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} \| \delta m (t) \| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2} \leq (- 2 + C _ {y} (t)) \| \delta m (t) \| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2}, \tag {F.2} +$$ + +where + +$$ +C _ {y} (t) := 3 \sup _ {z \in \Gamma_ {y}} \left| \Re \frac {\partial_ {z} m (t , z)}{m (t , z) ^ {2}} \right| + 2 M \sup _ {z \in \Gamma_ {y}} \left| \Im \frac {1}{m (t , z)} \right|. \tag {F.3} +$$ + +Proof. Let $\tilde{m} = m + \delta m$ , where $\delta m \in H^{2}(\mathbb{R}_{>0} \times \mathbb{C}^{+})$ (the usual Hardy space) is a small perturbation with the boundary condition $\delta m(t,z) = 0$ at $|z| \to \infty$ . Plugging $\tilde{m}$ into the PDE (1) and subtracting the unperturbed PDE gives the perturbation equation + +$$ +\frac {\partial \delta m}{\partial t} = - \delta m + \frac {1}{m + \delta m} \frac {\partial (m + \delta m)}{\partial z} - \frac {1}{m} \frac {\partial m}{\partial z}. +$$ + +Expanding the nonlinear term using a first-order approximation and rearranging, we obtain + +$$ +\frac {\partial \delta m}{\partial t} = \frac {1}{m} \frac {\partial \delta m}{\partial z} - \delta m - \frac {\delta m}{m ^ {2}} \frac {\partial m}{\partial z} + \mathcal {O} (| \delta m | ^ {2}). +$$ + +The linearized equation above is essentially a transport-type PDE, with the first, second, and third terms on the right-hand side representing the transport term, damping term, and perturbation term, respectively. + +To obtain the energy, we multiply both sides of the above by $\overline{\delta m}$ (the complex conjugate of $\delta m$ ), integrate over $\Gamma_y$ , and consider its real part, yielding + +$$ +\Re \int_ {\mathbb {R}} \frac {\partial \delta m}{\partial t} \overline {{\delta m}} \mathrm {d} x = \Re \int_ {\mathbb {R}} \frac {1}{m} \frac {\partial \delta m}{\partial z} \overline {{\delta m}} \mathrm {d} x - \Re \int_ {\mathbb {R}} \left(1 + \frac {1}{m ^ {2}} \frac {\partial m}{\partial z}\right) | \delta m | ^ {2} \mathrm {d} x. \tag {F.4} +$$ + +The term on the left-hand side of (F.4), after taking $\sup_{y > 0}$ , can be expressed as + +$$ +\Re \int_ {\mathbb {R}} \frac {\partial \delta m}{\partial t} \bar {\delta m} d x = \frac {1}{2} \frac {d}{d t} \| \delta m \| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2}. \tag {F.5} +$$ + +Since $\delta m$ is holomorphic, we have $\partial_z\delta m = \partial_x\delta m$ on every stripe $\Gamma_y$ . Writing $m^{-1} = a + ib$ with real valued $a,b$ , we have + +$$ +\Re \left(\frac {1}{m} (\partial_ {x} \delta m) \overline {{\delta m}}\right) = \frac {1}{2} \Re \left(\frac {\partial_ {x} m}{m ^ {2}}\right) | \delta m | ^ {2} - b \Im \left((\partial_ {x} \delta m) \overline {{\delta m}}\right) + \frac {1}{2} \partial_ {x} (a | \delta m | ^ {2}). +$$ + +Integrating over $x \in \mathbb{R}$ , the third term on the right hand side of the above vanishes. As for the other terms, we apply Cauchy-Schwarz and the Bernstein bound (F.1) to obtain + +$$ +\Re \int_ {\mathbb {R}} \frac {1}{m} \frac {\partial \delta m}{\partial z} \overline {{\delta m}} \mathrm {d} x \leq \left(\frac {1}{2} \sup _ {z \in \Gamma_ {y}} \left| \Re \frac {\partial_ {z} m}{m ^ {2}} \right| + M \sup _ {z \in \Gamma_ {y}} \left| \Im \frac {1}{m} \right|\right) \int_ {\mathbb {R}} | \delta m | ^ {2} \mathrm {d} x. \tag {F.6} +$$ + +Also, the second term on the right-had side of (F.4) satisfies + +$$ +- \Re \int_ {\mathbb {R}} \left(1 + \frac {1}{m ^ {2}} \frac {\partial m}{\partial z}\right) | \delta m | ^ {2} d x \leq - \int_ {\mathbb {R}} | \delta m | ^ {2} d x + \sup _ {z \in \Gamma_ {y}} \left| \frac {\partial_ {z} m}{m ^ {2}} \right| \int_ {\mathbb {R}} | \delta m | ^ {2} d x. \tag {F.7} +$$ + +Putting all together by substituting (F.5), (F.6), and (F.7) into (F.4), we obtain + +$$ +\begin{array}{l} \frac {1}{2} \frac {\mathrm {d}}{\mathrm {d} t} \| \delta m \| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2} \leq \left(- 1 + \frac {3}{2} \sup _ {z \in \Gamma_ {y}} \left| \Re \frac {\partial_ {z} m}{m ^ {2}} \right| + M \sup _ {z \in \Gamma_ {y}} \left| \Im \frac {1}{m} \right|\right) \int_ {\mathbb {R}} | \delta m | ^ {2} \mathrm {d} x \\ = \left(- 1 + C _ {y} (t)\right) \| \delta m \| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2}. \\ \end{array} +$$ + +This completes the proof. + +![](images/acecba66487bf4fd4c45b3c012ddcc44e267a4b1e93901751e2e52923a0db647.jpg) + +Remark F.1. To complete Proposition F.1, we verify that the constant $C_y(t)$ is finite for every $0 < y < \infty$ . Recall the representation of the Stieltjes transform, + +$$ +m (t, z) = \int_ {\mathbb {R}} \frac {\rho (t , \xi)}{\xi - z} d \xi , \tag {F.8} +$$ + +for $z = x + iy\in \mathbb{C}^{+}$ with $y > 0$ . Then, since $\rho (t,x)$ is supported on a compact interval of diameter $D\coloneqq \mathrm{diam}(\mathrm{supp}(\rho))$ , we have + +$$ +\left| m (t, z) \right| \geq \Im m (t, z) = \int_ {\mathbb {R}} \frac {y \rho (t , \xi)}{(\xi - x) ^ {2} + y ^ {2}} d \xi \geq \frac {y}{D ^ {2} + y ^ {2}} \int_ {\mathbb {R}} \rho (t, \xi) d \xi = \frac {y}{D ^ {2} + y ^ {2}}. +$$ + +It follows that + +$$ +\sup _ {z \in \Gamma_ {y}} | \Im m (t, z) ^ {- 1} | \leq \sup _ {z \in \Gamma_ {y}} | m (t, z) ^ {- 1} | \leq \frac {D ^ {2} + y ^ {2}}{y}. \tag {F.9} +$$ + +On the other hand, from the derivative of (F.8): + +$$ +\partial_ {z} m (t, z) = \int_ {\mathbb {R}} \frac {\rho (t , \xi)}{(\xi - z) ^ {2}} \mathrm {d} \xi , +$$ + +so + +$$ +\left| \partial_ {z} m (t, z) \right| \leq \int_ {\mathbb {R}} \frac {\rho (t , \xi)}{(\xi - x) ^ {2} + y ^ {2}} \mathrm {d} \xi \leq \frac {1}{y ^ {2}}. +$$ + +Combining the above with (F.9), we obtain + +$$ +\sup _ {z \in \Gamma_ {y}} \left| \Re \frac {\partial_ {z} m (t , z)}{m (t , z) ^ {2}} \right| \leq \sup _ {z \in \Gamma_ {y}} \left| \frac {\partial_ {z} m (t , z)}{m (t , z) ^ {2}} \right| \leq \frac {1}{y ^ {2}} \left(\frac {D ^ {2} + y ^ {2}}{y}\right) ^ {2} = \frac {(D ^ {2} + y ^ {2}) ^ {2}}{y ^ {4}}. \tag {F.10} +$$ + +Substituting the uniform bounds (F.9) and (F.10) into (F.3) confirms that $C_y(t) < \infty$ for all $0 < y < \infty$ . + +We note that this bound is not defined at $y = 0$ , which is expected: as $y \to 0^+$ , the Stieltjes transform approaches the real axis where it develops discontinuities, and quantities like $\partial_z m$ may diverge. However, in both our theory and implementation, we evaluate $m(t,z)$ slightly above the real axis (e.g., at $z = x + i\delta$ for fixed $0 < \delta \ll 1$ ), where all terms remain finite. In practice, the observed error propagation is significantly milder than this worst-case upper bound suggests. + +Proposition F.1 then implies that the perturbation $\delta m$ satisfies an exponential growth bound in time: + +$$ +\left\| \delta m (t) \right\| _ {L ^ {2} (\Gamma_ {y})} ^ {2} \leq \left\| \delta m _ {0} \right\| _ {L ^ {2} (\Gamma_ {y})} ^ {2} e ^ {\nu t}, +$$ + +with exponent $\nu = -2 + C_y < \infty$ for any fixed $y > 0$ . But since $e^t = \frac{n}{n_s}$ , the exponential bound translates to polynomial growth in $n$ , + +$$ +\left\| \delta m (t) \right\| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2} \leq \left\| \delta m _ {0} \right\| _ {L ^ {2} \left(\Gamma_ {y}\right)} ^ {2} \left(\frac {n}{n _ {s}}\right) ^ {\nu}. +$$ + +# Appendix G Background Material for the Neural Tangent Kernel + +The neural tangent kernel (NTK) is an important object in machine learning (Jacot et al., 2018), and has become a prominent theoretical and practical tool for studying the behavior of neural networks both during training and inference (Novak et al., 2022). The Gram matrix associated to the NTK has been used in lazy training (Chizat et al., 2019), and as a tool for uncertainty quantification and estimation (Immer et al., 2021; Wilson et al., 2025), exploiting the deep connections between neural networks and Gaussain processes. More recently, it has been shown to arise naturally as a measure of model quality in approximations of the marginal likelihood (Hodgkinson et al., 2023b; Immer et al., 2023), as a tool for quantification of model complexity (Vakili et al., 2022), and in estimation of generalization error in PAC-Bayes bounds (Hodgkinson et al., 2023c; Kim et al., 2023). + +While it was first derived in the context of neural networks, the quantity is well-defined for a broader class of functions. For clarity will consider the setting where the underlying function is $C^1$ . However, this can be relaxed in practice to include, as an important example, networks with ReLU activations. For such a function $f_{\theta}:\mathcal{X}\to \mathbb{R}^{d}$ parameterized by $\pmb {\theta}\in \mathbb{R}^{p}$ , we define the (empirical) NTK to be + +$$ +\kappa_ {\boldsymbol {\theta}} \left(x, x ^ {\prime}\right) := \mathbf {J} _ {\boldsymbol {\theta}} \left(f _ {\boldsymbol {\theta}} (x)\right) \mathbf {J} _ {\boldsymbol {\theta}} \left(f _ {\boldsymbol {\theta}} \left(x ^ {\prime}\right)\right) ^ {\intercal}. \tag {G.1} +$$ + +Here, $\mathbf{J}_{\theta}(f_{\theta}(x)) \in \mathbb{R}^{d \times p}$ is the Jacobian of $f_{\theta}$ with respect to its (flattened vector of) parameters $\pmb{\theta}$ , evaluated at the data point $x$ . Since $\kappa_{\theta}(x,x') \in \mathbb{R}^{d \times d}$ , evaluating the NTK over $n$ data points produces a fourth-order tensor of shape $(n,n,d,d)$ . For computational convenience, this tensor is typically reshaped into a two-dimensional block matrix of size $nd \times nd$ , where each $(i,j)$ -block contains the $d \times d$ matrix $\kappa(x_i, x_j)$ . + +The fact that the NTK arises in a variety of contexts makes it a natural object of study. For well-trained models, the Gram matrix is nearly singular, making accurate characterization of its spectral properties crucial for estimating quantities used in downstream tasks (Ameli et al., 2025). This also motivates considering the spectrum on a logarithmic scale. In Section 5, we analyze the empirical NTK of a ResNet50 model trained on the CIFAR10 dataset, a 10-class classification task. In this case, $10\%$ of the eigenvalues are extremely small $(\sim 10^{-14})$ , reflecting near-singular behavior likely tied to the model's class structure. + +To examine the scaling behavior of the non-singular bulk spectrum, we project the NTK matrix onto its non-null eigenspace, removing the low-rank null component. We then randomly sample orthogonal eigenvectors within this reduced space and reconstruct a corresponding Hermitian matrix, which is subsequently permuted to generate random subsets for use in our free decompression procedure. + +![](images/168d4cc6a2d7578a6c24dd17a555ce4f69fe787b260a35de988fd0eb855bf293.jpg) +Figure G.1: Analytic continuation of the Stieltjes transform of a density estimate of the spectral distribution of the log-NTK matrix described in Section 5. (a) The principal branch contains a branch cut along the support of the spectral density, while (b) the secondary branch is continuous in this region. + +![](images/2c4f107c4d0278ece190c204ebbb992990dd1e214f3d5be40c299cde01e490fc.jpg) + +The Stieltjes transform of the approximated spectrum of this matrix is depicted in Figure G.1, which shows its analytic continuation using the principal branch (left), as well as the secondary branch (right). + +# Appendix H Implementation and Reproducibility Guide + +We developed a Python package, freealg, which implements our algorithms and enables reproduction of all numerical results in this paper. A minimal example using the freealg.FreeForm class is shown in Listing 1. This example generates the results in Figures 1 and 2, illustrating the Marchenko-Pastur distribution with parameter $\lambda = \frac{1}{50}$ , starting from a matrix of size $n_s = 1000$ and decompressing to a matrix of size $n = 32,000$ . + +Hyperparameters used for the numerical experiments appearing in this document can be found in Table H.1 (notation summarized in the caption). Notebook files that reproduce the figures appearing in this work can be found in the codebase. Further details on function arguments and class parameters are available in the package documentation. + +Table H.1: Hyperparameters used in the numerical examples. Notation: $h$ smoothing kernel bandwidth; $\left( {\alpha ,\beta }\right)$ Jacobi shape parameters; $K$ Jacobi polynomial order; $\gamma$ regularization strength; Padé $\left( {p,q}\right)$ numerator/denominator degrees; $\delta$ imaginary shift for Stieltjes evaluation (Plemelj-Sokhotski). + +
ExampleReferenceSmoothingJacobi PolynomialsStieltjes Transform
Kernelh(α,β)KγDampingPadé (p,q)
Marchenko-PasturFigure 1Beta3 × 10-3(1/2, 1/2)500Jackson(1,1)
Wigner semicircleFigure B.1Beta10-2(1/2, 1/2)500Jackson(1,1)
Kesten-McKayFigure B.2Beta10-3(1/2, 1/2)500Jackson(1,2)
WachterFigure B.3Beta10-3(1/2, 1/2)200Jackson(1,2)
MeixnerFigure B.4Beta10-3(1/2, 1/2)500Jackson(7,8)
Page/PageFigure 3 (a)Beta10-5(0,0)1000Jackson(4,3)
NTKFigure 3 (b)Beta10-3(2,2)5010-2Jackson(3,2)
+ +Listing 1: A minimal usage example of the freealg package. +```txt +# Install freealg with "pip install freealg" +import freealg as fa +# Create an object for the Marchenko-Pastur distribution with the parameter $\lambda = \frac{1}{50}$ +mp = fa.distributions.MarchenkoPastur(1/50) +# Generate a matrix of size $n_s = 1000$ corresponding to this distribution +A = mp.matrix(size=1000) +# Create a free-form object for the matrix within the support $I = [\lambda_{-},\lambda_{+}]$ +ff = fa.FreeForm(A, support=(mp.lam_m, mp.lam_p)) +# Fit the distribution using Jacobi polynomials of degree $K = 20$ , with $\alpha = \beta = \frac{1}{2}$ +# Also fit the glue function via Pade of degree $[p/q]$ with $p = 1$ , $q = 1$ . +psi = ff.fit(method='jacobi', K=20, alpha=0.5, beta=0.5, reg=0.0, damp='jackson', pade_p=1, pade_q=1, optimizer='ls', plot=True) +# Estimate the empirical spectral density $\rho(x)$ , similar to Figure 1(a) +rho = ff_density(plot=True) +# Estimate the Hilbert transform $\mathcal{H}[\rho](x)$ +hilb = ff.hilbert(plot=True) +# Estimate the Stieltjes transform $m(z)$ (both branches $m^+$ and $m^-$ ), similar to Figure 2(a,b) +m1, m2 = ff.stieltjes(plot=True) +# Decompress the spectral density corresponding to a larger matrix of size $n = 2^5 \times n_s$ , +# similar to Figure 1(c) +rho_large, x = ff.decompress(size=32_000, plot=True) +``` + +# NeurIPS Paper Checklist + +# 1. 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By treating node features as functions on the graph and preserving their inner products, our method retains both structural and feature relationships, facilitating substantial benefits for downstream tasks. To formalize this, we introduce the Inner Product Error (IPE), which quantifies how the inner products between node features are preserved. Leveraging the underlying geometry of the problem on the Grassmann manifold, we formulate an optimization objective that minimizes the IPE, also for unseen smooth functions. We show that minimizing the IPE improves standard coarsening metrics, and illustrate our method's properties through visual examples that highlight its clustering ability. Empirical results on benchmarks for graph coarsening and node classification show that our approach outperforms existing state-of-the-art methods. + +# 1 Introduction + +Graph-structured data has become ubiquitous in a wide range of domains, including social networks [1], biological systems [2], and recommendation systems [3], due to its ability to model complex relationships and interactions. With the exponential increase in data availability, the size of graphs in many applications has also grown significantly. This surge in graph size presents major challenges, as traditional and even advanced graph processing techniques often become computationally infeasible or excessively time-consuming when applied to large-scale graphs. To address these issues, graph reduction techniques are developed with the aim of simplifying large graphs while retaining key structural features, thereby enhancing computational efficiency. There are three main strategies for graph reduction [4]: graph sparsification, graph condensation, and graph coarsening. + +Graph sparsification [5, 6] reduces graph size by selectively removing edges and nodes while maintaining overall structural properties. However, there is a limit to how much a graph can be sparsified without compromising its integrity. Graph condensation methods [7, 8] aim to reduce graph size by generating a smaller, synthetic graph that replicates the performance of the original graph on specific tasks, such as training a Graph Neural Network (GNN). Although condensation significantly lowers computational costs, it may not retain a clear structural interpretation, making it difficult to understand how or why certain nodes or edges are represented in the reduced version. + +In contrast, graph coarsening [9] is a traditional approach that reduces graph size by grouping similar nodes into super-nodes, aiming to approximate the original structure. These methods typically seek to preserve key structural properties such as spectral characteristics [10], connectivity [11], and community structure [12, 13]. Most existing coarsening methods focus primarily on structure and often overlook node features, which play a critical role in many graph learning tasks. These methods typically operate solely on the graph topology, neglecting the rich information encoded in the node + +features. Recently, the Featured Graph Coarsening (FGC) method was proposed to address this limitation by incorporating node features into the coarsening process [14]. FGC emphasizes the reconstruction of the node features after coarsening and promotes smoothness in the coarsened graph as part of its optimization objective. However, since FGC focuses on preserving the individual characteristics of the node features, it does not fully exploit their mutual relationships, which may encode valuable information. + +In this work, we propose a new approach to graph coarsening from a functorial perspective. We treat node features as functions, or signals, defined on the graph, focusing on preserving the relationships between these functions by maintaining their inner products during coarsening. Our method introduces a new coarsening metric, the Inner Product Error (IPE), which measures how the inner products between graph signals are preserved. We postulate that minimizing IPE ensures that the coarsened graph retains structural consistency and node feature relationships crucial for graph learning tasks. We exploit the geometry of the problem by recognizing that both the coarsening operator and the matrix spanning the node features (under a smoothness assumption) can be viewed as points on the Grassmann manifold. Leveraging the properties of the Grassmann manifold, we extend IPE minimization beyond observed node features, enabling our method to generalize to unseen features that satisfy the smoothness assumption. This approach is formulated as an optimization problem, and we compute the coarsening operator using gradient descent. Additionally, we theoretically show that minimizing our proposed approach leads to improvements in common graph coarsening metrics. + +To demonstrate the effectiveness of our method, we present visual and empirical results showing that, while our method focuses on the functional relationships between node features, it also captures the global structure of the graph. We validate its performance through extensive experiments on multiple graph coarsening and node classification benchmarks, where our method consistently outperforms state-of-the-art coarsening methods, demonstrating its practical utility. + +# 2 Background + +Grassman manifold The set of $n \times k$ matrices whose columns are orthonormal vectors forms a Riemannian manifold called the Stiefel manifold [15] defined by, + +$$ +\operatorname {S t} (n, k) := \left\{\boldsymbol {U} \in \mathbb {R} ^ {n \times k} \mid \boldsymbol {U} ^ {T} \boldsymbol {U} = \boldsymbol {I} _ {k \times k} \right\}. \tag {1} +$$ + +where $I_{k\times k}$ is a rank- $k$ identity matrix. The Grassmann manifold $\mathrm{Gr}(n,k)$ is a quotient manifold representing the set of $k$ -dimensional subspaces of the Euclidean space $\mathbb{R}^n$ . Two points on the Stiefel manifold that span the same subspace represent the same point on the Grassmann manifold [16, 17]. In general, a point on $\mathrm{Gr}(n,k)$ is represented by an equivalence class + +$$ +[ \boldsymbol {U} ] = \left\{\boldsymbol {U O}: \boldsymbol {O} \in \mathcal {S O} (k) \right\}, \tag {2} +$$ + +where $\mathbf{U} \in \mathrm{St}(n, k)$ , and $\mathcal{SO}(k)$ are all $k \times k$ rotation matrices (also known as the special orthogonal group), such that any $\mathbf{O} \in \mathcal{SO}(k)$ satisfies $\mathbf{OO}^T = \mathbf{O}^T\mathbf{O} = \mathbf{I}_{k \times k}$ . The principal angles between two subspaces $U_1$ and $U_2$ are the angles that measure the smallest angular separation between basis vectors in one subspace and basis vectors in the other subspace. We denote them by $\pmb{\theta} = [\theta^{(1)}, \theta^{(2)}, \dots, \theta^{(k)}]$ . Given two subspaces $U_1$ and $U_2$ on the Grassmann manifold $\operatorname{Gr}(n, k)$ , the cosine of the principal angles between them can be computed using the SVD decomposition of $U_1^T U_2 = A\Theta B^T$ , where the singular values on the diagonal of $\Theta$ are $[\cos(\theta^{(1)}), \cos(\theta^{(2)}), \dots, \cos(\theta^{(k)})]$ . + +The geodesic similarity on the Grassmann manifold is defined using the principal angles between subspaces [17]. The authors in [18] showed that this geodesic similarity can be computed by, + +$$ +G r \left(\boldsymbol {U} _ {1}, \boldsymbol {U} _ {2}\right) = \sum_ {i = 1} ^ {k} \cos^ {2} \left(\theta^ {(i)}\right) = t r \left(\boldsymbol {U} _ {1} \boldsymbol {U} _ {1} ^ {T} \boldsymbol {U} _ {2} \boldsymbol {U} _ {2} ^ {T}\right). \tag {3} +$$ + +Graph coarsening A graph with node features is denoted by the quadruplet $\mathcal{G} = (\mathcal{V},\mathcal{E},W,X)$ where $\mathcal{V}$ is a set of $n$ vertices, $\mathcal{E}$ is a set of edges, $W\in \mathbb{R}^{n\times n}$ is a weighted adjacency matrix, and $X\in \mathbb{R}^{n\times p}$ is a node features matrix such that each row specifies the values of the $p$ features for each node. Each node feature, represented as a column of $X$ , can also be considered as a graph signal $\boldsymbol {x}\in \mathbb{R}^n$ , assigning a real value to each vertex, namely, $\boldsymbol {x}:\mathcal{V}\to \mathbb{R}$ . The graph Laplacian matrix $\pmb{L}$ is + +defined by $\pmb{L} = \pmb{D} - \pmb{W}$ , where $\pmb{D} = \mathrm{diag}(\pmb{W}\mathbf{1})$ is the diagonal degree matrix. We define the inner product between two graph signals $\pmb{x}, \pmb{y} \in \mathbb{R}^n$ with respect to the graph $\mathcal{G}$ by: + +$$ +\langle \boldsymbol {x}, \boldsymbol {y} \rangle_ {L} = \boldsymbol {x} ^ {T} \boldsymbol {L} \boldsymbol {y} = \sum_ {(i, j) \in E} w _ {i j} (\boldsymbol {x} (i) - \boldsymbol {x} (j)) (\boldsymbol {y} (i) - \boldsymbol {y} (j)), \tag {4} +$$ + +where $w_{ij}$ are edge weights, and $\pmb{x}(i),\pmb{y}(i)$ are the values of the features at node $i$ . Since $\pmb{L}$ is a positive semi-definite matrix, $\pmb{x}^{\top}\pmb{L}\pmb{x}$ induces a semi-norm and defines an inner product on the subspace of $\mathbb{R}^n$ orthogonal to the constant vector $\mathbf{1}$ , as discussed in Von Luxburg [19]. We denote the graph Laplacian eigen decomposition by $\pmb{L} = \pmb{U}\pmb{\Lambda}\pmb{U}^{T}$ , where the columns of $\pmb{U}$ are the eigenvectors of $\pmb{L}$ , and $\pmb{\Lambda}$ is a diagonal matrix consisting of its corresponding eigenvalues. + +Given a graph $\mathcal{G} = (\mathcal{V},\mathcal{E},\boldsymbol {W},\boldsymbol {X})$ with $n$ nodes, the goal of graph coarsening is to construct a coarsened graph $\mathcal{G}_c = (\mathcal{V}_c,\mathcal{E}_c,\boldsymbol {W}_c,\boldsymbol {X}_c)$ with $k\ll n$ nodes, while preserving the main structural properties of $\mathcal{G}$ , thereby simplifying subsequent analysis and computations. The coarsening procedure is defined through a linear mapping $\pi :\mathcal{V}\to \mathcal{V}_c$ that maps nodes in $\mathcal{G}$ to nodes in $\mathcal{G}_c$ , termed 'supernodes'. This linear mapping is defined by the coarsening matrix $P\in \mathbb{R}_{+}^{k\times n}$ , such that $\boldsymbol {X}_c = \boldsymbol {P}\boldsymbol{X}$ . Each non-zero entry of $\pmb{P}$ indicates a mapping from a node in $\mathcal{G}$ to a super-node in $\mathcal{G}_c$ , i.e., if and only if the $j$ -th node in $\mathcal{G}$ is mapped to the $i$ -th super-node of $\mathcal{G}_c$ , then $P_{i,j} > 0$ . Let $\pmb {L}\in \mathbb{R}^{n\times n}$ and $\pmb {L}_c\in \mathbb{R}^{k\times k}$ be the respective Laplacian matrices of $\mathcal{G}$ and $\mathcal{G}_c$ , and let $\pmb {L}_l\in \mathbb{R}^{n\times n}$ and $\pmb {X}_l\in \mathbb{R}^{n\times p}$ be the lifted Laplacian and feature matrices, i.e., the reconstructed full graph matrices after the coarsening procedure. The relationships between the coarse graph Laplacian and features and the original graph Laplacian and features are [20]: + +$$ +\boldsymbol {L} _ {c} = \boldsymbol {C} ^ {\top} \boldsymbol {L} \boldsymbol {C}, \quad \boldsymbol {X} _ {c} = \boldsymbol {P} \boldsymbol {X} \tag {5} +$$ + +$$ +\boldsymbol {L} _ {l} = \boldsymbol {P} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {P}, \quad \boldsymbol {X} _ {l} = \boldsymbol {C} \boldsymbol {X} _ {c} \tag {6} +$$ + +where $\pmb{C} \in \mathbb{R}_{+}^{n \times k}$ is the pseudo-inverse of $\pmb{P}$ , i.e., $\pmb{C} = \pmb{P}^{\dagger}$ . The non-zero entries of $\pmb{C}$ also imply a node mapping from $\mathcal{G}$ to $\mathcal{G}_c$ , such that $C_{i,j} > 0$ if the $i$ -th node of $\mathcal{G}$ is mapped to the $j$ -th super-node of $\mathcal{G}_c$ . We note that the matrix $\pmb{C}$ belongs to the following set: + +$$ +\mathcal {C} = \left\{\boldsymbol {C} \geq 0 \mid \langle \boldsymbol {C} _ {:, i}, \boldsymbol {C} _ {:, j} \rangle = 0 \quad \forall i \neq j, \right. \tag {7} +$$ + +$$ +\langle \boldsymbol {C} _ {:; i}, \boldsymbol {C} _ {:; i} \rangle = d _ {i}, \| \boldsymbol {C} _ {:; i} \| _ {0} \geq 1, \| \boldsymbol {C} _ {i,:} \| _ {0} = 1 \rbrace +$$ + +where $C_{\cdot,i}$ is the $i$ -th orthogonal column of $C$ , $C_{i,\cdot}$ is the $i$ -th row of $C$ , $\langle \cdot, \cdot \rangle$ is the standard inner product, and $d_i$ is some positive number. Since the columns of a valid coarsening matrix $C$ are orthogonal, it can also be viewed as a point on the Grassmann manifold $\operatorname{Gr}(n,k)$ . + +Numerous evaluation metrics exist for graph coarsening, each assessing how well specific graph properties are preserved during reduction. Next, we review the main ones. + +Definition 2.1 (Relative Eigen Error (REE) [10]) The REE is defined as $REE = \frac{1}{k} \sum_{i=1}^{k} \frac{\lambda_{c,i} - \lambda_i}{\lambda_i}$ , where $\lambda_i$ and $\lambda_{c,i}$ are the $k$ dominant eigenvalues of the original graph Laplacian matrix $L$ and the coarsened graph Laplacian matrix $L_c$ , respectively. + +Definition 2.2 (Reconstruction Error (RE) [21]) The $RE$ between the original graph Laplacian $L$ and the lifted graph Laplacian $L_{l}$ is defined by $RE = \| L - L_{l}\|_{F}^{2}$ . + +The REE and RE are coarsening metrics independent of the graphs' node features. The REE measures the spectral similarity between graphs and how global properties such as important edges are preserved, while the RE quantifies how local information is preserved during coarsening. + +Definition 2.3 (Hyperbolic Error (HE) [22]) The HE between the original Laplacian matrix $\pmb{L}$ and lifted Laplacian matrix $\pmb{L}_l$ is defined as $HE = \operatorname{arccosh}\left(1 + \frac{\|\pmb{(L - L_l)}\pmb{X}\|_F^2\|\pmb{X}\|_F^2}{2tr(\pmb{X}^T\pmb{L}\pmb{X})tr(\pmb{X}^T\pmb{L}_l\pmb{X})}\right)$ , where $\pmb{X}$ is the node features matrix of the original graph. + +Definition 2.4 (Dirichlet Energy Error (DEE)) The Dirichlet Energy (DE) of a graph is defined by $DE = \operatorname{tr}(\mathbf{X}^\top \mathbf{L}\mathbf{X})$ , where $\mathbf{L}$ denotes the graph Laplacian and $\mathbf{X}$ denotes the node feature matrix of the graph [23]. We define the DEE between the original graph $\mathcal{G}$ and its coarsened version $\mathcal{G}_c$ as $DEE = \left|\log\left(\frac{DE_{\mathcal{G}}}{DE_{\mathcal{G}_c}}\right)\right|$ , where $DE_{\mathcal{G}}$ and $DE_{\mathcal{G}_c}$ are the DE of the original and coarsened graphs. + +The HE and DEE are coarsening measures that consider the node features. The HE measures distortion in the geometric structure of the data in hyperbolic space. This is useful when the graph + +has a hierarchical structure (e.g., trees), as hyperbolic spaces are suited for representing such data. The DE measures the smoothness of the node features on a graph; lower DE values suggest that the node features are closely aligned with the graph structure. Consequently, we define the DEE which quantifies the extent to which the intrinsic graph structure in the node features is preserved during the coarsening process. We note that the authors in Kumar et al. [14] suggest minimizing the DE of the coarsened graph as part of their graph coarsening optimization objective. + +# 3 Proposed method + +Our method adopts a functorial perspective for graph coarsening, focusing on maintaining the relationships between different functions defined on the graph. Specifically, it aims to preserve the inner products between functions defined on the graph. To achieve this, we first introduce the following new graph coarsening metric that quantifies how the inner products between given graph signals (i.e., node features) are preserved during the coarsening process. + +Definition 3.1 (Inner Product Error (IPE)) Let $\mathbf{L}$ and $\mathbf{X}$ be the original graph Laplacian and node features matrix, and let $\mathbf{L}_c$ and $\mathbf{X}_c$ be their respective coarsened graph Laplacian and features matrix. The Inner Product Error (IPE) is defined by $IPE = \| \mathbf{X}^\top \mathbf{L}\mathbf{X} - \mathbf{X}_c^\top \mathbf{L}_c\mathbf{X}_c\|_F^2$ . + +The motivation for this approach is based on the following: + +Proposition 3.2 Let $\pmb{L}$ and $L_{c}$ be the graph Laplacians of a graph $\mathcal{G}$ and its coarsened graph $\mathcal{G}_{c}$ , respectively. If for all pairs of graph signals $\pmb{x}, \pmb{y} \in \mathbb{R}^{n}$ , the inner product between the two signals is preserved under the coarsening process, i.e.: + +$$ +\boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} = \boldsymbol {x} _ {c} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {y} _ {c}, +$$ + +then the graph Laplacian of the original graph, $\pmb{L}$ , can be fully reconstructed from $L_{c}$ via: + +$$ +\boldsymbol {L} = \boldsymbol {L} _ {l} = \boldsymbol {P} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {P} +$$ + +See App. A.1 for proof. Prop. 3.2 shows that preserving inner products of graph signals maintains key structural properties, enabling graph reconstruction. We note that a necessary condition for the assumption in Proposition 3.2 to hold is that $\mathrm{rank}(L) < k$ , which implies that the graph has at least $n - k$ connected components, a condition that is rarely met in practice. Yet, we show in Sec. 4 that our approach achieves the lowest reconstruction error (RE) among baselines, even when this criterion is violated, demonstrating its broad applicability. In Sec. 3.1, we provide analytical evidence that minimizing IPE leads to the minimization of other coarsening metrics as well. + +Preserving inner products using the Grassmann manifold. We propose an optimization approach for graph coarsening to preserve inner products between graph signals. The objective function and constraints are given by: + +$$ +\min _ {\boldsymbol {C}} f (\boldsymbol {C}) = \left\| \boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - \boldsymbol {X} _ {c} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {X} _ {c} \right\| _ {F} ^ {2} - \beta G r (\boldsymbol {C}, \boldsymbol {U} ^ {(k)}) + \lambda g (\boldsymbol {C}) + \alpha h (\boldsymbol {L} _ {c}) \tag {8} +$$ + +$$ +\mathbf {s}. \mathbf {t}. \quad \boldsymbol {L} _ {c} = \boldsymbol {C} ^ {T} \boldsymbol {L} \boldsymbol {C}, \boldsymbol {X} _ {c} = \boldsymbol {C} ^ {\dagger} \boldsymbol {X}, \boldsymbol {C} \in \mathcal {C} +$$ + +where $C \in \mathbb{R}^{n \times k}$ is the coarsening operator; $L \in \mathbb{R}^{n \times n}$ and $X \in \mathbb{R}^{n \times p}$ are the given graph Laplacian and feature matrix of the original graph; $U^{(k)}$ is a matrix containing the $k$ leading eigenvectors of $L$ ; $L_{c} \in \mathbb{R}^{k \times k}$ and $X_{c} \in \mathbb{R}^{k \times p}$ are the Laplacian and feature matrix of the coarsened graph, and $\mathcal{C}$ is the set defined in (7). The function $Gr(\cdot, \cdot)$ is the Grassmann similarity score defined in (3), and functions $h(\cdot)$ and $g(\cdot)$ are regularization functions for $L_{c}$ and $C$ , while $\lambda, \alpha > 0$ are positive regularization parameters. + +The objective in (8) minimize the IPE using two complementary terms. The first involves directly minimizing the IPE on the given node features. Although this improves performance on the available data, it does not generalize well to new signals, as its effectiveness depends heavily on the specific information encoded in the feature matrix $\mathbf{X}$ . The second term aims to maximize the Grassmann similarity (3) between the coarsening matrix $\mathbf{C}$ and the leading eigenvectors $U^{(k)}$ of $\mathbf{L}$ . In Sec. 3.1, we show analytically that this alignment promotes IPE minimization for general unseen smooth signals, e.g., satisfying a smoothness assumption (Prop. 3.4), and preserves important structural properties (Prop. 3.5). The parameter $\beta$ balances the two terms, adjusting the emphasis between performance on the observed data and generalization to new signals. Our empirical results show that both terms contribute to the coarsening process. + +Proposed algorithms. One limitation of the objective in (8) is that the derivative of the first term does not have a closed-form expression with respect to $C$ . As a remedy, we adopt the multi-block optimization framework suggested by Kumar et al. [14], recasting our objective function as: + +$$ +\begin{array}{l} \min _ {\boldsymbol {C}} f \left(\boldsymbol {X} _ {c}, \boldsymbol {C}\right) = \left\| \boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - \boldsymbol {X} _ {c} ^ {\top} \boldsymbol {C} ^ {\top} \boldsymbol {L} \boldsymbol {C} \boldsymbol {X} _ {c} \right\| _ {F} ^ {2} - \beta t r \left(\boldsymbol {U} ^ {(k)} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {C} \boldsymbol {C} ^ {\top}\right) \tag {9} \\ + \lambda \| \boldsymbol {C} ^ {T} \| _ {1, 2} ^ {2} - \alpha \log \det (\boldsymbol {C} ^ {\top} \boldsymbol {L} \boldsymbol {C} + \boldsymbol {J}) \\ \begin{array}{c c} \text {s . t .} & \boldsymbol {X} _ {c} = \boldsymbol {C} ^ {\dagger} \boldsymbol {X}, \quad \boldsymbol {C} \in \mathcal {C} \end{array} \\ \end{array} +$$ + +where the terms involving Grassmann similarity and $L_{c}$ are written explicitly. We adopt the same regularization functions as in Kumar et al. [14], which promote balanced super-node assignment and connectivity in the coarsened graph. The function $g(\pmb{C}) = \| \pmb{C}^{\top}\|_{1,2}^{2} = \sum_{i=1}^{n}\left(\sum_{j=1}^{k}|C_{i,j}|\right)^{2}$ is an $l_{1,2}$ -based group penalty that, as shown in [24, 14], encourages valid coarsening operators. The second regularization term is $h(L_{c}) = \log \det(L_{c} + J)$ , where $J = \frac{1}{k}\mathbf{1}_{k\times k}$ . This term ensures that $L_{c} + J$ is full rank, implying $\mathrm{rank}(L_c) = k - 1$ , which guarantees that the coarsened graph $\mathcal{G}_c$ is connected [25, 26]. + +In this recast, the derivative of the modified objective function has a closed-form expression with respect to $C$ , and the gradient is presented in App. A.4. We optimize this objective by applying projected gradient descent [27] to estimate the matrix $C$ . Specifically, applying ordinary gradient descent steps could deviate from the feasible set $C$ on the Grassmann manifold. Therefore, we use projected gradient descent to periodically project $C$ back onto the Grassmann manifold after a fixed number of gradient descent steps using the operator $\operatorname{Hardmax}(C)$ . $\operatorname{Hardmax}(C)$ applies a hard maximum at each row of $C$ , ensuring column-wise orthogonality, which coincides with the structural constraints of the Grassmann manifold. + +A full description of this method is in Algorithm 1, termed INGC. In Algorithm 2, we present another version, where we omit the first term in (9). In this case, we denote the algorithm's objective function by $\tilde{f}(\boldsymbol{C})$ . This version focuses on minimizing the IPE via Grassmann similarity, removes dependence on the feature matrix $\boldsymbol{X}$ , and enables estimating $\boldsymbol{C}$ using standard gradient descent. We refer to this algorithm as We note that, for SINGC, we empirically observed that projecting only once at the end of the optimization yields performance similar to using intermediate projections, as done for INGC. Therefore, for simplicity and efficiency, we chose to apply standard gradient descent throughout the iterations and perform the projection using $\operatorname{Hardmax}(\boldsymbol{C})$ only at the end. + +
Algorithm 1 INGC AlgorithmAlgorithm 2 SINGC Algorithm
Input: L ∈ Rn×n, X ∈ Rn×pInput: L ∈ Rn×n
Parameters: β, λ, α, η, titer, citerParameters: λ, α, η, titer
Output: Lc ∈ Rk×k, Xc ∈ Rk×pOutput: Lc ∈ Rk×k
1: Compute U(k).1: Compute U(k).
2: Initialize C0, t = 0.2: Initialize C0, t = 0.
3: while ||Ct+1 - Ct||F < εC or t < titer do3: while ||Ct+1 - Ct||F < εC or t < titer do
4: Ct(0) = Ct.
5: Compute ∇Cf(Xc, C) (see (16))4: Compute ∇Cf(C) (see (17))
6: for i in range(citer) do
7: Update Ct+1 using the gradient descent step:5: Update Ct+1 using the gradient descent step:
8: Ct(i+1) ← Ct(i) - η∇Cf(Xc(t), Ct(i))Ct+1 ← Ct - η∇Cf(Ct)
9: end for
10: Ct+1 = Hardmax(Ct(citer))6: t = t + 1
11: Xc(t+1) = Ct†X, t = t + 1
12: end while7: end while
13: Hardmax(C)8: Hardmax(C)
return Lc = Ct↑LC, Xc = Ct†X9: return Lc = Ct↑LC
+ +# 3.1 Theoretical analysis + +Next, we present two key analytical results. First, we show that the second term in (8) minimizes the IPE for general smooth graph signals. Second, we connect this minimization and common graph coarsening metrics, such as DEE and REE. We begin by defining a smooth graph signal. Dong et al. [28] model a smooth graph signal generation mechanism as: + +$$ +\boldsymbol {x} = \boldsymbol {U} \boldsymbol {h} + \epsilon_ {\eta} \boldsymbol {\eta}, \tag {10} +$$ + +where $\pmb{L} = \pmb{U}\pmb{\Lambda}\pmb{U}^{\top}$ is the Laplacian of the respective graph signal, $h\sim \mathcal{N}(0,\pmb{\Lambda}^{\dagger})\in \mathbb{R}^{n}$ , $\eta \sim \mathcal{N}(0,\pmb{I}_{n\times n})\in \mathbb{R}^n$ , and $\epsilon_{\eta} > 0$ is the noise standard deviation. This model suggests that a smooth graph signal is a combination of the first eigenvectors of $\pmb{L}$ and scaled noise. + +Assumption 3.3 ( $k$ -smooth graph signal [29]) A graph signal $\pmb{x} \in \mathbb{R}^n$ is termed " $k$ -smooth" on the graph $\mathcal{G}$ if it can be fully expressed by the first $k$ eigenvectors of its corresponding Laplacian $\pmb{L}$ , i.e., $\pmb{x} = \sum_{i=1}^{k} c_i \pmb{u}^{(i)} = \pmb{c}(\pmb{U}^{(k)})^\top$ where the columns of $\pmb{U}^{(k)} = [\pmb{u}^{(1)}, \dots, \pmb{u}^{(k)}] \in \mathbb{R}^{n \times k}$ are the $k$ -leading eigenvectors of $\pmb{L}$ . + +Table 4 in App. E shows the extent to which this assumption holds in real datasets. Based on Assumption 3.3, the following result provides motivation to incorporate the Grassmann similarity score $Gr(C, U^{(k)})$ defined in (3) into our proposed objective. + +Proposition 3.4 Let $X$ be a feature matrix of a graph $\mathcal{G}$ with a Laplacian matrix $L$ , where each column of $X$ is $k$ -smooth on the graph $\mathcal{G}$ . Then, any mapping $L_{c} = CLC^{\top}, X_{c} = C^{\top}X$ , such that $C = U^{(k)}O$ satisfies: + +$$ +\left\| \boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - \boldsymbol {X} _ {c} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {X} _ {c} \right\| _ {F} ^ {2} = 0 +$$ + +where the columns of $\mathbf{U}^{(k)} \in \mathbb{R}^{n \times k}$ are the $k$ -leading eigenvectors of $\mathbf{L}$ , and $\mathbf{O} \in S\mathcal{O}(k)$ . + +See App. A.2 for proof. Proposition 3.4 implies that any coarsening operator $C$ whose columns span the same subspace as $U^{(k)}$ minimizes the IPE in (3.1) for any $k$ -smooth signals on the original graph. Thus, the second term in our objective (8) maximizes the Grassmann similarity (3) between $C$ and $U^{(k)}$ , aiming to find a valid coarsening operator (i.e., $C \in \mathcal{C}$ ) that satisfies $C = U^{(k)}O$ . + +Next, we present a theorem that provides bounds on the DEE (Def. 2.4) and REE (Def. 2.1) as functions of $\epsilon$ , which quantifies the deviation of the second term in our objective from its optimal value (if $C$ and $U^{(k)}$ span the same subspace, then $\mathrm{tr}(U^{(k)}(U^{(k)})^\top CC^\top) = k$ ). + +Proposition 3.5 Let $\pmb{L}$ be the Laplacian of a connected graph $\mathcal{G}$ , and let $\pmb{U}^{(k)}$ be the matrix containing its $k$ -leading eigenvectors. Suppose $\pmb{L}_c = \pmb{C}^\top \pmb{L}\pmb{C}$ is the Laplacian of a coarsened graph derived using a coarsening operator $\pmb{C}$ such that $\text{tr}(\pmb{U}^{(k)}(\pmb{U}^{(k)})^\top\pmb{C}\pmb{C}^\top) = k - \epsilon$ , and the constant vector in $\mathbb{R}^n$ is spanned by the columns of $\pmb{C}$ . Then, the eigenvalues of the original graphs $\{\lambda_i\}_{i=1}^k$ and the coarsened graph $\{\lambda_{c,i}^k\}_{i=1}^k$ satisfy, + +$$ +\frac {1}{\mu_ {1}} \lambda^ {(i)} \leq \lambda_ {c} ^ {(i)} \leq \frac {1}{\mu_ {k}} \frac {\left(1 + \epsilon \kappa\right) ^ {2}}{1 - (\epsilon \kappa) ^ {2} \left(\lambda^ {(i)} / \lambda^ {(2)}\right)} \lambda^ {(i)}, 2 \leq i \leq k +$$ + +and the Dirichlet energies of a $k$ -smooth graph signal $\pmb{x}$ on the graph, the inequalities: + +$$ +(1 - \epsilon \kappa) ^ {2} \| \boldsymbol {x} \| _ {L} \leq \| \boldsymbol {x} _ {c} \| _ {L _ {c}} \leq (1 + \epsilon \kappa) ^ {2} \| \boldsymbol {x} \| _ {L}, +$$ + +whenever $\epsilon \kappa < \frac{\lambda^{(2)}}{\lambda^{(i)}}$ . Here $\kappa = \frac{\lambda_{max}(\pmb{L})}{\lambda^{(2)}}$ , $\lambda_{max}(\pmb{L})$ is the maximum eigenvalue of $\pmb{L}, \pmb{P} = \pmb{C}^{\dagger}$ , $\mu_1, \mu_k$ and the first and $k$ eigenvalues of the matrix $\pmb{P}\pmb{P}^{\top}$ , $\| \pmb{x} \|_L = \pmb{x}^T\pmb{L}\pmb{x}$ , $\| \pmb{x}_c \|_{L_c} = \pmb{x}_c^T\pmb{L}_c\pmb{x}_c$ . + +See App.A.3 for proof. Prop. 3.5 shows that the bounds for both REE (Def. 2.1) and DEE (Def. 2.4) become tighter as $\epsilon$ decreases. Combining Prop. 3.4 and 3.5 implies that minimizing the IPE for general smooth signals reduces both the REE and DEE graph coarsening metrics. + +**Complexity.** Our approach formulates coarsening as an optimization problem, so its applicability depends on both computational complexity and convergence time. In App. C, we compare the complexity of our methods with FGC [14], an optimization-based coarsening approach considered + +![](images/5a5136930ad3dc5957e9e9404cdbd74fd5946f5fa15c4036e392d1542cbddf2e.jpg) +Figure 1: Node assignments of all methods on a synthetic graph generated from an SBM. Nodes with the same color belong to the same class (super-node). The bar plot on the right shows the Grassmann similarity between each method's coarsening matrix $C$ and the leading four eigenvectors $U^{(k)}$ of the graph Laplacian. The bottom bar shows the similarity for the ideal partitioning—i.e., between the block-model-based $C$ and $U^{(k)}$ . The maximum similarity in this case is 4. + +state-of-the-art on many benchmarks, which has been shown to accelerate GNNs training time. We show that our methods are similarly efficient while achieving better results. Moreover, we show that applying coarsening before a GCN is particularly advantageous for dense graphs with significantly more edges than nodes. Runtime comparisons supporting this claim are presented in App. C. Finally, App. D provides convergence plots illustrating the trade-off between speed and final objective value. + +# 4 Experimental results + +Visual illustration. A key aspect of graph coarsening is how well the global graph's structure is preserved. This can be assessed by how effectively the partitioning into super-nodes captures it. To illustrate this property, we provide a visual example showing how the super-nodes generated by our method align with the graph's global structure. This example highlights how our approach maintains a meaningful graph representation, despite focusing solely on functional relationships. + +The example is based on a synthetic graph generated using a Stochastic Block Model (SBM) [30] with four classes, each containing 10 vertices ( $N = 40$ ), an intra-class probability of $p = 0.9$ , and an inter-class probability of $q = 0.05$ . The feature matrix $\mathbf{X}$ is generated following the same graph signal generation mechanism described in (10). We set the target coarsened graph for all coarsening methods to have $k = 4$ super-nodes. In Figure 1, we present the results obtained by our methods (INGC/SINGC), alongside three other graph coarsening baselines: Feature-based Graph Coarsening (FGC) [14], which incorporates node features into the coarsening process, and the Local Variation Neighborhood (LVN) and Local Variation Edges (LVE) methods [10], which use the original graph Laplacian eigenvectors as part of their coarsening objectives. We observe that the super-nodes assignment of our methods closely aligns with the partitioning of the nodes to four classes according to the SBM, as indicated by the node colors in Figure 1. On the right-hand side of Figure 1, we present a bar plot comparing the Grassmann similarity between the coarsening matrices $C$ (which encode the vertex partitioning) produced by each method and the top four eigenvectors of the original graph Laplacian, $U^{(k)}$ . The bottom bar represents the matrix $C$ that encodes the ideal partition based on the underlying block model of the graph, serving as a baseline. We observe that our method achieves the highest similarity, closely approaching the ideal partitioning. Note that when the coarsening matrix $C$ and the eigenvector matrix $U^{(k)}$ span the same subspace, the Grassmann similarity reaches its maximum of 4. This comparison highlights our motivation for incorporating Grassmann similarity into our coarsening objective, as it preserves the graph's global structure. Numerically, this property is reflected in the REE metric; we present an extensive evaluation of this and other metrics in the following section. For more visual comparisons, see App.B. + +Table 1: Comparison of coarsening methods on four datasets using multiple metrics and coarsening ratios $(r)$ . For each method, we report REE, RE, HE, DEE, and INP on each dataset. Best results are in bold; second-best are underlined. The last two columns count how often each method achieved the best or second-best performance. + +
MethodKarate ClubLes MiserablesCoraCiteseer#Best #2-Best
r0.70.50.30.70.50.30.70.50.30.70.50.3
LVNREE0.301.523.150.361.397.820.571.314.230.681.584.1134
RE9.719.8710.3111.4211.9111.9111.4211.6211.7010.8511.0511.1400
HE1.741.892.251.922.602.541.892.503.171.932.523.4300
DEE36.247.846.665.199.490.239.170.790.240.066.3111.100
IPE0.710.721.092.022.561.912.872.231.730.930.981.0900
LVEREE0.820.482.261.054.567.600.811.945.140.791.624.2632
RE9.399.919.9711.5512.1412.4810.6211.5111.6910.5211.0211.1400
HE1.401.932.311.632.162.891.292.313.101.612.503.4000
DEE17.339.084.819.120.663.622.544.978.130.163.5109.000
IPE0.660.880.921.592.773.870.581.191.540.610.790.8800
FGCREE1.353.947.543.0810.3133.181.785.4015.991.588.9235.8800
RE8.708.819.269.979.9810.919.7010.8210.7610.4710.2310.3101
HE1.031.231.800.820.871.580.761.401.561.891.391.6102
DEE8.0511.5721.364.787.659.740.200.415.0119.57.871.3401
IPE0.550.580.941.052.173.670.411.013.671.100.490.6800
INGC (Ours)REE0.781.302.970.081.3010.400.860.840.760.710.620.4264
RE6.277.008.285.438.089.799.4910.1710.428.869.8510.1093
HE0.290.451.000.080.330.830.671.041.370.641.211.6193
DEE0.010.030.020.040.020.100.030.403.120.020.661.01111
IPE0.190.310.480.300.570.860.310.430.680.240.340.5884
SINGC (Ours)REE0.861.784.020.502.327.780.860.845.100.830.620.4230
RE6.098.058.749.079.6010.499.5410.1710.959.329.769.9247
HE0.270.851.460.510.721.290.691.041.840.831.141.2747
DEE0.020.516.560.470.350.100.030.408.121.011.450.1047
IPE0.190.430.590.210.631.070.310.430.690.270.250.4766
+ +Graph coarsening metrics. Here, we evaluate the performance of our methods on several benchmark datasets using the coarsening metrics from Sec. 2. We compare them with current SOTA coarsening methods—LVN and LVE—which are known to perform best on structural-preservation metrics (e.g., REE, RE), and FGC, which is considered the leading method for metrics that also consider the features of the nodes. We conduct experiments on four datasets: The Karate Club[31], Les Miserables[32], Cora[33], and CiteSeer[34]. Note that the Cora and CiteSeer datasets include node features, whereas the Karate Club and Les Miserables datasets do not. For the latter two, we generated node features using the signal generation mechanism presented in Section 3.1. + +Table 1 summarizes the performance of our methods (INGC and SINGC) and the baselines (FGC, LVN and LVE) across different datasets and coarsening ratios ( $r = \frac{k}{n} = 0.7$ , 0.5, and 0.3). The best performance for each metric is highlighted in bold, and the second-best is underlined. The last two columns summarize the number of settings in which each method achieved the lowest or second-lowest score compared to others. We observe that INGC achieves the best overall performance across all graph metrics. SINGC, a more efficient variant, is also highly competitive—often ranking second and occasionally achieving the best results. Baseline methods (LVN, LVE, and FGC) show mixed performance. LVN and LVE perform well on REE for some datasets, indicating good spectral preservation, but generally underperform on metrics involving node features. FGC, the only baseline that incorporates node features in coarsening, outperforms the others on related metrics. However, both our methods consistently surpass FGC. The strong RE score of our approach demonstrates its broader ability to preserve graph structure, even beyond the theoretical setting of Theorem 3.2, as the datasets used are connected or have far fewer than $n - k$ connected components. + +The hyperparameters in these experiments were selected via grid search, a standard approach in graph coarsening when aiming to minimize a specific metric. The best results for each metric were obtained using different hyperparameters, as reported in App. G.3. This variability highlights the + +Table 2: Node classification accuracy across datasets and coarsening ratios $(r)$ . Best results are bolded; second-best are underlined. The last two rows count best and second-best results per method. + +
DatasetrGCONDSCAL(LV)FGCMGCINGC (Ours)SINGC (Ours)
Cora0.381.56 ± 0.6079.42 ± 1.7185.79 ± 0.2484.56 ± 1.4087.55 ± 0.1684.51 ± 0.33
0.181.37 ± 0.4071.38 ± 3.6281.46 ± 0.7976.02 ± 0.9383.38 ± 0.4782.76 ± 0.32
0.0579.93 ± 0.4455.32 ± 7.0380.01 ± 0.51-77.42 ± 0.7877.81 ± 0.68
CiteSeer0.372.43 ± 0.9468.87 ± 1.3774.64 ± 1.3774.60 ± 1.2076.89 ± 0.2376.66 ± 0.27
0.170.46 ± 0.4771.38 ± 3.6273.36 ± 0.5370.57 ± 1.2572.63 ± 0.2569.71 ± 0.72
0.0564.03 ± 2.4055.32 ± 7.0371.02 ± 0.96-66.02 ± 0.3266.37 ± 0.57
Co-phy0.0593.05 ± 0.2673.09 ± 7.4194.27 ± 0.2594.52 ± 0.1994.29 ± 0.1094.04 ± 0.06
0.0392.81 ± 0.3163.65 ± 9.6594.02 ± 0.2093.64 ± 0.2594.20 ± 0.1393.52 ± 0.13
0.0192.79 ± 0.4031.08 ± 2.6593.08 ± 0.22-93.95 ± 0.2093.20 ± 0.10
Pubmed0.0578.16 ± 0.3072.82 ± 2.6280.73 ± 0.4481.89 ± 0.0183.59 ± 0.2283.55 ± 0.32
0.0378.04 ± 0.4770.24 ± 2.6379.91 ± 0.3080.70 ± 0.0181.93 ± 0.2283.19 ± 0.18
0.0177.20 ± 0.2054.49 ± 10.578.42 ± 0.43-79.09 ± 0.2679.96 ± 0.34
Co-CS0.0586.29 ± 0.6334.45 ± 10.089.60 ± 0.39-90.84 ± 0.1290.92 ± 0.22
0.0386.32 ± 0.4526.06 ± 9.2988.29 ± 0.79-89.59 ± 0.3889.99 ± 0.41
0.0184.01 ± 0.0214.42 ± 8.5086.37 ± 1.36-87.93 ± 0.3383.39 ± 0.33
#Best003174
#2-Best103154
+ +need to tune hyperparameters based on the specific application and the most relevant coarsening metric. For example, minimizing REE may be crucial for clustering, while tasks like graph pooling and node classification—which rely heavily on node features—benefit from prioritizing DEE and INP during tuning. App. F presents a hyperparameter study showing each parameter's contribution and sensitivity across metrics. + +Node classification. Next, we evaluate our method on node classification using several benchmark datasets. This task assesses how well coarsened graphs preserve structural and feature information for accurate label prediction [4]. Following Kumar et al. [14], we train a Graph Neural Network (GNN) on the coarsened graph and predict node labels for the original graph, reducing training time due to fewer nodes and edges. Specifically, we: (1) generate a coarsened graph using a selected method; (2) compute super-node labels via $\boldsymbol{y}_c = C^\dagger \boldsymbol{y}$ ; (3) train a GNN on the coarsened graph; and (4) evaluate predictions $\hat{\boldsymbol{y}} = \mathrm{GNN}(\boldsymbol{L},\boldsymbol{X})$ against the original labels $\boldsymbol{y}$ . Note that this task is used solely for evaluation, with all labels $\boldsymbol{y}$ available throughout. + +We replicated the experimental settings from Kumar et al. [14] and Huang et al. [35], both of which employ a Graph Convolutional Network (GCN) [36]. We then compared the performance of our methods (INGC and SINGC) with the top-performing methods in those papers that are considered to be SOTA, namely SCAL [35], Feature Graph Coarsening (FGC) [14], Graph Condensation (GCOND) [7], and Multi-Component Coarsening (MGC) [37]. The datasets include two medium-sized graphs (Cora and Citeseer) and three large-scale graphs (Co-Physics, Pubmed, and Co-CS). Each method's effectiveness was evaluated using 10-fold cross-validation. A detailed description of the experimental setup is in App. G.2. + +The results are reported in Table 2 with mean accuracy and standard deviation across the folds for each method at different coarsening ratios $r$ . We observe that INGC outperforms the SOTA methods on large datasets (Co-Physics, Pubmed, and Co-CS), and matches their performance on medium-sized datasets. SINGC outperforms baseline methods in most settings, despite having a simpler optimization objective. A key takeaway from the results is that the integration of node features into the coarsening process gives FGC, INGC and SINGC a competitive advantage over methods that primarily focus on structural properties, such as SCAL and GCOND, as was also indicated in [14]. This relationship is intuitive, as node classification relies not only on structural relationships but also on the meaningful preservation of node features. + +App. F presents an ablation study on $\beta$ , highlighting the role of the Grassmann similarity term. App. E discusses Assumption 3.3, evaluates its validity on real datasets, and shows our method remains effective even when it is not fully met. Moreover, our empirical results indicate that SINGC benefits when a larger fraction of signal energy resides in the low-frequency band, and its best performance reported on PubMed, where this assumption is most valid. + +The source code for all experiments is available at: Code. + +# 5 Conclusion + +In this paper, we introduced a new graph coarsening method that focuses on preserving the inner products of graph signals during the coarsening process. We demonstrated that, although primarily considering node features, our approach also maintains the global structure of the graph. Our methods, INGC and SINGC, outperform state-of-the-art techniques across various graph coarsening metrics and tasks (e.g., node classification), showcasing their versatility and effectiveness in preserving essential graph properties. These results highlight the potential of our approach for graph-based learning applications. However, similar to other optimization-based coarsening techniques, our method faces scalability limitations when applied to extremely large graphs. This challenge may be addressed in practical scenarios by partitioning the graph into manageable subgraphs, processing each subgraph independently, potentially in parallel, and subsequently reconnecting them [38]. A critical assumption underlying our work is that the node features exhibit smoothness with respect to the graph topology, an attribute closely related to homophily. Consequently, our evaluation is restricted to homophilic graphs. For future research, an extension to heterophilic datasets could involve adapting the second term in our objective to facilitate alignment with alternative frequency bands. For example, the low-frequency basis $U^{(k)}$ may be replaced with a different basis representing mid or high frequencies, guided by prior knowledge of label-frequency alignment. Future directions also include integrating our coarsening into graph pooling and evaluating its impact on improving GNN performance. + +# Acknowledgments + +We thank the anonymous reviewers for their insightful feedback. This work was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 802735-ERC-DIFFOP. + +# References + +[1] Stanley Wasserman and Katherine Faust. Social network analysis: Methods and applications. 1994. +[2] Georgios A Pavlopoulos, Maria Secrier, Charalampos N Moschopoulos, Theodoros G Soldatos, Sophia Kossida, Jan Aerts, Reinhard Schneider, and Pantelis G Bagos. Using graph theory to analyze biological networks. BioData mining, 4:1-27, 2011. +[3] Maarten Van Steen. Graph theory and complex networks. An introduction, 144(1), 2010. +[4] Mohammad Hashemi, Shengbo Gong, Juntong Ni, Wenqi Fan, B Aditya Prakash, and Wei Jin. A comprehensive survey on graph reduction: Sparsification, coarsening, and condensation. arXiv preprint arXiv:2402.03358, 2024. +[5] Joshua D Batson, Daniel A Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 255–262, 2009. +[6] Ryan Wickman, Xiaofei Zhang, and Weizi Li. A generic graph sparsification framework using deep reinforcement learning. In 2022 IEEE International Conference on Data Mining (ICDM), pages 1221-1226. IEEE, 2022. +[7] Wei Jin, Lingxiao Zhao, Shichang Zhang, Yozen Liu, Jiliang Tang, and Neil Shah. Graph condensation for graph neural networks. arXiv preprint arXiv:2110.07580, 2021. +[8] Yang Liu, Deyu Bo, and Chuan Shi. Graph condensation via eigenbasis matching. arXiv preprint arXiv:2310.09202, 2023. +[9] Jie Chen, Yousef Saad, and Zechen Zhang. Graph coarsening: from scientific computing to machine learning. SeMA Journal, 79(1):187-223, 2022. +[10] Andreas Loukas and Pierre Vandergheynst. Spectrally approximating large graphs with smaller graphs. In International conference on machine learning, pages 3237-3246. PMLR, 2018. + +[11] Kristen LeFevre and Evimaria Terzi. Grass: Graph structure summarization. In Proceedings of the 2010 SIAM International Conference on Data Mining, pages 454-465. SIAM, 2010. +[12] Yuanyuan Tian, Richard A Hankins, and Jignesh M Patel. Efficient aggregation for graph summarization. In Proceedings of the 2008 ACM SIGMOD international conference on Management of data, pages 567-580, 2008. +[13] Sorour E Amiri, Bijaya Adhikari, Aditya Bharadwaj, and B Aditya Prakash. Netgist: Learning to generate task-based network summaries. In 2018 IEEE International Conference on Data Mining (ICDM), pages 857-862. IEEE, 2018. +[14] Manoj Kumar, Anurag Sharma, Shashwat Saxena, and Sandeep Kumar. *Featured graph coarsening with similarity guarantees*. In International Conference on Machine Learning, pages 17953–17975. PMLR, 2023. +[15] Ioan Mackenzie James. The topology of Stiefel manifolds, volume 24. Cambridge University Press, 1976. +[16] Thomas Bendokat, Ralf Zimmermann, and P-A Absil. A grassmann manifold handbook: Basic geometry and computational aspects. arXiv preprint arXiv:2011.13699, 2020. +[17] Alan Edelman, Tomás A Arias, and Steven T Smith. The geometry of algorithms with orthogonality constraints. SIAM journal on Matrix Analysis and Applications, 20(2):303-353, 1998. +[18] Ido Cohen and Ronen Talmon. Functorial comparison of graph signals using the geodesic distance on the grassmann manifold. arXiv preprint arXiv:2406.06989, 2024. +[19] Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17:395-416, 2007. +[20] Andreas Loukas. Graph reduction with spectral and cut guarantees. Journal of Machine Learning Research, 20(116):1-42, 2019. +[21] Yike Liu, Tara Safavi, Abhilash Dighe, and Danai Koutra. Graph summarization methods and applications: A survey. ACM computing surveys (CSUR), 51(3):1-34, 2018. +[22] Gecia Bravo Hermsdorff and Lee Gunderson. A unifying framework for spectrum-preserving graph sparsification and coarsening. Advances in Neural Information Processing Systems, 32, 2019. +[23] Vassilis Kalofolias and Nathanaël Perraudin. Large scale graph learning from smooth signals. arXiv preprint arXiv:1710.05654, 2017. +[24] Di Ming, Chris Ding, and Feiping Nie. A probabilistic derivation of lasso and 112-norm feature selections. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pages 4586-4593, 2019. +[25] Fan RK Chung. Spectral graph theory, volume 92. American Mathematical Soc., 1997. +[26] Vassilis Kalofolias. How to learn a graph from smooth signals. In Artificial intelligence and statistics, pages 920-929. PMLR, 2016. +[27] Dimitri P Bertsekas. Nonlinear programming. Journal of the Operational Research Society, 48 (3):334-334, 1997. +[28] Xiaowen Dong, Dorina Thanou, Pascal Frossard, and Pierre Vandergheynst. Learning laplacian matrix in smooth graph signal representations. IEEE Transactions on Signal Processing, 64 (23):6160-6173, 2016. +[29] Felix Dietrich, Or Yair, Rotem Mulayoff, Ronen Talmon, and Ioannis G Kevrekidis. Spectral discovery of jointly smooth features for multimodal data. SIAM Journal on Mathematics of Data Science, 4(1):410-430, 2022. + +[30] Emmanuel Abbe. Community detection and stochastic block models: recent developments. The Journal of Machine Learning Research, 18(1):6446-6531, 2017. +[31] Wayne W Zachary. An information flow model for conflict and fission in small groups. Journal of anthropological research, 33(4):452-473, 1977. +[32] Donald Ervin Knuth. The Stanford GraphBase: a platform for combinatorial computing, volume 1. AcM Press New York, 1993. +[33] Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymour. Automating the construction of internet portals with machine learning. Information Retrieval, 3:127-163, 2000. +[34] C Lee Giles, Kurt D Bollacker, and Steve Lawrence. Citeseer: An automatic citation indexing system. In Proceedings of the third ACM conference on Digital libraries, pages 89-98, 1998. +[35] Zengfeng Huang, Shengzhong Zhang, Chong Xi, Tang Liu, and Min Zhou. Scaling up graph neural networks via graph coarsening. In Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining, pages 675-684, 2021. +[36] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016. +[37] Subhanu Halder, Manoj Kumar, and Sandeep Kumar. Multi-component coarsened graph learning for scaling graph machine learning. In *Companion Proceedings of the ACM on Web Conference* 2025, pages 1001-1004, 2025. +[38] George Karypis and Vipin Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on scientific Computing, 20(1):359-392, 1998. +[39] Daniel Spielman. Spectral and algebraic graph theory. Yale lecture notes, draft of December, 4: 47, 2019. +[40] Anil K Jain and Richard C Dubes. Algorithms for clustering data. Prentice-Hall, Inc., 1988. +[41] Rotem Mulayoff, Tomer Michaeli, and Daniel Soudry. The implicit bias of minima stability: A view from function space. Advances in Neural Information Processing Systems, 34:17749-17761, 2021. +[42] Antonio Ortega, Pascal Frossard, Jelena Kovacevic, José MF Moura, and Pierre Vandergheynst. Graph signal processing: Overview, challenges, and applications. Proceedings of the IEEE, 106 (5):808-828, 2018. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction accurately summarize our main contributions: a graph coarsening method that preserves inner products of node features, maintaining both structural and feature fidelity. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix - Theorems' proofs + +# A.1 Proof of Proposition 3.2 + +[Proof of Proposition 3.2] Given a graph $\mathcal{G}$ with a graph Laplacian $\pmb{L}$ and its coarsened graph $\mathcal{G}_c$ with a graph Laplacian $\pmb{L}_{c} = \pmb{C}^{T}\pmb{L}\pmb{C}$ , and assume that for any two graph signals $\pmb{x},\pmb{y}\in \mathbb{R}^n$ , the following condition is satisfied: + +$$ +\boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} = \boldsymbol {x} _ {c} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {y} _ {c} \tag {11} +$$ + +We plug in the definitions of $\pmb{x}_c = \pmb{P}\pmb{x}$ and $\pmb{y}_c = \pmb{P}\pmb{y}$ in (11) and obtain: + +$$ +\begin{array}{l} \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} = \boldsymbol {x} _ {c} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {y} _ {c} \\ = (\boldsymbol {P} \boldsymbol {x}) ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {P} \boldsymbol {y} \\ = \boldsymbol {x} ^ {\top} \boldsymbol {P} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {P} \boldsymbol {y} \\ = \boldsymbol {x} ^ {\top} \boldsymbol {L} _ {l} \boldsymbol {y}. \tag {12} \\ \end{array} +$$ + +where in the last equality we plug-in the definition of the lifted Laplacian (reconstructed Laplacian) $\pmb{L}_l = \pmb{P}^T\pmb{L}_c\pmb{P}$ . + +Assuming (12) holds for every pair of signals $\pmb{x}, \pmb{y} \in \mathbb{R}^n$ , one can choose specific signals such that $\pmb{L}[i,j] = \pmb{L}_l[i,j]$ for all $i,j = 1,\dots,n$ , allowing us to conclude: + +$$ +\boldsymbol {L} = \boldsymbol {L} _ {l} +$$ + +This implies that the full Laplacian $\pmb{L}$ can be fully reconstructed for $L_{c}$ . + +# A.2 Proof of Proposition 3.4 + +[Proof of Proposition 3.4] Let $\pmb{x}, \pmb{y} \in \mathbb{R}^n$ be two k-smooth signals on the graph $\mathcal{G}$ with graph Laplacian $L$ . Define $\pmb{x}_c = C^\top \pmb{x}, \pmb{y}_c = C^\top \pmb{y} \in \mathbb{R}^k$ , and let $L_c = C^\top LC$ , where $C = U^{(k)}O$ , and $O \in \mathcal{O}$ is some k-dimension rotation matrix. Then, the following relation holds: + +$$ +\begin{array}{l} \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} - \boldsymbol {x} _ {c} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {y} _ {c} = \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} - \left(\left(\boldsymbol {C} ^ {\top} \boldsymbol {x}\right) ^ {\top} \boldsymbol {C} ^ {\top} \boldsymbol {L} \boldsymbol {C} \boldsymbol {C} ^ {\top} \boldsymbol {y} \right. \\ = \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} - \left(\left(\boldsymbol {U} ^ {(k)} \boldsymbol {O}\right) ^ {\top} \boldsymbol {x}\right) ^ {\top} \left(\boldsymbol {U} ^ {(k)} \boldsymbol {O}\right) ^ {\top} \boldsymbol {L} \boldsymbol {U} ^ {(k)} \boldsymbol {O} \left(\boldsymbol {U} ^ {(k)} \boldsymbol {O}\right) ^ {\top} \boldsymbol {y} \\ = \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} - \boldsymbol {x} ^ {\top} \boldsymbol {U} ^ {(k)} \boldsymbol {O O} ^ {\top} (\boldsymbol {U} ^ {(k)}) ^ {\top} \boldsymbol {L} \boldsymbol {U} ^ {(k)} \boldsymbol {O O} ^ {\top} (\boldsymbol {U} ^ {(k)}) ^ {\top} \boldsymbol {y} \\ = \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} - \boldsymbol {x} ^ {\top} \boldsymbol {U} ^ {(k)} (\boldsymbol {U} ^ {(k)}) ^ {\top} L \boldsymbol {U} ^ {(k)} (\boldsymbol {U} ^ {(k)}) ^ {\top} \boldsymbol {y} \\ = \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} - \boldsymbol {x} ^ {\top} \boldsymbol {L} \boldsymbol {y} = 0 \\ \end{array} +$$ + +where the third equality holds because $\pmb{O}$ is a rotation matrix satisfying $\pmb{OO}^{\top} = \pmb{I}_{k\times k}$ . The fifth equality holds because $\pmb{x}$ and $\pmb{y}$ are k-smooth and satisfy $\pmb{x} = \pmb{U}^{(k)}(\pmb{U}^{(k)})^{\top}\pmb{x}$ and $\pmb{y} = \pmb{U}^{(k)}(\pmb{U}^{(k)})^{\top}\pmb{y}$ . + +Thus, for any matrix $X$ whose columns are k-smooth signals, we have: + +$$ +\left\| \boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - \boldsymbol {X} _ {c} ^ {\top} \boldsymbol {L} _ {c} \boldsymbol {X} _ {c} \right\| _ {F} ^ {2} = 0. +$$ + +# A.3 Proof of Proposition 3.5 + +The proof of Proposition 3.5 relies on the following definition and lemma. + +Definition A.1 (Restricted spectral approximation [10]) Let $R$ be a $k$ -dimensional subspace of $\mathbb{R}^n$ . Matrices $\pmb{L}_c$ and $\pmb{L}$ are $(R,\epsilon)$ -similar if there exists an $\epsilon > 0$ such that + +$$ +\| \boldsymbol {x} - \boldsymbol {x} _ {l} \| _ {L} \leq \epsilon \| \boldsymbol {x} \| _ {L}, \quad f o r a l l \boldsymbol {x} \in R, +$$ + +where $\pmb{x}_l = \pmb{C}\pmb{C}^\dagger\pmb{x}$ . + +Lemma A.2 Let $\pmb{L}$ be the Laplacian matrix of a connected graph $\mathcal{G}$ , and let $\pmb{U}^{(k)}$ be the matrix containing its $k$ -leading eigenvectors. Suppose $\pmb{L}_c = \pmb{C}^\top \pmb{L}\pmb{C}$ is the Laplacian matrix of a coarsened graph derived using a coarsening operator $\pmb{C}$ with normalized columns such that + +$$ +\operatorname {t r} \left(\boldsymbol {U} ^ {(k)} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {C} \boldsymbol {C} ^ {\top}\right) = k - \epsilon . +$$ + +Then, the matrices $\mathbf{L}$ and $\mathbf{L}_c$ are $(R,\epsilon \kappa)$ -similar, where $\kappa = \frac{\lambda_{\max}(\mathbf{L})}{\lambda_2(\mathbf{L})}$ , and $R = \text{span}(\mathbf{U}^{(k)})$ . + +[Proof of Lemma A.2] We note the projection matrix defined by $C$ as $\Pi_C = CC^\top$ . Given the trace condition $\operatorname{tr}(\boldsymbol{U}^{(k)}(\boldsymbol{U}^{(k)})^\top \Pi_C) = k - \epsilon$ , we can express it as: + +$$ +\operatorname {t r} \left(\boldsymbol {U} ^ {(k)} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {\Pi} _ {C}\right) = \operatorname {t r} \left(\left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {\Pi} _ {C} \boldsymbol {U} ^ {(k)}\right) = k - \epsilon . +$$ + +From this, we immediately obtain: + +$$ +\operatorname {t r} \left(\left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \left(\boldsymbol {I} - \boldsymbol {\Pi} _ {C}\right) \boldsymbol {U} ^ {(k)}\right) = \epsilon . \tag {13} +$$ + +This follows from the fact that: + +$$ +\begin{array}{l} k = \operatorname {t r} \left(\left(U ^ {(k)}\right) ^ {\top} U ^ {(k)}\right) = \operatorname {t r} \left(U ^ {(k)} \left(U ^ {(k)}\right) ^ {\top}\right) = \operatorname {t r} \left(U ^ {(k)} I _ {n \times n} U ^ {(k)}\right) \\ = \operatorname {t r} \left(\left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {\Pi} _ {C} \boldsymbol {U} ^ {(k)}\right) + \operatorname {t r} \left(\left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} (\boldsymbol {I} - \boldsymbol {\Pi} _ {C}) \boldsymbol {U} ^ {(k)}\right), \\ \end{array} +$$ + +where the first equality holds because $\pmb{U}^{(k)}$ is orthonormal matrix. + +Next, we express the term $\| \pmb{x} - \pmb{x}_l\|_L$ . Since the columns of $\pmb{C}$ are orthogonal, we can use $\pmb{x}_l = \pmb{CP}\pmb{x} = \pmb{CC}^\dagger\pmb{x} = \pmb{CC}^\top\pmb{x}$ , and obtain: + +$$ +\begin{array}{l} \left\| \boldsymbol {x} - \boldsymbol {x} _ {l} \right\| _ {L} = \left(\boldsymbol {x} - \boldsymbol {C C} ^ {\top} \boldsymbol {x}\right) ^ {\top} \boldsymbol {L} \left(\boldsymbol {x} - \boldsymbol {C C} ^ {\top} \boldsymbol {x}\right) \\ = \left(\left(\boldsymbol {I} - \boldsymbol {C C} ^ {\top}\right) \boldsymbol {x}\right) ^ {\top} \boldsymbol {L} \left(\boldsymbol {I} - \boldsymbol {C C} ^ {\top}\right) \boldsymbol {x}. \tag {14} \\ \end{array} +$$ + +Using the Rayleigh quotient [39], we can bound by: + +$$ +\begin{array}{l} \left\| \boldsymbol {x} - \boldsymbol {x} _ {l} \right\| _ {L} = \left(\left(\boldsymbol {I} - \boldsymbol {C C} ^ {\top}\right) \boldsymbol {x}\right) ^ {\top} \boldsymbol {L} \left(\left(\boldsymbol {I} - \boldsymbol {C C} ^ {\top}\right) \boldsymbol {x}\right) \\ \leq \lambda_ {\max } (\boldsymbol {L}) \| (\boldsymbol {I} - \boldsymbol {C C} ^ {\top}) \boldsymbol {x} \| _ {2} ^ {2}. \tag {15} \\ \end{array} +$$ + +Next, we proceed to bound the term $\| (\boldsymbol{I} - \boldsymbol{C}\boldsymbol{C}^{\top})\boldsymbol{x}\|_{2}^{2}$ . Since $\boldsymbol{x}$ is spanned by $U^{(k)}$ , we write $\boldsymbol{x} = \boldsymbol{U}^{(k)}\boldsymbol{z}$ . Therefore, we get: + +$$ +\left\| \left(\boldsymbol {I} - \boldsymbol {C C} ^ {\top}\right) \boldsymbol {x} \right\| _ {2} ^ {2} = \boldsymbol {z} ^ {\top} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \left(\boldsymbol {I} - \boldsymbol {C C} ^ {\top}\right) \boldsymbol {U} ^ {(k)} \boldsymbol {z}. +$$ + +From (13), we know that the maximum eigenvalue of $(\pmb{U}^{(k)})^{\top}(\pmb{I} - \pmb{\Pi}_C)\pmb{U}^{(k)}$ is bounded by $\epsilon$ . Thus, by applying the Rayleigh quotient, we obtain: + +$$ +\left\| \left(\boldsymbol {I} - \boldsymbol {C C} ^ {\top}\right) \boldsymbol {x} \right\| _ {2} ^ {2} \leq \epsilon \| \boldsymbol {z} \| _ {2} ^ {2} = \epsilon \| \boldsymbol {x} \| _ {2} ^ {2}. +$$ + +Substituting this bound into (15), we have: + +$$ +\left\| \boldsymbol {x} - \boldsymbol {x} _ {l} \right\| _ {L} \leq \epsilon \lambda_ {\max } (\boldsymbol {L}) \| \boldsymbol {x} \| _ {2} ^ {2}. +$$ + +Since $\pmb{L}$ is the graph Laplacian of a connected graph, it has only one zero eigenvalue, corresponding to the constant vector. Assuming $\pmb{x}$ is not a constant vector, we can bound $\| \pmb{x} \|_2^2$ using the Rayleigh quotient: + +$$ +\left\| \boldsymbol {x} \right\| _ {2} ^ {2} \geq \frac {\left\| \boldsymbol {x} \right\| _ {L}}{\lambda_ {2} (\boldsymbol {L})}. +$$ + +Substituting this into the previous inequality, we obtain: + +$$ +\left\| \boldsymbol {x} - \boldsymbol {x} _ {l} \right\| _ {L} \leq \epsilon \frac {\lambda_ {\max } (\boldsymbol {L})}{\lambda_ {2} (\boldsymbol {L})} \| \boldsymbol {x} \| _ {L} = \epsilon \kappa \| \boldsymbol {x} \| _ {L}, +$$ + +where $\kappa = \frac{\lambda_{\max}(\pmb{L})}{\lambda_2(\pmb{L})}$ is the condition number of $\pmb{L}$ . + +Finally, if $x$ is a constant vector, then since the columns of $C$ span the constant vector, we have: + +$$ +\left\| \boldsymbol {x} - \boldsymbol {x} _ {l} \right\| _ {L} = \left\| \boldsymbol {x} - \boldsymbol {C C} ^ {\top} \boldsymbol {x} \right\| _ {L} = 0. +$$ + +Thus, for all $\pmb{x} \in \mathrm{span}(U^{(k)})$ , we conclude that: + +$$ +\left\| \boldsymbol {x} - \boldsymbol {x} _ {l} \right\| _ {L} \leq \epsilon \kappa \left\| \boldsymbol {x} \right\| _ {L}. +$$ + +i.e $\pmb{L}$ and $L_{c}$ are $(R,\epsilon \kappa)$ similar. + +Then, according to Theorem 13 and Corollary 12 in [20], if the full graph Laplacian $\pmb{L}$ and the coarsen graph Laplacian are $\pmb{L}_c$ are $(U^{(k)},\epsilon \kappa)$ -similar, they satisfy the following inequalities: : + +$$ +\begin{array}{l} (1 - \epsilon \kappa) \| \boldsymbol {x} \| _ {L} \leq \| \boldsymbol {x} | _ {L _ {c}} \leq (1 + \epsilon \kappa) \| \boldsymbol {x} \| _ {L} \\ \frac {1}{\mu_ {1}} \lambda^ {(i)} \leq \lambda_ {c} ^ {(i)} \leq \frac {1}{\mu_ {2}} \frac {(1 + \epsilon \kappa) ^ {2}}{1 - (\epsilon \kappa) ^ {2} (\lambda^ {(i)} / \lambda^ {(2)})} \lambda^ {(i)}, \quad 2 \leq i \leq k \\ \end{array} +$$ + +according to Proposition 3.5, where $\kappa = \frac{\lambda_{\max}(\mathbf{L})}{\lambda_2(\mathbf{L})}$ , and $\mu_1,\mu_2$ and the first and $k$ eigenvalues of the matrix $\mathbf{P}\mathbf{P}^{\top}$ . + +![](images/4e3a4c5d38d3de824bde58d4597630245d6901ba4b0514c817f471a8b09cdde6.jpg) +(a) + +![](images/c6c37b5f1405364dd328473306932f125231b8ac3e75fa9f2e597e09c12ddf6f.jpg) +(b) +Figure 2: Clustering of the Karate Club network using (a) our SINGC method and (b) conventional $k$ -means clustering on the leading eigenvectors. Each color represents a distinct cluster, and the coarsened graph is obtained by mapping nodes from the same cluster (color) to a single super-node. We observe that the clusters produced by SINGC are more balanced, which is advantageous for downstream graph learning tasks. + +# A.4 Gradient Computation + +We start with a recap of our suggested objective function: + +$$ +\begin{array}{l} \min _ {\boldsymbol {X} _ {c}, \boldsymbol {C}} f (\boldsymbol {X} _ {c}, \boldsymbol {C}) = \| \boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - \boldsymbol {X} _ {c} ^ {\top} \boldsymbol {C} ^ {\top} \boldsymbol {L} \boldsymbol {C} \boldsymbol {X} _ {c} \| _ {F} - \beta t r (\boldsymbol {U} ^ {(k)} (\boldsymbol {U} ^ {(k)}) ^ {\top} \boldsymbol {C} \boldsymbol {C} ^ {\top}) \\ + \lambda \| \boldsymbol {C} ^ {T} \| _ {1, 2} ^ {2} - \alpha \log \det (\boldsymbol {L} _ {c} + \boldsymbol {J}) \\ \begin{array}{c c} \text {s . t .} & X _ {c} = C ^ {\dagger} X \end{array} \\ \end{array} +$$ + +The gradient of each term with respect to $C$ is: + +$$ +\begin{array}{l} \nabla_ {\boldsymbol {C}} \left(- t r \left(\boldsymbol {U} ^ {(k)} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {C} \boldsymbol {C} ^ {\top}\right)\right) = - 2 \boldsymbol {U} ^ {(k)} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {C} \\ \nabla_ {C} \left(\| X ^ {\top} L X - X _ {c} ^ {\top} C ^ {\top} L C X _ {c} \| _ {F}\right) = - \left(2 L C X _ {c} \left(X ^ {\top} L X - \left(L C X _ {c}\right) ^ {\top} C X _ {c}\right) X _ {c} ^ {\top} \right. \\ + 2 \boldsymbol {L} ^ {\top} \boldsymbol {C} \boldsymbol {X} _ {c} (\boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - (\boldsymbol {C} \boldsymbol {X} _ {c}) ^ {\top} \boldsymbol {L} \boldsymbol {C} \boldsymbol {X} _ {c}) \boldsymbol {X} _ {c} ^ {\top}) \\ \nabla_ {\boldsymbol {C}} \left(\| \boldsymbol {C} ^ {T} \| _ {1, 2} ^ {2}\right) = \boldsymbol {C} \mathbf {1} _ {k \times k} \\ \nabla_ {\boldsymbol {C}} \left(\operatorname {l o g d e t} \left(\boldsymbol {L} _ {c} + \boldsymbol {J}\right)\right) = \boldsymbol {L C} \left(\boldsymbol {C} ^ {\top} \boldsymbol {L} \boldsymbol {C} + \boldsymbol {J}\right) ^ {- 1} \\ \end{array} +$$ + +where the third was shown in [14], assuming all elements of $C$ to non-negative (since $C \in \mathcal{C}$ ). The full gradient with respect to $C$ of (9) is: + +$$ +\begin{array}{l} \nabla_ {C} f (\boldsymbol {C}, \boldsymbol {X} _ {c}) = 2 \beta \boldsymbol {U} ^ {(k)} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {C} + \lambda \boldsymbol {C} \\ - \left(2 \boldsymbol {L} \boldsymbol {C} \boldsymbol {X} _ {c} \left(\boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - \left(\boldsymbol {L} \boldsymbol {C} \boldsymbol {X} _ {c}\right) ^ {\top} \boldsymbol {C} \boldsymbol {X} _ {c}\right) \boldsymbol {X} _ {c} ^ {\top} \right. \\ + 2 \boldsymbol {L} ^ {\top} \boldsymbol {C} \boldsymbol {X} _ {c} (\boldsymbol {X} ^ {\top} \boldsymbol {L} \boldsymbol {X} - (\boldsymbol {C} \boldsymbol {X} _ {c}) ^ {\top} \boldsymbol {L} \boldsymbol {C} \boldsymbol {X} _ {c}) \boldsymbol {X} _ {c} ^ {\top}) \\ + \lambda C \mathbf {1} _ {k \times k} - \alpha \left(L C \left(C ^ {\top} L C + J\right) ^ {- 1}\right) \tag {16} \\ \end{array} +$$ + +We note that in case of the SINGC Algorithm the computed gradient is simpler and can be express as: + +$$ +\nabla_ {C} \tilde {f} (\boldsymbol {C}, \boldsymbol {X} _ {c}) = 2 \boldsymbol {U} ^ {(k)} \left(\boldsymbol {U} ^ {(k)}\right) ^ {\top} \boldsymbol {C} + \lambda \boldsymbol {C} \mathbf {1} _ {k \times k} - \alpha \left(\boldsymbol {L C} \left(\boldsymbol {C} ^ {\top} \boldsymbol {L C} + \boldsymbol {J}\right) ^ {- 1}\right) \tag {17} +$$ + +![](images/5bbdc332b2ad4201311f253ab4e0a0fc49decf1dbe9b46da1068046006bc0c7c.jpg) +Figure 3: Convergence rates of INGC and SINGC methods for Cora and CiteSeer datasets. CiteSeer results are shown in blue with varying line styles for different learning rates, while Cora results are shown in red. Gradient steps are on the x-axis, and normalized loss values are on the y-axis. + +![](images/1c0ebda29cafc2f7ab0366c26ed53c3996e8aa695ef900440af1afd230280a51.jpg) + +# B Methods Performance - Visual Comparison + +Table 3: Comparison of gradient expressions and time complexities for FGC, INGC, and SINGC. + +
FGCINGCSINGC
Gradient Expression∇CF(C,Xc)=2((CXc-X)+L(CXc))Xc+λC1k×k-α(LC(CTLC+J)-1)∇CF(C,Xc)=2βU(k)(U(k))TCC-[2L(CXc)(XTLX-(LCXc)T(CXc))XcT]+λC1k×k-α(LC(CTLC+J)-1)∇CF(C)=2U(k)(U(k))TCC+λC1k×k-α(LC(CTLC+J)-1)
Theoretical Time ComplexityO(n2(k+d)+k3)O(n2(k+d)+ndk+nk2+k3)O(n2k+nk2+k3)
+ +In Section 4, we demonstrated the global preservation property of our method on a specific task with a low number of super-nodes, similar to a clustering task. However, in typical coarsening scenarios, + +Table 4: Runtime comparison of our methods and FGC for a coarsening ratio of $r = 0.03$ , where all times $\tau$ are reported in seconds. Columns 2-5 show the runtime of each coarsening algorithm. Column 6 reports the training time of a GCN on the original graph, while Column 7 reports the training time on the coarsened graph. The final column shows in percentage how much training time was saved by applying coarsening before training, compared to training on the full graph. + +
DatasetFGCINGCSINGCINGC(β = 0)GCN fullGCN coarseSpeedup
CiteSeer04.6709.3804.0406.1142.0226.0314-29%
Pubmed20.5475.2749.4733.0593.1523.9221-38%
Co-CS70.80284.13180.33120.66369.1040.8612-56%
Co-Phy266.831200.76720.22540.50883.9652.2612-32%
+ +there are usually a larger number of super-nodes. In this section, we present the performance of this property in more practical coarsening scenarios, showing how our method continues to preserve the global structure of the graph. + +In Figures 5 and 7, we present the results obtained by our methods (INGC/SINGC), alongside the three baseline methods described in Section 4 on the Karate Club and Les Miserables datasets. Each row in the figures shows the partitioning produced by each method at different coarsening ratios. Nodes of the same color are grouped into the same super-node. We observe that our method groups adjacent nodes into super-nodes, thereby preserving the global structure of the graph. + +Since our method leverages the graph Laplacian eigenvectors to partition the vertices, we also compare it to the commonly used spectral clustering approach [19], which partitions the vertices by applying k-means [40] on the leading graph Laplacian eigenvectors. In Figure 2, we present the clustering results obtained by the SINGC method on the well-known Karate Club dataset[31], which consists of $n = 34$ nodes. We apply our method with a target of $k = \frac{n}{2} = 17$ super-nodes and compare the results to those obtained by spectral clustering[19] applied to the top $k$ leading eigenvectors. In the figure, each color represents a distinct cluster. We observe that the clusters produced by our method are more balanced compared to those generated by spectral clustering, which tends to form one large cluster alongside several smaller, single-node clusters. This balance is advantageous for downstream graph learning tasks, such as graph pooling. + +# C Complexity Analysis + +For an input graph with $n$ nodes, $e$ edges, and node features of dimension $p$ , the coarsened graph has $k$ nodes and $e_c$ edges. The dominant computational cost of the coarsening process arises from the gradient computation performed at each optimization step. Table 3 summarizes the gradient expressions and their time complexities, highlighting that SINGC is the most efficient, while INGC remains competitive with FGC, considering the gradient computation. + +Both INGC and SINGC have additional overhead beyond the optimization itself due to the computation of the top- $k$ Laplacian eigenvectors, which typically requires $\mathcal{O}(n^2 k)$ operations. This step can significantly increase the total runtime. However, this cost can be avoided by setting $\beta = 0$ , which removes the Grassmann similarity term from the objective. As shown in Table 6, even without this term, our methods still outperform the baselines. + +Table 4 reports the runtime of our methods and FGC, demonstrating their efficiency in reducing GNN training time. Specifically, we compare the time required to train a GCN on the original graph versus the coarse graph, including the coarsening time. The results confirm that applying coarsening before GCN training yields substantial time savings as summarized in the last column of the Table. + +All runtime values are reported in seconds. Experiments were conducted on a machine with a 12th-Gen Intel i7 CPU, an NVIDIA RTX A5000 GPU, and 64 GB of RAM. + +Table 5: Average fraction of signal energy captured by the first $k$ Laplacian eigenvectors (i.e., $k$ -smoothness) for each dataset and coarsening ratio. + +
Datasetr=1r=0.7r=0.5r=0.3r=0.1r=0.05r=0.03r=0.01
Cora10.850.740.600.380.270.190.08
Citeseer10.860.760.630.340.250.180.10
Pubmed10.870.770.650.500.430.390.32
Co-CS10.770.670.550.340.260.220.14
+ +# D Convergence Analysis + +Figure 3 illustrates the convergence rates of the INGC and SINGC methods on the Cora dataset for a coarsening ratio $r = 0.3$ . The left subplot shows the performance of INGC, while the right subplot depicts SINGC. For both methods, Citeseer results are in blue with varying line styles for different learning rates, and Cora results in red with corresponding line styles. The x-axis represents gradient steps, and the y-axis shows normalized loss values. + +The results reveal a typical convergence pattern for different learning rates. A trade-off is observed between convergence speed and final objective loss: higher learning rates lead to faster convergence but result in a higher final loss. This phenomenon is consistent across both datasets. We note that recent work has shown that lower learning rates can achieve lower minimal loss values but may risk unstable solutions [41]. + +# E Smoothness Assumption + +Assumption 3.3 is a standard premise in graph signal processing and learning on graphs, closely related to the notion of homogeneity and homophily in datasets [42]. To assess how frequently this assumption holds in practice, we evaluate the $k$ -smoothness of node features by computing the fraction of their energy captured by the subspace spanned by the first $k$ Laplacian eigenvectors. Table 5 reports the average energy concentration across all datasets used in our experiments for various values of $k$ . + +Notably, our method performs well even when the smoothness assumption is only partially satisfied. This highlights its robustness and broader applicability beyond the ideal smooth setting. + +# F Hyperparameters Discussion and Ablation Study + +We review the purpose of each term in our optimization and clarify the motivations behind selecting the hyperparameters values. The parameter $\beta$ promotes minimizing the IPE for general smooth signals. As shown in Proposition 3, minimizing the respective term also bounds the REE (related to preserving the graph's global structure) and DE (related to preserving the norm of node features). Therefore, $\beta$ is significant when these properties are prioritized in coarsening. The parameter $\lambda$ enforces group sparsity in each row, ensuring the validity of the obtained coarsening operator $C$ . Since $C$ lacks meaningful structure without this term, we did not perform an ablation study on $\lambda$ . Finally, the parameter $\alpha$ promotes connectivity in the coarsened graph, making it significant in scenarios where preserving graph connectivity is essential. + +Figure 4 presents an experiment analyzing each parameter's contribution and our method's sensitivity to their variations. The bars represent normalized scores for different values of a given hyperparameter, with distinct colors denoting specific values. For all metrics, lower values indicate better performance. The other two parameters are set to their optimal values for each metric as specified in Table 9. The figure illustrates the sensitivity of each parameter and evaluates the impact of deviations from optimal values on various metrics. + +In Figures 4(a) and 4(b), varying $\alpha$ shows minimal sensitivity across metrics, except for IPE, where changes up to an order of magnitude still yield similar results. Additionally, the figures include an ablation study on the parameter $\alpha$ illustrating its contribution to the optimization process. In Figures 4(c) and 4(d), varying $\lambda$ demonstrates that our methods are more sensitive to this parameter, highlighting its critical role in performance. + +![](images/de7ad37d6dd9a6ae688c01f5196554fee3058ee635204f97da38851257b3531e.jpg) +(a) Sensitivity to $\alpha$ (INGC) + +![](images/0e2ce9839d3146859ae7187f1a643e36fce5d5f9ecb14cc946b31b015d937da3.jpg) +(b) Sensitivity to $\alpha$ (SINGC) + +![](images/fc9b2c71ba9ca4d699db8eb10444fe9ecb5fa165b3ebb63d0d20f71fd9b870b1.jpg) +(c) Sensitivity to $\lambda$ (INGC) + +![](images/e51ca3abebe62745505f988d541e51c97b01547e6a498ac73f35464a20dcc4b4.jpg) +(d) Sensitivity to $\lambda$ (SINGC) +Figure 4: Ablation study on the sensitivity and contribution of the hyperparameters $\alpha$ and $\lambda$ across different metrics on the Cora dataset with a coarsening ratio $r = 0.3$ . (a) and (b) show the sensitivity of the parameter $\alpha$ across metrics. (c) and (d) illustrate the sensitivity of the methods to parameter $\lambda$ . The bars represent normalized scores for different values of the respective hyperparameter, with distinct colors denoting specific values. Lower bar values indicate better performance. + +Table 6: Ablation study of the parameter $\beta$ on node classification tasks. The table reports the accuracy on various datasets for different coarsening ratios $r$ using different coarsening methods. The third column presents the results of our INGC method with $\beta = 0$ , the fourth column corresponds to the optimal $\beta$ value, and the fifth column shows the results of SINGC. Best results are in bold; second-best results are underlined. The last two rows indicate the number of times each method achieved the best and second-best performance. + +
DatasetrINGC(β=0)INGCSINGC
Cora0.384.62 ± 0.5987.55 ± 0.1684.51 ± 0.33
0.183.01 ± 0.5383.38 ± 0.4782.76 ± 0.32
0.0576.92 ± 1.1177.42 ± 0.7877.81 ± 0.68
Citeseer0.376.25 ± 0.2876.89 ± 0.2376.66 ± 0.27
0.167.07 ± 0.5972.63 ± 0.2569.71 ± 0.72
0.0560.66 ± 1.5866.02 ± 0.3266.37 ± 0.57
Pubmed0.0583.60 ± 0.2383.59 ± 0.2283.55 ± 0.32
0.0381.62 ± 0.1481.93 ± 0.2283.19 ± 0.18
0.0179.08 ± 0.7279.09 ± 0.2679.96 ± 0.34
Co-CS0.0590.42 ± 0.1890.84 ± 0.1290.92 ± 0.22
0.0389.28 ± 0.2189.59 ± 0.3889.99 ± 0.41
0.0177.79 ± 1.1587.93 ± 0.3383.39 ± 0.33
#Best166
#2-Best165
+ +In Table 6, we present an ablation study on the parameter $\beta$ for the node classification task across various datasets and coarsening ratios $r$ . The comparison includes three methods: INGC with $\beta = 0$ (ignoring the term $\mathrm{tr}(U^{(k)}(U^{(k)})^\top CC^\top)$ for minimizing IPE for general smooth signals), INGC with the optimal $\beta$ , and SINGC (our second proposed method, which omits the first term of the objective entirely). The table reports node classification accuracy, with the best results highlighted in bold and the second-best results underlined. For each metric, other hyperparameters are set to their optimal values. The results demonstrate the importance of balancing the two complementary approaches to minimizing IPE. INGC with $\beta = 0$ generally underperforms compared to the other methods, emphasizing the significance of the smooth signal term in achieving high classification accuracy. + +# G Additional Details on the Experimental Study + +# Reproducibility Statement + +We ensure reproducibility by providing a detailed description of our methodology, including algorithmic steps (Algorithms 1, 2), evaluation procedures, and hyperparameter settings (Appendix G.3,G.2). The code used in this paper will be made available in a public repository upon acceptance. Full proofs of the theoretical results are included in the appendix, along with precise descriptions of our experimental setups. + +# G.1 Datasets Details + +Table 7: Graph coarsening metrics experimental setting: Chosen hyperparameters for the Karate Club dataset at different coarsening ratios $(r)$ and metrics. + +
MetricMethodKarate Club dataset
r=0.7r=0.5r=0.3
REEINGCβ=0, λ=100, α=0.1β=200, λ=10, α=1β=100, λ=100, α=0.1
SINGCλ=0.1, α=0.1λ=0.1, α=0.1λ=0.1, α=0.1
REINGCβ=0, λ=1, α=200β=200, λ=0.1, α=200β=200, λ=1, α=200
SINGCλ=0.1, α=200λ=0.1, α=200λ=1, α=200
HEINGCβ=0, λ=1, α=200β=200, λ=0.1, α=200β=200, λ=1, α=200
SINGCλ=0.1, α=200λ=0.1, α=200λ=10, α=200
DEEINGCβ=0, λ=1, α=200β=200, λ=1, α=200β=0.1, λ=1, α=200
SINGCλ=10, α=200λ=1, α=200λ=200, α=200
+ +Table 8: Graph coarsening metrics experimental setting: Chosen hyperparameters for the Les Miserables dataset at different coarsening ratios $(r)$ and metrics. + +
MetricMethodLes Miserables dataset
r=0.7r=0.5r=0.3
REEINGCβ=100, λ=200, α=0.1β=200, λ=10, α=0.1β=200, λ=200, α=1
SINGCλ=0.1, α=0.1λ=100, α=0.1λ=0.1, α=0.1
REINGCβ=200, λ=10, α=100β=200, λ=10, α=200β=200, λ=100, α=200
SINGCλ=0.1, α=100λ=1, α=200λ=100, α=100
HEINGCβ=200, λ=10, α=100β=200, λ=10, α=200β=200, λ=100, α=200
SINGCλ=0.1, α=100λ=0.1, α=200λ=100, α=100
DEEINGCβ=0, λ=200, α=200β=0.1, λ=0.1, α=10β=0.1, λ=0.1, α=200
SINGCλ=100, α=100λ=10, α=100λ=100, α=200
+ +# G.2 Node Classification Experiments Setting + +The GCN model used in our experiments consists of two graph convolutional layers and is implemented using PyTorch and PyTorch Geometric libraries. The architecture is as follows: + +- Layer 1: A Graph Convolutional Network (GCNConv) layer that takes the input node feature matrix $X$ (with $X$ .shape[1] features) and outputs a hidden representation of size 64. +- Layer 2: A second GCNConv layer that maps the 64-dimensional hidden representation to the number of output classes (NUM_OF_CLASSES). + +We use ReLU for non-linearity and dropout for regularization during training. + +Table 9: Graph coarsening metrics experimental setting: Chosen hyperparameters for the Cora dataset at different coarsening ratios $(r)$ and metrics. + +
MetricMethodCora dataset
r=0.7r=0.5r=0.3
REEINGCβ=10, λ=1, α=1β=100, λ=1, α=1β=200, λ=10, α=10
SINGCλ=1, α=10λ=100, α=100λ=200, α=0.1
REINGCβ=10, λ=100, α=200β=100, λ=200, α=200β=100, λ=10, α=200
SINGCλ=100, α=200λ=0.1, α=100λ=100, α=100
HEINGCβ=10, λ=100, α=200β=100, λ=200, α=200β=100, λ=10, α=200
SINGCλ=100, α=200λ=1, α=10λ=100, α=100
DEEINGCβ=0.1, λ=0.1, α=10β=0.1, λ=100, α=200β=0, λ=10, α=10
SINGCλ=100, α=200λ=10, α=100λ=100, α=200
+ +Table 10: Graph coarsening metrics experimental setting: Chosen hyperparameters for the CiteSeer dataset at different coarsening ratios $(r)$ and metrics. + +
MetricMethodCiteSeer dataset
r=0.7r=0.5r=0.3
REEINGCSINGCβ=200, λ=200, α=200λ=10, α=0.1β=100, λ=10, α=0.1λ=100, α=0.1β=100, λ=10, α=0.1λ=10, α=1
REINGCSINGCβ=100, λ=10, α=200λ=1, α=100β=100, λ=1, α=10λ=10, α=100β=0, λ=10, α=0.1λ=100, α=200
HEINGCSINGCβ=100, λ=10, α=200λ=1, α=100β=0.1, λ=100, α=200λ=1, α=100β=200, λ=0.1, α=0.1λ=100, α=200
DEEINGCSINGCβ=1, λ=0.1, α=0.1λ=100, α=200β=200, λ=1, α=10λ=100, α=200β=0.1, λ=0.1, α=10λ=100, α=100
+ +For SINGC, we set $t_{\mathrm{iter}} = 2000$ in all experiments, and for INGC, we set $t_{\mathrm{iter}} = 20$ and $c_{\mathrm{iter}} = 100$ . Tables 12 and 11 present the hyperparameters of our methods for each experiment. The results for the three baseline methods presented in Table 2 are sourced from Kumar et al. [14]. + +We note that tuning the hyperparameters in our methods is crucial for achieving optimal performance. By reviewing some of the corresponding setting in Tables 10, 9 and 11 we can observe that good performance often aligns with low values of REE and INP in this application. Therefore, we recommend that practitioners first optimize the hyperparameters by minimizing REE and INP. Once optimized, the coarsened graph can be used in the GNN for training and evaluation, leading to improved classification accuracy. + +The additional details of real datasets are as follows: + +Table 11: Node classification experimental setting: Chosen hyperparameters for the Cora Citeseer dataset at different coarsening ratios $(r)$ and metrics. + +
DatasetMethodNode Classification Parameters - Medium datasets
r = 0.3r = 0.1r = 0.05
CoraINGCβ=100, λ=100, α=10β=1, λ=100, α=1β=1, λ=100, α=100
SINGCλ=10, α=0.01λ=1000, α=1λ=1000, α=0.01
CiteSeerINGCβ=0, λ=100, α=10β=10, λ=1000, α=1000β=0, λ=20, α=10
SINGCλ=50, α=20λ=300, α=100λ=50, α=10
+ +Table 12: Node classification experimental setting: Chosen hyperparameters for the Co-phy, Pubmed, Co-CS dataset at different coarsening ratios $(r)$ and metrics. + +
DatasetMethodNode Classification Parameters - Large datasets
r = 0.05r = 0.03r = 0.01
Co-phyINGCβ=10, λ=10, α=0.01β=10, λ=1000, α=0.01β=10, λ=1000, α=100
SINGCλ=100, α=0.01λ=10, α=1λ=10, α=100
PubmedINGCβ=0, λ=1000, α=0.001β=0.1, λ=100, α=10β=0.1, λ=100, α=100
SINGCλ=1000, α=0.001λ=100, α=0.1λ=10, α=10
Co-CSINGCβ=1, λ=1000, α=10β=0.1, λ=1, α=10β=10, λ=100, α=100
SINGCλ=40, α=10λ=10, α=10λ=100, α=100
+ +- Karate Club - $n = 34$ , $p = 30$ , $|\mathcal{E}| = 78$ . Here, nodes represent members of a karate club, and edges represent friendships between them. Synthetic features generated using the signal model presented at Section 3.1. +- Les Miserables - $n = 77$ , $p = 50$ , $|\mathcal{E}| = 254$ - Nodes represent characters in the novel *Les Miserables*, and edges indicate co-occurrence in the same chapter. Synthetic features generated using the signal model presented at Section 3.1. +- Cora - $n = 2,708$ , $p = 1,433$ , $|\mathcal{E}| = 5,429$ - Nodes represent research papers, and edges represent citation links between them. Node features correspond to the presence of specific words in each paper, and class labels indicate the paper's research field. Number of classes $= 7$ . +- Citeseer - $n = 3,327$ , $p = 3,703$ , $|\mathcal{E}| = 4,732$ - Nodes represent research papers, and edges represent citation relationships. Node features are based on word occurrences in each paper, and class labels indicate the paper's topic. Number of classes = 6. +- Co-Physics - $n = 34,493$ , $p = 8,415$ , $|\mathcal{E}| = 247,962$ - Nodes represent physics research papers, and edges represent citations. Node features represent article keywords, and class labels indicate different fields of physics. Number of classes $= 5$ . +- PubMed - $n = {19},{717},p = {500},\left| \mathcal{E}\right| = {44},{338} -$ Nodes represent biomedical research papers, and edges represent citations. Node features are derived from TF-IDF scores of medical terms, and class labels indicate disease categories. Number of classes $= 3$ . +- Co-Computer - $n = 13,752$ , $p = 767$ , $|\mathcal{E}| = 245,861$ - Nodes represent products in a co-purchase network, and edges indicate products frequently purchased together. Node features describe product attributes, and class labels represent product categories. Number of classes $= 10$ . +- Co-CS - $n = 18,333$ , $p = 7005$ , $|\mathcal{E}| = 163,788$ . Here, nodes are authors, that are connected by an edge if they co-authored a paper; node features represent paper keywords for each author's papers, and class labels indicate most active fields of study for each author. Number of classes $= 15$ . + +# G.3 Graph Coarsening Metrics Experiments Setting + +Tables 7, 8, 9, and 10 present the hyperparameters of our methods for each experiment. For SINGC, we set $t_{\mathrm{iter}} = 2000$ in all experiments, and for INGC, we set $t_{\mathrm{iter}} = 20$ and $c_{\mathrm{iter}} = 100$ . + +Regarding the implementation of the baseline comparison methods, the FGC hyperparameters were selected based on their optimal values as reported in their paper. The LVN and LVE methods were implemented using their provided graph coarsening libraries, with the maximum value of the parameter $K = k = r \cdot n$ . + +# Karate Club + +![](images/3a59a4f6a5b1244af2f98288572b5b99691e9cda5f8459db5aeec9a7d31773a1.jpg) +Figure 5: Visual comparison of coarsening methods on the Karate Club dataset. Each row displays the partitioning produced by each method at a different coarsening ratio. Nodes of the same color are grouped into the same super-node. + +# Les Miserables + +![](images/76cc2b35e5457ff4842b416f40c093590f65d96b4bfe80699e1ba98a4c72450c.jpg) +Figure 6: Visual comparison of coarsening methods on the Les Miserables dataset. Each row displays the partitioning produced by each method at a different coarsening ratio. Nodes of the same color are grouped into the same super-node. + +# Planner Graph + +![](images/2a5f251269a846c4b1dbc89b46ce95e418937b24a60b74f60248ecf9b639eeb0.jpg) +Figure 7: Visual comparison of coarsening methods on a planner graph. Each row displays the partitioning produced by each method at a different coarsening ratio. 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One popular example is the usage of graph Laplacian eigenvectors for positional encoding in MPNNs and Graph Transformers. The expressive power of such Spectrally-enhanced GNNs (SGNNs) is usually evaluated via the $k$ -WL graph isomorphism test hierarchy and homomorphism counting. Yet, these frameworks align poorly with the graph spectra, yielding limited insight into SGNNs' expressive power. In this paper, we leverage a well-studied paradigm of classifying graphs by their largest eigenvalue multiplicity to introduce an expressivity hierarchy for SGNNs. We then prove that many SGNNs are incomplete even on graphs with distinct eigenvalues. To mitigate this deficiency, we adapt rotation equivariant neural networks to the graph spectra setting, yielding equiEPNN, a novel SGNN that provably improves upon contemporary SGNNs' expressivity on simple spectrum graphs. We then demonstrate that equiEPNN achieves perfect eigenvector canonicalization on ZINC, and performs favorably on image classification on MNIST-Superpixel and graph property regression on ZINC, compared to leading spectral methods. + +# 1 Introduction + +Graph Neural Networks (GNNs) have become a ubiquitous paradigm for learning on graph-structured data. The core principle of GNNs is to maintain a representation of each graph vertex and leverage the graph structure to iteratively refine each representation by its vertex's graph neighborhood [41]. To enhance the purview of the vertex's neighborhood, it is common to incorporate spectral features, such as Random Walk matrices, positional encoding, and graph distances, into the refinement operation of GNNs [8, 1, 45, 51]. Such GNNs, which systematically incorporate spectral features within their representation refinement procedure, or Spectrally-enhanced GNNs (SGNNs)[52], have gained significant traction in the graph learning community, due to their reasonable complexity and empirical benefits [17, 53, 52, 12]. + +Understanding the expressive power of GNNs provides researchers with a framework for comparing different models and identifying their deficiencies, often leading to improvements [14, 32, 35, 15, 49]. These frameworks ought to characterize which graphs the GNN can distinguish among, based on the GNNs' inner workings. For instance, the Weisfeiler-Leman (WL) test, which maintains and refines vertex representations similarly to Message Passing Neural Networks, a subclass of GNNs, completely determines which graphs these models can distinguish among [49]. + +![](images/27c4a2b9975e0405fad0057b80e8af3aab143a11a6c4a46de36a5492f0d813ff.jpg) +Figure 1: Hierarchy of 1-WL test variants. The arrows with $\square$ indicate strict inclusion relationships, meaning each variant can distinguish all graphs that the previous one can, plus additional graphs. Standard 1-WL is the least discriminative, while equiEPNN achieves the highest discriminative power, by incorporating both spectral invariant and equivariant refinement. + +To study the expressive power of SGNNs, recent papers [52, 12] proposed a spectrally enhanced GNN, called Eigenspace Projection GNN (EPNN), which generalizes many popular spectral graph neural networks, and analyze its expressivity via WL tests and homomorphism counting. This comparison is valuable in comparing the expressivity of SGNNs to that of their combinatorial GNN counterparts. Yet, this analysis does not yield insight into the role of the graph spectra in the distinguishing ability of these GNNs. + +To address this gap, we propose analyzing the expressive power of SGNNs via Spectral Graph Theory, and in particular via the maximal eigenvalue multiplicity of a graph. As isomorphism of graphs with bounded eigenvalue multiplicity can be determined in polynomial time, with the complexity depending exponentially on the eigenvalue multiplicity [2], this notion imposes a natural hierarchical classification of graphs, and SGNNs can potentially be complete on these graph classes, making this hierarchy a viable method for assessing their expressive power. + +Our analysis centers around the expressivity of EPNN on graphs with distinct eigenvalues. This model is at least as expressive as many commonly used SGNNs [52], making an upper bound on the expressivity of EPNN applicable to these models. Surprisingly, we find that EPNN is incomplete even on the class of graphs with distinct eigenvalues. On the positive side, EPNN achieves completeness on simple spectrum graphs whose eigenvectors exhibit certain sparsity patterns. Based on these theoretical insights, we propose equiEPNN, inspired by equivariant neural networks for point clouds, which attains provably improved expressivity on graphs with distinct eigenvalues. + +Our main contributions are summarized as follows: + +1. We prove the incompleteness of EPNN (in Subsection 3.2) on graphs with a simple spectrum. +2. We formulate a guarantee on the completeness of EPNN on graphs with a simple spectrum based on sparsity patterns of the eigenvectors. +3. We introduce equiEPNN (in Section 3.3), a modified EPNN variant, which integrates Euclidean message passing into the feature refinement procedure. +4. We benchmark equiEPNN on the ZINC and MNIST-Superpixel datasets, yielding favorable performance in comparison with popular spectral methods. Furthermore, equiEPNN performs perfect eigenvector canonicalization on the ZINC dataset. + +# 2 Related work + +# 2.1 Spectral invariant GNNs + +An enhancement to MPNNs and Transformer models is to incorporate spectral distances such as Random Walk, resistance, and shortest-path distances within the message passing operation [25, 50, 33, 11]. Zhang et al. [52] have compared among spectral GNNs and the WL hierarchy by proving EPNN is strictly more powerful than 1-WL yet strictly less powerful than 3-WL. Despite the important result that 3-WL strictly bounds the expressive power of EPNN, the large expressivity gap between 1- and 3-WL makes this determination difficult to conceptualize. Building on this work, Gai et al. [12] have characterized the expressive power of EPNN via graph homomorphism counting, showing spectral invariant GNNs can homomorphism-count a class of specific tree-like graphs. Despite providing a deeper understanding of EPNN's expressive power, it remains hard to conceptualize and propose more expressive models based on it. + +# 2.2 Spectral canonicalization methods + +The eigenvectors of a graph are used as positional encoding to improve the expressive power of message-passing and as positional encoding for Transformer [38, 20, 31] based models. Yet, positional encoding has an inherent ambiguity problem. An eigenvector corresponding to a unique eigenvalue can be represented as itself or its negation [43]. Canonicalization methods [29, 30] are used to address the ambiguity problems of eigenvectors, by choosing a unique representative for each eigenvector. + +Ma et al. [30] have uncovered an inherent limitation of canonicalization methods that process each eigenspace separately, which is that they cannot canonicalize eigenvectors with nontrivial self-symmetries. These models process each eigenbasis independently to obtain an orthogonal invariant and permutation-equivariant feature, and then use these features for downstream applications. Notable examples include SignNet and BasisNet [27], MAP [30] and OAP [29]. Ma et al. [29] have shown that these methods lose information when canonicalizing eigenvectors with self-symmetries, proving that the popular spectral invariant models SignNet and BasisNet are incomplete. In section 5.2, we provide a canonicalization scheme that bypasses this issue, and while not provable complete on all eigenvectors, empirically it canonicalizes all eigenvectors corresponding to distinct eigenvalues, in the ZINC [18] dataset. + +# 2.3 Expressivity on simple spectrum graphs + +An early study on the connection between GNNs and spectral features of the underlying graph studied the expressive power of CGNs [47]. They have proven that linear graph convolutional neural networks (GCNs) can map a graph signal to any chosen target vector, if the graph has distinct eigenvalues. Yet, this graph signal is sampled randomly and thus is not equivariant to permutations of the graph nodes, which may lead to degraded generalization, see Bechler-Speicher et al. [5]. + +For more related work see Appendix C. + +# 3 Problem statement + +# 3.1 Spectral graph decomposition + +Graphs are typically represented by a matrix $A \in \mathbb{R}^{n \times n}$ , where the $(i,j)$ entry of the matrix encodes the relationship between node $i$ and $j$ . This matrix could be the adjacency matrix, the normalized or un-normalized graph Laplacian, or a distance or Gram Matrix where the graph nodes have some underlying geometry. + +An important principle in the design of graph neural networks is the notion of permutation invariance: since graph nodes do not come with an intrinsic order, we would like to think of a matrix $A$ and its conjugation $PAP^T$ by a permutation matrix $P \in S_n$ , as being equivalent. Graph neural networks respect this invariance constraint and produce permutation invariant function $f$ satisfying $f(A) = f(PAP^T)$ . One popular method to design these functions exploits the eigendecomposition of the matrix $A$ . + +For the purpose of discussion, we assume that $A$ has an eigenbasis $v^{(1)},\ldots ,v^{(n)}$ of vectors of norm one, which corresponds to real eigenvalues $\lambda^{(1)},\dots,\lambda^{(n)}$ . This assumption always holds when in the typical case where $A$ is a symmetric matrix (e.g., adjacency and Laplacian matrices), and also often holds in other settings (e.g., random walk laplacian matrix). This gives us an alternative representation of the matrix $A$ with its own interesting symmetries. Firstly, we note that each vector $P v^{(q)}$ will be an eigenvector of $PAP^T$ with the same eigenvalue $\lambda^{(q)}$ . Secondly, if $v^{(q)}$ is an eigenvector of norm one, then so is $-v^{(q)}$ . When the eigenvalues of $f$ are pairwise-distinct, then these are all the relevant ambiguities. This is called the simple spectrum case. In the case of an eigenbasis of dimension $k$ , the eigendecomposition ambiguity is defined by orthogonal transformations in $O_k$ . In this paper, we will focus on the simple spectrum case. In this case, we define sign-invariant functions as follows + +Definition 1 (Sign Invariant functions). For fixed natural $n$ and $K \leq n$ , denote + +$$ +\mathcal {V} _ {\text {s i m p l e}} ^ {K} = \left\{\left(V, \vec {\lambda}\right) \in \mathbb {R} ^ {n \times K} \oplus \mathbb {R} ^ {K} \mid \begin{array}{l} \lambda_ {1} > \lambda_ {2} > \dots > \lambda_ {K} \end{array} \right\}. +$$ + +We say that $F:\mathcal{V}_{\mathrm{simple}}^{K}\to \mathbb{R}^{m}$ is sign invariant if + +$$ +F (V, \vec {\lambda}) = F (P V S, \vec {\lambda}), \quad \forall P \in S _ {n}, \quad S \in \{- 1, 1 \} ^ {K} +$$ + +We note that in this definition, $V$ represents a $n \times K$ matrix whose $K$ columns represent the first $K$ eigenvectors $v^{(1)}, \ldots, v^{(K)}$ of $A$ , and the notation $S \in \{-1, 1\}^K$ means that $S$ is a diagonal matrix whose diagonal is a vector in $\{-1, 1\}^K$ . + +The notion of sign invariant function was first introduced in [27], and was later discussed in [30, 29]. These papers discuss a collection of parametric functions $\mathcal{F} = \{f_{\theta}(V,\vec{\lambda})\mid \theta \in \Theta \}$ , such that for all parameters $\theta$ the function $f_{\theta}$ is sign invariant. To understand the expressiveness of these models, we formally define the notion of completeness on simple spectrum graphs. + +Definition 2 (Sign Invariant Separation). For $K \leq n$ , let $\mathcal{F}$ denote a collection of sign invariant functions defined on $\mathcal{V}_{\mathrm{simple}}^{K}$ , and let $\mathcal{D}$ be a subset of $\mathcal{V}_{\mathrm{simple}}^{K}$ . We say that $\mathcal{F}$ is complete on $\mathcal{D}$ if for any non-isomorphic pair $(V, \vec{\lambda})$ and $(U, \vec{\eta})$ in $\mathcal{D}$ , there exists a function $f \in \mathcal{F}$ such that + +$$ +f (V, \vec {\lambda}) \neq f (U, \vec {\eta}). +$$ + +Ideally, we would like $\mathcal{F}$ to be complete on all of the domain $\nu_{\mathrm{simple}}^{K}$ . If $\mathcal{F}$ is complete, then by applying it to eigendecompositions of graphs with simple spectrum, we will obtain models which can separate all graphs with simple spectrum, up to permutation equivalence. The goal of this paper is to understand whether existing sign-invariant functions are complete. + +# 3.2 EPNN + +We will focus on a large family of sign invariant functions named Eigenspace Projection GNNs (EPNN). This family of functions, introduced in Zhang et al. [52], was shown to generalize many spectral invariant methods such as Random Walk, resistance, and shortest-path distances [25, 50, 33, 11]. This method is based on a message passing like mechanism, where the spectral information is encoded by using the projection onto eigenspaces as edge features. In the simple spectrum case, this method can be formulated as follows: + +For a given eigendecomposition $(V,\vec{\lambda})\in \mathcal{V}_{\mathrm{simple}}^{K}$ , we we initialize a coloring for each 'node' $i\in [n]$ by + +$$ +h _ {i} ^ {(0)} = V _ {i} \odot V _ {i}, \tag {1} +$$ + +where $V_{i}$ is the $K$ dimensional vector $[V_{i}^{(1)},\dots ,V_{i}^{(k)}]$ obtained by sampling all eigenvectors at the $i$ -th node, and $\odot$ denotes elementwise multiplication. Importantly, this initialization is sign-invariant: while the global sign of each eigenvector is ambiguous, the product of two elements of the same eigenvector is not. + +We next iteratively refine the node features via the update rule: + +$$ +h _ {i} ^ {(t + 1)} = \operatorname {U P D A T E} _ {(t)} \left(h _ {i} ^ {(t)}, \vec {\lambda}, \left\{\left(h _ {j} ^ {(t)}, V _ {i} \odot V _ {j}\right) \mid j = 1, \dots , n \right\}\right) \tag {2} +$$ + +Here and throughout $\{\cdot\}$ denote multisets (multiplicities are allowed) and the multiset notation implies that $\mathrm{UPDATE}_{(t)}$ is required to be invariant to the order of the elements in the multiset. + +Finally, we apply a global pooling operation to obtain a final permutation invariant representation + +$$ +h _ {\text {g l o b a l}} = \operatorname {R E A D O U T} \left(\left\{h _ {i} ^ {(T)} \mid i = 1, \dots , n \right\}\right) \tag {3} +$$ + +Once $\mathrm{UPDATE}_{(t)}$ and READOUT functions are determined, this procedure determines a function $f(V,\vec{\lambda}) = h_{\mathrm{global}}$ which is sign-invariant as in Definition 1. The collection of all such functions obtained by all possible choices of $\mathrm{UPDATE}_{(t)}$ and READOUT functions is denoted by $\mathcal{F}_{\mathrm{EPNN}}$ . + +# 3.3 Equivariant EPNN + +In [12], the authors suggest methods based on higher order WL tests to boost the expressive power of spectral message passing neural networks. The complexity of these methods is considerably higher + +than EPNN. In contrast, we will now suggest a method for increasing the expressive power of EPNN without significantly changing model complexity. + +Our suggestions are based on constructions from neural networks for geometric point clouds. These neural networks operate on point clouds $X \in \mathbb{R}^{n \times d}$ (where in many applications $d = 3$ ) and each of the $n$ points in $\mathbb{R}^d$ represents a geometric coordinate. Models for such data are required to be invariant (or equivariant) to both permutations in $S_n$ and rotations in $O(d)$ . This equivariant structure is similar to, but not identical to, the situation we have for graph eigecomposition: under the simple spectrum assumption, the symmetry transformations we are interested in is a single global permutation, and $K$ sign changes, which are rotations in $O(1)^K$ . In the more general setting, we will have a single permutation and multiple rotations, whose dimension is determined by the multiplicity of each eigenvalue. + +Via this analogy, we can look at spectral models for graphs from the perspective of point cloud networks. From this perspective, EPNN resembles geometric invariant networks, such as Schnet [42], which are based on simple invariant features. In contrast, [19] and [44] showed that, at least for point clouds, expressivity can be increased by recursively updating a rotation equivariant (in our scenario, sign equivariant) feature $v_{i}^{(t)}$ in parallel with the invariant feature $h_{i}^{(t)}$ . Inspired by these observations, we suggest the following sign equivariant feature refinement procedure: + +We use the same initialization $h_i^{(0)}$ as in Equation 1, and we initialize the equivariant feature $v_i^{(0)}$ to $v_i^{(0)} = V_i$ . We then iteratively update these two features via + +$$ +h _ {i} ^ {(t + 1)} = \operatorname {U P D A T E} _ {(t, 1)} \left(h _ {i} ^ {(t)}, \vec {\lambda}, \left\{\left(h _ {j} ^ {(t)}, v _ {i} ^ {(t)} \odot v _ {j} ^ {(t)}\right) \mid j = 1, \dots , n \right\}\right) +$$ + +$$ +v _ {i} ^ {(t + 1)} = v _ {i} ^ {(t)} + \sum_ {j = 1} ^ {n} v _ {j} ^ {(t)} \odot \mathrm {U P D A T E} _ {(t, 2)} (h _ {i} ^ {(t)}, h _ {j} ^ {(t)}, v _ {i} ^ {(t)} \odot v _ {j} ^ {(t)}) +$$ + +where $\mathrm{UPDATE}_{(t,1)}$ is a multiset function, and $\mathrm{UPDATE}_{(t,2)}$ maps its input to $\mathbb{R}^K$ so that the elementwise product in the equation above is well defined. + +After running this procedure for $T$ iterations, we obtain an invariant global feature $h_{\mathrm{global}}$ by aggregating the invariant node features $h_i^{(T)}$ using a READOUT function, as in (3). This gives us a sign invariant function $f(V,\vec{\lambda}) = h_{\mathrm{global}}$ . We name the class of all functions obtained by running this procedure with all different choices of update and readout functions equiEPNN. + +We note that we can obtain EPNN models by setting $\mathrm{UPDATE}_{(t,2)}$ to be the constant mapping to the zero vector. Accordingly, $\mathcal{F}_{\mathrm{equi}}$ is at least as expressive as EPNN. In Section 4.4 we will show that it is strictly more expressive. + +# 4 On the incompleteness of spectral graph neural networks + +In this section, we analyze the expressive power of EPNN and equiEPNN on graphs with a simple spectrum. We first provide a counterexample to prove its incompleteness of EPNN on simple spectrum graphs. We then show that an equiEPNN can separate the counterexample, thus proving it is strictly more expressive than EPNN. Next we provide a subset of $\mathcal{V}_{\mathrm{simple}}^{K}$ on which EPNN is complete. Finally, we discuss how our results imply the incompleteness of popular spectral GNNs even in the simple spectrum case. + +# 4.1 EPNN is incomplete + +We first introduce a pair of non-isomorphic eigendecompositions, $(V,\vec{\lambda})$ and $(U,\vec{\lambda})$ in $\mathcal{V}_{\mathrm{simple}}^{K}$ , which EPNN cannot distinguish, that is, it assigns them the same final feature after any number of refinement steps. In this construction $n = 12$ , $K = 6$ , and we fix the same choice of distinct eigenvalues $\vec{\lambda}$ for both examples. To define $V,U$ , we denote + +$$ +z _ {0} = \left( \begin{array}{c} 1 \\ 1 \end{array} \right), \quad z _ {1} = \left( \begin{array}{c} - 1 \\ 1 \end{array} \right), \quad z _ {2} = \left( \begin{array}{c} 1 \\ - 1 \end{array} \right), \quad z _ {3} = \left( \begin{array}{c} - 1 \\ - 1 \end{array} \right), \quad 0 _ {2} = \left( \begin{array}{c} 0 \\ 0 \end{array} \right), +$$ + +and note that $z_0, \ldots, z_3$ are the four elements of the abelian group $\{-1, 1\}^2$ . Using these, we define $U, V$ via + +$$ +U ^ {T} = \left( \begin{array}{c c c c c c c c c c c c} z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {2} & z _ {3} \\ z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & z _ {1} & z _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {3} & z _ {2} & z _ {0} & z _ {2} & z _ {1} & z _ {3} \end{array} \right) +$$ + +$$ +V ^ {T} = \left( \begin{array}{c c c c c c c c c c c c} z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {2} & z _ {3} \\ z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {1} & z _ {0} & z _ {2} & z _ {3} & z _ {2} & z _ {0} & z _ {3} & z _ {1} \end{array} \right) +$$ + +We now show that $U, V$ are not isomorphic and cannot be separated by EPNN: + +Theorem 1. (Incompleteness of EPNN) The following statements hold: + +1. $U$ and $V$ are not isomorphic under the group action of $S_{12} \times \{-1, 1\}^6$ . +2. EPNN cannot separate $U$ and $V$ after any number of iterations. +3. $U$ and $V$ have no non-trivial automorphisms. + +Therefore, EPNN is incomplete on simple spectrum graphs. + +Proof Idea. To show $U, V$ are not isomorphic, we note that for any pair of permutation-sign matrices taking $U$ to $V$ , the first four columns of $U^T$ must be mapped the first four columns of $V^T$ . The same is true for columns $5 - 8$ and $9 - 12$ . Considering the first four columns, we see that any sign matrix mapping them from $U$ to $V$ will be of the form $\mathrm{diag}(z, z, z')$ for $z, z' \in \{-1, +1\}^2$ . The same argument for columns $5 - 8$ and $9 - 12$ gives sign patterns of the form $\mathrm{diag}(z', z, z_1 \cdot z)$ and $\mathrm{diag}(z, z', z_2 \cdot z)$ , respectively. But there is no sign pattern satisfying these three constraints simultaneously. + +We now explain the lack of separation of EPNN. We refer to the multiset of the multiplications of a column $i$ with all the other columns, as the column $i$ 's purview. In the initial step, the purview of each column in the first 4-column block in $V^T$ and $U^T$ , is identical, as the first 4 columns exhibit a group structure with the multiplication operation. Thus, the hidden states of the first 4 indices of $U^T$ and $V^T$ will be identical. By similar arguments, this holds for the remaining two blocks. Thus, after a refinement step, the nodes in each block cannot distinguish among those from other blocks, both in $U^T$ and $V^T$ . Therefore, additional refinement procedures maintain identical representations for members of each index 'block' and corresponding blocks in $U^T$ and $V^T$ . This implies EPNN cannot separate $U$ and $V$ . + +A full proof of the theorem is provided in the Appendix. + +![](images/bbf4a250b6f62472023ee3d1a96b4d569ba50b5d5a9f30af3b1e7aa0c2f42bae.jpg) + +Remark: In many cases we are interested in eigenvalue decompositions of symmetric matrices, in which case the columns of $V, U$ (the rows of $V^T, U^T$ ) should be orthonormal. While our $V, U$ do not satisfy this condition, in the Appendix we show how they can be enlarged to yield a counterexample that has the same properties, and does have orthonormal columns. + +# 4.2 When is EPNN complete? + +The counterexample proves that there is an inherent limit to the expressive power of contemporary spectral invariant networks. We note that in this example $U, V$ had a significant number of zero entries. We now show that when $U, V$ each have at least one row without any zeros, EPNN will be complete (in particular, this condition always holds when the matrices $U, V$ have less than $n$ zero entries): + +Theorem 2 (EPNN Can Distinguish Dense Graphs with Distinct Eigenvalues). Let $\mathcal{D} \subseteq \mathcal{V}_{\mathrm{simple}}^{K}$ denote the set of $(V, \vec{\lambda})$ where $V$ has a row without zero entries. Then EPNN is complete on $\mathcal{D}$ . + +Proof. By assumption, an index $i$ exists such that the $i$ -th row of $V$ has no zeros. The hidden state $h_i^{(1)}$ after a signal iteration of EPNN (see (2)) can encode the eigenvalues $\vec{\lambda}$ , the squared values of each coordinate of $V_i$ , and the multiset of pairwise products $V_i \odot V_j$ , as + +$$ +h _ {i} ^ {(1)} = \left(V _ {i} \odot V _ {i}, \left\{V _ {i} \odot V _ {j} \mid j = 1, \dots , n \right\}\right) \tag {4} +$$ + +To recover $V$ from $h_i^{(1)}$ up to symmetries, we can fix the sign ambiguity by choosing all coordinates of $V_i$ to be positive. We can then recover the remaining $V_j$ from the multiset in Equation 4. + +This uncovers the inner workings of EPNN in processing simple spectrum graphs. Essentially, each entry can be normalized to represent a group element in $O(1)$ , which acts as a local frame of reference, see [9] for more background, allowing us to reconstruct the eigenvectors up to sign symmetries. + +# 4.3 Unique node identification via EPNN + +A well-known mechanism for circumventing the limited expressive power of GNNs is by injecting unique node identifiers (IDs), which break the symmetries that hinder GNNs' separation ability [28, 13]. Popular approaches include random node initialization [5] and combinatorial methods [7], yet they are either limited by their discontinuity or break permutation equivariance. A natural question is whether the node features from EPNN are unique after finitely many iterations? If so, we have attained node IDs that do not break equivariance and change continuously with the eigendecomposition, alleviating the deficiencies of widely-used methods. We answer this question in the affirmative, provided the eigenvectors adhere to a sparsity pattern. + +Theorem 3. (EPNN for Unique Node Identifiers) Let $\mathcal{D} \subseteq \mathcal{V}_{\mathrm{simple}}^{K}$ denote the set of $(V, \vec{\lambda})$ where $V$ has no automorphisms, and has at most one zero per eigenvector. Then, one iteration of EPNN with injective UPDATE and READOUT functions assigns a unique identifier to each hidden node feature. + +Proof. By contradiction, assume that there exist distinct indices $i,j$ such that $h_i^{(1)} = h_j^{(1)}$ . By the definition of EPNN, we have that + +$$ +\left\{V _ {i} \odot V _ {k} \right\} _ {k = 1} ^ {n} = \left\{V _ {j} \odot V _ {k} \right\} _ {k = 1} ^ {n} \text {a n d} V _ {i} \odot V _ {i} = V _ {j} \odot V _ {j}. \tag {5} +$$ + +We deduce for the second equality that $|V_{i}^{(q)}| = |V_{j}^{(q)}|$ for all coordinates $q = 1, \ldots, K$ . If for some $q$ we had $V_{i}^{(q)} = 0$ , then also $V_{j}^{(q)} = 0$ , in contradiction to the assumption that the $q$ -th eigenvector has at most one zero entry. Thus all entries of $V_{i}$ and $V_{j}$ are non-zero. + +Next, we deduce from Equation 5 and the fact that $|V_{i}^{(q)}| = |V_{j}^{(q)}| > 0$ for all $q$ , that + +$$ +\{(s _ {i} ^ {(q)} V _ {k} ^ {(q)}) _ {q = 1} ^ {K} \} _ {k = 1} ^ {n} = \{(s _ {j} ^ {(q)} V _ {k} ^ {(q)}) _ {q = 1} ^ {K} \} _ {k = 1} ^ {n} +$$ + +where $s_i^{(q)} \in \{\pm 1\}$ and is defined as $\frac{V_i^{(q)}}{|V_i^{(q)}|}$ and $s_j^{(q)}$ is defined analogously. This means that there exists a permutation $\sigma$ which swaps $i$ with $j$ , such that $s_i^{(q)} V_k^{(q)} = s_j^{(q)} V_{\sigma(k)}^{(q)}$ for all $k = 1, \ldots, n$ and $q = 1, \ldots, K$ . Equivalently, + +$$ +P V S _ {1} = V S _ {2} \Longrightarrow P V S _ {1} S _ {2} = V \tag {6} +$$ + +where $S_{1}$ and $S_{2}$ are diagonal matrices with $s_i^{(q)}$ and $s_j^{(q)}$ , respectively, on the diagonals, and $P$ is the permutation matrix corresponding to $\sigma$ . Since $P$ swaps $i$ with $j$ , this is a non-trivial automorphism, in contradiction to the assumption. Thus $h_i^{(1)} \neq h_j^{(1)}$ , as required. + +# 4.4 equiEPNN is strictly more expressive than EPNN + +We show that equiEPNN is strictly more powerful than EPNN, as it separates the pair $U$ and $V$ from Subsection 4.1, which EPNN cannot separate: + +Corollary 1. equiEPNN (see Section 3.3) can separate $U$ and $V$ after 2 iterations. Thus equiEPNN is strictly stronger than EPNN. + +Proof Idea. We show that after a single iteration, the equivariant update step can yield new matrices $U^{(t)}, V^{(t)}, t = 1$ which have no zeros. From Theorem 2, we know that a single iteration of EPNN, and hence also equiEPNN, is complete for such $U^{(t)}, V^{(t)}$ , and thus two iterations of equiEPNN are sufficient for separation. + +While equiEPNN is stronger than EPNN, the following result (proven in the appendix) shows that equiEPNN is also incomplete over simple spectrum graphs: + +Theorem 4. (Incompleteness of Equivariant EPNN) There exist $X, Y \in \mathbb{R}^{16 \times 6}$ such that the following statements hold: + +1. $X$ and $Y$ are not isomorphic under the group action of $S_{16} \times \{-1, 1\}^{6}$ . +2. Equivariant EPNN cannot separate $X$ and $Y$ after any number of iterations. + +Therefore, Equivariant EPNN is incomplete on simple spectrum graphs. + +In the appendix we also explain how this counterexample can be extended so that $X, Y$ are orthogonal matrices which thus can form a full eigendecomposition of a real symmetric matrix. + +# 4.5 Incompleteness of spectral GNNs + +Theorem 1 proves that EPNN is incomplete on graphs with a simple spectrum. This spectral isomorphism test upper bounds the expressive power of many popular distance-based GNNs, which incorporate graph distances as edge features, such as Random Walk, PageRank, shortest path, or resistance distances [50, 25, 1, 45, 51]. Therefore, an immediate corollary of Theorem 1 follows: + +Corollary 2. Graphormer-GD [50], PRD-WL [25], DiffWire [1], and Random-Walk based GNNs[45, 51] are incomplete over graphs with a simple spectrum. + +In addition to this result, in the appendix we prove that the model proposed by Zhou et al. [53] is not universal on simple spectrum graphs. + +Proposition 3. Vanilla OGE-Aug [53] is incomplete over graphs with a simple spectrum. + +# 5 Experiments + +Our goal in the experiments section is twofold: (a) statistically evaluate the validity of our bounded eigenmultiplicity approach for measuring expressivity and (b) empirically exemplify the utility of equiEPNN ${}^{1}$ . To meet the first goal, we statistically analyze the eigenvalue multiplicity in real-world datasets, and the number of non-zero entries in the eigenvectors, to compare these with our theoretical conditions for EPNN completeness. We find that while the sparsity conditions for EPNN completeness are satisfied on some real-world datasets (MNIST-Superpixel), they are not satisfied on datasets with more intricate symmetries (ZINC). For the second goal, we evaluate the utility of the equivariant features derived from equiEPNN on the task of eigenvector canonicalization [29]. Finally, we benchmark equiEPNN against leading spectral methods on the popular ZINC and MNIST-Superpixel datasets. + +# 5.1 Dataset statistics + +We surveyed popular graph datasets and documented their graph spectral properties. The results are shown in Table 1. We find that the MNIST Superpixel [34] dataset is almost homogeneously composed of graphs with a simple spectrum, and we find that $(96.9\%)$ of the graphs in this dataset have a full row without zeros, implying that EPNN is complete on almost all graphs. + +Other datasets, such as MUTAG, ENZYMES, PROTEINS and ZINC [18, 36], contain a substantial amount of graphs with eigenvalue multiplicity 2 and 3. Despite this, the number of eigenspaces of dimensions 2 and 3 is very low, averaging at around 1 per graph. On datasets with highly symmetric graphs, such as ENZYMES and PROTEINS, the graphs do not meet the sparsity condition of Theorem 2, thus EPNN will not necessarily faithfully learn the graph structure. This exemplifies the need for more expressive models that are complete on graphs with higher maximal eigenvalue multiplicity and sparse eigenvectors. + +Table 1: Graph Statistics Analysis Across Different Datasets (Eigenvalue Tolerance: ${10}^{-4}$ ) + +
Dataset NameMUTAGENZYMESPROTEINSMNISTZINC
Dataset Overview
Number of Graphs1886001,11360,00010,000
Eigenvalue Characteristics
Graphs with Distinct Eigenvalues41.5% (78)34.8% (209)22.1% (246)99.9% (59,950)40.7% (4,072)
Graphs with Multiplicity 2 Eigenvalues58.5% (110)65.2% (391)77.9% (867)-59.3% (5,928)
Graphs with Multiplicity 3 Eigenvalues19.1% (36)46.2% (277)57.9% (644)-26.2% (2,617)
Avg. Number of Multiplicity 2 Eigenvalues0.741.011.24-1.282
Avg. Number of Multiplicity 3 Eigenvalues0.260.580.71-1.105
Eigenvector Properties
Average Ratio of Zeros1.674.286.390.312.52
Average Number of Zeros31.13172.93817.2023.1661.04
Graphs with a Full Row75.0% (141)35.8% (215)37.1% (413)96.9% (58,077)64.5% (6,447)
Graphs with ≤1 Zero per Eigenvector0.0% (0)6.3% (38)5.0% (56)20.2% (12,085)4.3% (430)
Graphs with Total Zeros < Vertices29.8% (56)16.3% (98)14.3% (159)89.9% (53,873)13.0% (1,295)
Graphs Meeting Any Condition75.0% (141)35.8% (215)37.1% (413)96.9% (58,077)64.5% (6,447)
+ +# 5.2 Eigenvector canonicalization + +Positional encoding is a cornerstone of graph learning using Transformer architectures, yet they suffer from the sign ambiguity problem [8]. It can be resolved by eigenvector canonicalization, which involves choosing a unique representation of each eigenvector. Yet, an inherent limitation of current canonicalization methods is that they are unable to canonicalize eigenvectors with nontrivial self-symmetries, often called uncanonicalizable eigenvectors [30, 29]. + +Table 2: Uncanonicalizable Graph Eigenvectors in ZINC (Subset) [18] as percentage of total eigenvectors of eigen-space dimension 1. + +
PropertyPercentage (%)
Sum to 011.15 %
Uncanonicalizable10.93 %
equiEPNN output sum to 00.0 %
Uncanonicalizable after equiEPNN0.0 %
+ +To overcome this limitation, we devise a method to choose a canonical representation of the original eigenvectors via the equivariant output of equiEPNN. The only requirement is that each vector in the equivariant output does not sum to 0. + +We test our hypothesis on a popular benchmark ZINC [18], and find that all the vectors in the equivariant output are canonicalizable and sum to zero, in contrast to the vectors from the eigendecomposition, where $10\%$ of them are uncanonicalizable. Furthermore, we devise a way to choose a canonical representation of the original eigenvectors via the equivariant output and describe this in the Appendix. The results are shown in Table 2. + +Table 3: Results on ZINC and MNIST-Superpixel datasets. The values are the MSE for ZINC (Subset) and the accuracy for MNIST-Superpixel. Edge features are not used even if they are available in the datasets. For ZINC, all models use node labels. For MNIST-Superpixel, the model uses superpixel-intensive values and node degree as node features. Models have a budget of $30\mathrm{K}$ free parameters for ZINC and $35\mathrm{K}$ for MNIST. + +
CategoryModelZINC (MAE ↓)MNIST-Superpixel (Acc.↑)
NNMLP0.5869 ± 0.02525.10% ± 0.12
MPNNGCN0.3322 ± 0.01052.80% ± 0.31
GAT0.3977 ± 0.00782.73% ± 0.21
GIN0.3044 ± 0.01075.23% ± 0.41
3-WLPPGN0.1589 ± 0.00790.04% ± 0.54
SpectralChebNet0.3569 ± 0.01292.08% ± 0.22
GNNML10.3140 ± 0.01584.21% ± 1.75
equiEPNN (Ours)0.2805 ± 0.01990.32% ± 0.7
+ +# 5.3 Benchmarks: ZINC and MNIST + +We evaluated equiEPNN on the image classification task MNIST-Superpixel [34], in which clustering of images is performed according to regions with similar pixel values, an algorithm creates a graph based on these regions, and each node is assigned a region-induced feature. We compared equiEPNN to leading spectral methods, all with a comparable parameter budget of $\approx 35\mathrm{K}$ (see Table 3). We observe that it outperforms PPGN [32], which has cubic complexity, and GNNML1, which also processes the eigendecomposition of the graph. ChebNet outperforms all other methods, perhaps due to its handcrafted polynomial features. + +We further evaluate equiEPNN via the standard regression task on the ZINC dataset of molecular graphs (we also tested eigenvector canonicalization on this same dataset). ZINC (Subset) has 12000 graphs with an average of 23.16 nodes per graph. We compare ourselves to leading methods with the standard $\approx 500K$ parameter budget and find that, out of the spectral methods, our method attains the best results, see Table 4. + +Table 4: Results on ZINC. + +
methodtest (MAE ↓)
GIN0.526±0.051
GraphSage0.398±0.002
GCN0.384±0.007
GCN0.367±0.011
GatedGCN-PE0.214±0.006
MPNN (sum)0.145±0.007
PNA0.142±0.010
GT0.226±0.014
SAN0.139±0.006
GraphormerSLIM0.122±0.006
MPNN.138 ± .006
EPNN.103 ± .006
equiEPNN (Ours)0.99 ± 0.001
Subgraph GNN.110 ± .007
Local 2-GNN.069 ± .001
+ +# 6 Future Work + +A key future goal is to devise spectral GNNs that achieve completeness on graphs with simple spectra, and higher eigenvalue multiplicities. One interesting direction is to use higher-order point cloud networks to process the eigenvectors [53]. We have shown that treating each eigenspace as a separate entity does not lead to universality (see Subsection 4.5). Thus, these high-order networks should process the eigenvectors as a single entity, but remain invariant only to the sign and basis symmetries. + +Acknowledgements N.D. and S.H. were supported by ISF grant 272/23. + +# References + +[1] Adrián Arnaiz-Rodríguez, Amine Begga, David Gutiérrez-Gómez, Rubén Bernardo-Gavito, Pierre Borgnat, and Alexandre Gramfort. Diffwire: Inductive graph rewiring via the Lovász bound. In Advances in Neural Information Processing Systems, volume 35, 2022. +[2] László Babai, D. Yu. Grigoryev, and David M. Mount. Isomorphism of graphs with bounded eigenvalue multiplicity. 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A complete expressiveness hierarchy for subgraph GNNs via subgraph weisfeiler-lehman tests. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, 2023. + +[52] Bohang Zhang, Lingxiao Zhao, and Haggai Maron. On the expressive power of spectral invariant graph neural networks. In Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, 2024. +[53] Junru Zhou, Cai Zhou, Xiyuan Wang, Pan Li, and Muhan Zhang. Towards stable, globally expressive graph representations with laplacian eigenvectors, 2024. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Our claims are properly stated in the abstract within their scope. We diligently wrote the assumptions. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: No usage of LLMs in core method. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# A Proofs 22 + +A.1 Proof of Incompleteness of EPNN 22 +A.2 Extension to orthonormal counterexamples 24 +A.3 Proofs for implications for real-world GNNs 24 +A.4 Proof for equiEPNN strictly more expressive 26 +A.5 Proof of Incompleteness of Equivariant EPNN 26 + +# B Experiments 33 + +B.1 Dataset statistics 33 +B.2 MNIST Superpixel 33 +B.3 Realizable Expressivity 35 +B.4 Eigenvector Canonicalization 35 + +# C Further Related Work 37 + +C.1 Expressive Power and the Weisfeiler-Lehman Hierarchy 37 +C.2 Higher-Order and Subgraph GNNs 37 +C.3 Spectral GNNs and Universality 37 +C.4 Equivariant Design and Generalization 37 +C.5 Unified Theories and GNN Limitations 37 + +# A Proofs + +# A.1 Proof of Incompleteness of EPNN + +Theorem 1. (Incompleteness of EPNN) The following statements hold: + +1. $U$ and $V$ are not isomorphic under the group action of $S_{12} \times \{-1, 1\}^6$ . +2. EPNN cannot separate $U$ and $V$ after any number of iterations. +3. $U$ and $V$ have no non-trivial automorphisms. + +Therefore, EPNN is incomplete on simple spectrum graphs. + +Proof. For convenience, we recall the definitions of the point clouds $U$ and $V$ : + +We denoted the four elements of the abelian group $\{-1, 1\}^2$ , and the zero vector in $\mathbb{R}^2$ , by + +$$ +z _ {0} = \left( \begin{array}{c} 1 \\ 1 \end{array} \right), \quad z _ {1} = \left( \begin{array}{c} - 1 \\ 1 \end{array} \right), \quad z _ {2} = \left( \begin{array}{c} 1 \\ - 1 \end{array} \right), \quad z _ {3} = \left( \begin{array}{c} - 1 \\ - 1 \end{array} \right), \quad 0 _ {2} = \left( \begin{array}{c} 0 \\ 0 \end{array} \right). +$$ + +Using these, we define $U, V$ via + +$$ +U ^ {T} = \left( \begin{array}{c c c c c c c c c c c c} z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {2} & z _ {3} \\ z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {3} & z _ {2} & z _ {0} & z _ {2} & z _ {1} & z _ {3} \end{array} \right) +$$ + +$$ +V ^ {T} = \left( \begin{array}{c c c c c c c c c c c c} z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {2} & z _ {3} \\ z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {1} & z _ {0} & z _ {2} & z _ {3} & z _ {2} & z _ {0} & z _ {3} & z _ {1} \end{array} \right) +$$ + +We first prove: 2. the inseparation of $U$ and $V$ by EPNN. + +Observe the purview of node $i$ of $U$ after the first refinement step of EPNN: + +$$ +h _ {i} ^ {(1)} (U) = \left(U _ {i} \odot U _ {i}, \{U _ {i} \odot U _ {j} \mid j \in [ 1 0 ] \}\right) \tag {7} +$$ + +We will show that point clouds can be partitioned into 'blocks' such that each point in the block obtains the same hidden state. This block structure is recognized by viewing each point as a group element and each block as a multiplicative group. We will then show that this multiplicative group structure allows us to prove the inseparability of EPNN. + +Concretely, our proof proceeds as follows: + +1. The column entries of $U^T$ and $V^T$ can be partitioned into 3 blocks: $B_1 \triangleq \{1, 2, 3, 4\}$ , $B_2 \triangleq \{5, 6, 7, 8\}$ , and $B_3 \triangleq \{9, 10, 11, 12\}$ , such that $h_i^{(1)}(U)$ and $h_j^{(1)}(U)$ are identical for every $i, j \in B_k$ , $k = 1, 2, 3$ . +2. It holds that $h_i^{(1)}(U) = h_i^{(1)}(V)$ for every $i = 1, 2, \ldots, 12$ . +3. For any $t \in \mathbb{N}$ , $h_i^{(t)}(U) = h_i^{(t)}(V)$ for every $i = 1,2,\ldots,12$ . + +1. We first focus on $B_{1}$ and then extend the argument to $B_{2}$ and $B_{3}$ . + +Since the elements $U_{j}$ for $j = 1,2,3,4$ admit a multiplicative group structure, then for every $i = 1,2,3,4$ , the respective entries $U_{i} \odot U_{j}$ for $j \in [4]$ are identical (closure of groups.) + +For $j = 5,6,7,8$ and $i = 1,2,3,4$ , the entries of the products $V_{i} \odot V_{j}$ , are zeros in two row entries and the non-zero entries in the remaining row, each element of the group $\mathbb{Z}_2^2 \cong \{z_0,z_1,z_2,z_3\}$ appears exactly once in the non-zero entries of the products, as it holds that $z_{i}\mathbb{Z}_{2}^{2} = \mathbb{Z}_{2}^{2}$ . + +Analogously, we can extend this argument to $j = 9, 10, 11, 12$ and $i = 1, 2, 3, 4$ . + +This means that $h_i^{(1)}(U)$ and $h_j^{(1)}(U)$ are identical for every $i, j \in B_1$ . + +Since, by definition of $U$ and symmetry, each four-index quadruple $B_{1} \triangleq 1, 2, 3, 4$ , $B_{2} \triangleq 5, 6, 7, 8$ , and $B_{3} \triangleq 9, 10, 11, 12$ is a multiplicative group, the analysis for the hidden states of the indices in $B_{1}$ holds for $B_{2}$ and $B_{3}$ . This concludes item 1. + +2. Up to now, we proved for indices $i,j\in B_k$ for $k = 1,2,3$ , it holds that $h_i^{(1)} = h_j^{(1)}$ . It remains to be proven that these hidden states are equivalent in both point clouds to conclude step 2. + +Since the point cloud $V^T$ is derived from $U^T$ by multiplying the columns in $B_2$ by $\mathrm{diag}(z_0, z_0, z_1)$ and the columns in $B_3$ by $\mathrm{diag}(z_0, z_0, z_2)$ , the purview (see Equation 7) of each index is identical in both point clouds, since $z_2 \cdot z_2 = z_1 \cdot z_1 = z_0$ which is the identity element, thus by definition of EPNN, this modification that maps $U^T$ to $V^T$ doesn't affect the pairwise multiplications in Equation 7. + +3. To prove this step, we only need to show that the hidden states remain identical within each block, since the fact that they are identical across the point clouds stems from the same justification of step 2. + +In the second update step, the arguments of Step 1 remain identical. Still, now we have updated hidden node information, but the hidden node information is identical across nodes belonging to the same block. Therefore, the only information this refinement yields is the categorization of nodes into blocks. Yet this information is already known in the initialized hidden states, $\{h_i^{(0)}i = 1,\dots ,n\}$ since the zero entries of multiplication $h_i^{(0)} = V_i\odot V_i$ determine the block that $i$ belongs to. Therefore, the hidden states don't supply the network with any supplementary information other than the initialization $h_i^{(0)} = V_i\odot V_i$ . Thus, after a second refinement step, the hidden states remain identical within each block, as they have after the first refinement step. Moreover, the corresponding hidden states of the two point clouds also remain equivalent due to the arguments in Step 2, which remain analogous, as the hidden states after a refinement only assign each node its respective block + +membership, which is exactly the information given in the first update step. This argument can then be applied recursively to any number of update steps. + +In conclusion, we have shown that for any $t \in \mathbb{N}$ , the hidden states of both point clouds are identical (in corresponding indices), therefore after a permutation invariant readout, we obtain the same output. + +We now prove 3. $U$ and $V$ have no nontrivial automorphisms. To show $U, V$ are not isomorphic, we note that for any pair of permutation-sign matrices taking $U$ to $V$ , the first four columns of $U^T$ must be mapped the first four columns of $V^T$ . The same is true for columns $5 - 8$ and $9 - 12$ . Considering the first four columns, we see that any sign matrix mapping them from $U$ to $V$ will be of the form $\mathrm{diag}(z, z, z')$ for $z, z' \in \{-1, +1\}^2$ . The same argument for columns $5 - 8$ and $9 - 12$ gives sign patterns of the form $\mathrm{diag}(z', z, z_1 \cdot z)$ and $\mathrm{diag}(z, z', z_2 \cdot z)$ , respectively. But there is no sign pattern satisfying these three constraints simultaneously. + +The automorphism group of this extended eigendecomposition is contained within that of $U$ and $U$ , respectively, and thus is also only the trivial group. + +The proof of 1. which states that $U$ and $V$ are not isomorphic, is analogous to the proof of 3, and yields that the only sign pattern taking each point cloud to itself is $(z_0, z_0, z_0)$ , which implies each point cloud has only a trivial automorphism.. + +# A.2 Extension to orthonormal counterexamples + +The rows of the above point clouds $U, V$ are not orthonormal. Thus they are not eigenvectors of an eigendecomposition of a symmetric matrix. We fix this misalignment via the following 'orthogonalization' matrices: + +Taking $\tilde{U}$ to be a concatenation of the previous $U$ and $\hat{U}$ defined by + +$$ +\hat {U} ^ {T} = \left( \begin{array}{c c c c c c c c c c c} 2 z _ {0} & 2 z _ {1} & 2 z _ {2} & 2 z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & - 2 z _ {0} & - 2 z _ {1} & - 2 z _ {2} & - 2 z _ {3} \\ - \frac {z _ {0}}{2} & - \frac {z _ {1}}{2} & - \frac {z _ {2}}{2} & - \frac {z _ {3}}{2} & 2 z _ {0} & 2 z _ {1} & 2 z _ {2} & 2 z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & - \frac {z _ {0}}{2} & - \frac {z _ {1}}{2} & - \frac {z _ {3}}{2} & - \frac {z _ {2}}{2} & \frac {z _ {0}}{2} & \frac {z _ {2}}{2} & \frac {z _ {1}}{2} & \frac {z _ {3}}{2} \end{array} \right) +$$ + +Then take $\hat{V}$ to be a concatenation of the previous $V$ and $\hat{V}$ defined by + +$$ +\hat {V} ^ {T} = \left( \begin{array}{c c c c c c c c c c c} 2 z _ {0} & 2 z _ {1} & 2 z _ {2} & 2 z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & - 2 z _ {0} & - 2 z _ {1} & - 2 z _ {2} & - 2 z _ {3} \\ - \frac {z _ {0}}{2} & - \frac {z _ {1}}{2} & - \frac {z _ {2}}{2} & - \frac {z _ {3}}{2} & 2 z _ {0} & 2 z _ {1} & 2 z _ {2} & 2 z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & - \frac {z _ {1}}{2} & - \frac {z _ {0}}{2} & - \frac {z _ {2}}{2} & - \frac {z _ {3}}{2} & \frac {z _ {2}}{2} & \frac {z _ {0}}{2} & \frac {z _ {3}}{2} & \frac {z _ {1}}{2} \end{array} \right) +$$ + +The columns of $\tilde{U}$ and $\tilde{V}$ are now orthogonal, and they can be made to have unit norm by normalizing each column. As these extensions exhibit the same symmetries of $U$ and $V$ , respectively, analogous arguments to the proof of inseparation of $U$ and $V$ by EPNN (Theorem 2) will apply to this new pair $\tilde{U}, \tilde{V}$ . Therefore, EPNN cannot distinguish $\tilde{U}$ and $\tilde{V}$ . + +# A.3 Proofs for implications for real-world GNNs + +Proposition 3. Vanilla OGE-Aug [53] is incomplete over graphs with a simple spectrum. + +Proof. The method proposed by Zhou et al. [53] consists of a permutation equivariant and orthogonal invariant function. We will show that a counterexample by [29] also applies to this network. + +Vanilla PGE-Aug relies on a permutation equivariant and orthogonal invariant set encoding to process each eigenspace separate. Lets revisit their separately definitions and theorem: + +Definition 4 (O(p)-invariant universal representation [53]). Let $f: \bigcup_{n=0}^{\infty} \mathbb{R}^{n \times p} \to \bigcup_{n=0}^{\infty} \mathbb{R}^n$ . Given an input $V \in \mathbb{R}^{n \times p}$ , $f$ outputs a vector $f(V) \in \mathbb{R}^n$ . The function $f$ is said to be an $O(p)$ -invariant universal representation if given $V, V' \in \mathbb{R}^{n \times p}$ and $P \in S_n$ , the following two conditions are equivalent: + +(i) $f(V) = Pf(V^{\prime})$ +(ii) $\exists Q\in O(p)$ , such that $V = PV^{\prime}Q$ + +Definition 5 (Universal set representation [53]). Let $X$ be a non-empty set. A function $f: 2^X \to \mathbb{R}$ is said to be a universal set representation if $\forall X_1, X_2 \in 2^X$ , $f(X_1) = f(X_2)$ if and only if the two sets $X_1$ and $X_2$ are equal. + +Proposition 3.5 (Zhou et al. [53]) For each $p = 1,2,\ldots$ , let $f_{p}$ be an $O(p)$ -invariant universal representation function. Further let $g:2^{\mathbb{R}^3}\to \mathbb{R}$ be a universal set representation. Then the following function + +$$ +r (G, X _ {G}) = \operatorname {G N N} \left(A _ {G}, \operatorname {c o n c a t} \left[ X _ {G}, g \left(\{\operatorname {c o n c a t} [ \mu_ {j} \mathbf {1} _ {n}, \lambda_ {j} \mathbf {1} _ {n}, f _ {\mu_ {j}} (V _ {j}) ] \} _ {j = 1} ^ {K}\right) \right]\right) \tag {8} +$$ + +is a universal representation. Here $n = |V(G)|$ , $((\lambda_1, \mu_1), \ldots, (\lambda_K, \mu_K))$ is the spectrum of $G$ , and $V_j \in \mathbb{R}^{n \times \mu_j}$ are the $\mu_j$ mutually orthogonal normalized eigenvectors of $L_G$ corresponding to $\lambda_j$ . We denote $\mathbf{1}_n$ an all-1 vector of shape $n \times 1$ . GNN is a maximally expressive MPNN. + +Then Zhou et al. [53] propose the following graph neural network: + +Definition 3.6 (Vanilla OGE-Aug). Let $f_{p}$ be an $O(p)$ -invariant universal representation, for each $p = 1,2,\ldots$ , and $g:2\mathbb{R}^3\to \mathbb{R}$ be a universal set representation. Define $Z:\mathcal{G}\to \bigcup_{n = 1}^{\infty}\mathbb{R}^{n}$ as + +$$ +Z (G) = g \left(\left\{\operatorname {c o n c a t} \left[ \mu_ {j} \mathbf {1} _ {| V (G) |}, \lambda_ {j} \mathbf {1} _ {| V (G) |}, f _ {\mu_ {j}} \left(V _ {j}\right) \right] \right\} _ {j = 1} ^ {K}\right), \tag {5} +$$ + +in which the notations follow Proposition 3.5. For $G \in \mathcal{G}$ , $Z(G)$ is called a vanilla orthogonal group equivariant augmentation, or Vanilla OGE-Aug on $G$ . + +We will show that architectures of the form of Proposition 3.5 and specifically Vanilla OGE-Aug are incomplete on simple spectrum graphs, contradicting the claim in Proposition 3.5 such a representation is universal. + +Consider the point clouds proposed by Ma et al. [29]: + +$$ +U _ {1} = \left[ u _ {1 1}, u _ {1 2} \right] = \left( \begin{array}{c c c c} 1 & - 1 & 1 & - 1 \\ 2 & 3 & 4 & 5 \end{array} \right) ^ {\top}, \tag {9} +$$ + +$$ +U _ {2} = \left[ u _ {2 1}, u _ {2 2} \right] = \left( \begin{array}{c c c c} - 1 & 1 & 1 & - 1 \\ 2 & 3 & 4 & 5 \end{array} \right) ^ {\top}. \tag {10} +$$ + +Suppose the first column eigenvector of $U_{1}$ and $U_{2}$ corresponds to eigenvalue $\lambda_{1} = 1$ , the second column eigenvector of $U_{1}$ and $U_{2}$ corresponds to eigenvalue $\lambda_{2} = 2$ , and other eigenvectors not shown corresponds to eigenvalue 0 (so we safely ignore them). Then the Laplacian matrices corresponding to $U_{1}$ and $U_{2}$ are: + +$$ +L _ {1} = \lambda_ {1} u _ {1 1} u _ {1 1} ^ {\top} + \lambda_ {2} u _ {1 2} u _ {1 2} ^ {\top} = \left( \begin{array}{c c c c} 9 & 1 1 & 1 7 & 1 9 \\ 1 1 & 1 9 & 2 3 & 3 1 \\ 1 7 & 2 3 & 3 3 & 3 9 \\ 1 9 & 3 1 & 3 9 & 5 1 \end{array} \right), \tag {11} +$$ + +$$ +L _ {2} = \lambda_ {1} u _ {2 1} u _ {2 1} ^ {\top} + \lambda_ {2} u _ {2 2} u _ {2 2} ^ {\top} = \left( \begin{array}{l l l l} 9 & 1 1 & 1 5 & 2 1 \\ 1 1 & 1 9 & 2 5 & 2 9 \\ 1 5 & 2 5 & 3 3 & 3 9 \\ 2 1 & 2 9 & 3 9 & 5 1 \end{array} \right). \tag {12} +$$ + +We will now demonstrate the model in Proposition 3.5 will be unable to distinguish $U_{1}$ and $U_{\mathbb{Q}}$ , regardless of the choice of the GNN. + +First, consider an arbitrary $O(1)$ -invariant representation $f: \mathbb{R}^n \to \mathbb{R}^n$ . We will show that $f(U_1)$ and $f(U_2)$ are identical. + +By the permutation equivariance and $O(1)$ invariance: + +$$ +f \left(u _ {1 1}\right) = f (- u _ {1 1}) = f \left(P _ {1 1} u _ {1 1}\right) = P _ {1 1} f \left(u _ {1 1}\right) \tag {13} +$$ + +where $P_{11}$ is any permutation that satisfies $P_{11}u_{11} = -u_{11}$ . Therefore $P_{11}$ can be chosen to be $\sigma_1\triangleq (12)(34)$ or $\sigma_2\triangleq (14)(23)$ . + +By Equation 13, and since equality is a transitive relation, it holds that $f(u_{11})(i) = f(u_{11})(j)$ for any $i$ and $j$ in the same orbit under the group $< \sigma_1, \sigma_2 >$ , the group generated by $\sigma_1$ and $\sigma_2$ . It is easy to check any pair $(i,j) \in \{1,2,3,4\}^2$ can be transposed under a group element in the generated group. Therefore, $f(u_{11})$ is a constant function. Analogous arguments yield $f(u_{21})$ is also constant. + +Note that for $P_{12} \triangleq (12)(34)$ , it holds that + +$$ +f (u _ {1 1}) \underset {f (u _ {1 1}) \text {i s c o n s t a n t}} {=} P f (u _ {1 1}) \underset {\text {p e r m . e q u i v a r i a n c e}} {=} f (P u _ {1 1}) = f (u _ {2 1}) +$$ + +Therefore, $f(u_{11}) = f(u_{21})$ . Moreover, the second eigenvectors, $u_{12}$ and $u_{22}$ of $U_{1}$ and $U_{2}$ , respectively, are identical therefore clearly $f(u_{12}) = f(u_{22})$ . + +This analysis naturally extends to a proper eigendecomposition (orthonormal eigenvectors of a graph as proposed by Ma et al. [29] in the proof of their Corollary 3.5 [29]. + +Therefore, as any universal, invariant set representation is the same on both $U_{1}$ and $U_{2}$ , the input to the network will be identical per its definition, and thus for their corresponding graphs $G_{1}$ and $G_{2}$ and identical node features $X_{G_1}$ and $X_{G_2}$ , respectively it holds that + +$$ +r \left(G _ {1}, X _ {G _ {1}}\right) = r \left(G _ {2}, X _ {G _ {2}}\right) +$$ + +yet $G_{1}$ and $G_{2}$ are non-isomorphic, thus Vanilla OGE-Aug is incomplete. + +![](images/569b432d4c67ef40b4f8f476b67688a5c9616019e5e0466f6f6d292158202701.jpg) + +# A.4 Proof for equiEPNN strictly more expressive + +Corollary 1. equiEPNN (see Section 3.3) can separate $U$ and $V$ after 2 iterations. Thus equiEPNN is strictly stronger than EPNN. + +Proof. We show that after a single iteration, the equivariant update step can yield new matrices $U^{(t)}, V^{(t)}, t = 1$ which have no zeros. We can choose the update function $\mathrm{UPDATE}_{(1,2)}$ such that $\mathrm{UPDATE}_{(1,2)}(v_5 \odot v_5, v_1 \odot v_1, v_5 \odot v_1) \triangleq (1, 1, 0, 0, 0, 0, 0)$ and for all other values we define it as $\vec{0}$ . + +After a single iteration $U^{(1)}$ and $V^{(1)}$ will be + +$$ +U ^ {(1) T} = \left( \begin{array}{c c c c c c c c c c c c} z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {2} & z _ {3} \\ z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {3} & z _ {2} & z _ {0} & z _ {2} & z _ {1} & z _ {3} \end{array} \right) +$$ + +$$ +V ^ {(1) T} = \left( \begin{array}{c c c c c c c c c c c c} z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {0} & z _ {1} & z _ {2} & z _ {3} \\ z _ {0} & z _ {1} & z _ {2} & z _ {3} & z _ {0} & z _ {1} & z _ {2} & z _ {3} & 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} \\ 0 _ {2} & 0 _ {2} & 0 _ {2} & 0 _ {2} & z _ {1} & z _ {0} & z _ {2} & z _ {3} & z _ {2} & z _ {0} & z _ {3} & z _ {1} \end{array} \right) +$$ + +Since there exists a column (the fifth column) such that all its entries are non-zero in both $U^{(1)T}$ and $V^{(1)T}$ , from Theorem 2, we know that a single iteration of EPNN, and hence also of equiEPNN, can separate $U^{(1)T}, V^{(1)T}$ . In conclusion, two iterations of equiEPNN are sufficient for separation. + +# A.5 Proof of Incompleteness of Equivariant EPNN + +The purpose of this section is to show that equivariant EPNN is also not complete on simple spectrum graphs. To show this, we will construct a counter-example of a pair $X,Y$ which are not isomorphic with respect to the joint action of permutations and sign multiplications, and yet cannot be distinguished by equiEPNN. We note that the columns of $X,Y$ are not orthonormal, and they do have automorphisms. + +For $X, Y \in \mathbb{R}^{n \times K}$ , we will say that $X \equiv Y$ , if there is some permutation matrix $P$ such that $PX = Y$ . We will say that $s \in \{-1, 1\}^K$ is an isomorphism between $X$ and $Y$ , if $X \operatorname{diag}(s) \equiv Y$ . Here $\operatorname{diag}(s)$ is the $K \times K$ diagonal matrix with $s$ on the diagonal. An automorphism of $X$ is an isomorphism from $X$ to $X$ . + +As a first step to construct our counter example, we consider the subgroup $H \leq \{-1, 1\}^3$ defined by + +$$ +H = \{s \in \{- 1, 1 \} ^ {3} \mid s _ {1} \cdot s _ {2} \cdot s _ {3} = 1 \}. +$$ + +Let $T$ be the $4 \times 3$ matrix whose rows are the four elements of $H$ , namely + +$$ +T = \left( \begin{array}{c c c} 1 & 1 & 1 \\ 1 & - 1 & - 1 \\ - 1 & - 1 & 1 \\ - 1 & 1 & - 1 \end{array} \right) +$$ + +Note that $\operatorname{Aut}(T) = H$ due to $H$ having a group structure. + +Next, we build the matrix $X$ to consist of four different copies of $T$ . Each copy will be not a $4 \times 3$ but a $4 \times 6$ matrix, where three of the columns are the columns of $T$ , and the rest are zero columns. Moreover, any two copies of $T$ will only have one non-zero column in common. + +To do this, we choose four index sets in $\{1,2,\ldots ,6\}$ , who have this intersection pattern, namely + +$$ +I _ {1} = \{1, 2, 3 \}, \quad I _ {2} = \{3, 4, 5 \}, \quad I _ {3} = \{2, 4, 6 \}, \quad I _ {4} = \{1, 5, 6 \}. +$$ + +One can verify that indeed $|I_j \cap I_k| = 1$ for all $j \neq k$ . We then define the matrix $T[I_j]$ to be the $4 \times 6$ matrix as described previously. For example + +$$ +T [ I _ {2} ] = \left( \begin{array}{c c c c c c} 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & - 1 & - 1 & 1 & 0 \\ 0 & 0 & - 1 & 1 & - 1 & 0 \end{array} \right) +$$ + +We define $X$ to be the block matrix + +$$ +X = \left( \begin{array}{c} T [ I _ {1} ] \\ T [ I _ {2} ] \\ T [ I _ {3} ] \\ T [ I _ {4} ] \end{array} \right) \in \mathbb {R} ^ {1 6 \times 6} +$$ + +or explicitly + +$$ +X = \left( \begin{array}{c c c c c c} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & - 1 & - 1 & 0 & 0 & 0 \\ - 1 & - 1 & 1 & 0 & 0 & 0 \\ - 1 & 1 & - 1 & 0 & 0 & 0 \\ - \frac {1}{0} & - \frac {1}{0} & - \frac {1}{1} & - \frac {1}{1} & - \frac {1}{0} & - \\ 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & - 1 & - 1 & 1 & 0 \\ 0 & 0 & - \frac {1}{1} & - \frac {1}{0} & - \frac {1}{0} & - \frac {1}{1} \\ 0 & 1 & 0 & - 1 & 0 & - 1 \\ 0 & - 1 & 0 & - 1 & 0 & 1 \\ 0 & - \frac {1}{1} & - \frac {1}{0} & - \frac {1}{0} & - \frac {1}{1} & - \frac {1}{1} \\ 1 & 0 & 0 & 0 & - 1 & - 1 \\ - 1 & 0 & 0 & 0 & - 1 & 1 \\ - 1 & 0 & 0 & 0 & 1 & - 1 \end{array} \right) +$$ + +We define $Y$ similarly, but we elementwise multiply the rows of $T[I_1]$ by the sign vector + +$$ +q = [ - 1, 1, 1, 1, 1, 1 ] +$$ + +to obtain + +$$ +Y = \left( \begin{array}{c} T [ I _ {1} ] \mathrm {d i a g} (q) \\ T [ I _ {2} ] \\ T [ I _ {3} ] \\ T [ I _ {4} ] \end{array} \right) \in \mathbb {R} ^ {1 6 \times 6}. +$$ + +This is our counterexample. We claim + +Theorem 4. (Incompleteness of Equivariant EPNN) There exist $X, Y \in \mathbb{R}^{16 \times 6}$ such that the following statements hold: + +1. $X$ and $Y$ are not isomorphic under the group action of $S_{16} \times \{-1, 1\}^{6}$ . +2. Equivariant EPNN cannot separate $X$ and $Y$ after any number of iterations. + +Therefore, Equivariant EPNN is incomplete on simple spectrum graphs. + +Proof. Remark: $X, Y$ are not isomorphic. By considering the zero patterns of $X$ and $Y$ , one sees that if $s \in \{-1, 1\}^6$ is an isomorphism mapping $X$ to $Y$ , then $s$ satisfies $P^T T[I_1] \mathrm{diag}(s) = T[I_1] \mathrm{diag}(q)$ , for some permutation matrix $P$ (acting on the rows of $T[I_1]$ ), and $s$ must also define an automorphism of $T[I_j]$ for $j = 2, 3, 4$ . Since each $T[I_j]$ 's rows (padded with zeros) form a group, its only automorphisms are elementwise multiplications of its rows by its group elements, which implies $\prod_{i \in I_j} s_i = 1$ . From these three automorphism conditions, we deduce: + +$$ +s _ {3} \cdot s _ {4} \cdot s _ {5} = 1 +$$ + +$$ +s _ {2} \cdot s _ {4} \cdot s _ {6} = 1 +$$ + +$$ +s _ {1} \cdot s _ {5} \cdot s _ {6} = 1 +$$ + +Multiplying these three equations with each other we deduce that + +$$ +s _ {1} \cdot s _ {2} \cdot s _ {3} = 1. +$$ + +Now, if this holds, then $s$ cannot satisfy $P^T T[I_1]\mathrm{diag}(s) = T[I_1]\mathrm{diag}(q)$ because the product of the first three entries of any row of $P^T T[I_1]\mathrm{diag}(s)$ is 1, while the product of the first three entries of any row of $T[I_1]\mathrm{diag}(q)$ is $-1$ . + +# $X$ and $Y$ cannot be separated by equiEPNN + +We prove by induction that for any number of layers in an equiEPNN, the hidden states for nodes within the same partition $B_{k}$ are identical, and this holds for both graph structures $X$ and $Y$ . This equivalence prevents the network from separating them. + +We introduce useful definitions: + +Definition 6 (Block Structure and Neighborhoods). We partition the $n = 16$ nodes (rows) into 4 disjoint blocks $B_{k}$ for $k = 1, \ldots, 4$ (e.g., $B_{1} = \{1, \ldots, 4\}$ , $B_{2} = \{5, \ldots, 8\}$ , etc.). The $4 \times 6$ matrix of initial equivariant features for block $B_{k}$ is $B_{k}^{(0)} \triangleq X[B_{k},:] = T[I_{k}]$ . The non-zero feature indices for this block are $I_{k}$ . For a node $i \in B_{k}$ , we define its neighbors: $\mathcal{N}_{\text{intra}}(i) \triangleq B_{k}$ and $\mathcal{N}_{\text{inter}}(i) \triangleq \{1, \ldots, n\} \setminus B_{k}$ . + +Definition 7 (Invariant Node Neighborhood). The message from a neighbor $j$ to a node $i$ of $X$ is a tuple containing the neighbor's invariant features and an invariant computed from their equivariant features, $(h_j^{(l)}, x_i^{(l)} \odot x_j^{(l)})$ . The Invariant Node Neighborhood of a node $i$ at layer $l$ is the multiset of invariant features $\mathcal{I}_i^{(l)} \triangleq \mathcal{I}_{i,\text{intra}}^{(l)} \cup \mathcal{I}_{i,\text{inter}}^{(l)}$ , where + +$\mathcal{I}_{i, \text{intra}}^{(l)} = \{(h_j^{(l)}, x_i^{(l)} \odot x_j^{(l)}) \mid j \in \mathcal{N}_{\text{intra}}(i)\}$ +$\mathcal{I}_{i,inter}^{(l)} = \{(h_j^{(l)},x_i^{(l)}\odot x_j^{(l)})\mid j\in \mathcal{N}_{inter}(i)\}$ + +where $\{\cdot\}$ denotes a multi-set. The update rule combines the node's own invariant state $h_i^{(l)}$ with aggregations of the messages from its neighborhoods: + +$$ +h _ {i} ^ {(l + 1)} = \phi_ {h} \left(h _ {i} ^ {(l)}, A G G \left(\mathcal {I} _ {i} ^ {(l)}\right)\right) +$$ + +where AGG is a permutation-invariant aggregation function (e.g., sum or mean). + +Definition 8 (Equivariant Node Neighborhood). The Equivariant Node Neighborhood of a node $i$ at layer $l$ is defined by $\mathcal{E}_i^{(l)}\triangleq \mathcal{E}_{i,\text{intra}}^{(l)}\cup \mathcal{E}_{i,\text{inter}}^{(l)}$ , where + +- Intra-block Neighborhood $\mathcal{E}_{i, \text{intra}}^{(l)} = \{\phi_v(h_i^{(l)}, h_j^{(l)}, x_i^{(l)}, x_j^{(l)}) \odot x_j^{(l)} \mid j \in \mathcal{N}_{\text{intra}}(i)\}$ +- Inter-block Neighborhood $\mathcal{E}_{i,inter}^{(l)} = \{\phi_v(h_i^{(l)},h_j^{(l)},x_i^{(l)},x_j^{(l)})\odot x_j^{(l)}\mid j\in \mathcal{N}_{inter}(i)\}$ + +where $\{\cdot\}$ denotes a multi-set. Also, define the messages arriving to a node $i \in B_p$ from a different block $B_k$ ( $k \neq p$ ) at layer $l$ by $\mathcal{E}_{i,k}^{(l)} = \{\phi_v(h_i^{(l)}, h_j^{(l)}, x_i^{(l)}, x_j^{(l)}) \odot x_j^{(l)} \mid j \in B_k\}$ . + +The equivariant feature is updated by summing over both neighborhoods: + +$$ +x _ {i} ^ {(l + 1)} = x _ {i} ^ {(l)} + \sum_ {m \in \mathcal {E} _ {i, \text {i n t r a}} ^ {(l)}} m + \sum_ {m \in \mathcal {E} _ {i, \text {i n t e r}} ^ {(l)}} m +$$ + +Proof outline: + +1. We show that the blocks of $X$ and $Y$ are a particular case of a generalized block structure. +2. We analyze the mechanics of equiEPNN when processing these generalized X and Y to prove that the invariant node neighborhoods of corresponding nodes in X and Y are equivalent. This is the base of our induction. +3. We show that the equivariant update step maintains this generalized block structure for both $X$ and $Y$ . The equivariant update maintaining the generalized block pattern of $X$ and $Y$ is the induction step of the proof. +4. Since an equivariant update maintains the generalized block structure of $X$ and $Y$ , and the subsequent invariant node neighborhoods of corresponding points in generalized $X$ and $Y$ are identical, by the base of induction, equiEPNN will output the same readout for both $X$ and $Y$ after arbitrarily many refinement iterations (the hidden states are equivalent as multisets for both point clouds). + +# Base Case (Generalized Block Pattern and Invariant Update) + +Generalized Block Pattern The initial invariant features $h_i^{(0)} = x_i^{(0)} \odot x_i^{(0)}$ are identical for all $i \in B_k$ , as they equal the indicator vector for the partition $I_k$ . We consider a generalized case, where the initial equivariant features for block $B_k$ (with non-zero columns $I_k$ ) form a matrix $B_k^{(0)}$ where $B_k^{(0)}[:, I_k]$ is: + +$$ +B _ {k} ^ {(0)} [:, I _ {k} ] = \left[ \begin{array}{c c c} s _ {k, 1} ^ {(0)} & s _ {k, 2} ^ {(0)} & s _ {k, 3} ^ {(0)} \\ - s _ {k, 1} ^ {(0)} & - s _ {k, 2} ^ {(0)} & s _ {k, 3} ^ {(0)} \\ s _ {k, 1} ^ {(0)} & - s _ {k, 2} ^ {(0)} & - s _ {k, 3} ^ {(0)} \\ - s _ {k, 1} ^ {(0)} & s _ {k, 2} ^ {(0)} & - s _ {k, 3} ^ {(0)} \end{array} \right] +$$ + +In our counterexample $X$ , the scalars $s_{k,j} \equiv 1$ for all $k, j$ . For $Y$ , $s_{1,1} = -1$ (from block $k = 1$ , column $j = 1$ ) and all other $s_{k,j} \equiv 1$ . We consider this generalized case because we will show that after an equivariant aggregation, this will be the format of the blocks. These are called generalized $X$ and $Y$ with a single scalar choice defining them, as the generalized $Y$ is equivalent to generalized $X$ up to a negation first row of the first block of $X$ . We refer to these generalized $X$ and $Y$ as simply $X$ and $Y$ in the remainder of the proof. + +These initial hidden states $h_i^{(0)}$ are identical for both point clouds $X$ and $Y$ , due to the invariance of squaring to sign changes. Additionally, $h_i^{(0)} = h_j^{(0)}$ for $i,j\in B_k$ and $h_i^{(0)}\neq h_j^{(0)}$ for $j\notin B_{k}$ due to the unique sparsity pattern of each block. This completes the first step of the outline. We now proceed to the second step of the outline, where we prove that an invariant update maintains the equivalence of the hidden states within each block. + +Invariant Update We formally define the aggregation steps for a node $i \in B_k$ at layer $l$ by splitting our analysis of its neighborhood into intra-block neighbors $\mathcal{N}_{\mathrm{intra}}(i)$ and inter-block neighbors $\mathcal{N}_{\mathrm{inter}}(i)$ . For any two nodes $i, j \in B_k$ , we show their invariant neighborhoods yield identical aggregations. + +- Intra-block: A message from a neighbor $m \in B_k$ is $(h_m^{(l)}, x_i^{(l)} \odot x_m^{(l)})$ . By the inductive hypothesis, $h_m^{(l)}$ is constant for all $m \in B_k$ . The set of vectors $\{x_m^{(l)} \mid m \in B_k\}$ forms a group under the Hadamard product (up to scalar multiples). By the group closure property, the multiset of products $\{x_i^{(l)} \odot x_m^{(l)} \mid m \in \mathcal{N}_{\mathrm{intra}}(i)\}$ is simply a permutation of $\{x_j^{(l)} \odot x_m^{(l)} \mid m \in \mathcal{N}_{\mathrm{intra}}(j)\}$ for any $i, j \in B_k$ . Therefore, any permutation-invariant aggregation over $\mathcal{I}_{i,\mathrm{intra}}^{(l)}$ and $\mathcal{I}_{j,\mathrm{intra}}^{(l)}$ is identical. +- Inter-block: The graph is constructed such that for any $k \neq p$ , $|I_k \cap I_p| = 1$ . The inter-block neighborhood for a node in $B_k$ consists of nodes from the other three blocks. Consider a neighbor $m \in B_p$ . The product $x_i^{(l)} \odot x_m^{(l)}$ is non-zero only at the single index $j = I_k \cap I_p$ . Due to this structure, the resulting multiset of invariants from block $B_p$ is of the form $\{\alpha_j \mathbf{e}_j, \alpha_j \mathbf{e}_j, -\alpha_j \mathbf{e}_j, -\alpha_j \mathbf{e}_j\}$ (where $\mathbf{e}_j$ is the standard basis vector and $\alpha_j$ is some scalar), which is identical for all $i \in B_k$ . + +Since both neighborhood aggregations are identical, and $h_i^{(l)} = h_j^{(l)}$ for $i,j\in B_k$ , the update yields $h_i^{(l + 1)} = h_j^{(l + 1)}$ for both $X$ and $Y$ . + +# Inductive Step + +Assume at layer $l$ , for any partition $B_{k}$ , $h_{i}^{(l)} = h_{j}^{(l)}$ for all $i,j \in B_k$ , and the equivariant feature matrix $B_{k}^{(l)} \triangleq X[B_{k},:]^{(l)}$ maintains the scaled pattern structure. + +**Equivariant Update** The update for the equivariant features $x_{i}^{(l + 1)}$ combines the original features $x_{i}^{(l)}$ with aggregations from intra-block and inter-block neighbors. + +Intra-block Aggregation: The aggregation of messages within a block $B_{k}$ can be compactly expressed via summation and Hadamard products. The message function $\phi_v$ produces scalar weights for each interaction. Since the invariant features $h^{(l)}$ are constant within the block, these weights depend only on the structural relationship between nodes $i$ and $j$ . Due to the graph's symmetries, there are only four unique interaction types within a block, resulting in four learned scalar vectors, with scalar dimension weights in each feature dimension in $\mathbb{R}^K$ . Since only $I_{k}$ are the indices with non-zero features, we focus on their aggregation, and the rest of the inputs along other feature dimensions will be aggregated to 0, therefore we denote by $a, b, c, d \in \mathbb{R}^3$ the reduction into the feature indices in $I_{k}$ . These form a symmetric weight matrix (in the node dimension) we denote by + +$$ +\Phi_ {k} ^ {(l)} = \left[ \begin{array}{c c c c} a & b & c & d \\ b & a & d & c \\ c & d & a & b \\ d & c & b & a \end{array} \right] \in \mathbb {R} ^ {4 \times 4 \times 3} +$$ + +This operation, which we denote by $\star$ , scales the columns of the feature matrix $B_{k}^{(l)}$ while preserving their sign-pattern structure: + +$$ +\begin{array}{l} \Phi_ {k} ^ {(l)} \star B _ {k} ^ {(l)} [:, I _ {k} ] = \left[ \begin{array}{l} a \odot B _ {k} ^ {(l)} [ 1, I _ {k} ] + b \odot B _ {k} ^ {(l)} [ 2, I _ {k} ] + c \odot B _ {k} ^ {(l)} [ 3, I _ {k} ] + d \odot B _ {k} ^ {(l)} [ 4, I _ {k} ] \\ b \odot B _ {k} ^ {(l)} [ 1, I _ {k} ] + a \odot B _ {k} ^ {(l)} [ 2, I _ {k} ] + d \odot B _ {k} ^ {(l)} [ 3, I _ {k} ] + c \odot B _ {k} ^ {(l)} [ 4, I _ {k} ] \\ c \odot B _ {k} ^ {(l)} [ 1, I _ {k} ] + d \odot B _ {k} ^ {(l)} [ 2, I _ {k} ] + a \odot B _ {k} ^ {(l)} [ 3, I _ {k} ] + b \odot B _ {k} ^ {(l)} [ 4, I _ {k} ] \\ d \odot B _ {k} ^ {(l)} [ 1, I _ {k} ] + c \odot B _ {k} ^ {(l)} [ 2, I _ {k} ] + b \odot B _ {k} ^ {(l)} [ 3, I _ {k} ] + a \odot B _ {k} ^ {(l)} [ 4, I _ {k} ] \end{array} \right] \in \mathbb {R} ^ {4 \times 3} (14) \\ = \left[ \begin{array}{c c c} \alpha & \beta & \gamma \\ - \alpha & - \beta & \gamma \\ \alpha & - \beta & - \gamma \\ - \alpha & \beta & - \gamma \end{array} \right] (15) \\ \end{array} +$$ + +where the entries of the resulting matrix (in the $I_{k}$ columns) are denoted by the scalars $\alpha, \beta, \gamma$ . These scalars are the result of applying the learned weights $a, b, c, d$ (which are vectors) to the corresponding + +columns of $B_{k}^{(l)}$ . Specifically, they are defined as: + +$$ +\alpha = s _ {k, 1} ^ {(l)} \left(a _ {i _ {1}} - b _ {i _ {1}} + c _ {i _ {1}} - d _ {i _ {1}}\right) +$$ + +$$ +\beta = s _ {k, 2} ^ {(l)} \left(a _ {i _ {2}} - b _ {i _ {2}} - c _ {i _ {2}} + d _ {i _ {2}}\right) +$$ + +$$ +\gamma = s _ {k, 3} ^ {(l)} \left(a _ {i _ {3}} + b _ {i _ {3}} - c _ {i _ {3}} - d _ {i _ {3}}\right) +$$ + +where $a_{j}$ is the $j$ -th component of $a$ , etc. and $i_1, i_2, i_3 \in I_k$ . This operation preserves the fundamental sign-pattern structure of each column, merely updating its overall scaling factor. + +Inter-block Aggregation: Let $i \in B_p$ be a node index in block $p$ , and let $k \neq p$ be a different block index. Consider the equivariant node neighborhood $\mathcal{E}_{i,k}^{(l)}$ . We first focus on the inter-block aggregation of $X$ and proceed to discuss that of $Y$ . Consider the contribution of the equivariant message passing to the features of node $i$ from block $B_k$ . There are 3 possible cases: + +Case 1: Aggregation of $\mathcal{E}_{i,k}^{(l)}$ along dimension $d = I_k\cap I_p$ . Along this single feature index $d$ , the only non-zero information in the product is $x_{i}^{(l)}[d]\cdot x_{j}^{(l)}[d]\in \mathbb{R}$ for $j\in B_k$ . This scalar, apart from the hidden states (which are constant within blocks), is the only structural value that determines $\phi_v(h_i^{(l)},h_j^{(l)},x_i^{(l)},x_j^{(l)})$ . This $\phi_v$ in turn determines the feature-wise weighing of $x_{j}^{(l)}$ . It follows that for any node $s\in B_p$ , the sum of $\mathcal{E}_{s,k}^{(l)}$ in the feature dimension $d$ precisely equals that of $i$ up to $\mathrm{sign}(x_i^{(l)}[d]\cdot x_s^{(l)}[d])$ . Because $x_{s}^{(l)}[d]$ follows the generalized block pattern for $B_{p}$ , the aggregation into dimension $d$ also follows this pattern. + +Case 2: Aggregation of $\mathcal{E}_{i,k}^{(l)}$ along feature indices in $I_{k} \setminus I_{p}$ . From Case 1, $\phi_v$ is determined by the sign of the product in dimension $d$ . This results in two possible weight vectors, say $\vec{a}$ and $\vec{b}$ . The set of messages is + +$$ +\mathcal {E} _ {i, k} ^ {(l)} (X) = \left\{\vec {a} \odot x _ {j _ {1}}, \vec {a} \odot x _ {j _ {2}}, \vec {b} \odot x _ {j _ {3}}, \vec {b} \odot x _ {j _ {4}} \right\} \tag {16} +$$ + +where $x_{j_1}, x_{j_2}$ are (w.l.o.g) the points with a positive scalar product with $x_i$ in dimension $d$ , and $x_{j_3}, x_{j_4}$ are those with a negative product. By construction of $T$ , the points $x_{j_1}, x_{j_2}$ satisfy $x_{j_1}(m) = -x_{j_2}(m)$ for each $m \in I_k \setminus I_p$ . An analogous result holds for $x_{j_3}, x_{j_4}$ . Therefore, summing all points in $\mathcal{E}_{i,k}^{(l)}$ yields zeros in feature entries $I_k \setminus I_p$ . + +Case 3: Aggregation of $\mathcal{E}_{i,k}^{(l)}$ along remaining indices. In all other indices, $\{1,2,\dots,6\} \setminus (I_p\cup I_k)$ , the features of $x_{j}$ (for $j\in B_k$ ) are 0. Thus, it trivially holds that after aggregating $\mathcal{E}_{i,k}^{(l)}$ , the resulting vector entries in those dimensions will also be 0. + +We now address the inter-block update of $Y$ in comparison with that of $X$ . The only structural difference is the negated first column in block $B_{1}$ of $Y$ . This affects aggregation for $i \in B_{1}$ and for $i \in B_{4}$ (since $I_{1} \cap I_{4} = \{1\}$ ). The sign of $x_{i}^{(l)}[1] \cdot x_{j}^{(l)}[1]$ is flipped. This means the roles of $\vec{a}$ and $\vec{b}$ are swapped. For $i \in B_{1}$ : + +$$ +\mathcal {E} _ {i, k} ^ {(l)} (Y) = \left\{\vec {b} \odot y _ {j _ {1}}, \vec {b} \odot y _ {j _ {2}}, \vec {a} \odot y _ {j _ {3}}, \vec {a} \odot y _ {j _ {4}} \right\} \tag {17} +$$ + +The sum is thus negated. This occurs only along the first column of the first block. For $i \in B_4$ , the aggregation from $B_1$ is: + +$$ +\begin{array}{l} \operatorname {A G G} \left(\mathcal {E} _ {i, 1} (X)\right) = \operatorname {A G G} \left(\left\{\vec {a} \odot x _ {j _ {1}}, \vec {a} \odot x _ {j _ {2}}, \vec {b} \odot x _ {j _ {3}}, \vec {b} \odot x _ {j _ {4}} \right\}\right) (18) \\ = \operatorname {A G G} \left(\left\{\vec {a} \odot \mathbf {e} _ {1} \odot x _ {j _ {1}}, \vec {a} \odot \mathbf {e} _ {1} \odot x _ {j _ {2}}, \vec {b} \odot - \mathbf {e} _ {1} \odot x _ {j _ {3}}, \vec {b} \odot - \mathbf {e} _ {1} \odot x _ {j _ {4}} \right\}\right) (19) \\ = \operatorname {A G G} \left(\left\{\vec {b} \odot - \mathbf {e} _ {1} \odot y _ {j _ {1}}, \vec {b} \odot - \mathbf {e} _ {1} \odot y _ {j _ {2}}, \vec {a} \odot \mathbf {e} _ {1} \odot y _ {j _ {3}}, \vec {a} \odot \mathbf {e} _ {1} \odot y _ {j _ {4}} \right\}\right) (20) \\ = \operatorname {A G G} \left(\left\{\vec {b} \odot y _ {j _ {1}}, \vec {b} \odot y _ {j _ {2}}, \vec {a} \odot y _ {j _ {3}}, \vec {a} \odot y _ {j _ {4}} \right\}\right) = \operatorname {A G G} \left(\mathcal {E} _ {i, 1} (Y)\right) (21) \\ \end{array} +$$ + +for some $\vec{a},\vec{b}\in \mathbb{R}^6$ . The equality holds. Therefore, the equivariant aggregation of $Y$ is equivalent to that of $X$ , except in the first column of the first block, where it is negated. The aggregations for both $X$ and $Y$ maintain the generalized block pattern. + +In conclusion of the inter-block aggregation, each $x_{i} \in B_{k}$ will be added with an equivariant feature of the form $c^{(l)} \odot x_{i}$ (where $c^{(l)}$ is a shared column vector), and $y_{i}$ will be added with $c^{(l)} \odot y_{i}$ . + +Full Update: The new feature matrix $B_{k}^{(l + 1)}$ is the sum of the original features and the intra- and inter-block aggregations. This process preserves the essential column structure. The update can be expressed as: + +$$ +B _ {k} ^ {(l + 1)} = \left(I + \Phi_ {k} ^ {(l)}\right) B _ {k} ^ {(l)} + c ^ {(l)} \odot B _ {k} ^ {(l)} \tag {22} +$$ + +where $c^{(l)}$ is a column vector. This operation simply updates the scalar multiples of each column. For example, the sum of the original features and the intra-block aggregation (for the $I_{k}$ columns) results in: + +$$ +(I + \Phi_ {k} ^ {(l)}) B _ {k} ^ {(l)} [:, I _ {k} ] = \left[ \begin{array}{c c c} s _ {k, 1} ^ {(l)} + \alpha & s _ {k, 2} ^ {(l)} + \beta & s _ {k, 3} ^ {(l)} + \gamma \\ - (s _ {k, 1} ^ {(l)} + \alpha) & - (s _ {k, 2} ^ {(l)} + \beta) & s _ {k, 3} ^ {(l)} + \gamma \\ s _ {k, 1} ^ {(l)} + \alpha & - (s _ {k, 2} ^ {(l)} + \beta) & - (s _ {k, 3} ^ {(l)} + \gamma) \\ - (s _ {k, 1} ^ {(l)} + \alpha) & s _ {k, 2} ^ {(l)} + \beta & - (s _ {k, 3} ^ {(l)} + \gamma) \end{array} \right] +$$ + +By Equation 22, the equivariant features remain in the generalized block form. + +In conclusion, at each layer, invariant features remain uniform within partitions, and equivariant features update symmetrically. Since the representations are structurally identical (up to the $s_{1,1}$ sign flip, which is preserved) for both graphs, they are indistinguishable. + +# $X$ and $Y$ can be extended to a proper eigendecomposition + +To form a complete basis of 16 eigenvectors, we construct the remaining 10 orthogonal vectors, $\tilde{X} \in \mathbb{R}^{16 \times 10}$ . Define the local orthogonal basis for $\mathbb{R}^4$ (along the rows of the matrix): + +$$ +\left[ \begin{array}{l} \mathbf {a} \\ \mathbf {b} \\ \mathbf {c} \\ \mathbf {1} \end{array} \right] \triangleq \left[ \begin{array}{r r r r} 1 & - 1 & 1 & - 1 \\ 1 & - 1 & - 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 1 & 1 & 1 & 1 \end{array} \right] \tag {23} +$$ + +Let $\mathbf{0} \in \mathbb{R}^4$ be the zero vector. We construct the matrix $\tilde{X}^T \in \mathbb{R}^{10 \times 16}$ as a block matrix (where each block $\mathbf{a}, \mathbf{b}, \ldots$ is a $1 \times 4$ row vector): + +$$ +\tilde {X} ^ {T} \triangleq \left[ \begin{array}{c c c c} \mathbf {a} & - \mathbf {a} & \mathbf {0} & \mathbf {0} \\ \mathbf {b} & \mathbf {0} & - \mathbf {a} & \mathbf {0} \\ \mathbf {c} & \mathbf {0} & \mathbf {0} & - \mathbf {a} \\ \mathbf {0} & \mathbf {b} & - \mathbf {b} & \mathbf {0} \\ \mathbf {0} & \mathbf {c} & \mathbf {0} & - \mathbf {b} \\ \mathbf {0} & \mathbf {0} & \mathbf {c c c c} & - \mathbf {c c c c} \\ \mathbf {1} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {1} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {1} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {1} \end{array} \right] \tag {24} +$$ + +The rows of $\tilde{X}^T$ (columns of $\tilde{X}$ ) are orthogonal to each other and to the columns of $X$ . Thus, after scaling, $X_{full} = [X, \tilde{X}] \in \mathbb{R}^{16 \times 16}$ forms an orthonormal basis. The block structure is maintained within the first 6 rows of $\tilde{X}^T$ . An analogous proof to Part 2 shows that these do not contribute new information to the hidden states, other than their block membership. The "all-ones" vectors (last 4 rows) are constant on each block and do not pass messages between blocks. This means the invariant and equivariant aggregation remain analogous to Part 2 when processing the full matrix $X_{full} = [X, \tilde{X}]$ . The hidden states only depend on the structural relations defined by $X$ . Therefore, for an analogous matrix $\tilde{Y}$ (with its first row a replaced by $-\mathbf{a}$ to maintain orthogonality with $Y$ ), equiEPNN will yield the same output on $[Y, \tilde{Y}]$ and $[X, \tilde{X}]$ , which form valid eigendecompositions for an equal simple spectrum. In conclusion, equiEPNN cannot separate $X$ and $\tilde{Y}$ . + +□ + +# B Experiments + +# B.1 Dataset statistics + +We surveyed popular graph datasets and documented their graph spectral properties. The results are shown in Table 5. We find that the MNIST Superpixel [34] dataset is almost homogeneously composed of graphs with a simple spectrum, and we find that (96.9%) of the graphs in this dataset have a full row without zeros, implying that EPNN is complete on almost all graphs. + +Other datasets, such as MUTAG, ENZYMES, PROTEINS and ZINC [18, 36], contain a substantial amount of graphs with eigenvalue multiplicity 2 and 3. Despite this, the number of eigenspaces of dimensions 2 and 3 is very few per graph, averaging at around 1 per graph. On datasets with highly symmetric graphs, such as ENZYMES and PROTEINS, the graphs do not meet the sparsity condition of Theorem 2, thus EPNN will not necessarily faithfully learn the graph structure. This exemplifies the need for more expressive models that are complete on graphs with higher maximal eigenvalue multiplicity and sparse eigenvectors. + +Table 5: Graph Statistics Analysis Across Different Datasets (Eigenvalue Tolerance: ${10}^{-4}$ ) + +
Dataset NameMUTAGENZYMESPROTEINSMNISTZINC
Dataset Overview
Number of Graphs1886001,11360,00010,000
Eigenvalue Characteristics
Graphs with Distinct Eigenvalues41.5% (78)34.8% (209)22.1% (246)99.9% (59,950)40.7% (4,072)
Graphs with Multiplicity 2 Eigenvalues58.5% (110)65.2% (391)77.9% (867)-59.3% (5,928)
Graphs with Multiplicity 3 Eigenvalues19.1% (36)46.2% (277)57.9% (644)-26.2% (2,617)
Avg. Number of Multiplicity 2 Eigenvalues0.741.011.24--
Avg. Number of Multiplicity 3 Eigenvalues0.260.580.71--
Eigenvector Properties
Average Ratio of Zeros1.674.286.390.312.52
Average Number of Zeros31.13172.93817.2023.1661.04
Graphs with a Full Row75.0% (141)35.8% (215)37.1% (413)96.9% (58,077)64.5% (6,447)
Graphs with ≤1 Zero per Eigenvector0.0% (0)6.3% (38)5.0% (56)20.2% (12,085)4.3% (430)
Graphs with Total Zeros < Vertices29.8% (56)16.3% (98)14.3% (159)89.9% (53,873)13.0% (1,295)
Graphs Meeting Any Condition75.0% (141)35.8% (215)37.1% (413)96.9% (58,077)64.5% (6,447)
+ +We surveyed the graph spectra of popular datasets to verify the need for more expressive architectures based on graph properties. We now further specify the meaning of each row of Table 5 in Table B.1. + +# B.2 MNIST Superpixel + +Below, in Tables 7, 8 and B.2, we list the experiment configurations and hyperparameters of the MNIST experiment. + +As a toy experiment to examine the potential benefit of using equiEPNN, We implemented equiEPNN via a modification of the EGNN architecture [40] and EPNN with the same architecture, but without the eigenvector update step. For precise hyperparameter configuration, see the Appendix. + +In our first experiment, we applied the proposed method on a classical task of handwritten digit classification in the MNIST dataset [23]. While almost trivial by today's standards, we use this example to verify the theoretical claims regarding expressivity on simple spectrum graphs. Our experimental setup employed both EPNN (coordinate updates disabled) and equiEPNN (coordinate updates enabled) as our models exclusively on the superpixel-based graph representation from the MNISTSuperpixels dataset. In this approach, each $28 \times 28$ image was converted into a graph where vertices correspond to superpixels and edges represent their spatial adjacency relations, each image was represented as a different graph. We tested our models with different positional encoding dimensions of $k = 3, 8, 16$ to evaluate performance across varying levels of spectral information. + +For details configurations see Tanbes 7, 8, and 9. + +# B.2.1 Ablation + +We examined the performance of both methods on the MNIST Superpixel datasets, where the task is classification of handwritten digits. We found that equiEPNN outperforms EPNN, with the same + +
Eigenvalue Characteristics
Graphs with Distinct EigenvaluesGraphs where all eigenvalues have multiplic-ity 1, meaning each eigenvalue appears ex-actly once in the spectrum
Graphs with Multiplicity 2 EigenvaluesGraphs that have at least one eigenvalue that appears exactly twice in the spectrum
Graphs with Multiplicity 3 EigenvaluesGraphs that have at least one eigenvalue that appears exactly three times in the spectrum
Avg. Number of Multiplicity 2 EigenvaluesThe average number of eigenbasis that have multiplicity exactly 2
Avg. Number of Multiplicity 3 EigenvaluesThe average number of eigenbasis that have multiplicity exactly 3
Eigenvector Properties
Average Ratio of ZerosThe average proportion of zero entries found in the eigenvectors across all analyzed graphs
Average Number of ZerosThe average count of zero entries in the eigen-vectors across all analyzed graphs
Graphs with a Full RowGraphs that have at least one eigenvector with no zero entries (i.e., a "full row" in the eigen-vector matrix)
Graphs with ≤ 1 Zero per EigenvectorGraphs where each eigenvector has at most one zero entry
Graphs with Total Zeros < VerticesGraphs where the total number of zero entries across all eigenvectors is less than the number of vertices in the graph
Graphs Meeting Any ConditionGraphs that satisfy at least one of the specified eigenvector properties listed above
+ +Table 6: Explanation of Surveyed Graph Spectral Properties +Table 7: MNIST Superpixel Experiment Configuration + +
ParameterDefault ValueDescription
k_values[3, 8, 16]List of k values for positional encoding dimensions
epochs30Number of training epochs
batch_size32Training batch size
data_dir‘data’Data directory path
device‘CUDA’Computing device (CUDA if available)
early_stopping10Early stopping patience
output_dir‘results’Output directory for results
coord_update_options[True, False]Coordinate update configurations
random(seed42Random seed for reproducibility
+ +Table 8: MNIST Superpixel Network Hyperparameters + +
ParameterDefault ValueDescription
num_features1Input node features (MNIST characteristic)
num_classeses10Output classes (MNIST digits 0-9)
hidden_dim64Hidden layer dimension
num_layers3Number of EGNN layers
pos_enc_dimkPositional encoding dimension (varies: 3, 8, 16)
dropout0.2Dropout rate
lr0.0005Learning rate
weight Decay1e-5Weight decay for regularization
norm_featuresTrueNormalize node features
norm_coordinatesTrueNormalize coordinates
coordWeights_clamp1.0Clamping value for coordinate weights
with_pos_encTrueUse positional encoding
with_projFalseUse edge projectors
with_virtual_nodeFalseUse virtual node
update_coordinatesTrue/FalseCoordinate update flag (both tested)
+ +Table 9: MNIST Superpixel Training Configuration + +
ParameterValueDescription
OptimizerAdamOptimization algorithm
Loss FunctionNLL LossNegative log-likelihood loss
SchedulerReduceLROOnPlateauLearning rate scheduler
LR Reduction Factor0.5Factor for LR reduction
LR Patience5Scheduler patience
Min LR1e-6Minimum learning rate
Gradient Clipping1.0Maximum gradient norm
Early Stopping Patience10Training patience
+ +number of model parameters and hyperparameter instantiations, in the setting with few known eigenvectors. With a sufficient number of eigenvectors EPNN and equiEPNN achieve comparable results, as expected, since they are both complete on almost all graphs in MNIST Superpixel. (see $k = 8$ and $k = 16$ in Table 10.) + +# B.3 Realizable Expressivity + +The BREC [48] dataset is a graph expressivity benchmark consisting of highly symmetric graphs that high-order GNNs struggle at distinguishing, which was used by [52] to check the expressivity of EPNN. We implemented EPNN and equiEPNN via the popular EGNN [40] framework and obtained statistically identical results shown in Table 12. + +# B.4 Eigenvector Canonicalization + +We specify the problem setup for eigenvector canonicalization and our proposed method. + +Table 10: Ablation study on MNIST Superpixel [34]. Accuracy percentage comparison with deviation over 3 trials, for different values of $K$ for EPNN and equiEPNN. + +
kEPNNEquiEPNN
348.45 ± 1.2 %60.95 ± 0.9 %
885.55 ± 2.1 %83.56 ± 2.5 %
1690.13 ± 2.3 %91.37 ± 2.2 %
+ +Table 11: Network Hyperparameters for Eigenvector canonicalization + +
ParameterDefault ValueDescription
num_layers5Number of message passing layers
emb_dim128Embedding dimension
in_dim128Input feature dimension
proj_dim10Projection dimension
coords_weight3.0Coordinate update weight
activationreluActivation function
normlayerNormalization type
aggrsumAggregation function
residualFalseUse residual connections
edge_attr_dim20Edge feature dimension (2 × k_projectors)
+ +Table 12: Empirical performance of different GNNs on BREC (in percentages.) (Using $k = 3$ spectral features, results of non-EPNN models from [52]) + +
ModelWL classBasicRegExtCFITotal
GraphormerSPD-WL26.710.041.010.019.8
NGNNSWL98.334.359.0041.5
ESANGSWL96.734.3100.015.055.2
PPGN3-WL100.035.7100.023.058.2
EPNNEPWL100.035.7100.04.053.5
Equi-EPNNN/A100.035.7100.04.053.5
+ +Definition 9 (Eigenvector Canonicalization). A canonicalization of an eigenvector $v \in \mathbb{R}^n$ is a map $\phi : \mathbb{R}^n \to \mathbb{R}^n$ such that for every $s \in O(1) \simeq \{-1,1\}$ , it holds that $\phi(sv) = \phi(v)$ and is permutation equivariant, that is for every permutation $\sigma$ , $\phi(\sigma v) = \sigma \phi(v)$ . + +We now define the following eigenvector canonicalization map via the steps + +1. For given eigenvectors $V \in \mathbb{R}^{n \times k}$ corresponding to distinct eigenvalues, we run equiEPNN for $T$ iterations, to obtain the equivariant output $V^{(T)}$ +2. We sum over the columns to obtain a matrix $S = \mathrm{diag}(s_1, s_2, \ldots, s_k)$ where $s_i \triangleq \mathrm{sign}(\sum_{j=1}^{n} V^{(T)}(i,j)) \in \{-1, +1\}$ . +3. Canonicalize the eigenvectors via $SV$ . + +This defines an eigenvector canonicalization map $\psi : \mathbb{R}^{n \times k} \to \mathbb{R}^{n \times k}$ where $\psi(V) = SV$ for the $S(V)$ defined above. This map is naturally permutation equivariant, and it is easy to check that it is sign invariant. + +As this maps canonicalized the original eigenvectors via aggregating global graph information that depends on the entire graph eigendecomposition and not each eigenvector separately, we obtain a map that practically achieves perfect canonicalization on ZINC [18]. + +See Tables 11 and 13 for experiment configurations. + +Table 13: Eigenvector Canonicalization Configuration + +
ParameterDefault ValueDescription
subset_size100Number of ZINC graphs to test
k_projectors10Number of top eigenvalue projectors to use
num_workers4Number of workers for data loading
deviceCUDA/CPUComputing device (CUDA if available)
precisionfloat64Default tensor precision
+ +# C Further Related Work + +# C.1 Expressive Power and the Weisfeiler-Lehman Hierarchy + +The expressive power of GNNs is commonly evaluated via the Weisfeiler-Lehman (WL) test, with standard Message Passing Neural Networks (MPNNs) being upper-bounded by the 1-WL test [49, 35]. This has motivated the development of more powerful models aligned with higher-order k-WL tests [32]. The WL hierarchy and its variants have been clarified in tutorials by [16, 37]. Other works have moved beyond the binary isomorphism objective to develop more continuous, fine-grained measures of expressivity based on graphons and tree distances [6]. Our work diverges from these combinatorial frameworks by proposing a hierarchy based on eigenvalue multiplicity, a natural concept in spectral graph theory. We demonstrate that even SGNNs considered powerful in the WL hierarchy (EPNN) can fail on spectrally-defined graph classes, revealing limitations not captured by combinatorial tests. + +# C.2 Higher-Order and Subgraph GNNs + +To overcome the 1-WL barrier, a prominent line of research has focused on architectures that process higher-order structures. Subgraph GNNs, which represent a graph as an equivariant collection of its subgraphs, have proven to be a particularly powerful paradigm [26]. A significant challenge has been the computational complexity of these models. Recent work by [3] introduces a flexible and scalable framework for Subgraph GNNs using graph products and coarsening to manage complexity. This line of research, including work by [10], has also explored novel methods to boost expressivity by leveraging high-order derivatives of a base GNN model, drawing deep connections between this calculus-based approach and the WL hierarchy. Our work provides a complementary perspective by showing that even highly expressive architectural paradigms can have fundamental blind spots, such as the inability to distinguish certain graphs with simple spectra. + +# C.3 Spectral GNNs and Universality + +Spectral GNNs define graph convolutions via spectral filters. Early work improved filter expressivity by moving from polynomials to complex rational functions, as in CayleyNets [24]. A key theoretical result from [47] established that linear spectral GNNs can achieve universal approximation on graphs with a simple spectrum. However, this universality relies on a crucial assumption: the use of a randomly sampled, non-equivariant node signal. This setting is distinct from the standard GNN expressivity analysis, which assumes permutation-equivariant operations on graph structure. Our work investigates the expressivity of permutation-equivariant SGNNs, such as EPNN, under the same simple spectrum condition. We prove that, in this more standard setting, these models are fundamentally incomplete. We construct explicit counterexamples of non-isomorphic graphs with simple spectra that EPNN cannot distinguish, revealing a critical limitation that was not apparent from prior analyses. + +# C.4 Equivariant Design and Generalization + +A core principle in modern GNN theory is designing architectures that respect the symmetries of graph data, i.e., permutation invariance and equivariance [46]. This has led to principled methods for handling spectral features, such as the sign and basis ambiguities of eigenvectors. Models like SignNet and BasisNet are designed to be invariant to these symmetries by processing eigenspaces independently [27]. Work by [21] has analyzed the implicit bias of such equivariant networks, showing that gradient descent favors solutions with specific structural properties in the Fourier domain. Other work has explored probabilistic frameworks for breaking symmetries when necessary [22]. Our work builds on these principles; we show that even a principled equivariant architecture like EPNN is incomplete, and our proposed solution, equiEPNN, is directly inspired by equivariant network designs. + +# C.5 Unified Theories and GNN Limitations + +One recent research direction is to move towards a more holistic understanding of GNNs by connecting expressivity, generalization, and universality. Work by [39] proposes a unified framework using pseudometrics based on optimal transport to derive both universal approximation theorems and + +generalization bounds for MPNNs on attributed graphs. Concurrently, critical work has highlighted the practical limitations of GNNs. For instance, [4] demonstrated that GNNs can 'overfit' the graph structure, using it even when it is detrimental to the task. This suggests that theoretical expressivity does not automatically translate to better performance. Our paper contributes to this line of inquiry by identifying a novel and unexpected failure mode for a class of GNNs that are already considered highly expressive. 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Traditionally, this problem has been approached using Spectral Decomposition (SD) methods applied to samples collected under non-adaptive policies, such as uniform or round-robin policies. Recently, SD techniques have been extended to accommodate a restricted class of adaptive policies such as memoryless policies. However, the use of adaptive policies has introduced challenges related to data inefficiency, as SD methods typically require all samples to be drawn from a single policy. In this work, we propose Mixed Spectral Estimation, which generalizes spectral estimation techniques to support a broader class of belief-based policies. We solve the open question of whether spectral methods can be applied to samples collected from multiple policies, and we provide finite-sample guarantees for our approach under standard observability and ergodicity assumptions. Building on this data-efficient estimation method, we introduce the Mixed Spectral UCRL algorithm. Through a refined theoretical analysis, we demonstrate that it achieves a regret bound of $\tilde{\mathcal{O}} (\sqrt{T})$ when compared to the optimal policy, without requiring full knowledge of either the transition or the observation model. Finally, we present numerical simulations that validate the theoretical analysis of both the proposed estimation procedure and the Mixed Spectral UCRL algorithm. + +# 1 Introduction + +In Reinforcement Learning (RL) [31], an agent interacts with an unknown or partially known environment to maximize the long-term sum of rewards. This approach has been successfully used in a variety of problems [23, 28, 8] under the assumption of fully observing the state of the environment. However, less attention has been paid to the more realistic scenario where the agent only receives partial and noisy observations from the environment, a problem which can be modeled through the Partially Observable Markov Decision Process (POMDP) [35] formalism. This setting can be used to represent various real-world applications such as autonomous driving [18], resource allocation [7], or financial settings [6]. Dealing with POMDPs is notably a challenging task both $(i)$ statistically since it requires estimating the latent model parameters, and $(ii)$ computationally since computing the optimal policy for a POMDP is intractable even when the model parameters are known [24]. + +In this work, we tackle the infinite-horizon average-reward POMDP formulation. In the past works, the learning problem in this setting has been addressed using Spectral Decomposition (SD) methods [2, 1]. In particular, the standard approach consists of deploying fully explorative policies (e.g., round-robin or uniform) for data collection and then leveraging SD techniques for subsequent model + +estimation [11, 32]. A different approach is proposed in [3] where spectral strategies are extended to samples collected from adaptive memoryless policies. However, the model estimation they propose requires all samples to be drawn from a unique policy, which introduces data inefficiency issues since samples collected with older policies cannot be reused for model estimation. In addition, their approach is limited to stochastic policies under which each action can be chosen with a minimum positive probability $\iota >0$ . By inspecting the limitations of current works, an important question arises: Can we apply spectral techniques on samples collected from multiple adaptive policies to improve the sample-efficiency of online learning algorithms for POMDPs? + +Contributions. In this paper, we address this question and we provide the following contributions: + +- We extend the spectral estimation procedure to the larger class of stationary belief-based policies. +- We answer the previous question affirmatively and propose a procedure, Mixed Spectral Estimation, with finite-sample guarantees for estimating the POMDP parameters (Section 5). +- We plug this novel estimation approach into a regret minimization algorithm, Mixed Spectral UCRL, and we show that we can indeed avoid using stochastic policies required in previous works. By focusing on instances satisfying the common one-step reachability assumption (Assumption 6.1), our algorithm is the first to achieve a regret of order $\widetilde{\mathcal{O}} (\sqrt{T})^2$ competing against the optimal belief-based policy, hence improving over the state-of-the-art regret of order $\widetilde{\mathcal{O}} (T^{2 / 3})$ (Section 6). +- We provide numerical simulations showing both the effectiveness of the estimation procedure and the performance of our Mixed Spectral UCRL algorithm (Section 7). + +# 2 Preliminaries + +In this section, we provide the necessary background for the subsequent discussion. In the following, we will use $\Delta(\mathcal{X})$ to denote the simplex over a finite set $\mathcal{X}$ , $\sigma_S(\mathbb{X})$ to denote the $S$ -th singular value of matrix $\mathbb{X}$ , and $\mathbb{X}^\dagger$ to denote its Moore-Penrose pseudo-inverse. + +Partially Observable MDP. A Partially Observable Markov Decision Process (POMDP) [35] is defined by a tuple $\mathcal{Q} \coloneqq (\mathcal{S}, \mathcal{A}, \mathcal{O}, \mathbb{T}, \mathbb{O}, \nu, r)$ with $\mathcal{S}$ being a finite state space ( $S \coloneqq |\mathcal{S}|$ ), $\mathcal{A}$ a finite action space ( $A \coloneqq |\mathcal{A}|$ ) and $\mathcal{O}$ a finite observation space ( $O \coloneqq |\mathcal{O}|$ ). $\mathbb{T} = \{\mathbb{T}_a\}_{a \in \mathcal{A}}$ denotes a collection of transition matrices $\mathbb{T}_a \in \mathbb{R}^{S \times S}$ for every $a \in \mathcal{A}$ . Each transition matrix $\mathbb{T}_a(\cdot | s) \in \Delta(S)$ defines the distribution of the next state when the agent takes action $a$ in state $s \in S$ . $\mathbb{O} \in \mathbb{R}^{O \times S}$ denotes the observation matrix $\mathbb{O}(\cdot | s) \in \Delta(\mathcal{O})$ that represents the distribution over observations when the agent is in state $s$ . $\nu \in \Delta(S)$ denotes the distribution over the initial state, while $r: \mathcal{O} \to [0,1]$ is the known reward function, mapping each observation to a finite reward such that $r(o)$ is the reward received when the agents observe $o \in \mathcal{O}$ . In a POMDP, states are hidden and the agent can only see its own actions and the observations. At each step $t \in \mathbb{N}$ , the agent is in an unknown state $s_t$ , it receives an observation $o_t$ determined by $\mathbb{O}(\cdot | s_t)$ and a reward $r(o_t)$ , then chooses an action $a_t$ and the environment transitions into a new state $s_{t+1}$ according to $\mathbb{T}_{a_t}(\cdot | s_t)$ . Then, the process repeats. + +Policies in POMDPs. A policy $\pi \coloneqq (\pi_t)_{t=0}^{\infty}$ is a sequence of decision rules prescribing the action to play. We use $\mathcal{H}_t \coloneqq (\mathcal{O} \times \mathcal{A})^{t-1} \times \mathcal{O}$ to denote the space of histories up to time $t$ . A deterministic policy $\pi_t : \mathcal{H}_t \to \mathcal{A}$ is such that $\pi_t(h) \in \mathcal{A}$ is the action chosen when history $h \in \mathcal{H}_t$ is observed. + +From POMDP to Belief MDP. When the observation and the transition models are known, it is possible to build a belief vector $b_{t} \in \mathcal{B}$ (with $\mathcal{B} \coloneqq \Delta(\mathcal{S})$ ) from the observed history $h_{t} \coloneqq (o_{j}, a_{j})_{j=0}^{t-1} \oplus o_{t}$ , where $\oplus$ denotes the sequence concatenation operator, as $b_{t}(s) \coloneqq \operatorname*{Pr}(s_{t} = s | h_{t})$ , representing the probability that the true state is $s$ having observed history $h_{t}$ . The update rule of the belief $b_{t}$ is determined using Bayes' theorem as: + +$$ +b _ {t} (s) = \frac {\sum_ {s ^ {\prime} \in \mathcal {S}} \mathbb {O} \left(o _ {t} | s\right) \mathbb {T} _ {a _ {t - 1}} \left(s \mid s ^ {\prime}\right) b _ {t - 1} \left(s ^ {\prime}\right)}{\sum_ {s ^ {\prime} , s ^ {\prime \prime} \in \mathcal {S}} \mathbb {O} \left(o _ {t} \mid s ^ {\prime}\right) \mathbb {T} _ {a _ {t - 1}} \left(s ^ {\prime} \mid s ^ {\prime \prime}\right) b _ {t - 1} \left(s ^ {\prime \prime}\right)}. \tag {1} +$$ + +By using this notion of belief, we can transform the POMDP into a belief MDP [17] (which is a continuous-state MDP even if the original POMDP is tabular), which is used to address the POMDP + +learning problem. For an initial belief $b \in \mathcal{B}$ , the average reward of the infinite-horizon belief MDP is defined as: $\rho_b^\pi := \lim \sup_{T \to +\infty} (1 / T) \mathbb{E}[\sum_{t=0}^{T-1} r(o_t) | b_0 = b]$ . When the underlying MDP is weakly-communicating, it has been shown [5] that the optimal average reward $\rho^* := \sup_{\pi : \mathcal{B} \to \Delta(\mathcal{A})} \rho_b^\pi$ is independent of the initial belief $b$ and the following Bellman equation admits a unique solution: + +$$ +\rho^ {*} + v (b) = g (b) + \max _ {a \in \mathcal {A}} \int_ {\mathcal {B}} P (\mathrm {d} b ^ {\prime} | b, a) v (\mathrm {d} b ^ {\prime}), \tag {2} +$$ + +where $g(b) \coloneqq \sum_{s \in S} \sum_{o \in \mathcal{O}} b(s) \mathbb{O}(o|s)r(o)$ denotes the expected reward under belief $b$ , while $P(\cdot | b, a)$ is a probability measure over the next belief. Finally, $v: \mathcal{B} \to \mathbb{R}$ represents the bias function and quantifies the cumulative deviation of rewards w.r.t. $\rho^*$ when starting from $b$ [21]. + +# 3 Related Works + +POMDP Learning. Learning in POMDPs is known to be challenging both from a statistical and a computational perspective. When the observation model does not provide enough information to identify the latent states, we refer to the POMDP as hard. These intractable instances can be ruled out by introducing a full-rank assumption on the observation model. A quantitative version of this assumption was first introduced in [16] and is formalized as a lower bound $\alpha > 0$ to the minimum singular value of the observation model, namely $\sigma_S(\mathbb{O}) \geqslant \alpha$ . The instances satisfying this assumption can be efficiently learned and define the class of $\alpha$ -weakly revealing instances. + +Weakly-Revealing POMDPs. The weakly-revealing assumption has been used both in the episodic [16, 19] and the infinite-horizon average-reward setting. By focusing on the latter, some works employed the simplifying assumption of having partial knowledge of the environment, in particular of the observation model. Among them, [13] provide a Bayesian regret of order $\mathcal{O}(T^{2/3})$ when compared against the optimal policy, while a recent work from [26] proposes the Action-wise OAS-UCRL algorithm, which employs an estimation procedure with finite-sample guarantees that leverages the knowledge of the observation model to learn the transition model. They reach a $\tilde{\mathcal{O}}(\sqrt{T})$ regret guarantee when compared against the optimal policy. Several works have instead addressed the problem of fully learning the model parameters [11, 3, 34]. The standard approach relies on SD methods [1] for learning the latent variable model. In particular, [3] are the first to adapt SD methods to samples collected under the adaptive class of memoryless policies. They consider stochastic policies where each action is chosen with a positive probability $\iota > 0$ at each step and propose the SM-UCRL algorithm, which achieves a $\tilde{\mathcal{O}}(\sqrt{T}/\iota^2)$ regret guarantee when compared against this (less powerful) policy class. A different approach is taken in [32] where the regret is computed against the stronger class of deterministic ( $\iota = 0$ ) belief-based policies. They present the SEEU algorithm, which alternates between purely exploratory and purely exploitative phases. During exploration, samples are collected using a round-robin policy over the available actions, after which SD is applied to recover model parameters. Their algorithm achieves $\tilde{\mathcal{O}}(T^{2/3})$ regret when compared against the optimal class of belief-based policies. + +The introduction of our estimation strategy addresses two limitations of the aforementioned works. First, unlike the SEEU algorithm [32], we do not need to separate exploration and exploitation phases, as we can leverage samples collected during the exploitation phase to refine model estimates. Second, unlike the SM-UCRL [3], we are able to reuse samples from different policies, hence eliminating the need for stochastic policies ( $\iota > 0$ ) that foster continuous coverage of the action space. We refer to Table 1 for a comparison of our work with those mentioned above and to Appendix H for a more extensive discussion on the matter. + +# 4 Problem Formulation + +We consider the infinite-horizon average-reward POMDP setting described in Section 2. Specifically, we consider the undercomplete setting [16], where the number of states is less than or equal to the number of observations ( $S \leqslant O$ ). Our focus is on learning the POMDP parameters represented by the observation model $\mathbb{O}$ and the transition model $\mathbb{T} = \{\mathbb{T}_a\}_{a \in \mathcal{A}}$ . We consider the class of + +Table 1: Table comparing the SM-UCRL, SEEU and the Mixed Spectral UCRL algorithm. + +
PropertySM-UCRLSEEUMixed Spectral UCRL
No assumption on minimum entry of obs. model✓✗✓
No assumption on minimum entry of trans. model✓✗✗
No assumption on minimum action probability✗✗✓
Works with memoryless policies✓✗✓
Works with belief-based policies✗✗✓
Sample reuse with different policies✗✗✓
Compares against the optimal belief-based policy✗✓✓
Regret w.r.t. optimal belief-based policyO(T)\( \tilde{O}(T^{2/3}) \)\( \tilde{O}(\sqrt{T}) \)
+ +belief-based policies $\pi : \mathcal{B} \to \mathcal{A}$ , and we use $\mathcal{P}$ to denote such a set of policies. Before stating the main assumptions, we introduce some relevant quantities. + +Let $d_t^{\pi, b_0}(s, a) \coloneqq \operatorname*{Pr}(s_t = s, a_t = a|\pi, b_0)$ be the $t$ -step state-action distribution induced by policy $\pi \in \mathcal{P}$ , with $b_0 \in \mathcal{B}$ being the initial belief. Under mild regularity conditions (e.g., when the underlying MDP is weakly-communicating), a unique limiting distribution $d_{\infty}^{\pi}(s, a) \coloneqq \lim_{t \to \infty} d_t^{\pi, b_0}(s, a) \in \Delta(\mathcal{S} \times \mathcal{A})$ exists (see Proposition 5.1 in [25]) and it is independent of the initial belief $b_0$ . From the quantity just defined, we derive the stationary action distribution $d_{\infty}^{\pi} \in \Delta(\mathcal{A})$ defined as $d_{\infty}^{\pi}(a) \coloneqq \sum_{s \in \mathcal{S}} d_{\infty}^{\pi}(s, a)$ . Let us now introduce the conditional state distribution $\omega^{(a, \pi)} \in \Delta(\mathcal{S})$ defined as $\omega_s^{(a, \pi)} \coloneqq d_{\infty}^{\pi}(s|a) = d_{\infty}^{\pi}(s, a)/d_{\infty}^{\pi}(a)$ , which is well-defined when $d_{\infty}^{\pi}(a) > 0$ . + +The following assumptions represent the natural extension to the POMDP setting of the assumptions commonly employed for learning in (uncontrolled) settings (i.e., Hidden Markov Models [1]). + +Assumption 4.1 ( $\alpha$ -weakly Revealing Condition). There exists $\alpha > 0$ such that $\sigma_S(\mathbb{O}) \geqslant \alpha$ . + +This assumption quantifies the extent to which the received observations help in identifying the underlying hidden states. It is equivalent to the more common full-rank assumption largely adopted in problems involving the learning of Latent Variable Models [3, 12, 34]. It was first introduced in this form in [16] and then extensively employed in successive related works [19, 20, 26]. It has been shown that, without this assumption, learning becomes intractable [9]. + +Assumption 4.2 (Invertibility). For every action $a \in \mathcal{A}$ , the transition matrix $\mathbb{T}_a$ is invertible. + +This second assumption implies that for any state-action pair $(s, a) \in S \times \mathcal{A}$ , its next-state distribution $\mathbb{T}_a(\cdot | s)$ cannot be recovered as a linear combination of the next-state distribution of the other state-action pairs. This condition is crucial for achieving identifiability and is widely used in the SD and POMDP literature [1, 3, 32, 34, 11]. + +Assumption 4.3 (Per-Action Ergodicity). For any policy $\pi \in \mathcal{P}$ , a unique limiting state-action distribution $d_{\infty}^{\pi}(s,a)$ exists. Moreover, for every action $a$ , if $d_{\infty}^{\pi}(a) > 0$ , then $\omega_s^{(a,\pi)} > 0 \forall s \in S$ . + +Assumption 4.3 extends the standard non-degeneracy assumption [1] employed under SD techniques. The motivation behind this assumption lies in the fact that SD approaches are applied for each action $a$ separately. Hence, in order to fully recover the transition model $\mathbb{T}_a$ , all states should be visited with positive probability when taking action $a$ (i.e., $\omega_s^{(a,\pi)} > 0$ ). In Appendix H, we show how related works [3, 32] tackling the POMDP setting rely on assumptions that subsume Assumption 4.3. A simple example when this assumption holds is when the transition matrices $\{\mathbb{T}_a\}_{a\in \mathcal{A}}$ have all positive entries, as we shall see in Section 6 (Assumption 6.1). + +A discussion on the reasons why some of these assumptions are instead not required in the episodic setting is provided in Appendix H.2. + +Learning Objective. Our goal is to find the policy attaining Equation (2) in the policy class $\mathcal{P}$ . Our learning objective is to minimize the cumulative regret after $T \in \mathbb{N}$ time steps, defined as: + +$$ +\mathcal {R} _ {T} := T \rho^ {*} - \sum_ {t = 0} ^ {T - 1} r \left(o _ {t}\right), \tag {3} +$$ + +where $\rho^{*}$ represents the average reward obtained by the policy satisfying Equation (2), while $r(o_{t})$ is the reward obtained from the observation received by playing policy $\pi_t$ played at time $t$ . + +We remark that solving Equation (2) and computing such an optimal policy is known to be computationally intractable. Various methods have been devised to provide an approximately optimal policy. Most of them focus on devising clever discretizations of the belief space and then solve the discretized instance [33, 27, 29]. In this work, however, we do not focus on this planning problem, but following a common approach in the POMDP literature [32, 3, 34, 13], we assume access to an optimization oracle capable of providing the optimal policy for a given POMDP model. + +# 5 The POMDP Estimation Procedure + +In this section, we present an adaptation of the common multi-view model employed for latent parameter estimation when using SD techniques [1, 3, 32]. + +# 5.1 The Multi-View Model + +We now introduce a model-based strategy to estimate the parameters of the unknown POMDP which adapts the approach of [3]. For each step $t \in [1, T - 2]^4$ in which $a_t = a \in \mathcal{A}$ , we construct three views containing the observations in three consecutive steps centered in $t$ , i.e., $o_{t-1}, o_t, o_{t+1} \in \mathcal{O}$ . Let us use (bold) $o_t \in \{0, 1\}^O$ to denote the one-hot encoded vector corresponding to observation $o_t$ and similarly for the two remaining views $o_{t-1}$ and $o_{t+1}$ . We further use vectors $v_{\nu,t}^{(a)} \in \mathbb{R}^O$ with $\nu \in \{1, 2, 3\}$ to refer to the three different view vectors when conditioned on $a_t = a$ , and such that $v_{1,t}^{(a)} = o_{t-1}, v_{2,t}^{(a)} = o_t$ and $v_{3,t}^{(a)} = o_{t+1}$ respectively. Given a policy $\pi \in \mathcal{P}$ , we define three view matrices $V_\nu^{(a,\pi)} \in \mathbb{R}^{O \times S}$ with $\nu \in \{1, 2, 3\}$ associated with action $a \in \mathcal{A}$ , as follows: + +$$ +V _ {\nu} ^ {(a, \pi)} (o, s) = \lim _ {t \to \infty} \operatorname * {P r} \left(\pmb {v} _ {\nu , t} ^ {(a, \pi)} = o | a _ {t} = a, s _ {t} = s\right) =: \operatorname * {P r} \left(\pmb {v} _ {\nu} ^ {(a, \pi)} = o | a _ {2} = a, s _ {2} = s\right). +$$ + +It can be observed that the three views are independent when conditioning on both $s_t$ and $a_t$ . We also denote with $\pmb{\mu}_{\nu, s}^{(a, \pi)} = V_{\nu}^{(a, \pi)}(\cdot, s)$ the $s$ -th column of matrix $V_{\nu}^{(a, \pi)}$ . + +Remark 5.1. By inspecting the three different view matrices separately, we can observe that for the second view matrix it holds that $V_{2}^{(a,\pi)} = \mathbb{O}$ , hence it does not depend on either action $a$ or policy $\pi$ . Differently, for the third view matrix, it can be shown that $V_{3}^{(a,\pi)} = \mathbb{O}\mathbb{T}_{a}^{\top}$ , hence it is independent of policy $\pi$ . Finally, the first view matrix $V_{1}^{(a,\pi)}$ depends on both the action and employed policy. + +Given this multi-view model, the following result from [1] applies: + +Proposition 5.2. (Adapted from [3]) Let $\nu, \nu' \in \{1, 2, 3\}$ , $\pi \in \mathcal{P}$ be a policy, and $K_{\nu,\nu'}^{(a,\pi)} = \mathbb{E}\left[\pmb{v}_{\nu}^{(a,\pi)} \otimes \pmb{v}_{\nu'}^{(a,\pi)}\right]$ be the covariance matrix between views $\pmb{v}_{\nu}^{(a,\pi)}$ and $\pmb{v}_{\nu'}^{(a,\pi)}$ , where $\otimes$ denotes the tensor product, and denote with the superscript $\dagger$ the Moore-Penrose pseudo-inverse. We define a modified version of the first and second views as: + +$$ +\widetilde {\boldsymbol {v}} _ {1} ^ {(a, \pi)} := K _ {3, 2} ^ {(a, \pi)} \left(K _ {1, 2} ^ {(a, \pi)}\right) ^ {\dagger} \boldsymbol {v} _ {1} ^ {(a, \pi)}, \quad \widetilde {\boldsymbol {v}} _ {2} ^ {(a, \pi)} := K _ {3, 1} ^ {(a, \pi)} \left(K _ {2, 1} ^ {(a, \pi)}\right) ^ {\dagger} \boldsymbol {v} _ {2} ^ {(a, \pi)}. \tag {4} +$$ + +Then, the second and third moments of the modified views have a spectral decomposition as: + +$$ +M _ {2} ^ {(a, \pi)} = \mathbb {E} \left[ \widetilde {\pmb {v}} _ {1} ^ {(a, \pi)} \otimes \widetilde {\pmb {v}} _ {2} ^ {(a, \pi)} \right] = \sum_ {s \in \mathcal {S}} \omega_ {s} ^ {(a, \pi)} \pmb {\mu} _ {3, s} ^ {(a, \pi)} \otimes \pmb {\mu} _ {3, s} ^ {(a, \pi)}, +$$ + +$$ +M _ {3} ^ {(a, \pi)} = \mathbb {E} \left[ \widetilde {\boldsymbol {v}} _ {1} ^ {(a, \pi)} \otimes \widetilde {\boldsymbol {v}} _ {2} ^ {(a, \pi)} \otimes \boldsymbol {v} _ {3} ^ {(a, \pi)} \right] = \sum_ {s \in \mathcal {S}} \omega_ {s} ^ {(a, \pi)} \boldsymbol {\mu} _ {3, s} ^ {(a, \pi)} \otimes \boldsymbol {\mu} _ {3, s} ^ {(a, \pi)} \otimes \boldsymbol {\mu} _ {3, s} ^ {(a, \pi)}. +$$ + +where the expectations are w.r.t. the conditional state distribution $\omega_s^{(a,\pi)}$ defined in Section 4. + +When Assumptions 4.1, 4.2 and 4.3 hold, the three view matrices $V_{\nu}^{(a,\pi)} \in \mathbb{R}^{O \times S}$ with $\nu \in \{1,2,3\}$ associated with each action $a \in \mathcal{A}$ and policy $\pi \in \mathcal{P}$ are full-column rank and a unique spectral decomposition exists [1]. As a consequence, the original model parameters can be recovered. In particular, this can be performed by exploiting the following known relations between the columns of + +the different view matrices: + +$$ +\boldsymbol {\mu} _ {3, s} ^ {(a, \pi)} = \mathbb {E} \left[ \tilde {\boldsymbol {v}} _ {1} ^ {(a, \pi)} \mid s _ {2} = s, a _ {2} = a \right] = K _ {3, 2} ^ {(a, \pi)} \left(K _ {1, 2} ^ {(a, \pi)}\right) ^ {\dagger} \boldsymbol {\mu} _ {1, s} ^ {(a, \pi)}, \tag {5} +$$ + +$$ +\boldsymbol {\mu} _ {3, s} ^ {(a, \pi)} = \mathbb {E} \left[ \widetilde {\boldsymbol {v}} _ {2} ^ {(a, \pi)} \mid s _ {2} = s, a _ {2} = a \right] = K _ {3, 1} ^ {(a, \pi)} \left(K _ {2, 1} ^ {(a, \pi)}\right) ^ {\dagger} \boldsymbol {\mu} _ {2, s} ^ {(a, \pi)}. \tag {6} +$$ + +By applying SD techniques for each action $a$ separately, we obtain estimates of the third view matrix $V_{3}^{(a,\pi)}$ , hence of its columns $\pmb{\mu}_{3,s}^{(a,\pi)}$ . Finally, when such estimates are available, the columns $\pmb{\mu}_{1,s}^{(a,\pi)}$ and $\pmb{\mu}_{2,s}^{(a,\pi)}$ of the remaining view matrices can be estimated by inverting Equations (5) and (6). + +# 5.2 The Mixed Spectral Estimation Procedure + +We now show how we combine samples coming from multiple policies, thus overcoming the limitations of existing approaches and leading to our novel Mixed Spectral Estimation. We define a set of $L$ different trajectories of samples $\Gamma \coloneqq \{\tau_l\}_{l = 0}^{L - 1}$ such that the $l$ -th trajectory is generated from policy $\pi_{l}\in \mathcal{P}$ and is defined as $\tau_{l} = \{(o_{j}^{l},a_{j}^{l})\}_{j = 0}^{N_{l} - 1}$ . Additionally, we introduce the related set $\mathcal{T}_l^{(a)} = \{t\in [1,N_l - 2]$ s.t. $a_{t}^{l} = a\}$ which contains the time steps when action $a$ is selected in the $l$ -th trajectory. Let $n_l^{(a)} = |\mathcal{T}_l^{(a)}|$ denote its cardinality. For each $t\in \mathcal{T}_l^{(a)}$ , we construct the three corresponding views $(\pmb{v}_{1,t}^{(a,l)},\pmb{v}_{2,t}^{(a,l)},\pmb{v}_{3,t}^{(a,l)}) = (\pmb{o}_{t - 1},\pmb{o}_t,\pmb{o}_{t + 1})$ , where the superscript $l$ refers to the trajectory collected using $\pi_{l}$ . Our approach uses views from all the $L$ trajectories to define new covariance matrices $\pmb{K}_{\nu ,\nu^{\prime}}^{(a,L)}$ with $\nu ,\nu^{\prime}\in \{1,2,3\}$ and $\nu \neq \nu^{\prime}$ . These are weighted versions of the original covariance matrices and are defined as follows: + +$$ +\boldsymbol {K} _ {\nu , \nu^ {\prime}} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \boldsymbol {v} _ {\nu} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}} ^ {(a, l)} \right] = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \sum_ {s \in \mathcal {S}} \omega_ {s} ^ {(a, l)} \boldsymbol {\mu} _ {\nu , s} ^ {(a, l)} \otimes \boldsymbol {\mu} _ {\nu^ {\prime}, s} ^ {(a, l)}, \tag {7} +$$ + +where $N_{L}^{(a)} \coloneqq \sum_{l=0}^{L-1} n_{l}^{(a)}$ , while $\omega^{(a,l)} \coloneqq \omega^{(a,\pi_l)} \in \Delta(\mathcal{S})$ denotes the conditional state distribution determined by policy $\pi_{l}$ and action $a$ . We show that the following result holds when combining multiple policies. Its proof is deferred to Appendix A. + +Theorem 5.3. Let $\Gamma \coloneqq \{\pi_l\}_{l = 0}^{L - 1}$ be a set of trajectories collected using the set of policies $\{\pi_l\}_{l = 0}^{L - 1}$ . We define a modified version of the first and second views as: + +$$ +\widetilde {\boldsymbol {v}} _ {1} ^ {(a, l)} := \boldsymbol {K} _ {3, 2} ^ {(a, L)} \left(\boldsymbol {K} _ {1, 2} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {v} _ {1} ^ {(a, l)}, \quad \widetilde {\boldsymbol {v}} _ {2} ^ {(a, l)} := \boldsymbol {K} _ {3, 1} ^ {(a, L)} \left(\boldsymbol {K} _ {2, 1} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {v} _ {2} ^ {(a, l)}, \tag {8} +$$ + +where the covariance matrices are defined in Equation (7). Let $\omega^{(a,L)}\coloneqq (1 / N_L^{(a)})\sum_{l = 0}^{L - 1}n_l^{(a)}\omega^{(a,l)}$ , then, the second and third moments of the modified views have a spectral decomposition as: + +$$ +\boldsymbol {M} _ {2} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \widetilde {\boldsymbol {v}} _ {1} ^ {(a, l)} \otimes \widetilde {\boldsymbol {v}} _ {2} ^ {(a, l)} \right] = \sum_ {s \in \mathcal {S}} \boldsymbol {\omega} _ {s} ^ {(a, L)} \boldsymbol {\mu} _ {3, s} ^ {(a)} \otimes \boldsymbol {\mu} _ {3, s} ^ {(a)}, +$$ + +$$ +\boldsymbol {M} _ {3} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \widetilde {\boldsymbol {v}} _ {1} ^ {(a, l)} \otimes \widetilde {\boldsymbol {v}} _ {2} ^ {(a, l)} \otimes \boldsymbol {v} _ {3} ^ {(a, l)} \right] = \sum_ {s \in \mathcal {S}} \boldsymbol {\omega} _ {s} ^ {(a, L)} \boldsymbol {\mu} _ {3, s} ^ {(a)} \otimes \boldsymbol {\mu} _ {3, s} ^ {(a)} \otimes \boldsymbol {\mu} _ {3, s} ^ {(a)}, +$$ + +where the expectations are w.r.t. the conditional state distributions $\omega_{s}^{(a,l)}$ . + +This theorem shows that when the views $\boldsymbol{v}_1^{(a,l)}$ and $\boldsymbol{v}_2^{(a,l)}$ are modified using the weighted covariance matrices $K_{\nu,\nu'}^{(a,L)}$ defined in Equation (7) instead of the covariance matrices $K_{\nu,\nu'}^{(a,l)}$ associated with policy $\pi_l$ , the new second and third order moments have a spectral decomposition whose conditional state distribution $\omega^{(a,L)}$ is an average of the original conditional state distributions, each one weighted proportionally by the cardinality $n_l^{(a)}$ . Importantly, as discussed in Remark 5.1, the columns $\boldsymbol{\mu}_{3,s}^{(a)}$ of the third view matrix do not depend on the employed policies but only on action $a$ , hence in Theorem 5.3, we do not report the dependence on the mixture of the $L$ policies. The independence of the third view matrix from the employed policies plays a crucial role in proving Theorem 5.3. + +Algorithm Pseudocode. The estimation procedure of the quantities described above, and of the estimated POMDP parameters, is described in the Mixed Spectral Estimation approach presented in Algorithm 1. For each action $a$ , the view vectors are computed for all the $L$ policies, and they are used to compute the mixture covariance matrices (Line 8). Given the new covariance matrices, the + +# Algorithm 1 Mixed Spectral Estimation. + +1: Input: Trajectory set $\Gamma \coloneqq \{\tau_l\}_{l = 0}^{L - 1}$ for each $l$ we have $\tau_{l} = \{(a_{j}^{l},a_{j}^{l})\}_{j = 0}^{N_{l} - 1}$ +2: Output: Estimated Observation model $\hat{\mathbb{O}}$ and Transition model $\{\widehat{\mathbb{T}}_a\}_{a\in \mathcal{A}}$ +3: for $a \in \mathcal{A}$ do +4: for $l \in [0, L - 1]$ do +5: Construct views $\pmb{v}_{1,t}^{(a,l)} = \pmb{o}_{t-1}$ , $\pmb{v}_{2,t}^{(a,l)} = \pmb{o}_t$ , $\pmb{v}_{3,t}^{(a,l)} = \pmb{o}_{t+1}$ for any $t \in \mathcal{T}_l^{(a)}$ +6: end for +7: Compute $N_{L}^{(a)} = \sum_{l=0}^{L-1} n_{l}^{(a)}$ +8: Compute covariance matrices for $\nu, \nu' \in \{1, 2, 3\}$ : + +$$ +\widehat {\pmb {K}} _ {\nu , \nu^ {\prime}} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \pmb {v} _ {\nu , t} ^ {(a, l)} \otimes \pmb {v} _ {\nu^ {\prime}, t} ^ {(a, l)}. +$$ + +9: Compute modified views: + +$$ +\widetilde {\boldsymbol {v}} _ {1, t} ^ {(a, l)} = \widehat {\boldsymbol {K}} _ {3, 2} ^ {(a, L)} \left(\widehat {\boldsymbol {K}} _ {1, 2} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {v} _ {1, t} ^ {(a, l)}, \qquad \widetilde {\boldsymbol {v}} _ {2, t} ^ {(a, l)} = \widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)} \left(\widehat {\boldsymbol {K}} _ {2, 1} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {v} _ {2, t} ^ {(a, l)}. +$$ + +10: Compute second and third moments: + +$$ +\widehat {\boldsymbol {M}} _ {2} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \widetilde {\boldsymbol {v}} _ {1, t} ^ {(a, l)} \otimes \widetilde {\boldsymbol {v}} _ {1, t} ^ {(a, l)} +$$ + +$$ +\widehat {\boldsymbol {M}} _ {3} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \widetilde {\boldsymbol {v}} _ {1, t} ^ {(a, l)} \otimes \widetilde {\boldsymbol {v}} _ {2, t} ^ {(a, l)} \otimes \boldsymbol {v} _ {3, t} ^ {(a, l)} +$$ + +11: $\widehat{V}_3^{(a)} = \text{TENSORDECOMPOSITION}(\widehat{\pmb{M}}_2^{(a,L)},\widehat{\pmb{M}}_3^{(a,L)})$ +12: Compute $\widehat{V}_2^{(a)}$ inverting Eq. (6) +13: end for +14: Define $a^* \in \operatorname{argmax}_{a \in \mathcal{A}} N_L^{(a)}$ +15: for $a \in \mathcal{A}$ do +16: Match the columns of each $\widehat{V}_2^{(a)}$ with $\widehat{V}_2^{(a^*)}$ +17: Permute the columns of $\hat{V}_3^{(a)}$ using the same permutation adopted for $\hat{V}_2^{(a)}$ +18: end for +19: Compute $\hat{\mathbb{O}}$ according to Eq. (9) +20: for $a \in \mathcal{A}$ do +21: Compute $\widehat{\mathbb{T}}_a$ according to Eq. (10) +22: end for + +modified views are computed for each $t \in \mathcal{T}_l^{(a)}$ with $l \in [0, L - 1]$ (Line 9). The modified views are then used to compute second and third-order moments (Line 10), and a tensor decomposition routine $^6$ (line 11) is run for each action separately, thus obtaining the estimated view matrix $\hat{V}_3^{(a)}$ . By inverting Equation (6), we are able to derive an estimate of the second view matrix $\hat{V}_2^{(a)}$ . As noted in Remark 5.1, the second view matrices are identical across all actions, thus satisfying $V_2^{(a)} = \mathbb{O}$ for any action $a$ . Since spectral methods recover the columns of the original view matrices up to a permutation of the hidden states $s$ [1], this equivalence allows us to align the columns of the different $\hat{V}_2^{(a)}$ by appropriately permuting them, thus ensuring that the represented states are ordered consistently, as also done in [3]. To do that, we define $a^* \in \mathrm{argmax}_{a \in \mathcal{A}} N_L^{(a)}$ and choose $\hat{V}_2^{(a*)}$ as the reference view that the other views should match. $^7$ It is possible to show that when the estimation of each view is sufficiently accurate, the correct permutation can be found for each $\hat{V}_2^{(a)}$ . When the permutation step is completed, the observation and transition model are computed as: + +$$ +\widehat {\mathbb {O}} = \frac {1}{N _ {L}} \sum_ {a \in \mathcal {A}} N _ {L} ^ {(a)} \widehat {V} _ {2} ^ {(a)}, \tag {9} +$$ + +$$ +\widehat {\mathbb {T}} _ {a} = \left(\widehat {\mathbb {O}} ^ {\dagger} \hat {V} _ {3} ^ {(a)}\right) ^ {\top}, \tag {10} +$$ + +where $N_{L} \coloneqq \sum_{a \in \mathcal{A}} N_{L}^{(a)}$ . Thus, the estimated observation matrix is obtained as a weighted combination of the second view matrices $\widehat{V}_{2}^{(a)}$ , while each transition matrix is recovered by inverting the relation presented in Remark 5.1 and using the observation matrix computed as in Equation (9). The computational complexity of the presented approach is discussed in Appendix I. Algorithm 1 enjoys the following guarantees, which are proved in Appendix B. + +Theorem 5.4. Let $\widehat{\mathbb{O}}$ and $\{\widehat{\mathbb{T}}_a\}_{a\in \mathcal{A}}$ be the observation and transition model estimated using Algorithm 1, respectively. Let Assumptions 4.1 and 4.2 hold and let Assumption 4.3 be true for any $\pi_l$ with $l\in [0,L - 1]$ . Let $\delta \in (0,1 / (3SA))$ , then for a sufficiently large number of samples $N_L^{(a)}$ holding for every action $a\in \mathcal{A}$ , with probability at least $1 - 3SA\delta$ , it holds that: + +$$ +\left\| \mathbb {O} - \hat {\mathbb {O}} \right\| _ {F} \leqslant \frac {C _ {\mathbb {O}}}{\zeta^ {(L)}} \sqrt {\frac {S A L \log (L O / \delta)}{N _ {L}}}, \quad \left\| \mathbb {T} _ {a} - \hat {\mathbb {T}} _ {a} \right\| _ {F} \leqslant \frac {C _ {\mathbb {T}} S}{\sigma_ {S} (\mathbb {O}) \zeta^ {(L)}} \sqrt {\frac {A L \log (L O / \delta)}{N _ {L} ^ {(a)}}}, +$$ + +where $\zeta^{(L)}\coloneqq \widetilde{\sigma}_{3,1}^{(L)}\left[\sqrt{\widetilde{\omega}_{\min}^{(L)}}\min_{\nu \in \{1,2,3\},a\in \mathcal{A}}\sigma_S(V_\nu^{(a,L)})\right]^3,\widetilde{\omega}_{\min}^{(L)}\coloneqq \min_{a\in \mathcal{A}}\pmb{\omega}_{\min}^{(a,L)},$ and $\tilde{\sigma}_{3,1}^{(L)}\coloneqq \min_{a\in \mathcal{A}}\sigma_S(\pmb {K}_{3,1}^{(a,L)})$ , while $C_{\mathbb{O}}$ and $C_{\mathbb{T}}$ are suitable constants. + +We highlight that Theorem 5.4 requires a minimum number of samples $N_{L}^{(a)}$ for each action $a$ (this number should satisfy Equation (38) reported in Appendix B), which depends on the set of $L$ trajectories. Nevertheless, it places no restrictions on the length of the individual trajectories $\tau_{l}$ , allowing for certain trajectories not to contain a specific action $a$ . This aspect will be significant for proving the regret guarantees of our Mixed Spectral UCRL approach. + +# 6 Mixed Spectral UCRL + +The Mixed Spectral Estimation procedure can be easily combined with an optimistic strategy resembling the UCRL approach for MDPs [14]. We call this new algorithm Mixed Spectral UCRL, and we describe its workflow in Algorithm 2. During the first episode, we use a uniform policy $\pi_0$ (Line 3) to collect a sufficient amount of samples for each action $a\in \mathcal{A}$ in order to provide a first estimate of the POMDP parameters. The whole interaction horizon is divided into episodes of different lengths. At the beginning of each new episode $l$ , all samples collected up to that moment are used to estimate the new POMDP parameters according to Algorithm 1 (Line 7). Based on the estimated POMDP $\hat{\mathcal{Q}}_l$ , we build a high-probability confidence set $\mathcal{C}_l(\delta_l)$ of admissible POMDPs according to the bounds defined in Theorem 5.4, using a varying confidence level $\delta_l\coloneqq \delta /(3SA l^3)$ (Line 8). The optimistic policy and the associated POMDP are then computed at the beginning of episode $l$ according to the program: + +$$ +\left(\pi_ {l}, \mathcal {Q} _ {l}\right) \in \underset {\pi \in \mathcal {P}, \tilde {\mathcal {Q}} \in \mathcal {C} _ {l} (\delta_ {l})} {\operatorname {a r g m a x}} \rho (\pi , \tilde {\mathcal {Q}}), \tag {11} +$$ + +where $\rho (\pi ,\widetilde{\mathcal{Q}})$ is the average reward of policy $\pi$ in the POMDP instance $\tilde{\mathcal{Q}}$ . As specified in Section 4, we assume access to an oracle to solve Equation (11). Then, each episode terminates + +# Algorithm 2 Mixed Spectral UCRL. + +1: Input: Confidence level $\delta$ , length of initial episode $T_0$ , total horizon $T$ +2: Initialize: $t \gets 0, l \gets 0$ , belief $b_0$ uniform over states, Trajectory set $\Gamma = \{\}$ +3: Build trajectory $\tau_0$ from uniform policy $\pi_0$ for $T_0$ steps +4: $\Gamma \gets \Gamma \cup \{\tau_0\}$ +5: $t\gets T_0,l\gets 1$ , Set $N_{1}^{(a)}\leftarrow n_{0}^{(a)}\quad \forall a\in \mathcal{A}$ +6: while $t < T$ do +7: Run Algorithm 1 using trajectory set $\Gamma$ and obtain estimates $\hat{\mathbb{O}}$ and $\widehat{\mathbb{T}} = \{\widehat{\mathbb{T}}_a\}_{a\in \mathcal{A}}$ +8: Build a confidence set $\mathcal{C}_l(\delta_l)$ of admissible POMDPs +9: Compute policy $\pi_{l}$ and optimistic $\mathcal{Q}_l$ (Eq. 11) +10: $\tau_{l}\gets ()$ $n_l^{(a)}\gets 0$ for all $a\in \mathcal{A}$ +11: Observe $o_t$ , get reward $r_t \gets r(o_t)$ +12: Update belief $b_{t}$ using Equation (1) +13: Set $a_{t}\gets \pi_{l}(b_{t})$ +14: while $t < T$ or $n_l^{(a_t)} < N_l^{(a_t)}$ do +15: Execute $a_{t}$ , Set $n_l^{(at)}\gets n_l^{(at)} + 1$ +16: Observe $o_{t+1}$ , get reward $r(o_{t+1})$ +17: Update belief to $b_{t+1}$ using Equation (1) and estimated $\hat{\mathbb{O}}$ and $\hat{\mathbb{T}}_{a_t}$ +18: Set $a_{t+1} \gets \pi_l(b_{t+1})$ +19: $\tau_{l}\gets \tau_{l}\oplus (o_{t},a_{t})$ +20: Set $t\gets t + 1$ +21: end while +22: $\Gamma \leftarrow \Gamma \cup \{\tau_l\}$ +23: Set $N_{l + 1}^{(a)}\gets N_l^{(a)} + n_l^{(a)}\quad \forall a\in \mathcal{A}$ +24: Set $l\gets l + 1$ +25: end while + +when there exists an action $a \in \mathcal{A}$ such that the number of times $n_l^{(a)}$ it has been chosen during the $l$ -th episode exceeds the total number of times $N_l^{(a)}$ it has been chosen since the beginning (Line 14). + +# 6.1 Regret Analysis + +Before proceeding with the analysis of the regret of the Mixed Spectral UCRL algorithm, we remark that when the estimates of the POMDP parameters are accurate enough, the belief vector $\widehat{b}_t$ computed at each step $t$ using the estimated parameters is close to the real belief $b_t$ . To the best of our knowledge, the results in the literature [32, 34, 26, 10, 15] that relate the belief error $\| \widehat{b}_t - b_t\|_1$ with the estimation error of the model parameters all hold under the following one-step reachability assumption. + +Assumption 6.1. (Minimum Value Transition Model) The smallest value in the transition matrices satisfies $\epsilon := \min_{s, s' \in \mathcal{S}} \mathbb{T}_a(s'|s) > 0$ . + +Note that Assumption 6.1 implies the Per-Action Ergodicity (Assumption 4.3). The regret for Mixed Spectral UCRL can be expressed as follows. Its proof is deferred to Appendix C. + +Theorem 6.2. Under Assumptions 4.1, 4.2 and 6.1, let $\delta \in (0,1/2)$ . If the Mixed Spectral UCRL algorithm is run for a sufficiently large number of steps $T$ , with probability at least $1 - 2\delta$ , it suffers regret bounded as: + +$$ +\mathcal {R} _ {T} \leqslant \mathcal {O} \left(\frac {D (S A) ^ {3 / 2}}{\sigma_ {S} (\mathbb {O}) \tilde {\zeta} ^ {(L)}} \sqrt {T O \log^ {2} \left(\frac {S A O T}{\delta}\right)}\right). +$$ + +where $\tilde{\zeta}^{(L)}\coloneqq \min_{l\in [0,L - 1]}\zeta^{(l)}$ and $\zeta^{(l)}$ is defined as in Theorem 5.4. $D$ bounds the span of the bias function appearing in Equation (2) and is defined in Proposition G.1. + +This algorithm overcomes the limitations of SM-UCRL since it does not require a constantly exploring policy, and removes the need for a phased algorithm as done for SEEU. By efficiently reusing samples from different policies, we enhance the online learning of POMDPs by improving the current regret guarantee of $\tilde{\mathcal{O}}(T^{2/3})$ established by the SEEU algorithm. + +# 7 Numerical Simulations + +In this section, we analyze the estimation error of the Mixed Spectral Estimation approach under different belief policies and we show the performance in terms of regret of the Mixed Spectral UCRL algorithm when compared against state-of-the-art approaches. Further experiments and simulation details are provided in Appendix J. $^9$ + +Mixed Spectral Estimation Algorithm. This first set of experiments studies the estimation error achieved by the Mixed Spectral Estimation algorithm. In particular, we evaluate our method on a POMDP instance with sizes described in Figure 1. The estimation error is measured using the Frobenius norm of the observation matrix and the transition matrices (one per action). Figure 1 reports the average results over 10 runs. The simulation splits the interaction horizon into 10 episodes of equal length, and for each episode, we use a different belief-based policy for data collection. As observed in the figure, the total error decreases as the number of collected samples increases, demonstrating that our approach is able to efficiently combine data from different policies. + +Regret Comparison with state-of-the-art Algorithms. In this second set of experiments, we compare our Mixed Spectral UCRL algorithm with SEEU [32] and SM-UCRL [3]. The regret is measured w.r.t. the oracle whose policy satisfies Equation (2) and has full knowledge of the model parameters. As observed in Figure 2, the SM-UCRL algorithm experiences the highest regret since $(i)$ it does not reuse samples across episodes, $(ii)$ it relies on the weaker class of stochastic $(\iota >0)$ memoryless policies. This forced exploration leads to constantly selecting suboptimal actions, hence + +![](images/cae0b4fa6c4cb7e268b0ebce4fa991555a00a1f67fd10d4f1e99a6230034d054.jpg) +Figure 1: Estimation error of the Mixed Spectral Estimation on a POMDP with $S = 4$ , $A = 3$ and $O = 4$ . (10 runs, 95%c.i.). + +![](images/c4158f6d2243484271fb670ba636ef2662d192d7ff628089e437eead1b18ffca.jpg) +Figure 2: Regret comparison on a POMDP with $S = 3$ , $A = 3$ , $O = 4$ (10 runs, $95\%$ c.i.). + +resulting in higher regret. We also observe that the Mixed Spectral UCRL algorithm outperforms the SEEU algorithm. This result is in line with the theoretical guarantees, as the regret of SEEU scales with $\tilde{\mathcal{O}}(T^{2/3})$ . Besides the alternating exploration-exploitation phases, the inferior performance of SEEU can also be attributed to its reduced sample efficiency since its estimates only rely on data collected during the exploration phase, hence discarding those collected during the exploitation phase. Finally, in Appendix J, we present a regret experiment where Assumption 6.1 is violated in order to show the robustness of our approach with respect to the failure of this assumption. + +# 8 Conclusions and Future Directions + +In this work, we tackled the problem of learning using spectral methods in the infinite-horizon average-reward POMDP setting. We showed that spectral techniques can be extended to belief-based policies and, through our Mixed Spectral Estimation approach, we answered positively to the open question of whether it is possible to combine samples coming from different adaptive policies. We provided finite-sample guarantees for the devised estimation algorithm, and we showed that the error of the different parameters conveniently scales with respect to the number of employed samples. We combined the new estimation algorithm with an optimistic approach, Mixed Spectral UCRL, and provided the first algorithm achieving a $\widetilde{\mathcal{O}} (\sqrt{T})$ regret order when compared against the optimal belief-based policy, by leveraging the new sample reuse strategy, and a suitable episode stopping condition. Finally, we validated both our approaches through numerical simulations, and we showed that our approach has improved performance over state-of-the-art algorithms. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA]. + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix Organization + +We provide here an outline of the Appendix. + +- Section A, B and C present the proofs of the three theorems reported in the main paper. +- Section D provides some auxiliary results employed for the proof of Theorem 5.4. They are mostly related to the guarantees derived from the application of Tensor Decomposition methods. +- Section E gives an overview of the Symmetrization and Whitening steps, which are implemented on the third-order tensor before applying Tensor Decomposition techniques. It also introduces useful quantities that are used throughout the appendix. +- Section F provides a new bound relating the sum of successive belief errors with the error in the estimated model parameters. +Section G presents a miscellanea of useful results. +- Section H compares our work from a theoretical perspective with the related works of [3] and [32], and compares spectral approaches with Maximum-likelihood estimation techniques. +- Section I discusses the computational complexity of the Mixed Spectral Estimation method. +- Finally, Section J provides experimental performances of POMDP instances of different characteristics, together with details about the numerical simulations presented in the main paper. + +# A Proof of Theorem 5.3 + +In this section, we provide the proof of Theorem 5.3. For clarity, we report its statement here. + +Theorem 5.3. Let $\Gamma \coloneqq \{\tau\}_{l=0}^{L-1}$ be a set of trajectories collected using the set of policies $\{\pi_l\}_{l=0}^{L-1}$ . We define a modified version of the first and second views as: + +$$ +\widetilde {\boldsymbol {v}} _ {1} ^ {(a, l)} := \boldsymbol {K} _ {3, 2} ^ {(a, L)} \left(\boldsymbol {K} _ {1, 2} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {v} _ {1} ^ {(a, l)}, \quad \widetilde {\boldsymbol {v}} _ {2} ^ {(a, l)} := \boldsymbol {K} _ {3, 1} ^ {(a, L)} \left(\boldsymbol {K} _ {2, 1} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {v} _ {2} ^ {(a, l)}, \tag {8} +$$ + +where the covariance matrices are defined in Equation (7). Let $\omega^{(a,L)}\coloneqq (1 / N_L^{(a)})\sum_{l = 0}^{L - 1}n_l^{(a)}\omega^{(a,l)}$ , then, the second and third moments of the modified views have a spectral decomposition as: + +$$ +\boldsymbol {M} _ {2} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \widetilde {\boldsymbol {v}} _ {1} ^ {(a, l)} \otimes \widetilde {\boldsymbol {v}} _ {2} ^ {(a, l)} \right] = \sum_ {s \in \mathcal {S}} \boldsymbol {\omega} _ {s} ^ {(a, L)} \boldsymbol {\mu} _ {3, s} ^ {(a)} \otimes \boldsymbol {\mu} _ {3, s} ^ {(a)}, +$$ + +$$ +\pmb {M} _ {3} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \widetilde {\pmb {v}} _ {1} ^ {(a, l)} \otimes \widetilde {\pmb {v}} _ {2} ^ {(a, l)} \otimes \pmb {v} _ {3} ^ {(a, l)} \right] = \sum_ {s \in \mathcal {S}} \pmb {\omega} _ {s} ^ {(a, L)} \pmb {\mu} _ {3, s} ^ {(a)} \otimes \pmb {\mu} _ {3, s} ^ {(a)} \otimes \pmb {\mu} _ {3, s} ^ {(a)}, +$$ + +where the expectations are w.r.t. the conditional state distributions $\omega_{s}^{(a,l)}$ . + +Proof. Before proceeding, it is relevant to highlight the relation between the view matrices. We use $V_{1}^{(a,l)}$ , $V_{2}^{(a,l)}$ and $V_{3}^{(a,l)}$ to define the views associated with policy $\pi_{l}$ and action $a$ . We further recall that under the $\alpha$ -weakly revealing assumption (4.1) and the invertibility assumption of the transition matrices (4.2), the view matrices are always full-column rank [3]. + +We define the following quantity: + +$$ +\mathbb {T} _ {a, \pi_ {l}} := \sum_ {a ^ {\prime} \in \mathcal {A}} p _ {\pi_ {l}} \left(a ^ {\prime} | a\right) \mathbb {T} _ {a ^ {\prime}} \tag {12} +$$ + +with $p_{\pi_l}(\cdot |a) \in \Delta (\mathcal{A})$ being a probability distribution induced by policy $\pi_l$ and conditioned on action $a$ . As observed in [25], this distribution always exists under the employed assumptions. In particular, $p_{\pi_l}(a'|a)$ denotes the probability of having chosen action $a'$ in a previous time step (say $t - 1$ ) conditioned on the fact that action $a$ is taken in the successive time step (say $t$ ). Intuitively, $\mathbb{T}_{a,\pi_l}$ represents the mixture transition matrix defining the state transition from a previous step $(t - 1)$ to a successive one $t$ when action $a'$ is chosen in $t - 1$ by policy $\pi_l$ and the next action chosen by the policy in step $t$ is $a$ . Let us also recall that $\omega^{(a,l)}$ represents the state distribution induced by policy $\pi_l$ and conditioned on action $a$ such that $\omega_s^{(a,l)}$ is the probability of being in state $s$ when choosing + +action $a$ . Using the definition in (12), we can also define the state distribution at the previous time step $(t - 1)$ as: + +$$ +\xi^ {(a, l)} := \left(\mathbb {T} _ {a, \pi_ {l}} ^ {\top}\right) ^ {- 1} \omega^ {(a, l)} \tag {13} +$$ + +with $\xi^{(a,l)}\in \Delta (\mathcal{S})$ and such that $\xi_s^{(a,l)}$ represents the probability that state $s$ is visited in the previous time step $(t - 1)$ conditioned on having chosen action $a$ in $t$ . + +Having defined the previous state distribution $\xi^{(a,l)}$ in Eq. (13) and inspired by the multi-view model on Markov Chains of [1], we can now express the views using the following relations: + +$$ +V _ {1} ^ {(a, l)} = \mathbb {O} \operatorname {d i a g} \left(\xi^ {(a, l)}\right) \mathbb {T} _ {a, \pi_ {l}} \operatorname {d i a g} \left(\omega^ {(a, l)}\right) ^ {- 1}, \tag {14} +$$ + +$$ +V _ {2} ^ {(a, l)} = \mathbb {O}, \tag {15} +$$ + +$$ +V _ {3} ^ {(a, l)} = \mathbb {O} \mathbb {T} _ {a} ^ {\top}. \tag {16} +$$ + +From the relations stated above, we observe that the second view $V_{2}^{(a,l)}$ corresponds to the observation model, thus it depends neither on the action nor on the employed policy. Hence, we may refer to it simply as $V_{2}$ . The third view depends on the action $a$ but not on the employed policy, so we may refer to it also using $V_{3}^{(a)}$ . Finally, the first view depends on both the action $a$ and on quantities related to the employed policy $\pi_{l}$ . + +Let us now recall the definition of the covariance matrix associated with a single policy $\pi_l$ , as reported in Proposition 5.2. In particular, we will use the notation $K_{\nu,\nu'}^{(a,l)}$ to highlight that the covariance matrix depends on policy $\pi_l \in \mathcal{P}$ , thus distinguishing it from the mixture covariance (in bold) $K_{\nu,\nu'}^{(a,L)}$ resulting from the combination of $L$ different policies. + +I) Analysis of Covariance Matrix $K_{3,2}^{(a,L)}$ . We start by considering the covariance matrix $K_{3,2}^{(a,l)} \in \mathbb{R}^{O \times O}$ obtained from a single policy $\pi_l$ : + +$$ +K _ {3, 2} ^ {(a, l)} = \underset {s \sim \omega^ {(a, l)}} {\mathbb {E}} \left[ \boldsymbol {v} _ {3} ^ {(a, l)} \otimes \boldsymbol {v} _ {2} ^ {(a, l)} \right] = V _ {3} ^ {(a, l)} \operatorname {d i a g} \left(\omega^ {(a, l)}\right) \left(V _ {2} ^ {(a, l)}\right) ^ {\top} = V _ {3} ^ {(a)} \operatorname {d i a g} \left(\omega^ {(a, l)}\right) V _ {2} ^ {\top}, +$$ + +where $\omega^{(a,l)}\in \Delta (\mathcal{S})$ is the state distribution conditioned on action $a$ and $\mathrm{diag}\left(\omega^{(a,l)}\right)\in \mathbb{R}^{S\times S}$ represents a diagonal matrix whose diagonal values correspond to $w^{(a,l)}$ . Let us now recall the definition of the mixed covariance matrix in Equation (7). The following holds: + +$$ +\begin{array}{l} \boldsymbol {K} _ {3, 2} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \underset {s \sim \omega^ {(a, l)}} {\mathbb {E}} \left[ \boldsymbol {v} _ {3} ^ {(a, l)} \otimes \boldsymbol {v} _ {2} ^ {(a, l)} \right] (17) \\ = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} K _ {3, 2} ^ {(a, l)} (18) \\ = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} V _ {3} ^ {(a)} \operatorname {d i a g} \left(\omega^ {(a, l)}\right) V _ {2} ^ {\top} (19) \\ = V _ {3} ^ {(a)} \operatorname {d i a g} \left(\frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \omega^ {(a, l)}\right) V _ {2} ^ {\top} (20) \\ = V _ {3} ^ {(a)} \operatorname {d i a g} \left(\boldsymbol {\omega} ^ {(a, L)}\right) V _ {2} ^ {\top}, (21) \\ = \mathbb {O} \mathbb {T} _ {a} ^ {\top} \operatorname {d i a g} \left(\omega^ {(a, L)}\right) \mathbb {O} ^ {\top}, \quad \text {(F o l l o w s f r o m l i n e s 1 5 a n d 1 6)} \\ \end{array} +$$ + +where in line 19 we used $V_{3}^{(a,l)} = V_{3}^{(a)}$ for any $l$ , and $V_{2}^{(a,l)} = V_{2}$ for any $a$ and $l$ , hence highlighting the independence of both view matrices from the used policy $\pi_{l}$ . In line 21 we introduced the new state distribution $\omega^{(a,L)} \in \Delta(\mathcal{S})$ such that $\omega^{(a,L)} := (1/N_{L}^{(a)})\sum_{l=0}^{L-1}n_{l}^{(a)}\omega^{(a,l)}$ . + +II) Analysis of Covariance Matrix $K_{3,1}^{(a,L)}$ . Let us now consider a similar relation for the covariance matrix $K_{3,1}^{(a,L)} \in \mathbb{R}^{O \times O}$ combining $L$ different policies. We have that: + +$$ +\begin{array}{l} \boldsymbol {K} _ {3, 1} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \underset {s \sim \omega^ {(a, l)}} {\mathbb {E}} \left[ \boldsymbol {v} _ {3} ^ {(a, l)} \otimes \boldsymbol {v} _ {1} ^ {(a, l)} \right] (22) \\ = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} K _ {3, 1} ^ {(a, l)} (23) \\ = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} V _ {3} ^ {(a, l)} \operatorname {d i a g} \left(\omega^ {(a, l)}\right) \left(V _ {1} ^ {(a, l)}\right) ^ {\top} (24) \\ = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {O} \mathbb {T} _ {a} ^ {\top} \operatorname {d i a g} \left(\omega^ {(a, l)}\right) \left[ \operatorname {d i a g} \left(\omega^ {(a, l)}\right) ^ {- 1} \mathbb {T} _ {a, \pi_ {l}} ^ {\top} \operatorname {d i a g} \left(\xi^ {(a, l)}\right) \mathbb {O} ^ {\top} \right] (From lines 14 and 16) \\ = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {O} \mathbb {T} _ {a} ^ {\top} \mathbb {T} _ {a, \pi_ {l}} ^ {\top} \operatorname {d i a g} \left(\xi^ {(a, l)}\right) \mathbb {O} ^ {\top} (25) \\ = \mathbb {O} \mathbb {T} _ {a} ^ {\top} \left(\frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {T} _ {a, \pi_ {l}} ^ {\top} \operatorname {d i a g} \left(\xi^ {(a, l)}\right)\right) \mathbb {O} ^ {\top} \quad (\text {A s s o c i a t i v e p r o p e r t y}) \\ = \mathbb {O} \mathbb {T} _ {a} ^ {\top} \operatorname {d i a g} \left(\boldsymbol {\omega} ^ {(a, L)}\right) \left[ \operatorname {d i a g} \left(\boldsymbol {\omega} ^ {(a, L)}\right) ^ {- 1} \left(\frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {T} _ {a, \pi_ {l}} ^ {\top} \operatorname {d i a g} \left(\xi^ {(a, l)}\right)\right) \mathbb {O} ^ {\top} \right] (26) \\ = V _ {3} ^ {(a)} \operatorname {d i a g} \left(\boldsymbol {w} ^ {(a, L)}\right) \left(\boldsymbol {V} _ {1} ^ {(a, L)}\right) ^ {\top}, \quad \text {(F r o m l i n e s 1 4 a n d 1 6)} \\ \end{array} +$$ + +where in the last line $\pmb{V}_1^{(a,L)}\coloneqq \mathbb{O}\left(\frac{1}{N_L^{(a)}}\sum_{l = 0}^{L - 1}n_l^{(a)}\mathbb{T}_{a,\pi_l}^\top \operatorname {diag}\left(\xi^{(a,l)}\right)\right)^\top \operatorname {diag}\left(\pmb{\omega}^{(a,L)}\right)^{-1}$ defines the mixed first view matrix. + +III) Analysis of Covariance Matrix $K_{2,1}^{(a,L)}$ . By applying similar steps to those employed for covariance matrix $K_{3,1}^{(a,L)}$ , we are able to show that: + +$$ +\boldsymbol {K} _ {2, 1} ^ {(a, L)} = V _ {2} \operatorname {d i a g} \left(\boldsymbol {w} ^ {(a, L)}\right) \left(\boldsymbol {V} _ {1} ^ {(a, L)}\right) ^ {T}. \tag {27} +$$ + +We are now ready to provide the proofs for the second and third moments. For simplicity, we will just provide the proof for the second moment matrix $M_2^{(a,L)}$ since the proof for the third moment tensor $M_3^{(a,L)}$ follows analogous steps. + +Proof for the second Moment matrix $M_2^{(a,L)}$ . The relation for the mixed second moment matrix is defined as follows. + +$$ +\begin{array}{l} \boldsymbol {M} _ {2} ^ {(a, L)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \widetilde {\boldsymbol {v}} _ {1} ^ {(a, l)} \otimes \widetilde {\boldsymbol {v}} _ {2} ^ {(a, l)} \right] (28) \\ = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \boldsymbol {K} _ {3, 2} ^ {(a, L)} \left(\boldsymbol {K} _ {1, 2} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {v} _ {1} ^ {(a, l)} \left(\boldsymbol {v} _ {2} ^ {(a, l)}\right) ^ {\top} \left(\left(\boldsymbol {K} _ {2, 1} ^ {(a, L)}\right) ^ {\dagger}\right) ^ {\top} \boldsymbol {K} _ {1, 3} ^ {(a, L)} \right] (29) \\ = \boldsymbol {K} _ {3, 2} ^ {(a, L)} \left(\boldsymbol {K} _ {1, 2} ^ {(a, L)}\right) ^ {\dagger} \left(\frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \boldsymbol {v} _ {1} ^ {(a, l)} \left(\boldsymbol {v} _ {2} ^ {(a, l)}\right) ^ {\top} \right]\right) \left(\left(\boldsymbol {K} _ {2, 1} ^ {(a, L)}\right) ^ {\dagger}\right) ^ {\top} \boldsymbol {K} _ {1, 3} ^ {(a, L)} (30) \\ = \boldsymbol {K} _ {3, 2} ^ {(a, L)} \left(\boldsymbol {K} _ {1, 2} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {K} _ {1, 2} ^ {(a, L)} \left(\left(\boldsymbol {K} _ {2, 1} ^ {(a, L)}\right) ^ {\dagger}\right) ^ {\top} \boldsymbol {K} _ {1, 3} ^ {(a, L)} (31) \\ = \boldsymbol {K} _ {3, 2} ^ {(a, L)} \left(\left(\boldsymbol {K} _ {2, 1} ^ {(a, L)}\right) ^ {\dagger}\right) ^ {\top} \boldsymbol {K} _ {1, 3} ^ {(a, L)} (32) \\ = \boldsymbol {K} _ {3, 2} ^ {(a, L)} \left(\boldsymbol {K} _ {1, 2} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {K} _ {1, 3} ^ {(a, L)} (33) \\ = \left[ V _ {3} ^ {(a)} \operatorname {d i a g} \left(\omega^ {(a, L)}\right) V _ {2} ^ {\top} \right] \left[ \left(V _ {2} ^ {\top}\right) ^ {\dagger} \operatorname {d i a g} \left(\omega^ {(a, L)}\right) ^ {- 1} \left(V _ {1} ^ {(a, L)}\right) ^ {\dagger} \right]. \\ \cdot \left[ \boldsymbol {V} _ {1} ^ {(a, L)} \operatorname {d i a g} \left(\boldsymbol {\omega} ^ {(a, L)}\right) \left(V _ {3} ^ {(a)}\right) ^ {\top} \right] (34) \\ = V _ {3} ^ {(a)} \operatorname {d i a g} \left(\boldsymbol {\omega} ^ {(a, L)}\right) \left(V _ {3} ^ {(a)}\right) ^ {\top} (35) \\ = \sum_ {s \in S} \omega_ {s} ^ {(a, L)} \boldsymbol {\mu} _ {3, s} ^ {(a)} \otimes \boldsymbol {\mu} _ {3, s} ^ {(a)}, (36) \\ \end{array} +$$ + +where line 29 holds since $\widetilde{\pmb{v}}_1^{(a,l)}\otimes \widetilde{\pmb{v}}_2^{(a,l)} = \widetilde{\pmb{v}}_1^{(a,l)}\left(\widetilde{\pmb{v}}_2^{(a,l)}\right)^\top$ , while line 34 holds for the relations of covariance matrices found in the above points. + +The simplification steps made from line 34 to line 35 are done considering that the multiplication of a matrix and its pseudoinverse while projecting along the smaller space of size $S$ produces $\mathbb{I}_S$ , an identity matrix of rank $S$ . In particular, by applying the definition of the Moore-Penrose inverse of a matrix, we have that $V_2^\dagger = (V_2^\top V_2)^{-1} V_2^\top$ . Since the pseudo-inverse of a transpose corresponds to the transpose of the pseudo-inverse, we get that $(V_2^\top)^\dagger = V_2(V_2^\top V_2)^{-1}$ . Hence, the expression in line 34 can be simplified as: + +$$ +V _ {2} ^ {\top} \left(V _ {2} ^ {\top}\right) ^ {\dagger} = (V _ {2} ^ {\top} V _ {2}) (V _ {2} ^ {\top} V _ {2}) ^ {- 1} = \mathbb {I} _ {S}. +$$ + +Similar steps also lead to $\left(\mathbf{V}_1^{(a,L)}\right)^\dagger \mathbf{V}_1^{(a,L)} = \mathbb{I}_S$ + +Finally, the last equivalence in line 36 concludes the proof. + +# B Proof of Theorem 5.4 + +Theorem 5.4. Let $\widehat{\mathbb{O}}$ and $\{\widehat{\mathbb{T}}_a\}_{a\in \mathcal{A}}$ be the observation and transition model estimated using Algorithm 1, respectively. Let Assumptions 4.1 and 4.2 hold and let Assumption 4.3 be true for any $\pi_l$ with $l\in [0,L - 1]$ . Let $\delta \in (0,1 / (3SA))$ , then for a sufficiently large number of samples $N_L^{(a)}$ holding for every action $a\in \mathcal{A}$ , with probability at least $1 - 3SA\delta$ , it holds that: + +$$ +\left\| \mathbb {O} - \widehat {\mathbb {O}} \right\| _ {F} \leqslant \frac {C _ {\mathbb {O}}}{\zeta^ {(L)}} \sqrt {\frac {S A L \log (L O / \delta)}{N _ {L}}}, \quad \left\| \mathbb {T} _ {a} - \widehat {\mathbb {T}} _ {a} \right\| _ {F} \leqslant \frac {C _ {\mathbb {T}} S}{\sigma_ {S} (\mathbb {O}) \zeta^ {(L)}} \sqrt {\frac {A L \log (L O / \delta)}{N _ {L} ^ {(a)}}}, +$$ + +$$ +w h e r e \zeta^ {(L)} := \tilde {\sigma} _ {3, 1} ^ {(L)} \left[ \sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}} \min _ {\nu \in \{1, 2, 3 \}, a \in \mathcal {A}} \sigma_ {S} (V _ {\nu} ^ {(a, L)}) \right] ^ {3}, \tilde {\omega} _ {\operatorname * {m i n}} ^ {(L)} := \min _ {a \in \mathcal {A}} \omega_ {\operatorname * {m i n}} ^ {(a, L)}, a n d \tilde {\sigma} _ {3, 1} ^ {(L)} := +$$ + +$\min_{a\in \mathcal{A}}\sigma_S(\pmb{K}_{3,1}^{(a,L)})$ while $C_{\mathbb{O}}$ and $C_{\mathbb{T}}$ are suitable constants. + +Proof. We recall that Spectral Decomposition techniques are separately applied for each action $a \in \mathcal{A}$ and each of them outputs estimates of the third view $V_3^{(a)}$ . From the columns $\mu_{3,s}$ of the third view, estimates of the columns $\mu_{2,s}$ of the second view matrix can be computed by inverting Equation (6). We remark that the second view is equal for all actions $a$ and it corresponds to the observation matrix ①. Since we require that the number of samples $N_L^{(a)}$ satisfies conditions in Equation (95) and (102), Lemma D.1 can be used to bound the error of the columns $\mu_{2,s}$ of the second view matrix, thus having: + +$$ +\left\| \boldsymbol {\mu} _ {2, s} ^ {(a, L)} - \hat {\boldsymbol {\mu}} _ {2, s} ^ {(a, L)} \right\| _ {2} \leqslant \frac {1 6 \epsilon_ {M} ^ {(a , L)}}{\sigma_ {S} \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right)}, \tag {37} +$$ + +holding with probability at least $1 - 3\delta$ , and with $\epsilon_{M}^{(a,L)}$ defined in Lemma D.1. + +Condition for Column Permutation The next step of algorithm 1 consists in permuting the view matrices $\widehat{V}_2^{(a,L)10}$ for each action $a$ in order to minimize the 1-norm error with respect to view matrix $\widehat{V}_2^{(a^*,L)}$ where $a^* \in \operatorname{argmax}_{a \in \mathcal{A}} N_L^{(a)11}$ . The permutation found for each estimated matrix $\widehat{V}_2^{(a,L)}$ is then applied as well to the associated third view $\widehat{V}_3^{(a,L)}$ . + +Guarantees on the permutation are achieved when each column $\pmb{\mu}_{2,s}^{(a,L)}$ is estimated sufficiently well. Let us denote with $d_O := \min_{s,s' \in S, s \neq s'} \| \mathbb{O}(\cdot | s) - \mathbb{O}(\cdot | s') \|_1$ the minimum distance between columns of $\mathbb{O}$ . As observed in [3], when the estimation error is lower than $d_O / 4$ , the columns can be permuted without error. Hence, we derive here the minimum sample condition such that the estimation error of each column (reported in D.1) is bounded by $d_O / 4$ : + +$$ +N _ {L} ^ {(a)} \geqslant \left(\frac {1 2 8 \sqrt {2} \tilde {G} / (1 - \tilde {\eta})}{d _ {O} \sigma_ {S} (\boldsymbol {K} _ {3 , 1} ^ {(a , L)}) \left(\sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} (V _ {\nu} ^ {(a , L)})\right) ^ {3}}\right) ^ {2} 8 L \log \left(\frac {(O ^ {2} + O) 2 L}{\delta}\right). +$$ + +By combining the condition above with those required for the bound of Lemma D.1, we obtain: + +$$ +N _ {L} ^ {(a)} \geqslant \Gamma^ {(a, L)} \frac {8 L \tilde {G} ^ {2}}{(1 - \tilde {\eta}) ^ {2}} \log \left(\frac {2 L (O ^ {2} + O)}{\delta}\right) \tag {38} +$$ + +where + +$$ +\begin{array}{l} \Gamma^ {(a, L)} := \max \left\{\left(\frac {1}{d _ {O} \sigma_ {S} (\pmb {K} _ {3 , 1} ^ {(a , L)}) \left(\sqrt {\tilde {\omega} _ {\mathrm {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} (V _ {\nu} ^ {(a , L)})\right) ^ {3}}\right) ^ {2}, \right. \\ \left. \left(\frac {2 \sqrt {\Omega}}{\omega_ {\mathrm {m i n}} ^ {(a , L)} \left[ \min _ {\nu} \sigma_ {S} (V _ {\nu} ^ {(a , L)}) \right] ^ {2}}\right) ^ {2}, \left(\frac {4}{\left[ \sigma_ {S} (\pmb {K} _ {3 , 1} ^ {(a , L)}) \right] ^ {2}}\right) ^ {2} \right\}. \\ \end{array} +$$ + +Bound on the Observation Model Error After the permutation operation, we can finally combine the obtained view matrices $\widehat{V}_2^{(a,L)}$ as shown in Equation (9) to obtain a unique matrix $\widehat{V}_2^{(L)}$ such that: + +$$ +\widehat {\mathbb {O}} := \widehat {V} _ {2} ^ {(L)} := \frac {1}{N _ {L}} \sum_ {a \in \mathcal {A}} N _ {L} ^ {(a)} \widehat {V} _ {2} ^ {(a, L)}, +$$ + +with $N_{L} = \sum_{a\in \mathcal{A}}N_{L}^{(a)}$ . Let us denote with $\hat{\pmb{\mu}}_{2,s}^{(L)}$ the $s$ -th column of view matrix $\hat{V}_2^{(L)}$ . From the bound defined in Equation (37) and using a union bound argument, we finally get with probability at least $1 - 3A\delta$ that: + +$$ +\left\| \boldsymbol {\mu} _ {2, s} ^ {(L)} - \hat {\boldsymbol {\mu}} _ {2, s} ^ {(L)} \right\| _ {2} \leqslant \frac {1 6 \sqrt {A} \tilde {\epsilon} _ {M} ^ {(L)}}{\tilde {\sigma} _ {3 , 1} ^ {(L)}}, \tag {39} +$$ + +where: + +$$ +\tilde {\epsilon} _ {M} ^ {(L)} \leqslant \frac {\frac {2 \sqrt {2} \tilde {G}}{1 - \tilde {\eta}} \sqrt {\frac {8 L \log ((O ^ {2} + O) 2 L / \delta)}{N _ {L}}}}{\left[ \sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}} \min _ {\nu , a} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right) \right] ^ {3}} + \frac {\left(\frac {\frac {4 \tilde {G}}{1 - \tilde {\eta}} \sqrt {\frac {8 L \log (4 O L / \delta)}{N _ {L}}}}{\left(\sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}} \min _ {\nu , a} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {2}}\right) ^ {3}}{\sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}}}, \tag {40} +$$ + +and $\widetilde{\omega}_{\mathrm{min}}^{(L)}\coloneqq \min_{a\in \mathcal{A}}\omega_{\mathrm{min}}^{(a,L)}$ and $\widetilde{\sigma}_{3,1}^{(L)}\coloneqq \min_{a\in \mathcal{A}}\sigma_S(\pmb{K}_{3,1}^{(a,L)})$ + +We notice that a further $\sqrt{A}$ term appears in (39) as a result of the union bound, and we stress that the minimization over the singular values is done considering both $\nu \in \{1, 2, 3\}$ and $a$ . + +Since the result in (39) is independent of the single column $s$ , we can easily extend it to the whole observation matrix and finally get: + +$$ +\left\| \mathbb {O} - \hat {\mathbb {O}} \right\| _ {F} = \sqrt {\sum_ {s \in S} \left\| \boldsymbol {\mu} _ {2 , s} ^ {(L)} - \hat {\boldsymbol {\mu}} _ {2 , s} ^ {(L)} \right\| _ {2} ^ {2}} \leqslant \frac {1 6 \sqrt {S A} \tilde {e} _ {M} ^ {(L)}}{\tilde {\sigma} _ {3 , 1} ^ {(L)}}, \tag {41} +$$ + +holding with probability at least $1 - 3SA\delta$ . By simplifying the notation and highlighting the most relevant terms in the bound, we get: + +$$ +\left\| \mathbb {O} - \hat {\mathbb {O}} \right\| _ {F} \leqslant \frac {C _ {\mathbb {O}}}{\zeta^ {(L)}} \sqrt {\frac {S A L \log (L O / \delta)}{N _ {L}}} \tag {42} +$$ + +with $C_{\mathbb{O}}$ being a suitable constant and $\zeta^{(L)}$ being defined as: + +$$ +\zeta^ {(L)} := \widetilde {\sigma} _ {3, 1} ^ {(L)} \left[ \sqrt {\widetilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}} \min _ {\nu , a} \sigma_ {S} \left(V _ {\nu} ^ {(a, L)}\right) \right] ^ {3}. \tag {43} +$$ + +Bound on the Transition Model Error By following Algorithm 1, the $s$ -th row of each estimated transition matrix $\widehat{\mathbb{T}}_a$ is computed as $\widehat{\mathbb{T}}_a(s,\cdot) = \widehat{\mathbb{O}}^\dagger \widehat{\pmb{\mu}}_{3,s}^{(a,L)}$ . Let us analyze its associated error. We have: + +$$ +\begin{array}{l} \left\| \mathbb {T} _ {a} (s, \cdot) - \widehat {\mathbb {T}} _ {a} (s, \cdot) \right\| _ {2} = \left\| \mathbb {O} ^ {\dagger} \boldsymbol {\mu} _ {3, s} ^ {(a, L)} - \widehat {\mathbb {O}} ^ {\dagger} \widehat {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} \right\| _ {2} \\ \leqslant \underbrace {\left\| \mathbb {O} ^ {\dagger} - \widehat {\mathbb {O}} ^ {\dagger} \right\| _ {2} \left\| \boldsymbol {\mu} _ {3 , s} ^ {(a , L)} \right\| _ {2}} _ {\text {(a)}} + \underbrace {\left\| \boldsymbol {\mu} _ {3 , s} ^ {(a , L)} - \widehat {\boldsymbol {\mu}} _ {3 , s} ^ {(a , L)} \right\| _ {2} \left\| \widehat {\mathbb {O}} ^ {\dagger} \right\| _ {2}} _ {\text {(b)}}. \\ \end{array} +$$ + +Let us now analyze the different terms separately. Concerning the term (a), we use i) $\left\| \pmb{\mu}_{3,s}^{(a,L)}\right\|_2 \leqslant 1$ and ii) we first apply Proposition D.6 on the spectral norm of the pseudo-inverse of matrix $\widehat{\mathbb{O}}$ and then we use Proposition D.7 to bound $\left\| \mathbb{O}^\dagger - \widehat{\mathbb{O}}^\dagger \right\|_2$ and obtain: + +$$ +\mathrm {(a)} \leqslant \frac {2 (1 + \sqrt {5})}{2} \frac {\| \mathbb {O} - \widehat {\mathbb {O}} \| _ {2}}{\sigma_ {S} (\mathbb {O})} \leqslant \frac {4 \| \mathbb {O} - \widehat {\mathbb {O}} \| _ {2}}{\sigma_ {S} (\mathbb {O})} \leqslant \frac {4 \| \mathbb {O} - \widehat {\mathbb {O}} \| _ {F}}{\sigma_ {S} (\mathbb {O})} \leqslant \frac {4 \cdot 1 6 \sqrt {S A} \tilde {e} _ {M} ^ {(L)}}{\sigma_ {S} (\mathbb {O}) \tilde {\sigma} _ {3 , 1} ^ {(L)}}. +$$ + +Analogously, for the second term (b), we apply i) Proposition D.6 to bound $\| \hat{O}^{\dagger}\|_{2}$ and ii) we use Lemma D.2 to bound the error of the estimated view vector, thus obtaining: + +$$ +(\mathbf {b}) \leqslant \frac {2}{\sigma_ {S} (\mathbb {O})} \cdot 1 4 \epsilon_ {M} ^ {(a, L)} \leqslant \frac {2 8 \epsilon_ {M} ^ {(a , L)}}{\sigma_ {S} (\mathbb {O})}. +$$ + +Since Proposition D.6 holds under the condition $\| \mathbb{O} - \widehat{\mathbb{O}}\| _2\leqslant (1 / 2)\sigma_S(\mathbb{O})$ , we require a minimum number of samples $N_{L}$ based on the bound in 42. It should satisfy: + +$$ +N _ {L} \geqslant \left(\frac {2 C _ {\mathbb {O}}}{\zeta^ {(L)} \sigma_ {S} (\mathbb {O})}\right) ^ {2} S A L \log \left(\frac {L O}{\delta}\right). \tag {44} +$$ + +The conditions defined in 38 together with the one just stated above on the total number of samples $N_{L}$ determine the sufficient conditions for the theorem to hold. + +Going back to the bound on the estimated transition matrix, by combining the results reported so far, we get with probability at least $1 - 3SA\delta$ : + +$$ +\left\| \mathbb {T} _ {a} (s, \cdot) - \widehat {\mathbb {T}} _ {a} (s, \cdot) \right\| _ {2} \leqslant \frac {6 4 \sqrt {S A} \widetilde {\epsilon} _ {M} ^ {(L)}}{\sigma_ {S} (\mathbb {O}) \widetilde {\sigma} _ {3 , 1} ^ {(L)}} + \frac {2 8 \epsilon_ {M} ^ {(a , L)}}{\sigma_ {S} (\mathbb {O})} \leqslant \frac {C _ {\mathbb {T}} ^ {\prime} \sqrt {S A} \widetilde {\epsilon} _ {M} ^ {(a , L)}}{\sigma_ {S} (\mathbb {O}) \widetilde {\sigma} _ {3 , 1} ^ {(L)}}, +$$ + +where $C_{\mathbb{T}}^{\prime}$ is a suitable constant term, while we used here a new quantity $\widetilde{\epsilon}_{M}^{(a,L)}$ for which it holds both $\widetilde{\epsilon}_{M}^{(L)} \leqslant \widetilde{\epsilon}_{M}^{(a,L)}$ and $\epsilon_{M}^{(a,L)} \leqslant \widetilde{\epsilon}_{M}^{(a,L)}$ since it is defined as: + +$$ +\tilde {\epsilon} _ {M} ^ {(a, L)} \leqslant \frac {\frac {2 \sqrt {2 \tilde {G}}}{1 - \tilde {\eta}} \sqrt {\frac {8 L \log ((O ^ {2} + O) 2 L / \delta)}{N _ {L} ^ {(a)}}}}{\left[ \sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}} \min _ {\nu , a ^ {\prime}} \sigma_ {S} \left(V _ {\nu} ^ {(a ^ {\prime} , L)}\right) \right] ^ {3}} + \frac {\left(\frac {\frac {4 \tilde {G}}{1 - \tilde {\eta}} \sqrt {\frac {8 L \log (4 O L / \delta)}{N _ {L} ^ {(a)}}}}{\left(\sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}} \min _ {\nu , a ^ {\prime}} \sigma_ {S} \left(V _ {\nu} ^ {(a ^ {\prime} , L)}\right)\right) ^ {2}}\right) ^ {3}}{\sqrt {\tilde {\omega} _ {\operatorname* {m i n}} ^ {(L)}}}, \tag {45} +$$ + +scaling with rate $1 / N_L^{(a)}$ differently from the rate $1 / N_{L}$ of $\widetilde{\epsilon}_{M}^{(L)}$ defined in Equation (40). Since this bound holds for any row of the transition matrix, we can derive the error on the whole transition matrix as: + +$$ +\left\| \mathbb {T} _ {a} - \widehat {\mathbb {T}} _ {a} \right\| _ {F} = \sqrt {\sum_ {s \in S} \left\| \mathbb {T} _ {a} (s , \cdot) - \widehat {\mathbb {T}} _ {a} (s , \cdot) \right\| _ {2} ^ {2}} \leqslant \frac {C _ {\mathbb {T}} ^ {\prime} S \sqrt {A} \widetilde {\sigma} _ {M} ^ {(a , L)}}{\sigma_ {S} (\mathbb {O}) \widetilde {\sigma} _ {3 , 1} ^ {(L)}} \tag {46} +$$ + +holding with probability at least $1 - 3SA\delta$ and presenting an additional $\sqrt{S}$ term. By simplifying notation and highlighting the most relevant terms in the bound, we get: + +$$ +\left\| \mathbb {T} _ {a} - \widehat {\mathbb {T}} _ {a} \right\| _ {F} \leqslant \frac {C _ {\mathbb {T}} S}{\sigma_ {S} (\mathbb {O}) \zeta^ {(L)}} \sqrt {\frac {A L \log (L O / \delta)}{N _ {L} ^ {(a)}}} +$$ + +where $C_{\mathbb{T}}$ is a suitable constant and $\zeta^{(L)}$ is defined as in Eq. (43). + +This last step concludes the proof. + +# C Proof of Theorem 6.2 + +This section will present the proof for Theorem 6.2, showing the regret guarantees of the Mixed Spectral UCRL algorithm. This result makes use of Theorem 5.4 related to the estimation guarantees of the Mixed Spectral Estimation approach presented in Algorithm 1, and it makes use of the new bound on the belief error provided in Lemma F.1. Some steps of this analysis are inspired by the work of [34]. + +# Notation + +Before proceeding, we need to define some useful quantities that will be employed throughout the proof. + +Let us define vector $\phi \in \mathbb{R}^S$ of expected rewards. Its elements are such that: + +$$ +\phi (s) = \sum_ {o \in \mathcal {O}} r (o) \mathbb {O} (o | s) = \boldsymbol {r} ^ {\top} \mathbb {O} (\cdot | s). \tag {47} +$$ + +From the quantity defined above, we have that the expected reward given a belief $b_{t}$ at time $t$ is: + +$$ +g \left(b _ {t}\right) = \sum_ {s \in \mathcal {S}} \phi (s) b _ {t} (s) = \phi^ {\top} b _ {t} = \boldsymbol {r} ^ {\top} \mathbb {O} b _ {t}. \tag {48} +$$ + +The real transition and observation model of the POMDP instance $\mathcal{Q}$ are defined respectively as $\mathbb{T} = \{\mathbb{T}_a\}_{a\in A}$ and $\mathbb{O}$ . + +We will use instead $\widehat{\mathbb{T}}_l = \{\widehat{\mathbb{T}}_{a,l}\}_{a\in \mathcal{A}}$ and $\widehat{\mathbb{O}}_l$ to denote the transition model and observation models estimated by the Mixed Spectral Estimation procedure at the beginning of episode $l$ , while we will use $\mathbb{T}_l = \{\mathbb{T}_{a,l}\}_{a\in \mathcal{A}}$ and $\mathbb{O}_l$ to denote the optimistic transition and observation model returned as output by the oracle and actually used during episode $l$ . In a similar way, we will denote the estimated and optimistic POMDP instances at episode $l$ with $\widehat{\mathcal{Q}}_l$ and $\mathcal{Q}_l$ respectively. We use $\rho^l$ to denote the optimal average reward for the optimistic POMDP $\mathcal{Q}_l$ . + +We introduce the deterministic function $H(b_{t},a_{t},o_{t + 1})$ which returns the belief at the next step $b_{t + 1}$ given the action $a_{t}$ and the next observation $o_{t + 1}$ according to the Bayes' rule defined in (1). We define a similar function $H_{l}(b_{t},a_{t},o_{t + 1})$ which transforms the belief using the optimistic observation model $\mathbb{O}_l$ and transition model $\mathbb{T}_{a_t,l}$ used during the $l$ -th episode. + +The probability distribution over the next observation $o_{t + 1}$ given belief $b_{t}$ and action $a_{t}$ is defined by: + +$$ +P \left(o _ {t + 1} \mid b _ {t}, a _ {t}\right) = \boldsymbol {e} _ {o} ^ {\top} \mathbb {O} \mathbb {T} _ {a _ {t}} ^ {\top} b _ {t}, +$$ + +where $e_{o}$ is the standard basis vector in $\{0,1\}^{O}$ corresponding to observation $o \in \mathcal{O}$ . The probabilities here are computed according to the transition model $\mathbb{T}_{a_t}$ related to the chosen action and the observation model $\mathbb{O}$ of POMDP $\mathcal{Q}$ . With $P_{l}(o_{t + 1}|b_{t},a_{t})$ we denote the analogous probability computed using the observation and transition models of the optimistic POMDP $\mathcal{Q}_l$ . + +The same probability distribution holds over the next belief given the current belief, and it is defined as: + +$$ +U (b _ {t + 1} | b _ {t}, a _ {t}) = P _ {\mathcal {Q}} (b _ {t + 1} | b _ {t}, a _ {t}) = \left\{ \begin{array}{l l} P (o _ {t + 1} | b _ {t}, a _ {t}) & \text {i f} b _ {t + 1} = H (b _ {t}, a _ {t}, o), \\ 0 & \text {o t h e r w i s e .} \end{array} \right. +$$ + +We will use $U_{l}$ to denote a similar measure defined with respect to the observation and transition models of the optimistic POMDP $\mathcal{Q}_l$ . + +We will use $E_{l}$ to characterize the time intervals belonging to the $l$ -th episode, from which we exclude the first and the last interval (this is done since the first and last samples of an interval are not used for SD). Hence, we will have that the number of samples from the $l$ -th episode that will be used for SD is $n_{l} = |E_{l}|$ . + +Having defined the employed notation, we report here the statement of the theorem. + +Theorem 6.2. Under Assumptions 4.1, 4.2 and 6.1, let $\delta \in (0,1/2)$ . If the Mixed Spectral UCRL algorithm is run for a sufficiently large number of steps $T$ , with probability at least $1 - 2\delta$ , it suffers regret bounded as: + +$$ +\mathcal {R} _ {T} \leqslant \mathcal {O} \left(\frac {D (S A) ^ {3 / 2}}{\sigma_ {S} (\mathbb {O}) \tilde {\zeta} ^ {(L)}} \sqrt {T O \log^ {2} \left(\frac {S A O T}{\delta}\right)}\right). +$$ + +where $\tilde{\zeta}^{(L)}\coloneqq \min_{l\in [0,L - 1]}\zeta^{(l)}$ and $\zeta^{(l)}$ is defined as in Theorem 5.4. $D$ bounds the span12 of the bias function appearing in Equation (2) and is defined in Proposition G.1. + +Proof. Let us recall here the definition of the regret as reported in (3): + +$$ +\mathcal {R} _ {T} := T \rho^ {*} - \sum_ {t = 0} ^ {T - 1} r \left(o _ {t}\right) = \sum_ {t = 0} ^ {T - 1} \left(\rho^ {*} - \mathbb {E} \left[ r \left(o _ {t}\right) \mid \mathcal {F} _ {t - 1} \right]\right) + \sum_ {t = 0} ^ {T - 1} \left(\mathbb {E} \left[ r \left(o _ {t}\right) \mid \mathcal {F} _ {t - 1} \right] - r \left(o _ {t}\right)\right), \tag {49} +$$ + +where we consider an expectation $\mathbb{E}$ taken w.r.t. the true transition model $\mathbb{T} = \{\mathbb{T}_a\}_{a\in A}$ and the true observation model $\mathbb{O} = \{\mathbb{O}_a\}_{a\in A}$ . The quantity $\mathcal{F}_{t - 1}$ denotes the filtration defined with respect to the events that occurred up to time $t - 1$ . The second term in the summation defines a martingale. Indeed, by denoting the stochastic process as: + +$$ +X _ {0} = 0, X _ {t} = \sum_ {l = 0} ^ {t - 1} (\mathbb {E} [ r (o _ {l}) | \mathcal {F} _ {l - 1} ] - r (o _ {l})), +$$ + +we observe that $X_{t}$ defines a martingale. By applying now the Azuma-Hoeffding inequality [4], with probability at least $1 - \delta / 4$ we have: + +$$ +\sum_ {t = 0} ^ {T - 1} \left(\mathbb {E} \left[ r \left(o _ {t}\right) \mid \mathcal {F} _ {t - 1} \right] - r \left(o _ {t}\right)\right) \leqslant \sqrt {2 T \log (4 / \delta)}. \tag {50} +$$ + +We can further observe that since the belief $b_{t}$ is conditioned on the filtration $\mathcal{F}_{t - 1}$ , we have: + +$$ +\mathbb {E} \left[ r \left(o _ {t}\right) \mid \mathcal {F} _ {t - 1} \right] = \sum_ {s \in \mathcal {S}} b _ {t} (s) \phi (s) = g \left(b _ {t}\right), +$$ + +where vector $\phi$ is defined in Equation (47), while function $g$ is defined in Equation (48). We recall that the belief $b_{t}$ is computed using the true model parameters. Using analogous notation, we will denote the expected instantaneous reward assuming to have updated the belief using the optimistic transition model $\mathbb{T}_{a,l}$ and observation model $\mathbb{O}_l$ as: + +$$ +\mathbb {E} _ {l} \left[ r \left(o _ {t}\right) \mid \mathcal {F} _ {t - 1} \right] = r ^ {\top} \mathbb {O} _ {l} b _ {t} ^ {l} = g \left(b _ {t} ^ {l}\right). +$$ + +From the quantities defined above, we can rewrite the first term of Equation (49) as: + +$$ +\sum_ {t = 0} ^ {T - 1} \left(\rho^ {*} - \mathbb {E} \left[ r \left(o _ {t}\right) \mid \mathcal {F} _ {t - 1} \right]\right) = \sum_ {t = 0} ^ {T - 1} \left(\rho^ {*} - g \left(b _ {t}\right)\right), \tag {51} +$$ + +where we recall that the belief is updated using the actions taken by the played policy. + +By following the procedure described in the Mixed Spectral UCRL algorithm, at the beginning of each episode $l$ , an optimistic POMDP $\mathcal{Q}_l$ is chosen from the set of possible POMDPs determined by the confidence region $\mathcal{C}_l(\delta_l)$ . We recall that the optimistic POMDP $\mathcal{Q}_l$ is defined by the optimistic transition model $\mathbb{T}_l = \{\mathbb{T}_{a,l}\}_{a\in A}$ and the optimistic observation model $\mathbb{O}_l$ provided by the oracle. + +Since the bound for the estimated transition and observation models provided in Theorem 5.4 holds jointly with probability at least $1 - 3SA\delta$ , we can also observe that $P(\mathcal{Q} \in \mathcal{C}_l(\delta_l)) \geqslant 1 - 3SA\delta_l$ . Let us now consider two possible events: the good event which considers the case where for all episodes $l$ , the true POMDP is contained in the confidence sets $\mathcal{C}_l(\delta_l)$ and the failure event which denotes the complementary event. + +By setting the confidence level used for the $l$ -th episode as $\delta_l \coloneqq \frac{\delta}{3SAl^3}$ , the probability of the failure event can now be bounded as: + +$$ +P \left(\mathcal {Q} \notin \mathcal {C} _ {l} (\delta_ {l}), \text {f o r s o m e l}\right) \leqslant \sum_ {l = 1} ^ {L - 1} 3 S A \delta_ {l} = \sum_ {l = 1} ^ {L - 1} 3 S A \frac {\delta}{3 S A l ^ {3}} = \sum_ {l = 1} ^ {L - 1} \frac {\delta}{l ^ {3}} \leqslant \frac {3}{2} \delta , \tag {52} +$$ + +From the result above, we can observe that the good event holds with probability at least $1 - \frac{3}{2}\delta$ . When this is the case, we have that $\rho^{*} \leqslant \rho^{l}$ for any $l$ since the optimal average reward is taken from the optimistic POMDP $\mathcal{Q}_l$ . + +We can now bound the regret under the good event during the different $L$ episodes as: + +$$ +\begin{array}{l} \sum_ {t = 0} ^ {T - 1} \left(\rho^ {*} - g (b _ {t})\right) \leqslant 2 L + \sum_ {l = 0} ^ {L - 1} \sum_ {t \in E _ {l}} \left(\rho^ {*} - g (b _ {t})\right) \\ \leqslant 2 L + \left(T _ {0} - 2\right) + \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(\rho^ {l} - g \left(b _ {t}\right)\right) \\ = 2 L + \sum_ {a \in \mathcal {A}} n _ {0} ^ {(a)} + \underbrace {\sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left[ \rho^ {l} - g _ {l} \left(b _ {t} ^ {l}\right) \right] + \left[ g _ {l} \left(b _ {t} ^ {l}\right) - g \left(b _ {t}\right) \right]} _ {(\Psi)}, \tag {53} \\ \end{array} +$$ + +where we have rewritten the summation by highlighting the different $L$ episodes. In particular, for each episode $l$ we use interval $E_{l}$ that excludes the first and the last timestamp of that episode, while the term $2L$ appearing in the first inequality is obtained by assuming to pay maximum regret for each pair of samples not contained in each $E_{l}$ . + +In the second inequality instead, we explicit the length $T_0$ of the first episode for which we assume to pay maximum regret: the $-2$ term is due to the fact that the first and the last timestamps of the first episode are already counted in the $2L$ term. Finally, the last equality expresses the length of the first episode as the sum of the counts of the chosen actions, and adds and subtracts the quantity $g_l(b_t^l) \coloneqq r^\top \mathbb{O}_l b_t^l$ . + +For what will follow, we will focus on the term $\Psi$ . + +# Analysis of $(\Psi)$ + +Let us restate the term $\Psi$ defined above. + +$$ +\left(\Psi\right) := \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left[ \rho^ {l} - g _ {l} \left(b _ {t} ^ {l}\right) \right] + \left[ g _ {l} \left(b _ {t} ^ {l}\right) - g \left(b _ {t}\right) \right] = \underbrace {\sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left[ \rho^ {l} - g _ {l} \left(b _ {t} ^ {l}\right) \right]} _ {\text {F i r s t T e r m}} + \underbrace {\sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left[ g _ {l} \left(b _ {t} ^ {l}\right) - g \left(b _ {t}\right) \right]} _ {\text {S e c o n d T e r m}} \tag {54} +$$ + +We will now focus on analyzing the first and the second term separately. + +Analysis of the First Term of $\Psi$ (line 54) Let us use the Bellman equation reported in Equation (2) for the optimistic belief MDP, and the definition of the probability distribution $U$ over the next belief defined in the Notation section. The following relations hold: + +$$ +\begin{array}{l} \rho^ {l} + v _ {l} \left(b _ {t} ^ {l}\right) = g _ {l} \left(b _ {t} ^ {l}\right) + \int_ {b _ {t + 1} \in \mathcal {B}} v _ {l} \left(b _ {t + 1}\right) U _ {l} \left(d b _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \\ = g _ {l} \left(b _ {t} ^ {l}\right) + \left\langle U _ {l} \left(\cdot \mid b _ {t} ^ {l}, a _ {t}\right), v _ {l} (\cdot) \right\rangle . \\ \end{array} +$$ + +The equation above allows us to write that: + +$$ +\begin{array}{l} \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(\rho^ {l} - g _ {l} \left(b _ {t} ^ {l}\right)\right) = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(- v _ {l} \left(b _ {t} ^ {l}\right) + \left\langle U _ {l} \left(\cdot \mid b _ {t} ^ {l}, a _ {t}\right), v _ {l} (\cdot) \right\rangle\right) \\ = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \underbrace {\left(- v _ {l} \left(b _ {t} ^ {l}\right) + \langle U (\cdot | b _ {t} ^ {l} , a _ {t}) , v _ {l} (\cdot) \rangle\right)} _ {(a)} + \underbrace {\left(\langle U _ {l} (\cdot | b _ {t} ^ {l} , a _ {t}) - U (\cdot | b _ {t} ^ {l} , a _ {t}) , v _ {l} (\cdot) \rangle\right)} _ {(b)}, \tag {55} \\ \end{array} +$$ + +where the first equality is obtained from the Bellman Equation, while the last equality derives from adding and subtracting the term $\langle U(\cdot |b_t^l,a_t),v_l(\cdot)\rangle$ for each time step $t$ . We recall that $U(\cdot |b_t^l,a_t)$ defines the probability distribution over the belief at the next step $t + 1$ under the true POMDP instance $\mathcal{Q}$ , while $U_{l}(\cdot |b_{t}^{l},a_{t})$ represents this probability distribution under the optimistic instance $\mathcal{Q}_l$ . For the term $(a)$ in 55, we have: + +$$ +\begin{array}{l} (a) = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(- v _ {l} \left(b _ {t} ^ {l}\right) + \left\langle U (\cdot \mid b _ {t} ^ {l}, a _ {t}), v _ {l} (\cdot) \right\rangle\right) \tag {56} \\ = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(- v _ {l} \left(b _ {t} ^ {l}\right) + v _ {l} \left(b _ {t + 1} ^ {l}\right)\right) + \left(- v _ {l} \left(b _ {t + 1} ^ {l}\right) + \langle U (\cdot | b _ {t} ^ {l}, a _ {t}), v _ {l} (\cdot) \rangle\right) \\ = \underbrace{\sum_{l = 1}^{L - 1}\Big(-v_{l}(b_{s_{l}}^{l}) + v_{l}(b_{e_{l} + 1}^{l})\Big)}_{(a.1)} + \underbrace{\sum_{l = 1}^{K - 1}\sum_{t\in E_{l}}\mathbb{E}[v_{l}(b_{t + 1}^{l})|\mathcal{F}_{t}] - v_{l}(b_{t + 1}^{l})}_{(a.2)}, \\ \end{array} +$$ + +where the term $(a.1)$ is obtained by observing that the sum on the first line reduces to a telescopic summation. For each episode $l$ , the terms appearing in this summation are respectively the difference between the value of the bias function of the belief in the first timestamp (denoted $s_l$ ) and the last plus one (denoted $e_l + 1$ ) timestamp appearing in $E_l$ . + +The term (a.2) is instead obtained by observing that: + +$$ +\langle U (\cdot | b _ {t} ^ {l}, a _ {t}), v _ {l} (\cdot) \rangle = \int_ {b _ {t + 1} \in \mathcal {B}} v _ {l} (d b _ {t + 1}) U (d b _ {t + 1} | b _ {t} ^ {l}, a _ {t}) = \mathbb {E} [ v _ {l} (b _ {t + 1} ^ {l} | b _ {t} ^ {l}) ] = \mathbb {E} [ v _ {l} (b _ {t + 1} ^ {l}) | \mathcal {F} _ {t} ]. +$$ + +By using Proposition G.1, we can easily see that the span of the bias function defined as $\text{span}(v_l) \coloneqq \max_{b \in \mathcal{B}} v_l(b) - \min_{b \in \mathcal{B}} v_l(b)$ can be bounded by $D/2$ with $D$ being a finite quantity. Hence, we can write that: + +$$ +(a. 1) = \sum_ {l = 1} ^ {L - 1} - v _ {l} \left(b _ {s _ {l}} ^ {l}\right) + v _ {l} \left(b _ {e _ {l} + 1} ^ {l}\right) \leqslant \sum_ {l = 1} ^ {L - 1} D = (L - 1) D. \tag {57} +$$ + +For the term (a.2), we can observe that it defines a martingale. By applying analogous results as those used for bounding 50, we get with probability at least $1 - \delta /4$ that: + +$$ +(a. 2) = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \mathbb {E} \left[ v _ {l} \left(b _ {t + 1} ^ {l}\right) \mid \mathcal {F} _ {t} \right] - v _ {l} \left(b _ {t + 1} ^ {l}\right) \leqslant D \sqrt {2 T \log \left(\frac {4}{\delta}\right)}. \tag {58} +$$ + +By combining the bounds for (a.1) and (a.2), we obtain with probability at least $1 - \delta / 4$ : + +$$ +(a) = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(- v _ {l} \left(b _ {t} ^ {l}\right) + \left\langle U (\cdot | b _ {t} ^ {l}, a _ {t}), v _ {l} (\cdot) \right\rangle\right) \leqslant (L - 1) D + D \sqrt {2 T \log \left(\frac {4}{\delta}\right)}. \tag {59} +$$ + +We can now proceed in bounding the term $(b)$ appearing in 55. Let us recall the definition of the function $H(b_{t},a_{t},o_{t + 1})$ and $P(o_{t + 1}|b_t,a_t)$ defined in the Notation section. The following relations hold: + +$$ +\begin{array}{l} \langle U _ {l} \left(\cdot \mid b _ {t} ^ {l}, a _ {t}\right) - U \left(\cdot \mid b _ {t} ^ {l}, a _ {t}\right), v _ {l} (\cdot) \rangle \tag {60} \\ \leqslant \left| \int_ {\mathcal {B}} v _ {l} \left(d b ^ {\prime}\right) U _ {l} \left(d b ^ {\prime} \mid b _ {t} ^ {l}, a _ {t}\right) - \int_ {\mathcal {B}} v _ {l} \left(b ^ {\prime}\right) U \left(d b ^ {\prime} \mid b _ {t} ^ {l}, a _ {t}\right) \right| \\ = \left| \sum_ {o _ {t + 1} \in \mathcal {O}} v _ {l} \left(H _ {l} \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) P _ {l} \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) - \sum_ {o _ {t + 1} \in \mathcal {O}} v _ {l} \left(H \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \right| \\ \leqslant \underbrace {\left| \sum_ {o _ {t + 1} \in \mathcal {O}} \left[ v _ {l} \left(H _ {l} \left(b _ {t} ^ {l} , a _ {t} , o _ {t + 1}\right)\right) - v _ {l} \left(H \left(b _ {t} ^ {l} , a _ {t} , o _ {t + 1}\right)\right) \right] P \left(o _ {t + 1} \mid b _ {t} ^ {l} , a _ {t}\right) \right|} _ {(b. 1)} + \\ + \underbrace {\left| \sum_ {o _ {t + 1} \in \mathcal {O}} v _ {l} \left(H _ {l} \left(b _ {t} ^ {l} , a _ {t} , o _ {t + 1}\right)\right) \left[ P _ {l} \left(o _ {t + 1} \mid b _ {t} ^ {l} , a _ {t}\right) - P \left(o _ {t + 1} \mid b _ {t} ^ {l} , a _ {t}\right) \right] \right|}. _ {(b. 2)} \\ \end{array} +$$ + +where in the first equality we have decoupled the stochasticity induced by the observation from the deterministic update of the belief $b'$ at the next step through the $H$ and $H_{l}$ functions. Let us now analyze the different terms separately. + +$$ +\begin{array}{l} (b. 1) = \left| \sum_ {o _ {t + 1} \in \mathcal {O}} \left[ v _ {l} \left(H _ {l} \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) - v _ {l} \left(H \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) \right] P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \right| \\ \leqslant \sum_ {o _ {t + 1} \in \mathcal {O}} \left| v _ {l} \left(H _ {l} \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) - v _ {l} \left(H \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) \right| P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \\ \leqslant \sum_ {o _ {t + 1} \in \mathcal {O}} \frac {D}{2} \left| H _ {l} \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right) - H \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right) \right| P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \\ \end{array} +$$ + +(Holder's inequality and Proposition G.1) + +$$ +\begin{array}{l} \leqslant \sum_ {o _ {t + 1} \in \mathcal {O}} \frac {D}{2} \left(C _ {2} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + C _ {3} \| \mathbb {T} _ {a _ {t}, l} - \mathbb {T} _ {a _ {t}} \| _ {F}\right) P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \quad (Corollary F.2) \\ = \frac {D}{2} \left(C _ {2} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + C _ {3} \| \mathbb {T} _ {a _ {t}, l} - \mathbb {T} _ {a _ {t}} \| _ {F}\right), (61) \\ \end{array} +$$ + +The last inequality is instead obtained from Corollary F.2 which bounds the one-step error of the belief vector when updated using the estimated observation and transition matrices. Constants $C_2$ and $C_3$ are instead defined in Lemma F.1. + +Concerning the term (b.2), we have: + +$$ +\begin{array}{l} (b. 2) = \left| \sum_ {o _ {t + 1} \in \mathcal {O}} v _ {l} \left(H _ {l} \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) \left[ P _ {l} \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) - P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \right] \right| \\ \leqslant \sum_ {o _ {t + 1} \in \mathcal {O}} \left| v _ {l} \left(H _ {l} \left(b _ {t} ^ {l}, a _ {t}, o _ {t + 1}\right)\right) \left[ P _ {l} \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) - P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \right] \right| \\ \leqslant \frac {D}{2} \sum_ {o _ {t + 1} \in \mathcal {O}} \left| P _ {l} \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) - P \left(o _ {t + 1} \mid b _ {t} ^ {l}, a _ {t}\right) \right| \tag {Proposition G.1} \\ = \frac {D}{2} \left\| \left(\mathbb {O} _ {l} \mathbb {T} _ {a _ {t}, l} ^ {\top} - \mathbb {O} \mathbb {T} _ {a _ {t}} ^ {\top}\right) b _ {t} ^ {l} \right\| _ {1} \\ \leqslant \frac {D}{2} \| \mathbb {O} _ {l} \mathbb {T} _ {a _ {t}, l} ^ {\top} - \mathbb {O} \mathbb {T} _ {a _ {t}} ^ {\top} \| _ {1} \| b _ {t} ^ {l} \| _ {1} \\ \leqslant \frac {D}{2} \left(\| \mathbb {O} _ {l} \left(\mathbb {T} _ {a _ {t}, l} ^ {\top} - \mathbb {T} _ {a _ {t}} ^ {\top}\right) \| _ {1} + \| (\mathbb {O} _ {l} - \mathbb {O}) \mathbb {T} _ {a _ {t}} ^ {\top} \| _ {1}\right) \\ \leqslant \frac {D}{2} \left(\| \mathbb {O} _ {l} \| _ {1} \| \mathbb {T} _ {a _ {t}, l} ^ {\top} - \mathbb {T} _ {a _ {t}} ^ {\top} \| _ {1} + \| \mathbb {O} _ {l} - \mathbb {O} \| _ {1} \| \mathbb {T} _ {a _ {t}} ^ {\top} \| _ {1}\right) \quad (\text {D e f . o f M a t r i x N o r m s}) \\ \leqslant \frac {D}{2} \left(\| \mathbb {T} _ {a _ {t}, l} ^ {\top} - \mathbb {T} _ {a _ {t}} ^ {\top} \| _ {1} + \| \mathbb {O} _ {l} - \mathbb {O} \| _ {1}\right) \quad (\text {S i n c e} \| \mathbb {O} _ {l} \| _ {1} = 1 \text {a n d} \| \mathbb {T} _ {a _ {t}} ^ {\top} \| _ {1} = 1) \\ = \frac {D}{2} \left(\| \mathbb {T} _ {a _ {t}, l} - \mathbb {T} _ {a _ {t}} \| _ {\infty} + \| \mathbb {O} _ {l} - \mathbb {O} \| _ {1}\right) \\ = \frac {D}{2} \left(\sqrt {S} \| \mathbb {T} _ {a _ {t}, l} - \mathbb {T} _ {a _ {t}} \| _ {F} + \sqrt {O} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F}\right). \\ \end{array} +$$ + +By combining the results obtained for (b.1) and (b.2), we are able to bound the term (b) as: + +$$ +(b) = \frac {D}{2} \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(\left(C _ {2} + \sqrt {O}\right) \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + \left(C _ {3} + \sqrt {S}\right) \| \mathbb {T} _ {a _ {t}, l} - \mathbb {T} _ {a _ {t}} \| _ {F}\right). \tag {62} +$$ + +Finally, we can combine the results defined in lines 59 and 62 on $(a)$ and $(b)$ to finally bound the first term of $\Psi$ (line 54) and obtain with probability at least $1 - \delta /4$ : + +$$ +\begin{array}{l} \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(\rho^ {l} - g _ {l} \left(b _ {t} ^ {l}\right)\right) \leqslant (L - 1) D + D \sqrt {2 T \ln \left(\frac {4}{\delta}\right)} + (63) \\ + \frac {D}{2} \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(\left(C _ {2} + \sqrt {O}\right) \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + \left(C _ {3} + \sqrt {S}\right) \| \mathbb {T} _ {a _ {t}, l} - \mathbb {T} _ {a _ {t}} \| _ {F}\right) \\ \leqslant (L - 1) D + D \sqrt {2 T \ln \left(\frac {4}{\delta}\right)} + \frac {D \left(C _ {2} + \sqrt {O}\right)}{2} \sum_ {l = 1} ^ {L - 1} n _ {l} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + \\ + \frac {D \left(C _ {3} + \sqrt {S}\right)}{2} \sum_ {l = 1} ^ {L - 1} \sum_ {a \in \mathcal {A}} n _ {l} ^ {(a)} \| \mathbb {T} _ {a, l} - \mathbb {T} _ {a} \| _ {F}, (64) \\ \end{array} +$$ + +where we used that $n_l = |E_l|$ denotes the cardinality of the interval $E_l$ , and we also recall that $\sum_{a \in \mathcal{A}} n_l^{(a)} = n_l$ . + +# Analysis of the Second Term of $\Psi$ (line 54) + +We can now focus on the second term appearing in the summation of 54. We have that: + +$$ +\begin{array}{l} \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(g _ {l} \left(b _ {t} ^ {l}\right) - g \left(b _ {t}\right)\right) = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \boldsymbol {r} ^ {\top} \mathbb {O} _ {l} b _ {t} ^ {l} - \boldsymbol {r} ^ {\top} \mathbb {O} b _ {t} \\ \leqslant \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \| \boldsymbol {r} ^ {\top} \| _ {\infty} \| \mathbb {O} _ {l} b _ {t} ^ {l} - \mathbb {O} b _ {t} \| _ {1} \quad (\| \boldsymbol {r} ^ {\top} \| _ {\infty} \leqslant 1) \\ \leqslant \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \| \mathbb {O} _ {l} b _ {t} ^ {l} - \mathbb {O} b _ {t} ^ {l} + \mathbb {O} b _ {t} ^ {l} - \mathbb {O} b _ {t} \| _ {1} \\ \leqslant \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \| (\mathbb {O} _ {l} - \mathbb {O}) b _ {t} ^ {l} \| _ {1} + \| \mathbb {O} (b _ {t} ^ {l} - b _ {t}) \| _ {1} \\ \leqslant \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {1} \| b _ {t} ^ {l} \| _ {1} + \| \mathbb {O} \| _ {1} \| b _ {t} ^ {l} - b _ {t} \| _ {1} \quad (\text {D e f . o f M a t r i x N o r m s}) \\ = \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {1} + \| b _ {t} ^ {l} - b _ {t} \| _ {1} \quad (\text {S i n c e} \| b _ {t} ^ {l} \| _ {1} = 1 \text {a n d} \| \mathbb {O} \| _ {1} = 1) \\ \leqslant \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \sqrt {O} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + \| b _ {t} ^ {l} - b _ {t} \| _ {1} \\ = \sum_ {l = 1} ^ {L - 1} n _ {l} \sqrt {O} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + \sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \| b _ {t} ^ {l} - b _ {t} \| _ {1}. \\ \end{array} +$$ + +Let us now consider the last term appearing in the last inequality. It can be bounded by using the result appearing in Lemma F.1. In particular, we have: + +$$ +\sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \| b _ {t} ^ {l} - b _ {t} \| _ {1} \leqslant \sum_ {l = 1} ^ {L - 1} \left[ C _ {1} + C _ {2} n _ {l} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + C _ {3} \sum_ {a \in \mathcal {A}} n _ {l} ^ {(a)} \| \mathbb {T} _ {a, l} - \mathbb {T} _ {a} \| _ {F} \right], +$$ + +with constants $C_1, C_2$ and $C_3$ defined in Lemma F.1. + +From the results above, we obtain the following result for the second term of $\Psi$ : + +$$ +\sum_ {l = 1} ^ {L - 1} \sum_ {t \in E _ {l}} \left(g _ {l} \left(b _ {t} ^ {l}\right) - g \left(b _ {t}\right)\right) \leqslant (L - 1) C _ {1} + \left(C _ {2} + 1\right) \sum_ {l = 1} ^ {L - 1} n _ {l} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + C _ {3} \sum_ {l = 1} ^ {L - 1} \sum_ {a \in \mathcal {A}} n _ {l} ^ {(a)} \| \mathbb {T} _ {a, l} - \mathbb {T} _ {a} \tag {65} +$$ + +# Merge of Obtained Results and Final Bound + +Let us recall the definition of the regret in line 49 and let us observe that it can be bounded using the bound on the martingale in line 50 and the bound on line 53. We have just seen how line 64 and 65 allow us to bound the term $\Psi$ in 53. By combining everything, we get: + +$$ +\begin{array}{l} \mathcal {R} _ {T} \leqslant \sum_ {t = 0} ^ {T - 1} \left(\rho^ {*} - g \left(b _ {t}\right)\right) + \sqrt {2 T \log (4 / \delta)} \\ \leqslant 2 L + \sum_ {a \in \mathcal {A}} n _ {0} ^ {(a)} + (L - 1) (D + C _ {1}) + \sqrt {2 T \log \left(\frac {4}{\delta}\right)} + D \sqrt {2 T \log \left(\frac {4}{\delta}\right)} + \\ + \frac {D (C _ {2} + \sqrt {O}) + 2 (C _ {2} + 1)}{2} \sum_ {l = 1} ^ {L - 1} n _ {l} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} + \frac {2 C _ {3} + D (C _ {3} + \sqrt {S})}{2} \sum_ {l = 1} ^ {L - 1} \sum_ {a \in \mathcal {A}} n _ {l} ^ {(a)} \| \mathbb {T} _ {a, l} - \mathbb {T} _ {a} \| _ {F}, \\ \leqslant 2L + \underbrace{\sum_{a\in\mathcal{A}}n_{0}^{(a)}}_{(c)} + (L - 1)\left(D + C_{1}\right) + \sqrt{2T\log\left(\frac{4}{\delta}\right)} +D\sqrt{2T\log\left(\frac{4}{\delta}\right)} + \\ + \frac {D C _ {2} \sqrt {O}}{2} \underbrace {\sum_ {l = 1} ^ {L - 1} n _ {l} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F}} _ {(d)} + \frac {D C _ {3} \sqrt {S}}{2} \underbrace {\sum_ {l = 1} ^ {L - 1} \sum_ {a \in \mathcal {A}} n _ {l} ^ {(a)} \| \mathbb {T} _ {a , l} - \mathbb {T} _ {a} \| _ {F}} _ {(e)}, \tag {66} \\ \end{array} +$$ + +Let us now focus on the quantities appearing in $(c)$ and $(d)$ . We have: + +(Lemma G.2) + +$$ +\begin{array}{l} (c) + (d) = \sum_ {a \in \mathcal {A}} n _ {0} ^ {(a)} + \sum_ {l = 1} ^ {L - 1} n _ {l} \| \mathbb {O} _ {l} - \mathbb {O} \| _ {F} \leqslant n _ {0} + \sum_ {l = 1} ^ {L - 1} n _ {l} \frac {C _ {\mathbb {O}}}{\zeta^ {(l)}} \sqrt {\frac {S A l \log (l O / \delta_ {l})}{N _ {l}}} (Theorem 5.4) \\ \leqslant \sum_ {l = 0} ^ {L - 1} n _ {l} \frac {C _ {\mathbb {O}}}{\zeta^ {(l)}} \sqrt {\frac {S A l \log (3 S A l ^ {4} O / \delta)}{\max \{1 , N _ {l} \}}} \\ \left(\text {F r o m} \delta_ {l} := \frac {\delta}{3 S A l ^ {3}}\right) \\ \leqslant \frac {C _ {\mathbb {O}}}{\tilde {\zeta} ^ {(L)}} \sqrt {S A L \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} \sum_ {l = 0} ^ {L - 1} n _ {l} \sqrt {\frac {1}{\max \{1 , N _ {l} \}}} (67) \\ \leqslant \frac {C _ {\mathbb {O}}}{\tilde {\zeta} ^ {(L)}} \sqrt {S A L \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} (\sqrt {2} + 1) \sqrt {N _ {L}} \\ \leqslant \frac {C _ {\mathbb {O}} (\sqrt {2} + 1)}{\widetilde {\zeta} ^ {(L)}} \sqrt {S A L T \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} (68) \\ \end{array} +$$ + +where for the first term of the inequality on the first line we used $\sum_{a\in \mathcal{A}}n_0^{(a)} = n_0$ and Theorem 5.4. Here, we recall that $N_{l}$ represents the number of samples used for the model estimation for the $l$ -th + +episode. In line 67 we defined $\widetilde{\zeta}^{(L)}\coloneqq \min_l\zeta^{(l)}$ , while the last line simply follows by observing that $N_{L}\leqslant T$ . + +We can apply similar considerations to bound the term $(e)$ . In particular: + +$$ +\begin{array}{l} (e) = \sum_ {l = 1} ^ {L - 1} \sum_ {a \in \mathcal {A}} n _ {l} ^ {(a)} \| \mathbb {T} _ {a, l} - \mathbb {T} _ {a} \| _ {F} \\ \leqslant \sum_ {l = 1} ^ {L - 1} \sum_ {a \in \mathcal {A}} n _ {l} ^ {(a)} \frac {C _ {\mathbb {T}} S}{\sigma_ {S} (\mathbb {O}) \zeta^ {(l)}} \sqrt {\frac {A l \log (l O / \delta_ {l})}{N _ {l} ^ {(a)}}} (Theorem 5.4) \\ \leqslant \frac {C _ {\mathbb {T}} S}{\sigma_ {S} (\mathbb {O}) \widetilde {\zeta} ^ {(L)}} \sqrt {A L \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} \sum_ {a \in \mathcal {A}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \sqrt {\frac {1}{\max \{1 , N _ {l} ^ {(a)} \}}} (69) \\ \leqslant \frac {C _ {\mathbb {T}} S (\sqrt {2} + 1)}{\sigma_ {S} (\mathbb {O}) \widetilde {\zeta} ^ {(L)}} \sqrt {A L \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} \sum_ {a \in \mathcal {A}} \sqrt {N _ {L} ^ {(a)}} \quad (Lemma G.2 for each a) \\ \leqslant \frac {C _ {\mathbb {T}} S (\sqrt {2} + 1)}{\sigma_ {S} (\mathbb {O}) \widetilde {\zeta} ^ {(L)}} \sqrt {A L \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} \sqrt {A N _ {L}} \quad \text {(C a u c h y - S c h w a r z i n e q u a l i t y)} \\ \leqslant \frac {C _ {\mathbb {T}} S A (\sqrt {2} + 1)}{\sigma_ {S} (\mathbb {O}) \widetilde {\zeta} ^ {(L)}} \sqrt {L T \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} (70) \\ \end{array} +$$ + +where the last but one inequality follows by recalling that $N_{L} = \sum_{a\in \mathcal{A}}N_{L}^{(a)}$ + +From the result obtained in 68 and 70, we rewrite the bound on the regret reported in line 66 as: + +$$ +\begin{array}{l} \mathcal {R} _ {T} \leqslant 2 L + (L - 1) (D + C _ {1}) + \sqrt {2 T \log \left(\frac {4}{\delta}\right)} + D \sqrt {2 T \log \left(\frac {4}{\delta}\right)} + \\ + \frac {3 D \sqrt {O} C _ {2} C _ {\mathbb {O}}}{2 \tilde {\zeta} ^ {(L)}} \sqrt {S A L T \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)} + \frac {3 D S ^ {3 / 2} A C _ {3} C _ {\mathbb {T}}}{2 \sigma_ {S} (\mathbb {O}) \tilde {\zeta} ^ {(L)}} \sqrt {L T \log \left(\frac {3 S A L ^ {4} O}{\delta}\right)}, \\ \end{array} +$$ + +holding with probability at least $1 - 2\delta$ , obtained by using a union bound on the bound of the two martingales (each one holding with probability at least $1 - \delta / 4$ ) and on the bound of the optimistic model which holds with probability at least $1 - (3/2)\delta$ , as reported in Eq. (52). + +The last step of the proof consists in observing that, for the stopping condition employed by the algorithm, the number of total episodes can be bounded as $L \leqslant A \log (T / A)$ . Finally, the regret expression can be simplified by highlighting the dependencies on the main terms as follows: + +$$ +\mathcal {R} _ {T} \leqslant \mathcal {O} \left(\frac {D (S A) ^ {3 / 2}}{\sigma_ {S} (\mathbb {O}) \tilde {\zeta} ^ {(L)}} \sqrt {T O \log^ {2} \left(\frac {S A O T}{\delta}\right)}\right). +$$ + +This final step concludes the proof. + +# D Auxiliary Results for the Proof of Theorem 5.4 + +In this section, we will provide auxiliary results required for the proof of Theorem 5.4. They are based on previous results on learning Hidden Markov Models (HMM) and POMDPs by [1] and [3]. We carefully adapt the results to the Mixed Spectral Estimation strategy presented in Algorithm 1. + +Lemma D.1 (Error Bound of $\pmb{\mu}_{2,s}^{(a,L)}$ ). Let $\hat{V}_2^{(a,L)}$ be the second view estimated using Algorithm 1 when the set of policies $\{\pi_l\}_{l=0}^{L-1}$ is used to interact with the environment, and let $\hat{\pmb{\mu}}_{2,s}^{(a,L)} \in \Delta(\mathcal{O})$ be its $s$ -th column. If $N_L^{(a)}$ satisfies the conditions in Equation (95) and (102), then with probability at + +least $1 - 3\delta$ we have: + +$$ +\left\| \boldsymbol {\mu} _ {2, s} ^ {(a, L)} - \hat {\boldsymbol {\mu}} _ {2, s} ^ {(a, L)} \right\| _ {2} \leqslant \frac {1 6 \epsilon_ {M} ^ {(a , L)}}{\sigma_ {S} \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right)}, +$$ + +with $\epsilon_{M}^{(a,L)}$ defined as in Equation (98) of Lemma D.4. + +Proof. Let us recall that each column $\pmb{\mu}_{2,s}^{(a,L)}$ of the second view matrix $V_{2}^{(a,L)}$ can be obtained from $\pmb{\mu}_{3}^{(a,L)}$ by inverting Equation (6). We can thus write the following: + +$$ +\begin{array}{l} \left\| \boldsymbol {\mu} _ {2, s} ^ {(a, L)} - \hat {\boldsymbol {\mu}} _ {2, s} ^ {(a, L)} \right\| _ {2} = \left\| K _ {2, 1} ^ {(a, L)} \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {\dagger} \boldsymbol {\mu} _ {3, s} ^ {(a, L)} - \widehat {\boldsymbol {K}} _ {2, 1} ^ {(a, L)} \left(\widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)}\right) ^ {\dagger} \hat {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} \right\| _ {2} (71) \\ \leqslant \left\| \boldsymbol {K} _ {2, 1} ^ {(a, L)} - \widehat {\boldsymbol {K}} _ {2, 1} ^ {(a, L)} \right\| _ {2} \left\| \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {\dagger} \right\| _ {2} \left\| \boldsymbol {\mu} _ {3, s} ^ {(a, L)} \right\| _ {2} + \\ + \left\| \boldsymbol {K} _ {2, 1} ^ {(a, L)} \right\| _ {2} \left\| \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {\dagger} - \left(\widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)}\right) ^ {\dagger} \right\| _ {2} \left\| \boldsymbol {\mu} _ {3, s} ^ {(a, L)} \right\| _ {2} + \\ + \left\| \boldsymbol {K} _ {2, 1} ^ {(a, L)} \right\| _ {2} \left\| \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {\dagger} \right\| _ {2} \left\| \boldsymbol {\mu} _ {3, s} ^ {(a, L)} - \hat {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} \right\| _ {2}. (72) \\ \end{array} +$$ + +The terms in 72 can be bounded by using i) Lemma D.3 for the concentration bound of empirical estimates for $\left\| K_{2,1}^{(a,L)} - \widehat{K}_{2,1}^{(a,L)}\right\|_2$ , ii) Proposition D.5 for $\left\| \left(K_{3,1}^{(a,L)}\right)^{\dagger} - \left(\widehat{K}_{3,1}^{(a,L)}\right)^{\dagger}\right\|_2$ , iii) Lemma D.2 for $\left\| \pmb{\mu}_{3,s}^{(a,L)} - \widehat{\pmb{\mu}}_{3,s}^{(a,L)}\right\|_2$ , iv) $\left\| K_{2,1}^{(a,L)}\right\|_2 \leqslant 1$ , v) $\left\| \left(K_{3,1}^{(a,L)}\right)^{\dagger}\right\|_2 \leqslant 1 / \sigma_S(K_{3,1}^{(a,L)})$ and vi) $\left\| \pmb{\mu}_{3,s}^{(a,L)}\right\|_2 \leqslant 1$ . Thus we have: + +$$ +\begin{array}{l} \| \boldsymbol {\mu} _ {2, s} ^ {(a, L)} - \hat {\boldsymbol {\mu}} _ {2, s} ^ {(a, L)} \| _ {2} \leqslant \frac {\widetilde {G}}{\sigma_ {S} (\boldsymbol {K} _ {3 , 1} ^ {(a , L)}) (1 - \widetilde {\eta})} \sqrt {\frac {8 L \log (2 O L / \delta)}{N _ {L} ^ {(a)}}} + \\ + \frac {2 \tilde {G}}{\left[ \sigma_ {S} \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right) \right] ^ {2} (1 - \tilde {\eta})} \sqrt {\frac {8 L \log (2 O L / \delta)}{N _ {L} ^ {(a)}}} + \frac {1 4 \epsilon_ {M} ^ {(a , L)}}{\sigma_ {S} \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right)} \\ \leqslant \frac {1 6 \epsilon_ {M} ^ {(a , L)}}{\sigma_ {S} (\boldsymbol {K} _ {3 , 1} ^ {(a , L)})}, \\ \end{array} +$$ + +holding with probability at least $1 - 3\delta$ . The last inequality follows from observing that each of the first two terms is $\leqslant \epsilon_{M}^{(a,L)} / \sigma_{S}(K_{3,1}^{(a,L)})$ . + +Lemma D.2 (Error Bound for $\pmb{\mu}_{3,s}^{(a,L)}$ ). Let $\widehat{V}_3^{(a,L)}$ be the third view estimated in Algorithm 1 when the set $\{\pi_l\}_{l = 0}^{L - 1}$ of policies is used to interact with the environment, and let $\widehat{\pmb{\mu}}_{3,s}^{(a,L)}\in \Delta (\mathcal{O})$ be its $s$ -th column. If $N_L^{(a)}$ satisfies the condition in Equation (95) reported in Lemma D.4 then, with probability at least $1 - 2\delta$ , we have: + +$$ +\left\| \boldsymbol {\mu} _ {3, s} ^ {(a, L)} - \hat {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} \right\| _ {2} \leqslant 1 4 \epsilon_ {M} ^ {(a, L)}, +$$ + +with $\epsilon_M^{(a,L)}$ defined as in Equation (98) of Lemma D.4. + +Proof. The theoretical guarantees on the estimation quality of the third view $\widehat{V}_3^{(a,L)}$ are related to the guarantees provided by Spectral Decomposition approaches. + +In past works such as [1] and [30], it has been shown that among the different spectral algorithms, those relying on tensor decomposition are more sample efficient. Our approach relies on the Robust Tensor Power (RTP) method presented in [1], which is applied to the symmetrized and whitened third-order moment tensor. We will now denote the steps required to transform the empirical estimates and + +provide them to the RTP algorithm. The definition of some of the quantities that are used throughout this proof, together with the employed notation, is discussed in Section E. + +Let us consider now the empirical matrices and tensors (without symmetrization) defined as: + +$$ +\widetilde {\boldsymbol {M}} _ {2} ^ {(a, L)} := \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \boldsymbol {v} _ {1} ^ {(a, l)} \otimes \boldsymbol {v} _ {2} ^ {(a, l)} \right] +$$ + +$$ +\widetilde {\boldsymbol {M}} _ {3} ^ {(a, L)} := \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \mathbb {E} \left[ \boldsymbol {v} _ {1} ^ {(a, l)} \otimes \boldsymbol {v} _ {2} ^ {(a, l)} \otimes \boldsymbol {v} _ {3} ^ {(a, l)} \right]. \tag {73} +$$ + +We observe that the definition of the non-symmetrized matrix $\widetilde{M}_2^{(a,L)}$ coincides with the one of $K_{1,2}^{(a,L)}$ . These non-symmetrized versions13 indeed differ from the symmetrized one $M_3^{(a,L)}$ and $M_3^{(a,L)}$ presented in Theorem 5.3. + +Using the multilinear map notation introduced in Section E, we define the symmetrized and whitened tensor as $\widetilde{M}_3^{(a,L)}(W_1^{(a,L)},W_2^{(a,L)},W_3^{(a,L)})\in \mathbb{R}^{S\times S\times S}$ , where $W_{1}^{(a,L)}\in \mathbb{R}^{O\times S}$ , $W_{2}^{(a,L)}\in \mathbb{R}^{O\times S}$ and $W_{3}^{(a,L)}\in \mathbb{R}^{O\times S}$ are the corresponding symmetrization-whitening matrices for each of the tensor dimensions. By using Lemma D.4, it is possible to show that for a sufficient number of samples $N_L^{(a)}$ , the error $\epsilon_M^{(a,L)}$ on the estimated symmetrized and whitened tensor $\widehat{\widetilde{M}}_3^{(a,L)}(\widehat{W}_1^{(a,L)},\widehat{W}_2^{(a,L)},\widehat{W}_3^{(a,L)})$ can be bounded with probability at least $1 - \delta$ as: + +$$ +\epsilon_ {M} ^ {(a, L)} \leqslant \frac {\frac {2 \sqrt {2} \widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log ((O ^ {2} + O) 2 L / \delta)}{N _ {L} ^ {(a)}}}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {3}} + \frac {\left(\frac {\frac {4 \widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log (4 O L / \delta)}{N _ {L} ^ {(a)}}}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {2}}\right) ^ {3}}{\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}}}. \tag {74} +$$ + +From Lemma D.4, we can also observe that when a sufficient number of samples $N_{L}^{(a)}$ is used, the estimation properties of the RTP method are guaranteed. In particular, let us denote with $\left(\widehat{\widetilde{\mu}}_{3,s}^{(a,L)},\widehat{\widetilde{\omega}}_s^{(a,L)}\right)_{s\in S}$ the set of robust eigenvector/eigenvalue pairs provided as output by RTP. Then, from [1], with probability at least $1 - 2\delta$ the following holds: + +$$ +\left\| \widetilde {\boldsymbol {M}} _ {3} ^ {(a, L)} \left(W _ {1} ^ {(a, L)}, W _ {2} ^ {(a, L)}, W _ {3} ^ {(a, L)}\right) - \sum_ {s \in \mathcal {S}} \widehat {\widetilde {\omega}} _ {s} ^ {(a, L)} \left(\widehat {\widetilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)}\right) ^ {\otimes 3} \right\| _ {2} \leqslant 5 5 \epsilon_ {M} ^ {(a, L)}, \tag {75} +$$ + +$$ +\left\| \tilde {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} - \hat {\tilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} \right\| _ {2} \leqslant \frac {8 \epsilon_ {M} ^ {(a , L)}}{\tilde {\omega} _ {s} ^ {(a , L)}}, \quad \left| \tilde {\omega} _ {s} ^ {(a, L)} - \hat {\tilde {\omega}} _ {s} ^ {(a, L)} \right| \leqslant 5 \epsilon_ {M} ^ {(a, L)}. \tag {76} +$$ + +Let us now denote with $\epsilon_3^{(a,L)}\coloneqq \| \pmb{\mu}_{3,s}^{(a,L)} - \widehat{\pmb{\mu}}_{3,s}^{(a,L)}\| _2$ the error of the $s$ -th column of the third view matrix $V_{3}^{(a,L)}$ . + +We recall that in order to obtain the estimate $\widehat{\pmb{\mu}}_{3,s}^{(a,L)}$ from the corresponding robust eigenvector/eigenvalue pair $(\widehat{\pmb{\mu}}_{3,s}^{(a,L)},\widehat{\pmb{\omega}}_s^{(a,L)})$ given as output by RTP, we have to de-whiten vector $\widehat{\pmb{\mu}}_{3,s}^{(a,L)}$ which can be done by the following relation: + +$$ +\widehat {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} = \widehat {\widetilde {\omega}} _ {s} ^ {(a, L)} \widehat {B} \widehat {\widetilde {\mu}} _ {3, s}, +$$ + +where we defined $\widehat{B} \in \mathbb{R}^{O \times S}$ as the Moore-Penrose inverse of $\left(\widehat{W}_3^{(a,L)}\right)^\top$ . The equation above is obtained by inverting the first Equation appearing in (116), which relates the robust eigenvector/eigenvalue pair of the whitened tensor with that of the non-whitened counterpart. + +Let us now analyze the error $\epsilon_3^{(a,L)}$ : + +$$ +\begin{array}{l} \epsilon_ {3} ^ {(a, L)} = \left\| \boldsymbol {\mu} _ {3. s} ^ {(a, L)} - \hat {\boldsymbol {\mu}} _ {3. s} ^ {(a, L)} \right\| _ {2} (77) \\ \leqslant \left\| \widetilde {\omega} _ {s} ^ {(a, L)} B \widetilde {\mu} _ {3, s} ^ {(a, L)} - \widehat {\widetilde {\omega}} _ {s} ^ {(a, L)} \widehat {B} \widehat {\widetilde {\mu}} _ {3, s} ^ {(a, L)} \right\| _ {2} (78) \\ = \left\| \tilde {\omega} _ {s} ^ {(a, L)} B \tilde {\mu} _ {3, s} ^ {(a, L)} - \tilde {\omega} _ {s} ^ {(a, L)} \hat {B} \tilde {\mu} _ {3, s} ^ {(a, L)} + \tilde {\omega} _ {s} ^ {(a, L)} \hat {B} \tilde {\mu} _ {3, s} ^ {(a, L)} - \hat {\tilde {\omega}} _ {s} ^ {(a, L)} \hat {B} \hat {\tilde {\mu}} _ {3, s} ^ {(a, L)} \right\| _ {2} (79) \\ = \underbrace {\left\| \tilde {\omega} _ {s} ^ {(a , L)} \tilde {\mu} _ {3 , s} ^ {(a , L)} \right\| _ {2} \left\| B - \hat {B} \right\| _ {2}} _ {\text {(a)}} + \underbrace {\left\| \hat {B} \right\| _ {2} \left\| \tilde {\omega} _ {s} ^ {(a , L)} \tilde {\mu} _ {3 , s} ^ {(a , L)} - \hat {\tilde {\omega}} _ {s} ^ {(a , L)} \hat {\tilde {\mu}} _ {3 , s} ^ {(a , L)} \right\| _ {2}} _ {\text {(b)}}. (80) \\ \end{array} +$$ + +We can bound the error of each term separately. Let us start with (a). For the first term of (a), we have: + +$$ +\left\| \widetilde {\omega} _ {s} ^ {(a, L)} \widetilde {\mu} _ {3, s} ^ {(a, L)} \right\| _ {2} \leqslant \widetilde {\omega} _ {s} ^ {(a, L)} \left\| \widetilde {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} \right\| _ {2} = \widetilde {\omega} _ {s} ^ {(a, L)} = \frac {1}{\sqrt {\omega_ {s} ^ {(a , L)}}}, \tag {81} +$$ + +where the first equality follows from the fact that $\widetilde{\mu}_{3,s}^{(a,L)}$ is a unit vector, while the last equality follows from the definition in Equation (116) linking the original eigenvalue $\omega_s^{(a,L)}$ with the one of the whitened tensor $\widetilde{\omega}_s^{(a,L)}$ . For the second term of (a), we have: + +$$ +\left\| B - \hat {B} \right\| _ {2} \leqslant \frac {4 \| \widetilde {\boldsymbol {M}} _ {2} ^ {(a , L)} - \widehat {\widetilde {\boldsymbol {M}}} _ {2} ^ {(a , L)} \| _ {2}}{\omega_ {\min } ^ {(a , L)} \left[ \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right) \right] ^ {2}}, \tag {82} +$$ + +where the result directly follows from Equation (112) in Proposition D.8. + +Let us now consider the term (b). We have: + +$$ +\begin{array}{l} \left(\mathbf {b}\right) = \left\| \widehat {B} \right\| _ {2} \left\| \widetilde {\omega} _ {s} ^ {(a, L)} \widetilde {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} - \widehat {\widetilde {\omega}} _ {s} ^ {(a, L)} \widehat {\widetilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} \right\| _ {2} (83) \\ \leqslant \left\| \widetilde {\omega} _ {s} ^ {(a, L)} \tilde {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} - \widehat {\tilde {\omega}} _ {s} ^ {(a, L)} \widehat {\tilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} \right\| _ {2} (84) \\ \leqslant \left\| \widetilde {\omega} _ {s} ^ {(a, L)} \widetilde {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} - \widetilde {\omega} _ {s} ^ {(a, L)} \widehat {\widetilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} + \widetilde {\omega} _ {s} ^ {(a, L)} \widehat {\widetilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} - \widehat {\widetilde {\omega}} _ {s} ^ {(a, L)} \widehat {\widetilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} \right\| _ {2} (85) \\ \leqslant \widetilde {\omega} _ {s} ^ {(a, L)} \left\| \widetilde {\boldsymbol {\mu}} _ {3, s} ^ {(a, L)} - \widehat {\widetilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} \right\| _ {2} + \left\| \widetilde {\omega} _ {s} ^ {(a, L)} - \widehat {\widetilde {\omega}} _ {s} ^ {(a, L)} \right\| _ {2} \left\| \widehat {\widetilde {\boldsymbol {\mu}}} _ {3, s} ^ {(a, L)} \right\| _ {2} (86) \\ \leqslant \tilde {\omega} _ {s} ^ {(a, L)} \frac {8 \epsilon_ {M}}{\tilde {\omega} _ {s} ^ {(a , L)}} + 5 \epsilon_ {M} ^ {(a, L)} \quad \text {(F r o m r e s u l t s i n E q u a t i o n (7 6))} \\ = 1 3 \epsilon_ {M} ^ {(a, L)}. (87) \\ \end{array} +$$ + +where the inequality in line 84 follows from $\| \hat{B}\| _2\leqslant 1$ + +By combining the expressions in 81, 82 and 87, with probability at least $1 - 2\delta$ , we get: + +$$ +\| \boldsymbol {\mu} _ {3, s} - \widehat {\boldsymbol {\mu}} _ {3, s} \| _ {2} \leqslant \frac {4 \| \widetilde {\boldsymbol {M}} _ {2} ^ {(a , L)} - \widehat {\widetilde {\boldsymbol {M}}} _ {2} ^ {(a , L)} \| _ {2}}{\sqrt {\omega_ {s} ^ {(a , L)}} \omega_ {\min} ^ {(a , L)} \left[ \min _ {\nu} \sigma_ {S} (V _ {\nu} ^ {(a , L)}) \right] ^ {2}} + 1 3 \epsilon_ {M} ^ {(a, L)} \leqslant 1 4 \epsilon_ {M} ^ {(a, L)}, +$$ + +where the last inequality is obtained by observing that the first term of the summation is $\leqslant \epsilon_{M}^{(a,L)}$ . This last expression completes the proof. + +Lemma D.3 (Concentration Bounds for Covariance Matrices obtained from Multiple Policies). Let $\{\pi_l\}_{l=0}^{L-1}$ policies interact with a POMDP $\mathcal{Q}$ generating trajectories $\Gamma = \{\tau_l\}_{l=0}^{L-1}$ . Let Assumption 4.3 hold for each action $a \in \mathcal{A}$ and for each policy $\pi_l \in \mathcal{P}$ . Then, for any $\nu, \nu' \in \{1,2,3\}$ and $\nu \neq \nu'$ , with probability at least $1 - \delta$ , the following holds: + +$$ +\left\| \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} \left(\sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \left[ \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \right] - \mathbb {E} \left[ \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \right]\right) \right\| _ {2} \leqslant \frac {\widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log (2 O L / \delta)}{N _ {L} ^ {(a)}}}. +$$ + +For the tensor case, for $[\nu, \nu', \nu'']$ being any permutation of the set $\{1, 2, 3\}$ , with probability at least $1 - \delta$ , it holds: + +$$ +\begin{array}{l} \left\| \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} \left(\sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \left[ \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime \prime}, t} ^ {(a, l)} \right] - \mathbb {E} \left[ \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime \prime}, t} ^ {(a, l)} \right]\right) \right\| _ {2} \\ \leqslant \frac {\widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log ((O ^ {2} + O) L / \delta)}{N _ {L} ^ {(a)}}}, \\ \end{array} +$$ + +where $\widetilde{G} \coloneqq \max_{l \in [0, L-1]} G(\pi_l)$ and $\widetilde{\eta} \coloneqq \min_{l \in [0, L-1]} \eta(\pi_l)$ . Here, $1 \leqslant G(\pi_l) < \infty$ is the geometric ergodicity constant of the Markov Chain obtained from policy $\pi_l$ and $0 \leqslant \eta(\pi_l) < 1$ represents the related contraction coefficient. + +Proof. The proof of this lemma follows from standard concentration bounds on HMM when adapted to the observations conditioned on a specific action $a$ . Let us first observe that the covariance matrix obtained from policy $\pi_l$ is exactly defined as: + +$$ +K _ {\nu , \nu^ {\prime}} ^ {(a, l)} := \frac {1}{n _ {l} ^ {(a)}} \mathbb {E} \left[ \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \right], \tag {88} +$$ + +and we can define an analogous quantity for the tensor case as: + +$$ +K _ {\nu , \nu^ {\prime}, \nu^ {\prime \prime}} ^ {(a, l)} := \frac {1}{n _ {l} ^ {(a)}} \mathbb {E} \left[ \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime \prime}, t} ^ {(a, l)} \right], \tag {89} +$$ + +where we recall that $n_{l}^{(a)} \coloneqq |\mathcal{T}_{l}^{(a)}|$ . By applying Theorem 13 in [3], when a single policy $\pi_{l}$ is used, the error on the quantities defined above can be bounded as: + +$$ +\| K _ {\nu , \nu^ {\prime}} ^ {(a, l)} - \widehat {K} _ {\nu , \nu^ {\prime}} ^ {(a, l)} \| _ {2} \leqslant \frac {G (\pi_ {l})}{1 - \eta (\pi_ {l})} \sqrt {8 \frac {\log (2 O / \delta)}{n _ {l} ^ {(a)}}}, +$$ + +$$ +\| K _ {\nu , \nu^ {\prime}, \nu^ {\prime \prime}} ^ {(a, l)} - \widehat {K} _ {\nu , \nu^ {\prime}, \nu^ {\prime \prime}} ^ {(a, l)} \| _ {2} \leqslant \frac {G (\pi_ {l})}{1 - \eta (\pi_ {l})} \sqrt {8 \frac {\log \left((O ^ {2} + O) / \delta\right)}{n _ {l} ^ {(a)}}}, +$$ + +with probability at least $1 - \delta$ . In this version of the proof, differently from what done in [3], we bound the distance by assuming that the expectation defining both $K_{\nu,\nu'}^{(a,l)}$ and $K_{\nu,\nu',\nu''}^{(a,l)}$ is defined with respect to the initial (arbitrary) state distribution, which may be different from the stationary one15. + +Since we assume to have multiple policies interacting with the environment, our objective is to provide a bound for a mixing covariance matrix and a mixing tensor, respectively denoted as: + +$$ +\boldsymbol {K} _ {\nu , \nu^ {\prime}} ^ {(a, L)} := \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} K _ {\nu , \nu^ {\prime}} ^ {(a, l)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} \mathbb {E} \left[ \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \right], \tag {90} +$$ + +$$ +\boldsymbol {K} _ {\nu , \nu^ {\prime}, \nu^ {\prime \prime}} ^ {(a, L)} := \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} K _ {\nu , \nu^ {\prime}, \nu^ {\prime \prime}} ^ {(a, l)} = \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} \mathbb {E} \left[ \sum_ {t \in \mathcal {T} _ {l} ^ {(a)}} \boldsymbol {v} _ {\nu , t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime}, t} ^ {(a, l)} \otimes \boldsymbol {v} _ {\nu^ {\prime \prime}, t} ^ {(a, l)} \right]. \tag {91} +$$ + +We will study the error for the mixed covariance matrices. The same steps will hold for the tensor case. We have: + +$$ +\begin{array}{l} \left\| \boldsymbol {K} _ {\nu , \nu^ {\prime}} ^ {(a, L)} - \widehat {\boldsymbol {K}} _ {\nu , \nu^ {\prime}} ^ {(a, L)} \right\| _ {2} \leqslant \left\| \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \left(K _ {\nu , \nu^ {\prime}} ^ {(a, l)} - \widehat {K} _ {\nu , \nu^ {\prime}} ^ {(a, l)}\right) \right\| _ {2} (92) \\ \leqslant \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \| K _ {\nu , \nu^ {\prime}} ^ {(a, l)} - \hat {K} _ {\nu , \nu^ {\prime}} ^ {(a, l)} \| _ {2} \quad \text {(T r i a n g l e I n e q u a l i t y)} \\ \leqslant \frac {1}{N _ {L} ^ {(a)}} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \frac {G (\pi_ {l})}{1 - \eta (\pi_ {l})} \sqrt {8 \frac {\log (2 O L / \delta)}{n _ {l} ^ {(a)}}} \quad \text {(U n i o n B o u n d)} \\ \leqslant \frac {\widetilde {G}}{N _ {L} ^ {(a)} (1 - \widetilde {\eta})} \sqrt {8 \log (2 O L / \delta)} \sum_ {l = 0} ^ {L - 1} n _ {l} ^ {(a)} \sqrt {\frac {1}{n _ {l} ^ {(a)}}} (93) \\ = \frac {\widetilde {G}}{N _ {L} ^ {(a)} (1 - \widetilde {\eta})} \sqrt {8 \log (2 O L / \delta)} \sum_ {l = 0} ^ {L - 1} \sqrt {n _ {l} ^ {(a)}} (94) \\ \leqslant \frac {\widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log (2 O L / \delta)}{N _ {L} ^ {(a)}}} \quad (\text {C a u c h y - S c h w a r z}) \\ \end{array} +$$ + +where in line 93, we use the new terms $\widetilde{G} \coloneqq \max_{l \in [0,L-1]} G(\pi_l)$ and $\widetilde{\eta} \coloneqq \min_{l \in [0,L-1]} \eta(\pi_l)$ . We finally observe that the bound on the mixture covariance matrix presents a further term $\sqrt{L}$ in the bound due to the application of the union bound. + +The final result follows by substituting the definition of the covariance matrix in the statement of the lemma. + +# D.1 Minimum Number of Samples Required for Applying Tensor Decomposition + +Lemma D.4. Let $\widetilde{M}_2^{(a,L)}$ and $\widetilde{M}_3^{(a,L)}$ be defined as in Equations (73). Let Assumptions 4.1, 4.2 and 4.3 hold. Then, if the number of samples satisfies: + +$$ +N _ {L} ^ {(a)} \geqslant \left(\frac {2 \tilde {G} / (1 - \tilde {\eta})}{\omega_ {\min } ^ {(a , L)} \left[ \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right) \right] ^ {2}}\right) ^ {2} 8 L \log \left(\frac {2 L \left(O ^ {2} + O\right)}{\delta}\right) \Omega \tag {95} +$$ + +where + +$$ +\Omega = \max \left\{1, \frac {8 S}{C ^ {2} \omega_ {\min } ^ {(a , L)} \left[ \min _ {\nu} \sigma_ {S} (V _ {\nu} ^ {(a , L)}) \right] ^ {2}}, 1 6 \left(\frac {S}{C ^ {2} \omega_ {\min } ^ {(a , L)}}\right) ^ {1 / 3} \right\} +$$ + +then the following relation holds: + +$$ +\left\| \widetilde {M} _ {2} ^ {(a, L)} - \widehat {\widetilde {M}} _ {2} ^ {(a, L)} \right\| _ {2} \leqslant (1 / 2) \sigma_ {S} \left(\widetilde {M} _ {2} ^ {(a, L)}\right). \tag {96} +$$ + +Hence, this condition allows applying the RTP approach on the estimated tensor $\widehat{\widetilde{M}}_3^{(a,L)}\left(\widehat{W}_1^{(a,L)},\widehat{W}_2^{(a,L)},\widehat{W}_3^{(a,L)}\right)$ , as prescribed in Proposition D.8. + +Proof. We recall here the result in Proposition D.8 which allows us to provide a bound on the estimation error $\epsilon_{M}^{(a,L)}$ of matrix $\widehat{\widetilde{M}}_3^{(a,L)}\left(\widehat{W}_1^{(a,L)},\widehat{W}_2^{(a,L)},\widehat{W}_3^{(a,L)}\right)$ . We have: + +$$ +\begin{array}{l} \epsilon_ {M} ^ {(a, L)} \leqslant \frac {2 \sqrt {2} \left\| \widetilde {\boldsymbol {M}} _ {3} ^ {(a , L)} - \widehat {\widetilde {\boldsymbol {M}}} _ {3} ^ {(a , L)} \right\| _ {2}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {3}} + \frac {\left(\frac {4 \left\| \widetilde {\boldsymbol {M}} _ {2} ^ {(a , L)} - \widehat {\widetilde {\boldsymbol {M}}} _ {2} ^ {(a , L)} \right\| _ {2}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {2}}\right) ^ {3}}{\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}}} (97) \\ \leqslant \underbrace {\frac {\frac {2 \sqrt {2} \widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log ((O ^ {2} + O) 2 L / \delta)}{N _ {L} ^ {(a)}}}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {3}}} _ {\text {F i r s t T e r m}} + \underbrace {\frac {\left(\frac {\frac {4 \widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log (4 O L / \delta)}{N _ {L} ^ {(a)}}}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {2}}\right) ^ {3}}}{\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}}}, (98) \\ \end{array} +$$ + +where this last inequality uses concentration results on the empirical estimates of $\widetilde{M}_2^{(a,L)}$ and $\widetilde{M}_3^{(a,L)}$ (Lemma D.3), and holds with probability at least $1 - \delta$ . + +In order to successfully apply the RTP method on the estimated tensor, the estimation error $\epsilon_{M}^{(a,L)}$ should be reasonably small. In particular, the result in Equation (97) holds under the assumption that i) $\left\| \widetilde{M}_2^{(a,L)} - \widehat{\widetilde{M}}_2^{(a,L)}\right\|_2 \leqslant \frac{1}{2}\sigma_S(\widetilde{M}_2^{(a,L)})$ , as prescribed in Proposition D.8. In addition, from [2], it is required that ii) $\epsilon_{M}^{(a,L)} \leqslant \frac{C}{\sqrt{S}}$ for some constant $C$ . From condition i), we require that: + +$$ +N _ {L} ^ {(a)} \geqslant \left(\frac {2 \tilde {G} / (1 - \tilde {\eta})}{\omega_ {\min } ^ {(a , L)} \left[ \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right) \right] ^ {2}}\right) ^ {2} 8 L \log (4 O L / \delta), \tag {99} +$$ + +while for condition ii), it surely holds when each of the terms appearing in (98) is upper bounded by $C / (2\sqrt{S})$ under a suitable constant $C$ , namely: + +$$ +\text {(F i r s t T e r m i n (9 8))} \leqslant \frac {C}{2 \sqrt {S}} \quad \text {(S e c o n d T e r m i n (9 8))} \leqslant \frac {C}{2 \sqrt {S}}. +$$ + +From the previous bounds, we obtain respectively: + +$$ +N _ {L} ^ {(a)} \geqslant \left(\frac {4 \sqrt {2} \widetilde {G} / (1 - \widetilde {\eta})}{C \left[ \sqrt {\omega_ {\operatorname* {m i n}} ^ {(a , L)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right) \right] ^ {3}}\right) ^ {2} 8 S L \log \left(2 L \left(O ^ {2} + O\right) / \delta\right), \tag {100} +$$ + +$$ +N _ {L} ^ {(a)} \geqslant \left(\frac {8 \widetilde {G} / (1 - \widetilde {\eta})}{\left[ C \left(\omega_ {\min } ^ {(a , L)}\right) ^ {7 / 2} \right] ^ {1 / 3} \left(\min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a , L)}\right)\right) ^ {2}}\right) ^ {2} 8 S ^ {1 / 3} L \log (4 O L / \delta). \tag {101} +$$ + +By rearranging the results reported in Equations (99), (100) and (101), we get the final result of the lemma on the minimum number of samples required for the condition 96 to hold. + +# D.2 Auxiliary Propositions + +Proposition D.5. Let $\widehat{\pmb{K}}_{3,1}^{(a,L)}$ be an empirical estimate of $\pmb{K}_{3,1}^{(a,L)}$ obtained using $N_L^{(a)}$ samples. Then if: + +$$ +N _ {L} ^ {(a)} \geqslant \left(\frac {2 \widetilde {G}}{\sigma_ {S} \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right) (1 - \widetilde {\eta})}\right) ^ {2} 8 L \log (2 O L / \delta). \tag {102} +$$ + +then with probability at least $1 - \delta$ , the covariance matrix $\hat{K}_{3,1}^{(a,L)}$ is invertible and it holds that: + +$$ +\left\| \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {- 1} - \left(\widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)}\right) ^ {- 1} \right\| _ {2} \leqslant \frac {2 \widetilde {G}}{\left[ \sigma_ {S} (\boldsymbol {K} _ {3 , 1} ^ {(a , L)}) \right] ^ {2} (1 - \widetilde {\eta})} \sqrt {\frac {8 L \log (2 O L / \delta)}{N _ {L} ^ {(a)}}} +$$ + +Proof. Since $\widehat{\pmb{K}}_{3,1}^{(a,L)} = \frac{1}{N_L^{(a)}}\sum_{l = 0}^{L - 1}n_l^{(a)}\mathbb{E}\left[\pmb {v}_3^{(a,l)}\otimes \pmb {v}_1^{(a,l)}\right]$ , we can apply lemma D.3 and get + +$$ +\left\| \boldsymbol {K} _ {3, 1} ^ {(a, L)} - \widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)} \right\| _ {2} \leqslant \frac {\widetilde {G}}{1 - \widetilde {\eta}} \sqrt {\frac {8 L \log (2 O L / \delta)}{N _ {L} ^ {(a)}}}. \tag {103} +$$ + +Let us consider the condition: + +$$ +\left\| \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {- 1} \right\| _ {2} \left\| \boldsymbol {K} _ {3, 1} ^ {(a, L)} - \widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)} \right\| _ {2} \leqslant 1 / 2. \tag {104} +$$ + +By denoting with $\sigma_S(K_{3,1}^{(a,L)})$ the minimum singular value of matrix $K_{3,1}^{(a,L)}$ we have $\left\| \left(K_{3,1}^{(a,L)}\right)^{-1}\right\|_2 = 1 / \sigma_S(K_{3,1}^{(a,L)})$ . By using the bound in 103, it is easy to show that this condition (104) is verified with probability $1 - \delta$ when: + +$$ +N _ {L} ^ {(a)} \geqslant \left(\frac {2 \widetilde {G}}{\sigma_ {S} \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right) (1 - \widetilde {\eta})}\right) ^ {2} 8 L \log (2 O L / \delta). \tag {105} +$$ + +Under condition (104), we can state the following: + +$$ +\begin{array}{l} \left\| \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {- 1} - \left(\widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)}\right) ^ {- 1} \right\| _ {2} \leqslant \frac {\left\| \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right) ^ {- 1} \right\| _ {2} ^ {2} \left\| \boldsymbol {K} _ {3 , 1} ^ {(a , L)} - \widehat {\boldsymbol {K}} _ {3 , 1} ^ {(a , L)} \right\| _ {2}}{1 - \left\| \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right) ^ {- 1} \right\| _ {2} \left\| \boldsymbol {K} _ {3 , 1} ^ {(a , L)} - \widehat {\boldsymbol {K}} _ {3 , 1} ^ {(a , L)} \right\| _ {2}} (106) \\ \leqslant 2 \left\| \left(\boldsymbol {K} _ {3, 1} ^ {(a, L)}\right) ^ {- 1} \right\| _ {2} ^ {2} \left\| \boldsymbol {K} _ {3, 1} ^ {(a, L)} - \widehat {\boldsymbol {K}} _ {3, 1} ^ {(a, L)} \right\| _ {2} (107) \\ \leqslant \frac {2 \widetilde {G}}{\left[ \sigma_ {S} \left(\boldsymbol {K} _ {3 , 1} ^ {(a , L)}\right) \right] ^ {2} (1 - \widetilde {\eta})} \sqrt {\frac {8 L \log (2 O L / \delta)}{N _ {L} ^ {(a)}}} (108) \\ \end{array} +$$ + +where line 106 derives from Lemma E.4 in [2], while line 107 is obtained by substituting at the denominator the condition in 104. + +Proposition D.6. (From [3]) Let $W \in \mathbb{R}^{Y \times X}$ and $\widehat{W} \in \mathbb{R}^{Y \times X}$ with $Y \geqslant X$ be any pair of matrices such that $\widehat{W} = W + E$ for a suitable error matrix $E$ and let $\sigma_{X}(\widehat{W})$ be the $X$ -th singular value of matrix $\widehat{W}$ . If the error matrix is such that: + +$$ +\| E \| _ {2} \leqslant \frac {\sigma_ {X} (W)}{2}, \tag {109} +$$ + +then we can derive the following: + +$$ +\| \widehat {W} ^ {\dagger} \| _ {2} \leqslant \frac {1}{\sigma_ {X} (\widehat {W})} \leqslant \frac {2}{\sigma_ {X} (W)} +$$ + +Proof. Given that $\widehat{W}$ is a perturbation of the true matrix $W$ , we can use Weyl inequality to have a bound on the difference of the minimum singular value: + +$$ +\left| \sigma_ {X} (\widehat {W}) - \sigma_ {X} (W) \right| \leqslant \| \widehat {W} - W \| _ {2} +$$ + +which leads to + +$$ +\sigma_ {X} (\widehat {W}) \geqslant \sigma_ {X} (W) - \| \widehat {W} - W \| _ {2}. +$$ + +Since we have assumed that the perturbation is not too large, we can safely invert this bound to obtain: + +$$ +\frac {1}{\sigma_ {X} (\widehat {W})} \leqslant \frac {1}{\sigma_ {X} (W) - \| \widehat {W} - W \| _ {2}} \leqslant \frac {1}{\sigma_ {X} (W) / 2} +$$ + +where the last inequality follows from the precondition on the perturbation error (109). + +Hence, we can derive the final result as: + +$$ +\| \widehat {W} ^ {\dagger} \| _ {2} \leqslant \frac {1}{\sigma_ {X} (\widehat {W})} \leqslant \frac {2}{\sigma_ {X} (W)}. +$$ + +![](images/9210f13e5eb2f0e38215596f6e072f57339b469c3acc71b6e2aaa2c35360181b.jpg) + +Proposition D.7. (From [22]) Let $W$ and $\widehat{W}$ be any pair of matrices such that $\widehat{W} = W + E$ for a suitable error matrix $E$ . Then we have: + +$$ +\| W ^ {\dagger} - \widehat {W} ^ {\dagger} \| _ {2} \leqslant \frac {1 + \sqrt {5}}{2} \max \left\{\| W ^ {\dagger} \| _ {2}, \| \widehat {W} ^ {\dagger} \| _ {2} \right\} \| E \| _ {2}, +$$ + +with $\| \cdot \| _2$ denoting the spectral norm. + +Proposition D.8 (From [3]). Let $\widetilde{M}_2^{(a)}\coloneqq \mathbb{E}[\pmb {v}_1^{(a)}\otimes \pmb {v}_2^{(a)}]$ and $\widetilde{M}_3^{(a)}\coloneqq \mathbb{E}[\pmb {v}_1^{(a)}\otimes \pmb {v}_2^{(a)}\otimes \pmb {v}_3^{(a)}]$ be the matrices associated with action $a\in \mathcal{A}$ , with the expectations defined by policy $\pi \in \mathcal{P}$ . Let also denote with $\widetilde{M}_3^{(a)}(W_1^{(a)},W_2^{(a)},W_3^{(a)})$ the symmetrized and whitened third-moment tensor, as defined in Section E. If Assumptions 4.1, 4.2 and 4.3 hold, then, under the condition16 + +$$ +\left\| \widetilde {M} _ {2} ^ {(a)} - \widehat {\widetilde {M}} _ {2} ^ {(a)} \right\| _ {2} \leqslant (1 / 2) \sigma_ {S} \left(\widetilde {M} _ {2} ^ {(a)}\right), \tag {110} +$$ + +the two following statements hold: + +$$ +\begin{array}{l} (i) \qquad \epsilon_ {M} := \left\| \widetilde {M} _ {3} ^ {(a)} \left(W _ {1} ^ {(a)}, W _ {2} ^ {(a)}, W _ {3} ^ {(a)}\right) - \widehat {\widetilde {M}} _ {3} ^ {(a)} \left(\widehat {W} _ {1} ^ {(a)}, \widehat {W} _ {2} ^ {(a)}, \widehat {W} _ {3} ^ {(a)}\right) \right\| _ {2} \\ \leqslant \frac {2 \sqrt {2} \left\| \widetilde {M} _ {3} ^ {(a)} - \widehat {\widetilde {M}} _ {3} ^ {(a)} \right\| _ {2}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a)}\right)\right) ^ {3}} + \frac {\left(\frac {4 \left\| \widetilde {M} _ {2} ^ {(a)} - \widehat {\widetilde {M}} _ {2} ^ {(a)} \right\| _ {2}}{\left(\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a)}} \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a)}\right)\right) ^ {2}}\right) ^ {3}}{\sqrt {\omega_ {\operatorname* {m i n}} ^ {(a)}}}, \tag {111} \\ \end{array} +$$ + +$$ +\left. \right.\left.\left(\widehat {W} _ {3} ^ {(a)}\right) ^ {\dagger} - \left(\widehat {W} _ {3} ^ {(a)}\right) ^ {\dagger} \right\| _ {2} \leqslant \frac {4 \left\| \widetilde {M} _ {2} ^ {(a)} - \widehat {\widetilde {M}} _ {2} ^ {(a)} \right\| _ {2}}{\omega_ {\min } ^ {(a)} \left[ \min _ {\nu} \sigma_ {S} \left(V _ {\nu} ^ {(a)}\right)\right] ^ {2}}. \tag {112} +$$ + +# E Symmetrization and Whitening + +This section shows how the symmetrization and the whitening steps can be used for the quantities defined in this work. To reduce clutter, we will avoid using the apices $a$ and $L$ in this section. + +# Notation + +We will stick here with the notation used in [1]. Let us denote a $p$ -th order tensor as $A \in \bigotimes_{i=1}^{p} \mathbb{R}^{n_i}$ . When $n_1 = n_2 = \dots = n_p = n$ , we can simply write $A \in \bigotimes^p \mathbb{R}^n$ . For a vector $v \in \mathbb{R}^n$ let us use + +$v^{\otimes p} \coloneqq v \otimes v \otimes \dots \otimes v \in \bigotimes^{p} \mathbb{R}^{n}$ to denote its $p$ -th order tensor. + +We can consider $A$ to be a multilinear map when it holds that for a set of matrices $\{V_i \in \mathbb{R}^{n \times m_i} : i \in [p]\}$ , the $(i_1, i_2, \ldots, i_p)$ -th entry in the tensor $A(V_1, V_2, \ldots, V_p) \in \mathbb{R}^{m_1 \times m_2 \times \cdots \times m_p}$ is + +$$ +\left[ A \left(V _ {1}, V _ {2}, \dots , V _ {p}\right) \right] _ {i _ {1}, i _ {2}, \dots , i _ {p}} := \sum_ {j _ {1}, j _ {2}, \dots , j _ {p} \in [ n ]} A _ {j _ {1}, j _ {2}, \dots , j _ {p}} \left[ V _ {1} \right] _ {j _ {1}, i _ {1}} \left[ V _ {2} \right] _ {j _ {2}, i _ {2}} \dots \left[ V _ {p} \right] _ {j _ {p}, i _ {p}}. +$$ + +So, if $A$ is a matrix $(p = 2)$ , then we have: + +$$ +A \left(V _ {1}, V _ {2}\right) = V _ {1} ^ {\top} A V _ {2}. \tag {113} +$$ + +# Symmetrization + +Let us now denote with $\pmb{v}_1\in \mathbb{R}^{d_1}$ , $\pmb{v}_2\in \mathbb{R}^{d_2}$ and $\pmb{v}_3\in \mathbb{R}^{d_3}$ the three view vectors, and let $V_{1}\in \mathbb{R}^{d_{1}\times k}$ , $V_{2}\in \mathbb{R}^{d_{2}\times k}$ and $V_{3}\in \mathbb{R}^{d_{3}\times k}$ be the associated view matrices, with $k\leqslant d_{\nu}$ for $\nu \in \{1,2,3\}^{17}$ . We use $\mu_{\nu ,i}$ to denote the $i$ th column of the view matrix $V_{\nu}$ . Let us consider the second moment $\widetilde{M}_2\in \mathbb{R}^{d_1\times d_2}$ and third moment $\widetilde{M}_3\in \mathbb{R}^{d_1\times d_2\times d_3}$ of the three views as follows: + +$$ +\widetilde {M} _ {2} := \mathbb {E} [ \boldsymbol {v} _ {1} \otimes \boldsymbol {v} _ {2} ] = \sum_ {i = 1} ^ {k} \omega_ {i} \boldsymbol {\mu} _ {1, i} \otimes \boldsymbol {\mu} _ {2, i} \quad \widetilde {M} _ {3} := \mathbb {E} [ \boldsymbol {v} _ {1} \otimes \boldsymbol {v} _ {2} \otimes \boldsymbol {v} _ {3} ] = \sum_ {i = 1} ^ {k} \omega_ {i} \boldsymbol {\mu} _ {1, i} \otimes \boldsymbol {\mu} _ {2, i} \otimes \boldsymbol {\mu} _ {3, i}. \tag {114} +$$ + +Our objective is to represent these views as the second-order tensor and the third-order tensor with respect to view $\boldsymbol{v}_3$ . In order to achieve this result, we need to modify the views $\boldsymbol{v}_1$ and $\boldsymbol{v}_2$ by making use of the covariance matrices as follows: + +$$ +\widetilde {\boldsymbol {v}} _ {1} = \underbrace {K _ {3 , 2} \left(K _ {1 , 2}\right) ^ {\dagger}} _ {R _ {1} ^ {\top}} \boldsymbol {v} _ {1} \qquad \qquad \widetilde {\boldsymbol {v}} _ {2} = \underbrace {K _ {3 , 1} \left(K _ {2 , 1}\right) ^ {\dagger}} _ {R _ {2} ^ {\top}} \boldsymbol {v} _ {2}, +$$ + +with $R_{1} \in \mathbb{R}^{d_{1} \times d_{3}}$ and $R_{2} \in \mathbb{R}^{d_{2} \times d_{3}}$ being the rotation matrices of the views $\pmb{v}_{1}$ and $\pmb{v}_{2}$ respectively. Using notation in Equation (113), it is possible to show that the symmetrized version $M_{2} \in \mathbb{R}^{d_{3} \times d_{3}}$ can be defined as: + +$$ +M _ {2} := \widetilde {M} _ {2} (R _ {1}, R _ {2}) = R _ {1} ^ {\top} \widetilde {M} _ {2} R _ {2} = \mathbb {E} [ \boldsymbol {v} _ {3} \otimes \boldsymbol {v} _ {3} ] = \sum_ {i = 1} ^ {k} \omega_ {i} \boldsymbol {\mu} _ {3, i} \otimes \boldsymbol {\mu} _ {3, i}. +$$ + +# Whitening + +When the symmetrization step is concluded, the third-order matrix needs to be whitened in order to run the Robust Tensor Power (RTP) method on it. The whitening transformation is defined through the matrix $W \in \mathbb{R}^{d_3 \times k}$ and is such that: + +$$ +M _ {2} (W, W) = W ^ {\top} M _ {2} W = I, +$$ + +with $M_2$ being the symmetrized matrix defined above and $I \in \mathbb{R}^{k \times k}$ is the identity matrix. + +From the relations above, we also have: + +$$ +M _ {2} (W, W) = W ^ {\top} M _ {2} W = W ^ {\top} R _ {1} ^ {\top} \widetilde {M} _ {2} R _ {2} W = \widetilde {M} _ {2} \left(\underbrace {R _ {1} W} _ {W _ {1}}, \underbrace {R _ {2} W} _ {W _ {2}}\right), \tag {115} +$$ + +which introduces the symmetrization-whitening matrices $W_{1} \in \mathbb{R}^{d_{1} \times k}$ and $W_{2} \in \mathbb{R}^{d_{2} \times k}$ . Since the third view does not need to be symmetrized but only whitened, we have $W_{3} \coloneqq W \in \mathbb{R}^{d_{3} \times k}$ . + +Let us now define: + +$$ +\tilde {\boldsymbol {\mu}} _ {3, i} := \sqrt {\omega_ {i}} W ^ {\top} \boldsymbol {\mu} _ {3, i}, \quad \tilde {\omega} _ {i} := \frac {1}{\sqrt {\omega_ {i}}} \tag {116} +$$ + +and we observe that: + +$$ +\widetilde {M} _ {2} (W _ {1}, W _ {2}) = M _ {2} (W, W) = \sum_ {i = 1} ^ {k} W ^ {\top} (\sqrt {\omega_ {i}} \boldsymbol {\mu} _ {3, i}) (\sqrt {\omega_ {i}} \boldsymbol {\mu} _ {3, i}) ^ {\top} W = \sum_ {i = 1} ^ {k} \widetilde {\boldsymbol {\mu}} _ {3, i} \widetilde {\boldsymbol {\mu}} _ {3, i} ^ {\top} = I, +$$ + +from which we also observe that $\widetilde{\mu}_{3,i} \in \mathbb{R}^k$ are orthonormal vectors. + +We can now define the symmetrized and whitened tensor $\widetilde{M}_3(W_1, W_2, W_3) \in \mathbb{R}^{k \times k \times k}$ as: + +$$ +\widetilde {M} _ {3} \left(W _ {1}, W _ {2}, W _ {3}\right) = M _ {3} \left(W, W, W\right) = \sum_ {i = 1} ^ {k} \omega_ {i} \left(W ^ {\top} \boldsymbol {\mu} _ {3, i}\right) ^ {\otimes 3} = \sum_ {i = 1} ^ {k} \frac {1}{\sqrt {\omega_ {i}}} \tilde {\boldsymbol {\mu}} _ {3, i} ^ {\otimes 3} = \sum_ {i = 1} ^ {k} \tilde {\omega} _ {i} \tilde {\boldsymbol {\mu}} _ {3, i} ^ {\otimes 3}, \tag {117} +$$ + +where the first equality follows from analogous considerations as those in Equation (115). + +The decomposition expressed in the last equality allows representing tensor $\widetilde{M}_3(W_1, W_2, W_3)$ in terms of the orthonormal eigenvectors $\widetilde{\mu}_{3,i}$ and the related eigenvalues $\widetilde{\omega}_i$ . In this form, the tensor can be provided as input to the RTP method [1]. The RTP method will then provide as output an estimate of the robust eigenvector/eigenvalue pairs $(\widetilde{\mu}_{3,i}, \widetilde{\omega}_i)$ for each $i \in [k]$ . + +Finally, the original eigenvector/eigenvalue pairs $(\mu_{3,i},\omega_i)$ can be recovered by inverting the Equations in (116). + +# F Belief Vector Concentration Bound + +We present here Lemma F.1 that will be fundamental for proving the regret result of the Mixed Spectral UCRL algorithm. + +Lemma F.1. Let $\mathcal{Q}$ be a POMDP instance satisfying Assumption 6.1. Let $\widehat{\mathbb{O}}$ and $\widehat{\mathbb{T}} = \{\widehat{\mathbb{T}}_a\}_{a\in \mathcal{A}}$ be the estimate of the observation and transition model and let $\mathcal{T} = \{(o_t,a_t)\}_{t = 0}^T$ be a trajectory generated while interacting with the environment. We have that: + +$$ +\sum_ {t = 0} ^ {T} \| \widehat {b} _ {t} - b _ {t} \| _ {1} \leqslant C _ {1} + C _ {2} T \| \mathbb {O} - \widehat {\mathbb {O}} \| _ {F} + C _ {3} \sum_ {a \in \mathcal {A}} n ^ {(a)} \| \mathbb {T} _ {a} - \widehat {\mathbb {T}} _ {a} \| _ {F}, +$$ + +where $C_1, C_2, C_3$ are finite constants, while $n^{(a)}$ represents the number of times each action $a \in \mathcal{A}$ is chosen during the interaction. + +Proof. We denote with $\widehat{b}_t$ and $b_{t}$ the estimated and real belief vector at time $t$ updated using Equation 1, using respectively the estimated and real transition model. From the belief decomposition reported in [10], we derive that the belief error bound at time $t$ is: + +$$ +\left\| \widehat {b} _ {t} - b _ {t} \right\| _ {1} \leqslant 4 \eta^ {t} \left(\frac {\left\| \widehat {b} _ {0} - b _ {0} \right\| _ {2}}{\epsilon}\right) + \frac {4 (1 - \epsilon)}{\epsilon} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \left(\frac {\left\| \widehat {\mathbb {T}} _ {a _ {l}} - \mathbb {T} _ {a _ {l}} \right\| _ {F}}{\epsilon} + \sqrt {S O} \frac {\left\| \widehat {\mathbb {O}} - \mathbb {O} \right\| _ {F}}{c _ {o}}\right) \tag {118} +$$ + +where $\eta \coloneqq 1 - \frac{\epsilon}{1 - \epsilon}$ , while $c_{o}$ is a finite constant based on both the transition and the observation model such that $c_{o} \coloneqq \min_{o \in \mathcal{O}} \min_{a \in \mathcal{A}} \min_{s \in \mathcal{S}} \sum_{s' \in \mathcal{S}} \mathbb{T}_{a}(s'|s) \mathbb{O}(o|s')$ which is always positive thanks to Assumption 6.1. + +We proceed by bounding 118 as: + +$$ +\| \widehat {b} _ {t} - b _ {t} \| _ {1} \leqslant \frac {8 \eta^ {t}}{\epsilon} + \frac {4 (1 - \epsilon)}{\epsilon^ {2}} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \left(\| \widehat {\mathbb {T}} _ {a _ {l}} - \mathbb {T} _ {a _ {l}} \| _ {F}\right) + \frac {4 \sqrt {S O} (1 - \epsilon)}{\epsilon c _ {o}} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \| \widehat {\mathbb {O}} - \mathbb {O} \| _ {F}, +$$ + +where the inequality simply follows by observing that $\| \widehat{b}_0 - b_0\| _2\leqslant \| \widehat{b}_0 - b_0\| _1\leqslant 2$ + +This bound shows that the error in the belief vector depends on the sequence of actions and the contribution in the error of each action scales geometrically with time. Using the relations above, let + +us now bound the sum of belief errors over $T + 1$ different time steps: + +$$ +\begin{array}{l} \sum_ {t = 0} ^ {T} \left\| \widehat {b} _ {t} - b _ {t} \right\| _ {1} \leqslant 2 + \sum_ {t = 1} ^ {T} \left[ \frac {8 \eta^ {t}}{\epsilon} + \frac {4 (1 - \epsilon)}{\epsilon^ {2}} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \left(\left\| \widehat {\mathbb {T}} _ {a _ {l}} - \mathbb {T} _ {a _ {l}} \right\| _ {F}\right) + \right. \\ \left. + \frac {4 \sqrt {S O} (1 - \epsilon)}{\epsilon c _ {o}} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \| \widehat {\mathbb {O}} - \mathbb {O} \| _ {F} \right] \\ = \underbrace {2 + \sum_ {t = 1} ^ {T} \left[ \frac {8 \eta^ {t}}{\epsilon} \right]} _ {\text {(a)}} + \underbrace {\sum_ {t = 1} ^ {T} \left[ \frac {4 \sqrt {S O} (1 - \epsilon)}{\epsilon c _ {o}} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \| \widehat {\mathbb {O}} - \mathbb {O} \| _ {F} \right]} _ {\text {(b)}} + \\ + \underbrace {\sum_ {t = 1} ^ {T} \left[ \frac {4 (1 - \epsilon)}{\epsilon^ {2}} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \left(\| \widehat {\mathbb {T}} _ {a _ {l}} - \mathbb {T} _ {a _ {l}} \| _ {F}\right) \right]} _ {\text {(c)}}, \\ \end{array} +$$ + +where the constant 2 is obtained by bounding the first term $\| \widehat{b}_0 - b_0\| _1\leqslant 2$ Let us now focus on the terms $(a)$ and $(b)$ + +$$ +(a) = 2 + \frac {8}{\epsilon} \sum_ {t = 1} ^ {T} \eta^ {t} \leqslant 2 + \frac {8}{\epsilon} \left(\frac {1}{1 - \eta}\right) \leqslant \frac {1 0}{\epsilon} \left(\frac {1}{1 - \eta}\right) +$$ + +$$ +(b) = \frac {4 \sqrt {S O} (1 - \epsilon) \| \widehat {\mathbb {O}} - \mathbb {O} \| _ {F}}{\epsilon c _ {o}} \sum_ {t = 1} ^ {T} \sum_ {l = 0} ^ {t - 1} \eta^ {t - l - 1} \leqslant \frac {4 \sqrt {S O} (1 - \epsilon) \| \widehat {\mathbb {O}} - \mathbb {O} \| _ {F}}{\epsilon c _ {o}} \cdot \frac {T}{1 - \eta}. +$$ + +Differently, the term $c$ can be bounded by using the result from [26] (see their Lemma D.1) and we obtain that: + +$$ +(c) \leqslant \frac {4 (1 - \epsilon)}{(1 - \eta) \epsilon^ {2}} \sum_ {a \in \mathcal {A}} n ^ {(a)} \| \mathbb {T} _ {a} - \widehat {\mathbb {T}} _ {a} \| _ {F} = \frac {4 (1 - \epsilon) ^ {2}}{\epsilon^ {3}} \sum_ {a \in \mathcal {A}} n ^ {(a)} \| \mathbb {T} _ {a} - \widehat {\mathbb {T}} _ {a} \| _ {F} +$$ + +where $n^{(a)}$ represents the number of times action $a \in \mathcal{A}$ is chosen during the interaction, while the last step follows by using the definition of $\eta$ . + +By combining the results in $(a)$ , $(b)$ and $(c)$ , we get: + +$$ +\begin{array}{l} \sum_ {t = 0} ^ {T} \| \widehat {b} _ {t} - b _ {t} \| _ {1} \leqslant \frac {1 0}{\epsilon} \left(\frac {1}{1 - \eta}\right) + \frac {4 \sqrt {S O} (1 - \epsilon) \| \widehat {\mathbb {O}} - \mathbb {O} \| _ {F}}{\epsilon c _ {o}} \cdot \frac {T}{1 - \eta} + \frac {4 (1 - \epsilon) ^ {2}}{\epsilon^ {3}} \sum_ {a \in \mathcal {A}} n ^ {(a)} \| \widehat {\mathbb {T}} _ {a} - \mathbb {T} _ {a} \| _ {F} \\ = \frac {1 0 (1 - \epsilon)}{\epsilon^ {2}} + \frac {4 \sqrt {S O} (1 - \epsilon) ^ {2} \| \hat {\mathbb {O}} - \mathbb {O} \| _ {F} T}{\epsilon^ {2} c _ {o}} + \frac {4 (1 - \epsilon) ^ {2}}{\epsilon^ {3}} \sum_ {a \in \mathcal {A}} n ^ {(a)} \| \hat {\mathbb {T}} _ {a} - \mathbb {T} _ {a} \| _ {F} \tag {119} \\ \end{array} +$$ + +where in the last line we simply substituted the definition of $\eta$ into the bound. The final result of the lemma simply follows by defining the constants + +$$ +C _ {1} := \frac {1 0 (1 - \epsilon)}{\epsilon^ {2}}, \quad C _ {2} := \frac {4 \sqrt {S O} (1 - \epsilon) ^ {2}}{\epsilon^ {2} c _ {o}}, \quad C _ {3} := \frac {4 (1 - \epsilon) ^ {2}}{\epsilon^ {3}}. \tag {120} +$$ + +![](images/051c1e96cb1406f264e6260a82d6294878c8213ad3117a62cd801724090ebf93.jpg) + +From the considerations reported above, we can derive the following corollary for the one-step belief error. + +Corollary F.2. (One-step Belief Bound) Let $\mathcal{Q}$ be a POMDP instance satisfying Assumption 6.1. Let us denote with $(\mathbb{O},\mathbb{T}_a)$ and $(\hat{\mathbb{O}},\hat{\mathbb{T}}_a)$ respectively the real and estimated model parameters related to action $a$ . Starting from a common belief vector $b_{0}$ , and choosing action $a\in \mathcal{A}$ , the one-step error in the estimated belief vector can be bounded as: + +$$ +\left\| \widehat {b} _ {1} - b _ {1} \right\| _ {1} \leqslant C _ {2} \left\| \widehat {\mathbb {O}} - \mathbb {O} \right\| _ {F} + C _ {3} \left\| \widehat {\mathbb {T}} _ {a} - \mathbb {T} _ {a} \right\| _ {F}. +$$ + +where constants $C_2$ and $C_3$ are defined in line 120. + +Proof. The proof of this corollary easily follows from the bound in 118 by using $t = 1$ and having that $b_{0} = \widehat{b}_{0}$ . + +# G Miscellanea of Useful Results + +This section is devoted to the presentation of some useful results used throughout the work. + +The first one is taken from [34] and relates the maximum span of the bias function $\text{span}(v)$ with a finite constant $D$ . + +Proposition G.1 (Uniform bound on the bias span from [34]). Let us assume to have a POMDP instance that can be rewritten as a belief MDP. If Assumption 6.1 holds, then for $\rho, v$ satisfying the Bellman Equation (2), we have the span of the bias function $\text{span}(v) \coloneqq \max_{b \in \mathcal{B}} v(b) - \min_{b \in \mathcal{B}} v(b)$ is bounded by $D(\epsilon)$ , where: + +$$ +D (\epsilon) := \frac {8 \left(\frac {2}{(1 - \alpha) ^ {2}} + (1 + \alpha) \log_ {\alpha} \left(\frac {1 - \alpha}{8}\right)\right)}{1 - \alpha}, \qquad w i t h \qquad \alpha = \frac {1 - 2 \epsilon}{1 - \epsilon} \in (0, 1). +$$ + +Hence, this proposition ensures that $\text{span}(v)$ is bounded by $D = D(\epsilon / 2)$ for any bias functions $v$ associated with a belief MDP derived from a POMDP instance $\mathcal{Q}$ . + +This second result is used in the bound of Theorem 6.2. + +Lemma G.2 (Lemma 19 in [14]). For any sequence of numbers $y_0, \ldots, y_{n-1}$ with $0 \leqslant y_k \leqslant Y_k$ and $Y_k \coloneqq \max \{1, \sum_{i=0}^{k-1} y_i\}$ : + +$$ +\sum_ {k = 0} ^ {n - 1} \frac {y _ {k}}{\sqrt {Y _ {k}}} \leqslant (\sqrt {2} + 1) \sqrt {Y _ {n}}. +$$ + +# H Comparison with Related Literature + +We provide here a detailed comparison of our Mixed Spectral UCRL with respect to the SEEU and the SM-UCRL algorithms tackling the infinite-horizon average reward setting (Section H.1), while we devote Section H.2 to a discussion on the differences of our formulation with respect to Maximum-Likelihood approaches typically used in episodic settings. + +# H.1 Comparison with Algorithms in the Infinite-horizon setting + +We provide here a comparison in terms of assumptions and theoretical guarantees of our Mixed Spectral UCRL algorithm with other algorithms in the literature that tackle this setting. Some key aspects are reported in Table 1. In particular: + +Comparison with SEEU [32]. Our approach strictly improves over the SEEU algorithm both in terms of assumptions and results. Indeed, unlike SEEU, our algorithm does not require an assumption on the minimum values of the observation model. Additionally, we introduce the sample reuse strategy for adaptive policies, leading to an improved sample efficiency which, together with a more refined theoretical analysis, also translates to an improved regret bound with respect to the interaction horizon, from $\widetilde{\mathcal{O}}(T^{2/3})$ to $\widetilde{\mathcal{O}}(\sqrt{T})$ . + +Comparison with SM-UCRL [3]. Similarly, we also make improvements over the SM-UCRL algorithm. Indeed, the SM-UCRL algorithm employs stochastic memoryless policies which are known to suffer linear regret when compared against the optimal POMDP policy. The employed policy class includes those policies for which each action can be chosen with a minimum probability $\iota > 0$ at + +every time step. By introducing our sample reuse strategy, we improve sample efficiency, and we are not obliged to continuously choose every action since we can use those observed in the past, hence being able to eliminate stochastic policies and allowing for $\iota = 0$ . + +On the other hand, our approach employs the stronger class of belief-based policies. This comes at the cost of requiring an assumption on the minimum value of the transition model (as also done in SEEU) in order to bound the error of the estimated belief vector, as explained in Section 6 of the main paper. + +Both SEEU and SM-UCRL subsume Assumption 4.3. We show here how both the SEEU and the SM-UCRL algorithms rely on assumptions that imply our Assumption 4.3. In particular: + +- the SEEU algorithm directly employs the one-step reachability assumption (our Assumption 6.1) for learnability. Differently, we use the weaker Assumption 4.3 for learning the model parameters, and then require the stronger one-step reachability assumption to ensure guarantees for the Mixed Spectral UCRL algorithm. +- the SM-UCRL algorithm assumes standard ergodicity assumptions (not conditioned on action) but restricts to the class of stochastic policies $(\iota > 0)$ . Under this set of stochastic policies and the ergodicity assumption, the state-action distribution $d_{\infty}^{\pi}(s,a)$ always exists and satisfies $d_{\infty}^{\pi}(s,a) > 0$ for any $(s,a) \in S \times \mathcal{A}$ . Consequently, the conditional state distribution $\omega^{(a,\pi)}$ is always well-defined (since, under the considered policy class, $d_{\infty}^{\pi}(a) > 0$ for any $a \in \mathcal{A}$ ) and its elements are always strictly positive, hence satisfying Assumption 4.3. + +Finally, we remark that the set of Assumptions 4.1, 4.2 and 4.3 employed in our work constitute the minimum working assumptions for learning in the infinite-horizon average-reward POMDP setting. + +# H.2 Comparison between Spectral Decomposition and Maximum-likelihood Approaches + +Besides Spectral Decomposition techniques, other methods can be used for parameter estimation. Among the most common, we highlight those based on Maximum-Likelihood estimation mainly adopted in the episodic setting, such as the OOM-UCB [19] or the Optimistic-MLE [20] algorithms. We describe below the two key differences between these approaches: + +1. MLE-based methods lack Estimation Guarantees for Latent Variable Models, differently from Spectral Methods. + +MLE-based methods are not guaranteed to recover the original parameters $(\mathbb{O},\mathbb{T})$ when estimating latent variable models, such as HMMs or POMDPs. In contrast, Spectral Decomposition methods provide finite-sample guarantees for such models and represent the most computationally efficient methods for estimating such models. Notably, MLE-based approaches are used to learn an alternative POMDP parametrization known as the Observable Operator Model (OOM) for which finite-sample guarantees can be derived by only employing the $\alpha$ -weakly revealing condition. Crucially, it is important to highlight that knowledge of the Observable Operators does not alone allow recovering the original POMDP parameters $(\mathbb{O},\mathbb{T})$ for which instead different techniques (Spectral Decomposition) and further assumptions (invertibility of the transition matrices and ergodicity-like conditions) are needed to ensure estimation guarantees. + +2. MLE-based approaches typically addresses the finite-horizon setting, while our focus is on the infinite-horizon one. + +The difference between the two settings also lies in the class of optimal policies. Indeed, while the best policy in the finite-horizon case depends on the sequence of observations and actions of limited length (bounded by the episode length $H$ ) and does not rely on a notion of belief state, the optimal policy for the infinite-horizon case depends on maintaining and updating a belief vector over the hidden states. Since belief updates rely on the Bayes' rule, which in turn requires estimates of both the observation and transition models, we need to use estimation methods with finite-sample guarantees (such as Spectral Methods) to recover the model parameters. This is in contrast to the finite-horizon setting, where guarantees on the policy suboptimality can be related to the quality of OOM estimates. + +# I Discussion on Computational Complexity + +We discuss here the computational complexity of the Mixed Spectral Estimation procedure. The computational complexity of this approach is comparable with the estimation approaches used both by SEEU and SM-UCRL since all of them rely on the underlying tensor decomposition. The overall computational complexity of the method scales as $\mathcal{O}(A\max \{O^3,S^5\log S\})$ , where: + +- The complexity scales linearly with the number of actions since SD is performed separately for each action $a \in \mathcal{A}$ , +- The first term in the max arises from inverting the covariance matrices having order $O$ appearing in Equation (6), +- The second term comes from the RTP strategy introduced in [1], which is used as a subroutine by the Mixed Spectral Estimation strategy. This method operates on a symmetric and whitened three-order tensor $^{18}$ with dimension $\mathbb{R}^{S\times S\times S}$ . Hence, each operation requires $\mathcal{O}(S^3)$ computations, and, assuming each eigenvector is computed from roughly $\mathcal{O}(S)$ initializations, with $\mathcal{O}(\log S)$ power iterations per initialization, the total time for obtaining the $S$ different eigenvector/eigenvalue pairs is $\mathcal{O}(S^5\log S)$ . Some optimization techniques can reduce this complexity to $\mathcal{O}(S^4)$ . + +We refer to [1] for a more detailed discussion on this matter. + +# J Additional Simulations and Simulation Details + +This section provides details about the numerical simulations reported in the main paper. The simulations illustrated in this work have been run on an 88 Intel(R) Xeon(R) CPU E7-8880 v4 @ 2.20GHz CPUs with 94 GB of RAM. + +The code can be found at https://github.com/alesnow97/Spectral_Learning_POMDP.git. + +Transition and Observation Model Generation. For the generation of the different POMDPs, we adopted a similar approach to the one followed in [25]. The matrices of both the observation and transition models are randomly generated, and successive modifications are applied: + +- Transition model $\mathbb{T}_a$ : we set a minimum value for each cell of the matrix that should be at least $\epsilon = 1 / (10S)$ . +- Observation model $\mathbb{O}$ : for each state, we define a subset of observations that may be observed with higher probability with respect to the others. This caveat improves the informativeness of the observation model, hence avoiding matrices with zero (or close to zero) minimum singular values. + +# J.1 Simulations on Estimation Error of the Mixed Spectral Estimation Algorithm + +In this section, we report further experiments on estimation errors of POMDP instances of different sizes. In particular, we analyze the behavior of our estimation approach with both smaller and larger instances with respect to the one presented in the main paper. The results are presented in Figure 3 and are expressed in terms of the Frobenius norm. + +For the experiment on the left, we measured the estimation error over 10 different episodes, each one having size $10^{5}$ steps. Since the considered POMDP is smaller with respect to the others ( $S = 3$ , $A = 2$ , $O = 5$ ), fewer samples are required to achieve good model estimates. + +For the experiment on the right, we consider a larger POMDP instance $(S = 5, A = 5, O = 5)$ and we run our simulation across 30 episodes, each one of length $1.2 * 10^{6}$ steps. As expected, the estimation process in this case has more noise, but a decrease in the estimation error is evident across the different episodes. + +How Policies Vary across Episodes. The change of belief-policies across the different episodes is implemented in the following way. + +![](images/27b6b261b5647e99477343de5192b67d3350241dd8274b78358c974a0298e81a.jpg) +Figure 3: Frobenius norm of the estimation error of two different POMDP instances. For the instance on the left we have $S = 3$ , $A = 2$ , $O = 5$ , for the one on the right $S = 5$ , $A = 3$ , $O = 5$ . (10 runs, 95 %c.i.). + +![](images/68da9799a9eaed4e164caa10231a2406c8d9a7575fe86e762e4cb37a38781619.jpg) + +![](images/e09ba943b9c63c60f59b4eb36947376dce91097c43afdbb50dcb0c11e907d7ce.jpg) +Figure 4: Regret comparison on a POMDP with $S = 3$ , $A = 3$ , $O = 4$ violating Assumption 6.1 (10 runs, 95 %c.i.). + +(i) Each policy has an internal transition and observation model that it uses to update its belief. When the episode changes, we change as well these components. We remark that these models are only used for the internal update of the belief and are independent of the transition and observation model of the interacting POMDP instance. +(ii) Each policy has an internal vector $\boldsymbol{r} \in \mathbb{R}^{O}$ of rewards associated to each observation. At each step, the chosen action is the one maximizing the expected reward in the next time step. When the episode changes, we change as well the internal reward vector $\boldsymbol{r}$ . As a last point, in order to ensure enough exploration of all actions, the policy has a minimum probability of choosing every action at each time step. + +# J.2 Simulations and Details on Regret Experiments + +For the experiments on the regret, we adopted the following hyperparameters for the different algorithms. + +- Mixed Spectral UCRL: length of initial episode $T_{0} = 3 * 10^{5}$ +- SM-UCRL: length of initial episode $T_0 = 3 * 10^5$ , minimum action probability $\iota = 0.02$ +- SEEU: length of exploration phase $\tau_{1} = 10^{5}$ , length of initial exploitation phase $\tau_{2} = 3 * 10^{5}$ . At each new episode $l$ , the length of the exploitation phase is computed as $\sqrt{l + 1} \tau_{2}$ , as defined in the original work. + +Concerning the computation of the optimal policy, for both the SEEU and the Mixed Spectral UCRL algorithm, we adopted the following approach. Since there is uncertainty in the model parameters, the Extended Value Iteration algorithm [14] should be used to find a robust policy. However, in practice, since we are in the POMDP setting, our approach consists in sampling multiple POMDPs within the confidence region $C_l(\delta_l)$ , discretize the belief space of each of the corresponding belief MDPs, find the corresponding best policy by using Value Iteration on each discretized MDP, and finally return the best among them. Similar approaches are also employed in [3]. For the considered simulations, + +we adopted a discretization step size of 0.04. + +Since the SM-UCRL algorithm relies on memoryless policies, we applied a similar sampling procedure and then directly the Value Iteration algorithm, replacing the state space with the observation space. + +By following the suggestions in [3], we replaced the theoretical bounds with smaller values. This approach is commonly used in experimental comparisons in these settings and generally results in either a regret with larger multiplicative constants or guarantees holding with a lower probability. + +Regret Experiment Violating Assumption 6.1. Our belief is that Assumption 6.1 can be relaxed in practice while still guaranteeing sublinear regret, however it is hard to remove it from a technical perspective. + +To corroborate our intuition, we run new regret experiments on a POMDP instance that violates Assumption 6.1. 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Vishnoi* + +Yale University + +Van H. Vu + +Yale University + +# Abstract + +A central challenge in machine learning is to understand how noise or measurement errors affect low-rank approximations—particularly in the spectral norm. This question is especially important in differentially private low-rank approximation, where one aims to preserve the top- $p$ structure of a data-derived matrix while ensuring privacy. Prior work often analyzes Frobenius norm error or changes in reconstruction quality, but these metrics can over- or under-estimate true subspace distortion. The spectral norm, by contrast, captures worst-case directional error and provides the strongest utility guarantees. We establish new high-probability spectral-norm perturbation bounds for symmetric matrices that refine the classical Eckart-Young-Mirsky theorem and explicitly capture interactions between a matrix $A \in \mathbb{R}^{n \times n}$ and an arbitrary symmetric perturbation $E$ . Under mild eigengap and norm conditions, our bounds yield sharp estimates for $\|(A + E)_p - A_p\|$ , where $A_p$ is the best rank- $p$ approximation of $A$ , with improvements of up to a factor of $\sqrt{n}$ . As an application, we derive improved utility guarantees for differentially private PCA, resolving an open problem in the literature. Our analysis relies on a novel contour bootstrapping method from complex analysis and extends it to a broad class of spectral functionals, including polynomials and matrix exponentials. Empirical results on real-world datasets confirm that our bounds closely track the actual spectral error under diverse perturbation regimes. + +# 1 Introduction + +Low-rank approximation is a foundational technique in machine learning, data science, and numerical linear algebra, with applications ranging from dimensionality reduction and clustering to recommendation systems and privacy-preserving data analysis [1, 4, 5, 14, 21, 23, 24, 42, 45]. A common setting involves a real symmetric matrix $A \in \mathbb{R}^{n \times n}$ , such as a sample covariance matrix derived from high-dimensional data. Let $\lambda_1 \geq \dots \geq \lambda_n$ denote the eigenvalues of $A$ , with corresponding orthonormal eigenvectors $u_1, \ldots, u_n$ . The best rank- $p$ approximation of $A$ is denoted by $A_p := \sum_{i=1}^p \lambda_i u_i u_i^\top$ . This approximation solves the optimization problem $A_p = \arg \min_{\mathrm{rank}(B) \leq p} \|A - B\|$ , where the norm can be any unitarily invariant norm [7, 10]. In particular, $A_p$ minimizes both the spectral norm $\| \cdot \|$ , measuring worst-case error, and the Frobenius norm $\| \cdot \|_F$ , measuring average deviation. + +In many applications, the matrix $A$ is not directly available—it may be corrupted by noise, compressed for efficiency, or randomized to preserve privacy. A standard model introduces a symmetric perturbation $E$ , yielding the observed matrix $\tilde{A} := A + E$ . The approximation $\tilde{A}_p$ , computed from $\tilde{A}$ , is often used in downstream learning and inference. This leads to a central question: How does the perturbation $E$ affect the top- $p$ approximation $A_p$ ? Understanding the deviation $\| \tilde{A}_p - A_p \|$ is critical for ensuring the reliability and robustness of low-rank methods under noise. + +Motivating application: differential privacy. The stability under perturbations is especially important when the matrix $A$ encodes sensitive information, such as user behavior or medical data. In such settings, even low-rank approximations of $A$ can inadvertently leak private information [6]. To address this risk, differential privacy (DP) [14] has become the standard framework for designing privacy-preserving algorithms. Several mechanisms have been developed to release private low-rank approximations while satisfying DP guarantees [8, 9, 15, 25, 29, 31, 34, 39]. A canonical method, introduced in [15], adds a symmetric noise matrix $E$ with i.i.d. Gaussian entries to the input matrix $A$ , yielding the perturbed matrix $\tilde{A} = A + E$ . The algorithm then releases $\tilde{A}_p$ as the privatized output. The utility of such mechanisms is typically assessed by comparing $\tilde{A}_p$ to the ideal (non-private) approximation $A_p$ . Two standard metrics are: (1) the Frobenius norm error $\| \tilde{A}_p - A_p \|_F$ , and (2) the change in reconstruction error $\| A - A_p \|_\star - \| A - \tilde{A}_p \|_\star$ , which measures how much the quality of low-rank approximation degrades due to noise, for a norm $\| \cdot \|_\star$ [3, 11, 15, 29]. These metrics offer insight into the effect of noise on overall variance or total reconstruction error. However, as we explain next, they may fail to capture worst-case directional misalignment, which is often critical for downstream tasks and algorithmic guarantees. + +Limitations of existing utility metrics. The Frobenius norm error and reconstruction error may not be appropriate in applications that rely on the geometry of the top- $p$ eigenspace. In particular, the Frobenius norm may overestimate the impact of noise by up to a factor of $\sqrt{p}$ when the perturbation $E$ lies largely in directions orthogonal to the top- $p$ subspace. The reconstruction error metric can underestimate subspace deviation—sometimes dramatically. In some cases, it remains small (or even zero) despite substantial rotation in the top- $p$ eigenspace. (See Sections B for concrete illustrations.) These limitations motivate the use of the spectral norm $\| \tilde{A}_p - A_p\|$ , which captures the worst-case directional deviation between the two low-rank approximations. The spectral norm also governs algorithmic robustness in many downstream applications, such as PCA-based learning, private clustering, and subspace tracking. + +A classical spectral norm bound, derived from the Eckart-Young-Mirsky theorem [7, 16], states that $\| \tilde{A}_p - A_p\| \leq 2(\lambda_{p + 1} + \| E\|)$ , which holds for arbitrary matrices and noise. However, such bounds are often pessimistic and fail to exploit the structure of $A$ and $E$ . More refined bounds exist in the Frobenius norm setting. For example, recent work [29, 30] shows that when $A$ is positive semidefinite and has a nontrivial eigengap $\delta_p\coloneqq \lambda_p - \lambda_{p + 1}\geq 4\| E\|$ , and when $E$ is drawn from a complex Gaussian ensemble, one obtains: $\mathbb{E}\| \tilde{A}_p - A_p\| _F = \tilde{O} (\sqrt{p}\cdot \| E\| \cdot \frac{\lambda_p}{\delta_p})$ , which improves on the earlier reconstruction-error-based bounds of [15] by a factor of $\sqrt{p}$ . However, these bounds have important limitations: They hold only in expectation and do not yield high-probability guarantees; They often assume Gaussian noise distributions; They are not spectral norm bounds and therefore do not directly quantify the worst-case impact on the eigenspace. These limitations prompt the following open question, raised in [29, Remark 5.3]: Can one obtain high-probability spectral norm bounds for $\| \tilde{A}_p - A_p\|$ under natural structural assumptions on $A$ and realistic noise models? + +Our contributions. We resolve the open question posed in [29, Remark 5.3], proving new high-probability spectral norm bounds for low-rank approximation under symmetric perturbations. Our results rely on natural structural assumptions on $A$ and $E$ and yield the first such guarantees for differentially private PCA (DP-PCA). + +- Two high-probability spectral norm bounds. Under the same eigengap condition as [29], $\delta_p \coloneqq \lambda_p - \lambda_{p+1} \geq 4\|E\|$ , we prove $\|\tilde{A}_p - A_p\| = O\left(\|E\| \cdot \frac{\lambda_p}{\delta_p}\right)$ and $\|\tilde{A}_p - A_p\| = \tilde{O}\left(\|E\| + r^2 x \cdot \frac{\lambda_p}{\delta_p}\right)$ , where $r$ is the halving distance (a measure of spectral decay) and $x \coloneqq \max_{i,j \leq r} |u_i^\top E u_j|$ quantifies noise-eigenspace alignment (Theorems 2.1-2.2). In addition, our contour-based framework extends to a broader class of spectral functionals $f(A)$ (beyond $f(A) = A$ ), encompassing matrix powers, exponentials, and trigonometric transforms; see Theorem 2.3. +- Spectral utility bounds for DP-PCA. Our first bound yields a high-probability spectral norm utility guarantee for differentially private PCA under sub-Gaussian noise, improving existing Frobenius-norm bounds by up to a factor of $\sqrt{p}$ (Corollary 2.4). While prior work has achieved spectral norm guarantees in iterative or multi-pass settings [17, 18], our contribution concerns the direct noise-addition model, where this appears to be the first such result. For matrices with low stable rank and weak eigenspace-noise interaction, our second bound further improves by up to $\sqrt{n}$ . + +- Novel analytical technique: contour bootstrapping. Our proof relies on a contour bootstrapping argument (Lemma 3.1), which provides a new way to analyze the contour representation of perturbations [19, 26, 35], enabling analysis of a broader class of spectral functionals (Theorem 2.3). The bootstrapping argument here is a generalization of the argument used to handle eigenspaces perturbation introduced in [38]. +- Empirical validation. We benchmark our bounds on real covariance matrices under both Gaussian and Rademacher noise. Across datasets and noise regimes, the predicted error closely matches empirical behavior and consistently surpasses classical baselines, confirming the sharpness and robustness of our theoretical results (Section 4). + +# 2 Main results + +Main spectral norm bound. For clarity, we state our main bounds assuming $A \in \mathbb{R}^{n \times n}$ is positive semi-definite (PSD); extensions to symmetric matrices appear in Section D. Let $\lambda_1 \geq \dots \geq \lambda_n \geq 0$ be the eigenvalues of $A$ , with corresponding orthonormal eigenvectors $u_1, \ldots, u_n$ , and define the eigengap $\delta_k := \lambda_k - \lambda_{k+1}$ . Given a real symmetric perturbation matrix $E$ , we let $\tilde{A} := A + E$ , and define $A_p$ and $\tilde{A}_p$ as the best rank- $p$ approximations of $A$ and $\tilde{A}$ , respectively. Our goal is to bound the spectral error $\| \tilde{A}_p - A_p \|$ . + +Theorem 2.1 (Main spectral bound - PSD). If $4\| E\| \leq \delta_p$ , then: $\| \tilde{A}_p - A_p\| \leq O(\| E\| \cdot \frac{\lambda_p}{\delta_p})$ + +The $O(\cdot)$ notation here hides a small universal constant (less than 7), which we have not optimized; see Section D.1 for the proof of the generalization to the symmetric setting, of which this theorem is a special case. For Wigner noise—i.e., a symmetric matrix $E$ with i.i.d. sub-Gaussian entries of mean 0 and variance 1—we have $\| E \| = (2 + o(1)) \sqrt{n}$ with high probability [41, 43], so Theorem 2.1 reduces to $\| \tilde{A}_p - A_p \| = O\left(\sqrt{n} \frac{\lambda_p}{\delta_p}\right)$ . The right-hand side is explicitly noise-dependent, addressing a key limitation of the classical Eckart-Young-Mirsky bound. Moreover, in many widely studied structured models (e.g., spiked covariance, stochastic block, and graph Laplacian models), one typically has $\lambda_p = O(\delta_p)$ , yielding the clean bound $O(\| E \|)$ . This rate is theoretically tight: for instance, when $A$ is a PSD diagonal matrix and $E = \mu I_n$ for some $\mu > 0$ , we have $\| \tilde{A}_p - A_p \| = \mu = \| E \|$ . + +Gap condition. Our assumption $4\| E\| < \delta_p$ aligns with standard conditions in prior work, including [29, 30], and is satisfied in many well-studied matrix models—such as spiked covariance (Wishart) models, deformed Wigner ensembles, stochastic block models, and kernel matrices for clustering. It also arises naturally in classical perturbation theory [12, 26, 28]. Empirical analyses [29, Section B] further show that this condition holds for real-world datasets commonly used in private matrix approximation (e.g., the 1990 U.S. Census and the UCI Adult dataset [3, 11]). Hence, Theorem 2.1 operates under a mild and broadly applicable assumption, satisfied across both theoretical models and practical benchmarks. + +Comparison to the Eckart-Young-Mirsky bound. Using $\lambda_p = \delta_p + \lambda_{p+1}$ , Theorem 2.1 rewrites as $\| \tilde{A}_p - A_p \| = O(\| E \| + \lambda_{p+1} \cdot \frac{\| E \|}{\delta_p})$ . This improves on the E-Y-M bound $O(\| E \| + \lambda_{p+1})$ when $\lambda_{p+1} \gg \| E \|$ , by a factor of $\min \left\{ \frac{\lambda_{p+1}}{\|E \|}, \frac{\delta_p}{\|E \|} \right\}$ . For example, consider a matrix with spectrum $\{10n, 9n, \dots, n, n/2, 1, \dots, 1\}$ and $p = 10$ . For Gaussian noise with $\| E \| = O(\sqrt{n})$ , E-Y-M yields $O(n)$ error, while our bound gives $O(\sqrt{n})$ , a $\sqrt{n}$ -factor gain. + +Comparison to Mangoubi-Vishnoi bounds [29, 30]. Our bound also improves upon the Frobenius norm bounds of [29, 30], which under the same gap assumption yield: $\mathbb{E}\| \tilde{A}_p - A_p\| _F = \tilde{O} (\sqrt{p}\| E\| \cdot \frac{\lambda_p}{\delta_p})$ . We eliminate the $\sqrt{p}$ factor, upgrade from expectation to high probability, and support real-valued, non-Gaussian noise models. A more detailed comparison appears later in this section (Corollary 2.4), where we analyze implications for differentially private PCA. + +Proof technique: contour bootstrapping. Unlike prior analyses [29, 30], which rely on Dyson Brownian motion and tools from random matrix theory (see Section A, our proof of Theorem 2.1 uses a contour-integral representation of the rank- $p$ projector. This approach, which we call contour bootstrapping, isolates the top- $p$ eigenspace via complex-analytic techniques and avoids power-series or Davis-Kahan-type expansions. It enables tighter, structure-aware spectral bounds and ex + +tends naturally to refined perturbation results (Theorem 2.2) and general spectral functionals (Theorem 2.3). Full details appear in Section 3. + +Refined bound via eigenspace interaction. To sharpen our analysis, we incorporate fine-grained structure of the eigenspace and its interaction with the noise. Inspired by the recent works [33, 37], we start with the observation that the rank- $p$ perturbation is primarily influenced by the cluster of eigenvalues near $\lambda_{p}$ , and the interaction between $E$ and the corresponding eigenvectors. To control these factors, we define the halving distance $r$ (w.r.t the index $p$ ) as the smallest integer such that $\lambda_{r+1} \leq \lambda_{p}/2$ , and interaction term $x := \max_{1 \leq i,j \leq r} |u_i^\top Eu_j|$ , measuring the alignment between the noise $E$ and the top- $r$ eigenvectors of $A$ . This yields a refined spectral norm bound: + +Theorem 2.2 (Interaction-aware bound). If $4\| E\| \leq \delta_p$ , then $\| \tilde{A}_p - A_p\| \leq \tilde{O} (\| E\| +r^2 x\cdot \frac{\lambda_p}{\delta_p})$ + +See Section D.2 for the proof and its generalization to the symmetric setting. This bound improves upon the basic eigengap bound $O\left(\| E\| \cdot \frac{\lambda_p}{\delta_p}\right)$ when the interaction term $r^2 x$ is small. This occurs, for instance, when (i) $A$ has low stable rank or clustered eigenvalues (e.g., spiked models, multi-cluster Laplacians), (ii) the noise $E$ is random and approximately orthogonal to the leading eigenspace, or (iii) $\lambda_{p} / \delta_{p}$ is large but $x = \tilde{O}(1)$ and $r = \tilde{O}(1)$ . In such regimes, the bound simplifies to $\tilde{O}\left(\| E\| + \frac{\lambda_p}{\delta_p}\right)$ , yielding up to a $\sqrt{n}$ -factor improvement over Theorem 2.1. This highlights the benefit of explicitly incorporating spectral decay and noise-eigenspace alignment when analyzing noise-robust low-rank approximations. + +In practice, many public DP datasets (e.g., Census, Adult, KDD) have small dimensions and modest eigenspace decay, the simple bound is more effective. However, the refined bound becomes especially informative in large-scale or synthetically structured settings. Thus, the two bounds are best viewed as complementary: the first is robust and broadly applicable, while the second highlights structural regimes where stronger stability is provable. + +Extension to spectral functionals. Beyond approximating $A$ itself, many applications involve low-rank approximations of spectral functions $f(A)$ , such as $A^k$ , $\exp(A)$ , or $\cos(A)$ ; see [7, 44]. Our contour-based analysis extends naturally to this broader setting. Let $f_p(A) := \sum_{i=1}^{p} f(\lambda_i) u_i u_i^\top$ denote the best rank- $p$ approximation of $f(A)$ . We obtain the following general perturbation bound. + +Theorem 2.3 (Perturbation bounds for general functions). If $4\| E\| \leq \delta_p$ , then + +$$ +\left\| f _ {p} (\tilde {A}) - f _ {p} (A) \right\| \leq O \left(\max _ {z \in \Gamma_ {1}} \| f (z) \| \cdot \frac {\| E \|}{\delta_ {p}}\right), +$$ + +where $\Gamma_{1}$ is the rectangle with vertices $(x_0,T),(x_1,T),(x_1, - T),(x_0, - T)$ with + +$$ +x _ {0} := \lambda_ {p} - \frac {\delta_ {p}}{2}, x _ {1} := 2 \lambda_ {1}, T := 2 \lambda_ {1}. +$$ + +The $O(\cdot)$ notation hides a small universal constant (less than 4), which we have not attempted to optimize; see Section F for details. For example, let $f(z) = z^3$ , so that $f_{p}(\tilde{A})$ and $f_{p}(A)$ correspond to the best rank- $p$ approximations of $\tilde{A}^3$ and $A^3$ , respectively. Since $\max_{z \in \Gamma_1} \|f(z)\| \leq 64\|A\|^3$ , Theorem 2.3 yields $\| \tilde{A}_p^3 - A_p^3 \| = O\left(\| A\|^3 \cdot \| E\| / \delta_p\right)$ . This result applies to many important classes of functions—e.g., polynomials, exponentials, and trigonometric functions—and hence we expect it to be broadly useful. However, Theorem 2.3 does not apply to non-entire functions such as $f(z) = z^c$ for non-integer $c$ , where singularities obstruct the contour representation (1). In particular, when $c < 0$ , the expression $f_{p}(A)$ is no longer the best rank- $p$ approximation to $f(A)$ , so the conclusion of Theorem 2.3 is not meaningful in that setting. We note that in a related work [36], the first two authors present an extension of the setting $f(z) = z^{-1}$ . + +Application: differentially private low-rank approximation. We now apply our spectral norm bound to analyze a standard differentially private (DP) mechanism for releasing a low-rank approximation of a sensitive matrix $A$ , commonly assumed to be a sample covariance matrix and hence PSD. Under $(\varepsilon, \delta)$ -DP [14], the Gaussian mechanism releases $\tilde{A} \coloneqq A + E$ , where $E$ is a symmetric matrix with i.i.d. Gaussian entries scaled to sensitivity $\Delta = O(\sqrt{\log(1 / \delta)} / \varepsilon)$ . A common post-processing step is to compute $\tilde{A}_p$ , the best rank- $p$ approximation of $\tilde{A}$ . Prior analyses [3, 15, 30] focused primarily on Frobenius norm or reconstruction error. For instance, [30] showed that under complex Wigner noise and a moderate eigengap, $\mathbb{E}\| \tilde{A}_p - A_p\|_F \leq \frac{\sqrt{pn}\lambda_p}{\delta_p}$ up to lower-order terms. + +Since $\| \tilde{A}_p - A_p \| \leq \| \tilde{A}_p - A_p \|_F$ , the above inequality implies an expected spectral norm error of $\tilde{O}\left(\sqrt{pn} \frac{\lambda_p}{\delta_p}\right)$ . In contrast, our bound yields the following high-probability spectral norm guarantee: + +Corollary 2.4 (Application to differential privacy). Let $A$ be PSD and $E$ be a real or complex Wigner matrix. If $\delta_p \geq 8.01\sqrt{n}$ , then with probability $1 - o(1)$ , $\| \tilde{A}_p - A_p \| \leq O(\sqrt{n} \cdot \frac{\lambda_p}{\delta_p})$ . + +This follows directly from Theorem 2.1, using the fact that $\| E \| = O(\sqrt{n})$ with high probability for Wigner matrices [40, 43]. Compared to [30], this result provides a spectral norm (rather than Frobenius) guarantee, holds with high probability instead of in expectation, applies to both real and complex Wigner noise, removes the $\log^{\log \log n} n$ factor, and eliminates restrictive assumptions such as $\lambda_1 \leq n^{50}$ . It also improves the dependence on $p$ by a factor of $\sqrt{p}$ , thereby resolving the open question posed in [30, Remark 5.3]. + +The spectral norm better captures subspace distortion, which is critical in applications like private PCA. Unlike Frobenius or reconstruction error—both of which may remain small even when $\tilde{A}_p$ deviates significantly from the true top- $p$ eigenspace—the spectral norm reflects worst-case directional error and is thus a more reliable utility metric. This distinction is empirically validated in Figure 3. Moreover, Corollary 2.4 further yields high-probability Frobenius norm and reconstruction error bounds on the perturbation of low-rank approximations: + +$$ +\| \tilde {A} _ {p} - A _ {p} \| _ {F} \leq O \left(\sqrt {p n} \cdot \frac {\lambda_ {p}}{\delta_ {p}}\right), \text {a n d} | \| \tilde {A} _ {p} - A \| - \| A _ {p} - A \| | \leq O \left(\sqrt {n} \cdot \frac {\lambda_ {p}}{\delta_ {p}}\right). +$$ + +Finally, while Corollary 2.4 is stated for sub-Gaussian noise, Theorem 2.1 extends to any symmetric perturbation satisfying the norm and gap conditions, including subsampled or quantized Gaussians and Laplace noise. We leave the detailed analysis of these settings to future work. + +Table 1: Summary table of perturbation bounds on ${\widetilde{A}}_{p} - {A}_{p}$ for noise $E$ . + +
Bound typeNormNoise modelAssumptionExtra factor vs ||E||
EYM boundHigh-probabilitySpectralReal and ComplexNoneO(1 + λp+1/||E||)
M-V bound [29]ExpectationFrobeniusGOE (real)δi>4||E|| ∀1≤i≤pO(√pλp/δp)
M-V bound [30]ExpectationFrobeniusGUE (complex)δp>2||E||, λ1<n50O(√pλp/δp)
Thm. 2.1High-probabilitySpectralReal and Complexδp>4||E||O(λp/δp)
Thm. 2.2High-probabilitySpectralsub-Gaussianδp>4||E||, rankA=O(1)O(1 + λp/δp||E||)
+ +"EYM" and "M-V" denote the Eckart-Young-Mirsky and [29, 30] bounds, respectively. + +Alternative methods for approximating $A_p$ . Hardt and Price [17, 18] proposed a random iterative method which, under the condition $\delta_p \gg \sqrt{n} \log n$ , produces a rank- $k$ approximation $A'$ of $A_p$ with $k = p + O(1)$ , satisfying the trade-off bound $\| A' - A_p \| = \tilde{O}\left(\sqrt{n} \frac{\lambda_1}{\delta_p} \max_{1 \leq i \leq n} \| u_i \|_{\infty}\right)$ , where $u_i$ denotes the eigenvectors of $A$ . + +If at least one eigenvector $u_{i}$ is localized (i.e., $\max_{1\leq i\leq n}\| u_i\|_\infty = 1 / \tilde{O} (1))$ , this simplifies to $\tilde{O}\Big(\sqrt{n}\frac{\lambda_1}{\delta_p}\Big)$ . In this regime, Theorem 2.1 achieves a smaller bound by a factor of $\tilde{O} (\lambda_1 / \lambda_p)$ up to $\sqrt{n}$ when $\lambda_{1} = \Theta (n)$ and $\lambda_{p} = \Theta (\sqrt{n})$ . Furthermore, Theorem 2.2 provides an additional improvement by a factor of $O\Big(\min \Big\{\frac{\sqrt{n}}{r^2},\frac{\lambda_1}{\delta_p}\Big\} \Big)$ , which can reach $\sqrt{n}$ when $r = \tilde{O} (1)$ and $\delta_p = \Theta (\sqrt{n})$ a common regime in high-dimensional data. + +If all eigenvectors $u_{i}$ are delocalized (i.e., $\max_{1\leq i\leq n}\| u_i\|_\infty = \tilde{O}(1) / \sqrt{n}$ ), the Hardt-Price bound reduces to $\tilde{O}(\lambda_1/\delta_p)$ . Theorem 2.1 achieves a comparable rate when $\sigma_{1} = \Theta(n)$ and $\lambda_{p} = c\delta_{p} = \Theta(\sqrt{n})$ , while Theorem 2.2 yields an improvement by a factor of $\lambda_{1}/\lambda_{p}$ whenever $r = \tilde{O}(1)$ , i.e., when $A$ is approximately low-rank. + +# 3 Proof outline + +In the preceding section, we stated our main results—Theorems 2.1, 2.2, and 2.3. Here, we first sketch the key ideas behind the proof of Theorem 2.1, then adapt the same framework, with minor refinements, to derive Theorems 2.2 and 2.3. + +The proof of Theorem 2.1 proceeds in three main steps. First, using the contour method, we obtain the contour-based bound of our perturbation $\| \tilde{A}_p - A_p\| \leq F(z)\coloneqq \frac{1}{2\pi\mathbf{i}}\| \int_{\Gamma}z[(zI - \tilde{A})^{-1} - (zI - A)^{-1}]\| dz$ . Here $\Gamma$ is a contour on the complex plane, isolating the $p$ -leading eigenvalues of $A$ and $\tilde{A}$ . This contour step captures the $A - E$ interaction that the Eckart-Young-Mirsky bound omits (see Appendix A). Secondly, we develop the contour bootstrapping technique (Lemma 3.1), which under the gap assumption $4\| E\| \leq \delta_{p}$ , yields $F(z)\leq 2F_{1}(z)$ with $F_{1}(z)\coloneqq \int_{\Gamma}\| z(zI - A)^{-1}E(zI - A)^{-1}\| |dz|$ . This technique (valid for any entire function $f$ ) replaces the traditional series expansions and the heavy analysis of the matrix-derivative operator (the limitation of the Mangoubi-Vishnoi approach [29, 30], Appendix A) with a computable quantity. Third, we construct a bespoke contour $\Gamma$ — one specifically tailored so that the top- $p$ eigenvalues of $A$ and $\tilde{A}$ lie at prescribed distances from its sides. This precise alignment makes the integral defining $F_{1}(z)$ both tractable and essentially optimal, yielding a tight perturbation bound. + +Step 1: Representing $\| f_p(\tilde{A}) - f_p(A)\|$ via the classical contour method. Let $\lambda_1\geq \dots \geq$ $\lambda_{n}$ be the eigenvalues of $A$ with the corresponding eigenvectors $\{u_i\}_{i = 1}^n$ . We now present the contour method to bound matrix perturbations in the spectral norm. Let $\Gamma$ be a contour in $\mathbb{C}$ that encloses $\lambda_1,\lambda_2,\ldots ,\lambda_p$ and excludes $\lambda_{p + 1},\lambda_{p + 2},\ldots ,\lambda_n$ . Let $f$ be any entire function and recall $f_{p}(A) = \sum_{i = 1}^{p}f(\lambda_{p})u_{i}u_{i}^{\top}$ . Since $f$ is analytic on the whole plane $\mathbb{C}$ , the well-known contour integral representation [19, 26, 35] gives us: + +$$ +\frac {1}{2 \pi \mathbf {i}} \int_ {\Gamma} f (z) (z I - A) ^ {- 1} d z = \sum_ {i = 1} ^ {p} f (\lambda_ {i}) u _ {i} u _ {i} ^ {\top} = f _ {p} (A). +$$ + +Let $\tilde{\lambda}_1 \geq \dots \geq \tilde{\lambda}_n$ denote the eigenvalue of $\tilde{A}$ with the corresponding eigenvectors $\tilde{u}_1, \tilde{u}_2, \ldots, \tilde{u}_n$ . The construction of $\Gamma$ (presented later) and the gap assumption $4\|E\| < \delta_p$ ensure that the eigenvalues $\tilde{\lambda}_i$ for $1 \leq i \leq p$ lie inside $\Gamma$ , while all $\tilde{\lambda}_j$ for $j > p$ remain outside. Then, similarly, we have $\frac{1}{2\pi\mathrm{i}} \int_{\Gamma} f(z)(zI - \tilde{A})^{-1} dz = \sum_{i=1}^{p} f(\tilde{\lambda}_i) \tilde{u}_i \tilde{u}_i^\top \coloneqq f_p(\tilde{A})$ . Thus, we obtain the following contour identity for the perturbation: + +$$ +f _ {p} (\tilde {A}) - f _ {p} (A) = \frac {1}{2 \pi \mathbf {i}} \int_ {\Gamma} f (z) \left[ \left(z I - \tilde {A}\right) ^ {- 1} - \left(z I - A\right) ^ {- 1} \right] | d z |. \tag {1} +$$ + +Now we bound the perturbation by the corresponding integral + +$$ +\left\| f _ {p} (\tilde {A}) - f _ {p} (A) \right\| \leq \frac {1}{2 \pi} \int_ {\Gamma} \| f (z) [ (z I - \tilde {A}) ^ {- 1} - (z I - A) ^ {- 1} ] \| d z =: F (f). \tag {2} +$$ + +This inequality makes the interaction of $A$ and $E$ explicit and is widely used in functional perturbation analysis, e.g., [19, 26, 28, 32, 33, 38]. However, obtaining a sharp bound on its right-hand side remains a formidable analytical challenge. + +Step 2: Bounding $F \leq 2F_{1}$ via the contour bootstrapping method. Attempts to control $F(f)$ , the right-hand side of (2), often use series expansion and analytical tools. By repeatedly applying the resolvent formula, one can expand $f(z)[(zI - \tilde{A})^{-1} - (zI - A)^{-1}]$ into $\sum_{s=1}^{\infty} f(z)(zI - A)^{-1}[E(zI - A)^{-1}]^{s}$ . This yields the bound: + +$$ +F (f) \leq \sum_ {s = 1} ^ {\infty} F _ {s} (f), \text {w h e r e} F _ {s} (f) = \frac {1}{2 \pi} \int_ {\Gamma} \left\| f (z) (z I - A) ^ {- 1} [ E (z I - A) ^ {- 1} ] ^ {s} \right\| | d z |. +$$ + +One needs to estimate $F_{s}(f)$ for each $s$ . For example, when $f(z) = 1$ , [26, Part 2] bounds $F_{s}(1)$ by $O\left(\| E\|^{s}\int_{\Gamma}\frac{|dz|}{\min_{i\in[n]}|z - \lambda_{i}|^{s + 1}}\right) = O\left[(||E|| / \delta_{p})^{s}\right]$ , where $\Gamma$ is a union of vertical lines isolating $\{\lambda_i,i\in p\}$ , yielding the Davis-Kahan bound $O\left(\| E\| /\delta_p\right)$ . However, for $f(z) = z$ (relevant for low-rank perturbations), this approach fails as $|z|\to \infty$ . These estimates are highly nontrivial and rely on deep analytical techniques, making generalization to arbitrary $f$ challenging. + +Moreover, for $f(z) = 1$ , under certain conditions, the dominant term is $F_{1}(f)$ , i.e., $F(f) = O(F_{1}(f))$ ; see, e.g., [22, 27, 32, 33, 38]. In particular, using contour-boostrapping technique, the authors in [38] proved $F(f(z) = 1) \leq 2F_{1}(f(z) = 1)$ . Inspired by this technique, we prove that $F(f) \leq 2F_{1}(f)$ for any entire function $f$ . + +Lemma 3.1 (Contour bootstrapping for entire function $f$ ). If $\delta_p \geq 4\| E \|$ , then + +$$ +F (f) \leq 2 F _ {1} (f), w h e r e F _ {1} (f) := \frac {1}{2 \pi} \int_ {\Gamma} \left\| f (z) (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \right\| | d z |. +$$ + +Our contour bootstrapping argument is designed to prove Lemma 3.1. Our argument is concise and novel, avoiding the need for series expansion and convergence analysis. In the context of standard low-rank approximations, where $f(z) \equiv z$ and $f_{p}(A) = A_{p}$ , we write $F(z)$ and $F_{1}(z)$ instead of $F(f)$ and $F_{1}(f)$ respectively. + +Step 3: Construction of $\Gamma$ , $F_{1}(z)$ -estimation, and proof completion of Thm. 2.1. Given Lemma 3.1, we now need to carefully choose the contour $\Gamma$ and estimate $F_{1}(f)$ . Constructing $\Gamma$ (so that the perturbation analysis via contour integration provides a sharp bound) is delicate; for example, the classical pick of two vertically parallel lines and any $\Gamma$ placed too near any $\lambda_{i}$ can blow up $F_{1}(z)$ to infinity. Indeed, we tailor $\Gamma$ w.r.t $F_{1}(z)$ as follows. First, we choose $\Gamma$ to be rectangular as this simplifies integration. To control the factor $(zI - A)^{-1}$ in the expression of $F_{1}(f)$ , we need to ensure that the distance $|z - \lambda_{i}|$ for any $z \in \Gamma$ and $i \in [n]$ are relatively large. Since $\Gamma$ separates $\lambda_{p}$ and $\lambda_{p+1}$ , this minimal distance $\min_{z \in \Gamma, i \in [n]} |z - \lambda_{i}|$ cannot exceed $\Theta(\delta_{p})$ . Thus, we simply construct $\Gamma$ through the midpoint $x_{0} = \frac{\lambda_{p} + \lambda_{p+1}}{2}$ . Finally, by setting the contour sufficiently high in the complex plane (while avoiding excessive height to prevent $|f(z)|$ from diverging), we ensure that the primary contribution to the integral is from the vertical segments of $\Gamma$ . This is because the distance $|z - \lambda_{i}|$ is minimized on these segments. Note that, under the assumption $4 \| E \| < \delta_{p}$ , this construction ensures that the $p$ -leading eigenvalues of $A$ and $\tilde{A}$ are well aligned inside the contour. + +Now, in particular, to prove Theorem 2.1, we will estimate + +$$ +2 \pi F _ {1} (z) = \int_ {\Gamma} \| z (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \| | d z |, +$$ + +in which the contour $\Gamma$ is set to be a rectangle with vertices $(x_0, T), (x_1, T), (x_2, -T), (x_3, -T)$ , where $x_0 := \lambda_p - \delta_p / 2$ , $x_1 := 2\lambda_1$ , $T := 2\lambda_1$ . Then, we split $\Gamma$ into four segments: $\Gamma_1 := \{(x_0, t) | -T \leq t \leq T\}$ ; $\Gamma_2 := \{(x, T) | x_0 \leq x \leq x_1\}$ ; $\Gamma_3 := \{(x_1, t) | T \geq t \geq -T\}$ ; $\Gamma_4 := \{(x, -T) | x_1 \geq x \geq x_0\}$ . + +![](images/b408c65b9dc52e83f902af94e7cff1280ab5ca6118fc90308c9b89848adc1dad.jpg) + +Given the construction of $\Gamma$ , we have $2\pi F_{1} = \sum_{k=1}^{4} M_{k}$ , where + +$$ +M _ {k} := \int_ {\Gamma_ {k}} \left\| z (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \right\| | d z |. +$$ + +Intuitively, we set $T, x_{1}$ large ( $= 2\|A\|$ ) so that the main term is the integral along $\Gamma_{1}$ , i.e., $M_{1}$ . Indeed, factoring our $E$ and using the fact that $|z - \lambda_{i}| \geq |z - \lambda_{p}| = \sqrt{\delta_{p}^{2} + t^{2}}$ for all $1 \leq i \leq n$ and $z \in \Gamma_{1} := \{(x_{0},t) | - T \leq t \leq T\}$ , we have $M_{1} \leq \int_{\Gamma_{1}} \|E\| \cdot \frac{|z|}{\min_{i \in [n]} |z - \lambda_{i}|^{2}} |dz| \leq \|E\| \cdot \int_{-T}^{T} \frac{\sqrt{x_{0}^{2} + t^{2}}}{(\delta_{p}/2)^{2} + t^{2}} dt$ . Directly compute the integral $\int_{-T}^{T} \frac{\sqrt{x_{0}^{2} + t^{2}}}{(\delta_{p}/2)^{2} + t^{2}} dt$ (see Section E.3), we obtain: + +$$ +M _ {1} \leq \| E \| \cdot O \left(x _ {0} / \delta_ {p}\right) = O \left(\| E \| \lambda_ {p} / \delta_ {p}\right). +$$ + +By a similar manner, replace $\Gamma_{1}$ by $\Gamma_3\coloneqq \{(x_1,t)| - T\leq t\leq T\}$ , we have + +$$ +M _ {3} \leq \| E \| \cdot \int_ {\Gamma_ {3}} \frac {| z |}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} | d z | \leq \| E \| \cdot \int_ {\Gamma_ {3}} \frac {\sqrt {x _ {1} ^ {2} + t ^ {2}}}{\lambda_ {1} ^ {2} + t ^ {2}} d t, +$$ + +where the last inequality follows the fact that $\min_{i\in [n]}|z - \lambda_i| = |z - \lambda_1| = \sqrt{(x_1 - \lambda_1)^2 + t^2} = \sqrt{\lambda_1^2 + t^2}$ . Directly compute the integral $\int_{-T}^{T}\frac{\sqrt{x_1^2 + t^2}}{\lambda_1^2 + t^2} dt$ (see Section E.3), we obtain: + +$$ +M _ {3} \leq \| E \| \cdot O \left(x _ {1} / \lambda_ {1}\right) = O \left(\| E \|\right). +$$ + +Similarly, $M_2, M_4 = O(\|E\|)$ (Section E.2). These estimates on $M_1, M_2, M_3, M_4$ imply $F_1(z) = O\left(\|E\| \cdot \frac{\lambda_p}{\delta_p}\right)$ , which together with Lemma 3.1 proves Theorem 2.1. + +Proving the contour bootstrapping lemma (Lemma 3.1). The first observation is that using the Sherman-Morrison-Woodbury formula $M^{-1} - (M + N)^{-1} = (M + N)^{-1}NM^{-1}$ [20] and the fact that $\tilde{A} = A + E$ , we obtain + +$$ +(z I - A) ^ {- 1} - (z I - \tilde {A}) ^ {- 1} = (z I - A) ^ {- 1} E (z I - \tilde {A}) ^ {- 1}. +$$ + +Using this, we can rewrite + +$$ +F (f) = \frac {1}{2 \pi} \int_ {\Gamma} \| f (z) (z I - A) ^ {- 1} E (z I - \tilde {A}) ^ {- 1} \| | d z | a s +$$ + +$$ +\frac {1}{2 \pi} \int_ {\Gamma} \| f (z) (z I - A) ^ {- 1} E (z I - A) ^ {- 1} - f (z) (z I - A) ^ {- 1} E [ (z I - A) ^ {- 1} - (z I - \tilde {A}) ^ {- 1} ] \| | d z |. +$$ + +Using triangle inequality, we first see that $F(f)$ is at most + +$$ +\frac {\int_ {\Gamma} \| f (z) (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \| | d z |}{2 \pi} + \underbrace {\int_ {\Gamma} \| f (z) (z I - A) ^ {- 1} E [ (z I - A) ^ {- 1} - (z I - \tilde {A}) ^ {- 1} ] \| | d z |} _ {2 \pi}. +$$ + +Next is the key observation that the second term in the equation above can be rearranged and upper-bounded as follows so that the original perturbation appears again: + +$$ +\frac {\operatorname* {m a x} _ {z \in \Gamma} \left\| (z I - A) ^ {- 1} E \right\|}{2 \pi} \int_ {\Gamma} \| f (z) [ (z I - A) ^ {- 1} - (z I - \tilde {A}) ^ {- 1} ] \| | d z |. +$$ + +Thus, we have + +$$ +F (f) \leq F _ {1} (f) + \max _ {z \in \Gamma} \| (z I - A) ^ {- 1} E \| \cdot F (f). \tag {3} +$$ + +Now we need our gap assumption that $4\| E\| < \delta_p$ and the construction of $\Gamma$ , which imply $\min_{z\in \Gamma ,i\in [n]}|z - \lambda_i|\geq \delta_p / 2\geq 2\| E\|$ . Therefore, we have + +$$ +\max _ {z \in \Gamma} \left\| (z I - A) ^ {- 1} E \right\| \leq \max _ {z \in \Gamma} \| (z I - A) ^ {- 1} \| \cdot \| E \| = \frac {\| E \|}{\min _ {z \in \Gamma , i \in [ n ]} | z - \lambda_ {i} |} \leq \frac {\| E \|}{2 \| E \|} = \frac {1}{2}. +$$ + +Together with (3), it follows that $F(f) \leq F_1(f) + \frac{1}{2} F(f)$ . Therefore, $\frac{1}{2} F(f, S) \leq F_1(f, S)$ . This proves Lemma 3.1.² + +Remark 3.2. Using a similar strategy, one can prove that + +$$ +F _ {1} (f) \leq \max _ {z \in \Gamma} \| f (z) \| \cdot \frac {1}{2 \pi} \int_ {\Gamma} \| (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \| | d z | \leq \max _ {z \in \Gamma} \| f (z) \| \cdot \frac {2 \| E \|}{\delta_ {p}}; +$$ + +see Appendix F. Together, this estimate and Lemma 3.1 prove Theorem 2.3. + +Second upper bound of $M_1$ and proof of Theorem 2.2. The key idea of the second bound is to replace $(zI - A)^{-1}$ by its spectral expansion $\sum_{i=1}^{n} \frac{u_i u_i^\top}{z - \lambda_i}$ . Hence, $M_1$ is rewritten as $\int_{\Gamma_1} \| \sum_{1 \leq i,j \leq n} \frac{z}{(z - \lambda_i)(z - \lambda_j)} u_i u_i^\top E u_j u_j^\top \| dz$ . + +There are $n^2$ terms in the expression, and the direct use of the triangle inequality cannot provide a good estimate. The next key trick is grouping up the $r$ -top eigenvectors $\{u_i\}_{i=1}^r$ . Formally, $M_1$ is at most + +$$ +\begin{array}{l} \int_ {\Gamma_ {1}} \| \sum_ {1 \leq i, j \leq r} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \| | d z | + \int_ {\Gamma_ {1}} \| \sum_ {n \geq i, j > r} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \| | d z | \\ + \int_{\Gamma_{1}}\| \sum_{\substack{i\leq r < j\\ i > r\geq j}}\frac{z}{(z - \lambda_{i})(z - \lambda_{j})} u_{i}u_{i}^{\top}Eu_{j}u_{j}^{\top}\| |dz|. \\ \end{array} +$$ + +To estimate the first term, we apply the triangle inequality. For each term, we factor out components independent of $z$ and carefully evaluate the integral. Specifically, by the triangle inequality, the first term is at most + +$$ +\sum_ {1 \leq i, j \leq r} \int_ {\Gamma_ {1}} \| \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \| | d z | = \sum_ {1 \leq i, j \leq r} \int_ {\Gamma_ {1}} \frac {| u _ {i} ^ {\top} E u _ {j} | \cdot \| u _ {i} u _ {j} ^ {\top} \| \cdot | z |}{| (z - \lambda_ {i}) (z - \lambda_ {j}) |} | d z |. +$$ + +Since $\max_{1\leq i,j\leq r}|u_i^\top Eu_j|\leq x,\| u_iu_j^\top \| = 1$ , and $\Gamma_1\coloneqq \{z\mid z = x_0 + \mathbf{i}t, - T\leq t\leq T\}$ , the r.h.s. is at most + +$$ +\sum_ {i, j \leq r} x \int_ {- T} ^ {T} \frac {\sqrt {x _ {0} ^ {2} + t ^ {2}}}{\sqrt {((x _ {0} - \lambda_ {i}) ^ {2} + t ^ {2}) ((x _ {0} - \lambda_ {j}) ^ {2} + t ^ {2})}} d t \leq \sum_ {i, j \leq r} x \int_ {- T} ^ {T} \frac {| x _ {0} | + | t |}{\sqrt {((x _ {0} - \lambda_ {i}) ^ {2} + t ^ {2}) ((x _ {0} - \lambda_ {j}) ^ {2} + t ^ {2})}} d t. +$$ + +By the construction of $\Gamma_{1}$ , we have $|x_0 - \lambda_i| \geq \frac{\delta_p}{2}$ for all $i \in [n]$ . Thus, the r.h.s. is bounded by $r^2 x \int_{-T}^{T} \frac{|x_0| + |t|}{t^2 + (\delta_p / 2)^2} dt$ , which by direct computation (see Appendix E.1 for full details) is less than or equals + +$$ +r ^ {2} x \left(\frac {2 \pi x _ {0}}{\delta_ {p}} + 2 \log \left(\frac {3 T}{\delta_ {p}}\right)\right) = \tilde {O} \left(r ^ {2} x \frac {\lambda_ {p}}{\delta_ {p}}\right). +$$ + +To estimate the second term, we apply matrix-norm inequalities to factor out $E$ from the integral: $\int_{\Gamma_1} \left\| \sum_{i,j=r}^{n} \frac{z}{(z-\lambda_i)(z-\lambda_j)} u_i u_i^\top Eu_j u_j^\top \right\|\|dz\| \leq \int_{\Gamma_1} |z| \cdot \left\| \sum_{n \geq i>r} \frac{u_i u_i^\top}{z-\lambda_i} \right\|$ . $\|E\| \cdot \left\| \sum_{n \geq i>r} \frac{u_i u_i^\top}{z-\lambda_i} \right\|\|dz\|$ , which is at most $\|E\| \int_{\Gamma_1} \frac{|z|}{\min_{n \geq i>r} |z-\lambda_i|^2} |dz| = \|E\| \int_{-T}^T \frac{\sqrt{x_0^2 + t^2}}{\min_{n \geq i>r} [(x_0 - \lambda_i)^2 + t^2]} dt$ . Moreover, by the construction of $\Gamma_1$ and the definition of $r$ , $|x_0 - \lambda_i| = |(\lambda_p + \lambda_{p+1}) / 2 - \lambda_i| \geq |(\lambda_p + \lambda_{p+1}) / 2 - \lambda_{r+1}| \geq \frac{\lambda_p - \lambda_{r+1}}{2} \geq \frac{\lambda_p}{4}$ , where the first inequality follows the fact $i > r$ . Thus, the second term is at most + +$$ +\| E \| _ {- T} ^ {T} \frac {\sqrt {x _ {0} ^ {2} + t ^ {2}}}{t ^ {2} + (\lambda_ {p} / 4) ^ {2}} d t \leq \tilde {O} (\| E \|); +$$ + +see Section E.1 for the detailed estimation. + +Similar to estimating the second term, the last term is also $\tilde{O} (\| E\|)$ . Combining the estimates on three parts of $M_{1}$ , we obtain $M_1\leq \tilde{O}\left(r^2 x\frac{\lambda_p}{\delta_p} +\| E\|\right)$ . Consequently, by Lemma 3.1, we finally have + +$$ +F (z) \leq 2 F _ {1} (z) = O (M _ {1}) = \tilde {O} \left(\| E \| + r ^ {2} x \frac {\lambda_ {p}}{\delta_ {p}}\right) \text {a s d e s i r e d}. +$$ + +# 4 Empirical results + +In this section, we empirically evaluate the sharpness of our spectral-gap bound (Theorem 2.1) in real-world settings central to privacy-preserving low-rank approximation. We compare: (1) the actual spectral error $\| \tilde{A}_p - A_p\|$ , (2) our theoretical bound $^3 7\| E\| \cdot \frac{\lambda_p}{\delta_p}$ , (3) and the classical Eckart-Young-Mirsky (EYM) bound $2(\| E\| +\lambda_{p + 1})$ . Each quantity is computed over 100 trials and 20 noise levels. Because prior bounds [15, 29, 30] apply only to Gaussian noise and involve unspecified constants, we exclude them from this evaluation. + +Setting. We study three covariance matrices $A$ from the UCI Machine Learning Repository [13]: the 1990 US Census ( $n = 69$ ), the 1998 KDD-Cup network-intrusion data ( $n = 416$ ), and the Adult dataset ( $n = 6$ ). These matrices—henceforth Census, KDD, and Adult—are standard benchmarks in DP PCA [3, 11, 29]. The low-rank parameter $p$ is chosen so that the Frobenius norm of $A_p$ contains $> 99\%$ of the Frobenius norm of $A$ , giving $p = 10$ for $A =$ Census, $p = 2$ for $A =$ KDD, and $p = 4$ for $A =$ Adult [29, Section B]. + +Each matrix is perturbed with either GOE noise $E_{1}$ or Rademacher noise $E_{2}$ , scaled by twenty evenly spaced factors ranging from 0 to 1. Note that with high probability [41, 43], $\| E_1\| = \| E_2\| = (2 + o(1))\sqrt{n}$ , so the gap condition $4\| E_k\| < \delta_p$ simplifies to $8\sqrt{n} < \delta_p$ . For Census $(n = 69,p = 10)$ , we have $\delta_p \approx 1433.99 > 8\sqrt{69} \approx 66.45$ . For KDD $(n = 416,p = 2)$ , we get $\delta_p \approx 351.3 > 8\sqrt{416} \approx 163.2$ . For Adult $(n = 6,p = 4)$ , we find $\delta_p \approx 37.02 > 8\sqrt{6} \approx 19.6$ . Hence $4\| E_k\| < \delta_p$ holds in all tested configurations. + +Evaluation. Each data matrix is preprocessed as follows: non-numeric entries are replaced with 0; rows shorter than the maximum length are padded with zeros; each row is scaled to unit Euclidean norm; and each column is centered to have zero mean. We compute the covariance matrix $A \coloneqq M^\top M$ , where $M$ is the processed data matrix. For each configuration $(A, E_k, p)$ , we run 100 independent trials. In each trial, we perturb $A$ with noise $E_k \in \{E_1, E_2\}$ to form $\tilde{A} = A + E_k$ , compute its best rank- $p$ approximation $\tilde{A}_p$ , and measure the spectral error $\| \tilde{A}_p - A_p \|$ . We compare this with our bound $7\| E_k\| \cdot \lambda_p / \delta_p$ and the classical EYM bound $2(\| E_k\| + \lambda_{p+1})$ . Following standard practice, all reported values are averaged over 100 trials, with error bars shown for Actual Error and Our Bound (cap width = 3pt). + +Result and conclusion. Across all experiments—the $69 \times 69$ US Census, the $416 \times 416$ KDD-Cup, and the $6 \times 6$ Adult matrix—our bound closely matches the empirical error for both Gaussian and Rademacher noise (Figs. 1-2), consistently outperforming the classical EYM estimate. (Note: the error bars for Census and KDD are too small to see.) Over all three benchmark datasets, two distinct noise models, and twenty escalation levels per model, our spectral-gap estimate never deviates from the observed error by more than a single order of magnitude. This uniform tightness, achieved without any dataset-specific tuning, demonstrates that the bound of Theorem 2.1 is not merely sufficient but practically sharp across matrix sizes spanning two orders of magnitude and privacy-motivated perturbations spanning the entire operational range. Consequently, the bound can serve as a reliable, application-agnostic error certificate for low-rank covariance approximation in both differential-privacy pipelines and more general noisy-matrix workflows. + +![](images/cfc78bc7d7e6a99d630ecdefcfa7d59de6b8e40e68a14ce05fe78caea55aeea0.jpg) +Figure 1: From Left to Right: perturbation of the Census, KDD and Adult covariance matrices by Gaussian noise. Each panel plots the actual error, our bound, and the EYM bound; error bars indicate standard deviation over 100 trials. + +![](images/7eb0a5b29152614a1466761ed827bacdbb976b7c07ef32b71f5c8d4f52434659.jpg) + +![](images/6ef6750050b9185f4d2903455bbd10b38ed9b230a2309cac5860a4fd4d6475a0.jpg) + +![](images/1fd9af12d595cecb2824251fb165c85684ffd9b71bfc9925fb6502406ae88984.jpg) +Figure 2: Low-rank approximation errors under Rademacher perturbations. From left to right: the Census, KDD and Adult covariance matrices. + +![](images/992908ba08225c14d6a192c7ae05364728315db38fb8464fec53e966bd4f609f.jpg) + +![](images/90caf95b7d0835960b4911d8d8f56205987ee08acc16168fdefd53e6320b4d8d.jpg) + +# 5 Conclusion and future work + +We established new spectral norm perturbation bounds for low-rank approximations that explicitly account for the interaction between a matrix $A$ and its perturbation $E$ . Our results extend the Eckart—Young-Mirsky theorem, improving upon prior Frobenius-norm-based analyses. A key contribution is a novel application of the contour bootstrapping technique, which simplifies spectral perturbation arguments and enables refined estimates. Our bounds provide sharper guarantees for differentially private low-rank approximations with high probability spectral norm bounds that improve upon prior results. We also extended our approach to general spectral functionals, broadening its applicability. + +Several limitations and open questions remain. While spectral norm error bounds are standard and widely used in both theoretical and applied settings, can we extend our analysis to other structured metrics such as Schatten- $p$ norm, the Ky Fan norm, or subspace affinity norm? Can our bounds be further refined for matrices with specific spectral structures, such as polynomial or exponential decay? What can be the threshold for the gap assumption so that one still obtains a meaningful bound beyond the Eckart-Young-Mirsky theorem? Additionally, real-world noise often exhibits structured dependencies—can our techniques be adapted to handle sparse or correlated perturbations? + +# Acknowledgments + +This work was funded in part by NSF Award CCF-2112665, Simons Foundation Award SFI-MPS-SFM-00006506, and NSF Grant AWD 0010308. + +# References + +[1] Dimitris Achlioptas and Frank McSherry. Fast computation of low-rank matrix approximations. Journal of the ACM (JACM), 54(2):9-es, 2007. +[2] U. Alon, N. Barkai, D. A. Notterman, K. Gish, S. Ybarra, D. Mack, and A. J. Levine. Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays. Proceedings of the National Academy of Sciences of the United States of America, 96(12):6745-6750, 1999. +[3] Kareem Amin, Travis Dick, Alex Kulesza, Andres Munoz, and Sergei Vassilvitskii. Differentially private covariance estimation. Advances in Neural Information Processing Systems, 32, 2019. +[4] Y. Azar, A. 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Mathematische Annalen, 71(4):441-479, 1912. + +# Contents + +1 Introduction 1 +2 Main results 3 +3 Proof outline 6 +4 Empirical results 9 +5 Conclusion and future work 10 + +A Limitations of prior approaches 15 +B Comparison of error metrics 16 +C Empirical evaluation beyond gap assumption 17 +D Extensions of Theorem 2.1 and Theorem 2.2 to the symmetric matrices 18 + +D.1 Extension of Theorem 2.1 to the symmetric matrices 18 +D.2 Extension of Theorem 2.2 to the symmetric matrices 20 + +E Estimating integrals over segments 21 + +E.1 Estimating integrals over vertical segments for interaction-dependent bound 21 +E.2 Estimating integrals over horizontal segments 24 +E.3 Estimating integrals over vertical segments for non-interaction bound 24 + +F Perturbation of matrix functionals - Theorem 2.3 24 +G Some classical perturbation bounds 26 +H Notation 26 + +# A Limitations of prior approaches + +This section explains why existing perturbation methods fail to yield spectral norm bounds of the form $\| A_p - A_p\|$ that incorporate interaction between $A$ and the perturbation $E$ . + +Eckart-Young-Mirsky: lack of interaction sensitivity. Let $\sigma_{1} \geq \sigma_{2} \geq \dots \geq \sigma_{n} \geq 0$ denote the singular values of $A$ . The Eckart-Young-Mirsky theorem gives $\| A - A_{p}\| = \sigma_{p + 1}$ , and by the triangle inequality: + +$$ +\| \tilde {A} _ {p} - A _ {p} \| \leq \| A - A _ {p} \| + \| \tilde {A} - A \| + \| \tilde {A} - \tilde {A} _ {p} \| \leq \sigma_ {p + 1} + \| E \| + \tilde {\sigma} _ {p + 1} \leq 2 (\sigma_ {p + 1} + \| E \|), +$$ + +where the final step uses Weyl's inequality [46]. While this bound is assumption-free, it is uninformative in regimes where $\sigma_{p + 1}\gg \| E\|$ , which are common in practice. The key limitation is that the triangle inequality treats $A$ and $E$ independently, failing to capture how structure or spectral gaps in $A$ might mitigate the effect of $E$ . + +Mangoubi-Vishnoi: Frobenius only, spectral norm intractable. The strategy of [29, 30] models noise as a continuous-time matrix-valued Brownian motion: + +$$ +A (t) := A + t E = A + B (t), +$$ + +with eigen-decomposition + +$$ +A (t) = U (t) \operatorname {D i a g} [ \lambda_ {1} (t), \dots , \lambda_ {n} (t) ] U (t) ^ {\top}, +$$ + +where $U(t) = [u_i(t)]$ and $\lambda_1(t) \geq \dots \geq \lambda_n(t)$ . The rank- $p$ approximation at time $t$ is + +$$ +A _ {p} (t) = U (t) \operatorname {D i a g} \left[ \lambda_ {1} (t), \dots , \lambda_ {p} (t), 0, \dots , 0 \right] U (t) ^ {\top}. +$$ + +The total perturbation is then expressed as an integral: + +$$ +\tilde {A} _ {p} - A _ {p} = \int_ {0} ^ {1} d A _ {p} (t). +$$ + +Using properties of Dyson Brownian motion and Itô calculus, they derive a Frobenius-norm identity: + +$$ +\mathbb {E} \left\| \int_ {0} ^ {1} d A _ {p} (t) \right\| _ {F} ^ {2} = \sum_ {i = 1} ^ {n} \int_ {0} ^ {1} \left(\mathbb {E} \left[ \sum_ {j \neq i} \frac {(\lambda_ {i} - \lambda_ {j}) ^ {2}}{(\lambda_ {i} (t) - \lambda_ {j} (t)) ^ {2}} \right] + \left(\sum_ {j \neq i} \frac {\lambda_ {i} - \lambda_ {j}}{(\lambda_ {i} (t) - \lambda_ {j} (t)) ^ {2}}\right) ^ {2}\right) d t. +$$ + +Bounding these expressions depends on repulsion properties of the eigenvalues; for GOE matrices, Weyl's inequality suffices, while for GUE matrices, stronger gap estimates are used. + +Although this method captures the spectral structure of $A$ and interaction with $E$ , it only yields Frobenius-norm bounds. Extending it to the spectral norm would require controlling + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| = \left\| \int_ {0} ^ {1} d A _ {p} (t) \right\|, +$$ + +which entails bounding the operator norm of the full stochastic process. This requires detailed control over the dynamics of $U(t)$ and $\lambda(t)$ , including their correlations—none of which are tractable with current techniques. + +Moreover, for generalized functionals such as $\| f_p(\tilde{A}) - f_p(A)\|$ , the problem becomes even harder: one must analyze $\int_0^1 df_p(A(t))$ , which involves matrix-valued analytic functions under random perturbation, a setting far beyond existing random matrix tools. + +In contrast, our approach bypasses these limitations by using a complex-analytic representation of spectral projectors that directly captures interaction between $A$ and $E$ , yielding sharp spectral norm bounds under broad assumptions. + +![](images/83053e787680f487f80181d08e4f48ac9a6e1c2103123f7a6618827441d157c8.jpg) +Figure 3: Comparison of error metrics under Gaussian perturbation. Left: Synthetic PSD matrix with exponentially decaying spectrum ( $n = 50$ , $p = 5$ ); Center: 1990 US Census covariance matrix ( $n = 69$ , $p = 5$ ); Right: 1998 KDD-Cup covariance matrix ( $n = 416$ , $p = 5$ ). Each plot reports the spectral norm error $\| \tilde{A}_p - A_p\|$ , Frobenius norm error $\| \tilde{A}_p - A_p\|_F$ , and change-in-error $\left\| A - A_p\right\| - \left\| A - \tilde{A}_p\right\|$ , as functions of Gaussian noise level $\sigma$ . Error bars reflect standard deviation over 20 trials. + +![](images/524362db7d905373ed5f52d4ecbc28a43177de87dd99dd051010e0265996bd83.jpg) + +![](images/eb2f1304f537fbb0bdd0fc4edab47b71046fd3ef33d2aa9fcc9cf6ad670abe82.jpg) + +# B Comparison of error metrics + +This section studies three common metrics for low-rank approximation under perturbation—namely: - the spectral-norm error $\| \tilde{A}_p - A_p\|$ , - the Frobenius-norm error $\| \tilde{A}_p - A_p\|_F$ , and - the "change-in-error" $\left\| A - A_p\| - \| A - \tilde{A}_p\| \right\|$ . + +We compare these metrics both empirically (through Monte Carlo simulations) and theoretically. Empirically, we examine how the metrics behave under Gaussian noise applied to both synthetic and real-world matrices (Figure 3). Theoretically, we analyze their interpretability and limitations, highlighting that while Frobenius norms capture aggregate error and change-in-error quantifies residual shifts, only the spectral norm controls worst-case subspace distortion. + +A simple $2 \times 2$ example (Example B.1) further illustrates how residual-based measures can completely mask subspace drift, underscoring the robustness and interpretability of the spectral norm for tasks such as private low-rank approximation. + +Empirical comparison of utility metrics. We perform three Monte Carlo experiments under additive Gaussian perturbations. The first uses a synthetic PSD matrix $A \in \mathbb{R}^{50 \times 50}$ with exponentially decaying eigenvalues $\lambda_{i} = 0.8^{i}$ , and sets $p = 5$ . The second and third use real-world covariance matrices derived from: - the 1990 US Census dataset ( $n = 69$ ), - the 1998 KDD-Cup dataset ( $n = 416$ ). + +All datasets are drawn from the UCI Machine Learning Repository [13] and have been widely used in private matrix approximation and PCA [30, 29, 11]. + +In each setting, we compute the best rank- $p$ approximation $A_{p}$ , perturb $A$ with symmetric Gaussian noise of varying standard deviation $\sigma$ , and measure: + +1. Spectral norm deviation: $\| \tilde{A}_p - A_p\|$ +2. Frobenius norm deviation: $\| \tilde{A}_p - A_p\| _F$ +3. Change-in-error: $\left|\left\| A - A_p\right\| -\left\| A - \tilde{A}_p\right\| \right|$ . + +As shown in Figure 3, the Frobenius norm error grows fastest, reflecting total energy deviation. The change-in-error metric remains much smaller and, in the real-world cases, nearly flat, suggesting it may fail to capture meaningful distortion. Notably, in the synthetic case (left), the spectral norm error closely tracks the change-in-error—despite their differing intent—which may result from near-alignment of the top subspaces. However, such behavior is not guaranteed in general. + +Theoretical distinction between utility metrics. Frobenius norm bounds of the form $\| \tilde{A}_p - A_p\| _F\leq \varepsilon_F$ aggregate squared deviations across all directions, but may hide large errors in in + +dividual components. Spectral norm bounds $\| \tilde{A}_p - A_p\| \leq \varepsilon$ directly constrain the worst-case deviation and are thus more reliable in sensitive applications such as differentially private PCA. + +In contrast, residual-error metrics such as $\| A - A_p \| - \| A - \tilde{A}_p \|$ are commonly used for their analytical convenience. However, they reflect only changes in residual energy and are insensitive to subspace movement. In particular, this metric can be nearly zero even when the top- $p$ eigenspaces have shifted significantly. + +Given the spectral decompositions + +$$ +A _ {p} = U _ {p} \operatorname {D i a g} (\lambda_ {1}, \dots , \lambda_ {p}, 0, \dots , 0) U _ {p} ^ {\top}, \quad \tilde {A} _ {p} = \tilde {U} _ {p} \operatorname {D i a g} (\tilde {\lambda} _ {1}, \dots , \tilde {\lambda} _ {p}, 0, \dots , 0) \tilde {U} _ {p} ^ {\top}, +$$ + +the change-in-error vanishes whenever $U_pU_p^\top \approx \tilde{U}_p\tilde{U}_p^\top$ and $\lambda_{p + 1}$ is large. Such conditions are typical when noise $E$ is small and $p \leq \operatorname{sr}(A) := \sum_{i = 1}^{n}\lambda_{i} / \lambda_{1}$ . Moreover, standard perturbation results imply + +$$ +\left\| U _ {p} U _ {p} ^ {\top} - \tilde {U} _ {p} \tilde {U} _ {p} ^ {\top} \right\| = \tilde {O} \left(\frac {\| E \|}{\lambda_ {p}} + \frac {1}{\delta_ {p}}\right) [ 3 3, 3 7 ]. +$$ + +Example B.1 (Rank-1 rotation in $\mathbb{R}^2$ ). Let + +$$ +A = \left( \begin{array}{c c} 1 & 0 \\ 0 & 0 \end{array} \right), \quad p = 1, +$$ + +so that $A_{p} = A$ . Define the rotated matrix + +$$ +\tilde {A} = R _ {\theta} A R _ {\theta} ^ {\top}, \quad w h e r e \quad R _ {\theta} = \left( \begin{array}{c c} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right). +$$ + +Then $\tilde{A}_p = \tilde{A}$ , and although the top eigenspace has rotated by $\theta$ , the change-in-error is zero: + +$$ +\left\| A - A _ {p} \right\| = \left\| A - \tilde {A} _ {p} \right\| = 0. +$$ + +Yet the true subspace drift is visible in: + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| = | \sin \theta |, \quad \left\| \tilde {A} _ {p} - A _ {p} \right\| _ {F} = \sqrt {2} | \sin \theta |. +$$ + +This example highlights the limitations of residual-based utility metrics and illustrates why spectral norm deviation provides a more reliable and interpretable signal of approximation quality under perturbation. + +In summary, both our analysis and experiments support the use of the spectral norm as the most informative and robust error metric for evaluating private low-rank approximations. Unlike Frobenius and residual metrics, it captures the worst-case directional distortion and provides a tighter connection to subspace stability. + +# C Empirical evaluation beyond gap assumption + +In this section, we empirically compare (1) the actual spectral error $\| \tilde{A}_p - A_p\|$ , (2) our theoretical bound $7\| E\| \cdot \frac{\lambda_p}{\delta_p}$ , (3) and the classical Eckart-Young-Mirsky (EYM) bound $2(\| E\| +\lambda_{p + 1})$ in the setting beyond the gap assumption that $4\| E\| < \delta_{p}$ . + +Setup. We conducted a simulation on a covariance matrix $A$ with $n = 2000$ , derived from the Alon colon-cancer microarray dataset [2]. The low-rank parameter $p$ is chosen so that the Frobenius norm of $A_{p}$ contains $>95\%$ of the Frobenius norm of $A$ , giving $p = 9$ with $\lambda_{p} \approx 46.29$ . We first computed $\delta_{p}$ . Gaussian noise was then added in the form $E = \alpha \cdot \mathcal{N}(0, I_n)$ , with $\alpha$ chosen over 11 evenly spaced values such that + +$$ +\frac {\| E \|}{\delta_ {p}} \in \{0. 0 5, 0. 1 0, \dots , 0. 5 0 \}. +$$ + +For each $\alpha$ , we computed the following quantities: + +the true error: $\| \tilde{A}_p - A_p\|$ +- the classical EYM bound: $2(\|E\| + \sigma_{p+1})$ +our bound: $7\| E\| \cdot \frac{\lambda_p}{\delta_p}$ +the ratios our bound and our bound true error classical bound. + +Results. Table 2 summarizes the results. The ratio $\frac{\text{our bound}}{\text{true error}}$ remains remarkably stable even beyond the regime $4\| E \| < \delta_{p}$ (i.e., $\frac{\|E\|}{\delta_p} < .25$ ), and our bound outperforms the classical bound precisely when $4\| E \| < \delta_{p}$ (i.e., $\frac{\|E\|}{\delta_p} < .25$ ). + +Table 2: Comparison of bounds under increasing noise levels. + +
||E||/δp0.050.100.150.200.250.300.350.400.450.50
our bound90.1788.2787.0289.8389.4487.8188.3989.2987.0887.26
true error
our bound0.200.400.600.790.981.171.361.531.701.88
classical bound
+ +# D Extensions of Theorem 2.1 and Theorem 2.2 to the symmetric matrices + +In this section, we extend Theorem 2.1 and Theorem 2.2 to the setting where $A$ is a symmetric matrix. These extensions are naturally important since the data in real-world applications is often arbitrary, making it natural for the eigenvalues of $A$ to span both signs. While singular value decomposition (SVD) could be used to apply Theorem 2.1 or Theorem 2.2, singular value gaps are typically small. By working directly with eigenvalues, we exploit the fact that the eigenvalue gap $\delta_{k} = \lambda_{k} - \lambda_{k + 1}$ is significantly large when $\lambda_{k}\cdot \lambda_{k + 1} < 0$ . + +# D.1 Extension of Theorem 2.1 to the symmetric matrices + +To simplify the presentation, we assume that the eigenvalues (singular values) are different, so the eigenvectors (singular vectors) are well-defined (up to signs). However, our results hold for matrices with multiple eigenvalues. Let $A, E$ be $n \times n$ real symmetric matrices, and let $1 \leq p \leq n$ denote the rank of approximation. Let $\lambda_{k}$ be the $k$ th largest eigenvalue of $A$ and $u_{k}$ be the corresponding orthonormal eigenvector. Let $\tilde{A} := A + E$ . Let $A_{p}, \tilde{A}_{p}$ denote the best rank- $p$ approximations of $A$ and $\tilde{A}$ respectively. Define $1 \leq k \leq p$ such that the set of the top $p$ singular values corresponds to $\{\lambda_{\pi(1)}, \ldots, \lambda_{\pi(p)}\} = \{\lambda_1, \ldots, \lambda_k > 0 \geq \lambda_{n-(p-k)+1}, \ldots, \lambda_n\}$ . In other words, the $p$ th singular value of $A$ is either $\lambda_{k}$ or $|\lambda_{n-(p-k)+1}|$ . Let $\delta_{i} := \lambda_{i} - \lambda_{i+1}$ , for $i \in [n-1]$ . Theorem 2.1 is extended to the following result. + +Theorem D.1 (Extension of Theorem 2.1 to the symmetric matrices). If $4\| E\| \leq$ $\min \{\delta_k,\delta_{n - (p - k)}\}$ , and $2\| E\| < \sigma_p - \sigma_{p + 1}$ , then + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| \leq 6 \| E \| \left(\log \left(\frac {6 \sigma_ {1}}{\delta_ {k}}\right) + \frac {\lambda_ {k}}{\delta_ {k}} + \log \left| \frac {6 \sigma_ {1}}{\delta_ {n - (p - k)}} \right| + \frac {\left| \lambda_ {n - (p - k)} + 1 \right|}{\delta_ {n - (p - k)}}\right). +$$ + +Note that when $A$ is not PSD, $\{|\tilde{\lambda}_1|,\ldots ,|\tilde{\lambda}_k|,|\tilde{\lambda}_{n - (p - k) + 1}|,\ldots ,|\tilde{\lambda}_n|\}$ may not correspond to the $p$ leading singular values of $\tilde{A}$ . This issue is resolved by enforcing the singular-value gap condition $\sigma_{p} - \sigma_{p + 1} > 2\| E\|$ . Indeed, by Weyl's inequality, given $\sigma_p - \sigma_{p + 1} > 2\| E\|$ , we have + +$$ +\begin{array}{l} \tilde {\lambda} _ {k} \geq \lambda_ {k} - \| E \| \geq \sigma_ {p} - \| E \| = \sigma_ {p + 1} + \delta - \| E \| \\ \geq \left| \lambda_ {n - (p - k)} \right| + \delta - \| E \| \geq \left| \tilde {\lambda} _ {n - (p - k)} \right| + \delta - 2 \| E \| > \left| \tilde {\lambda} _ {n - (p - k)} \right|, \\ \end{array} +$$ + +here $\delta = \sigma_p - \sigma_{p + 1}$ . By a similar argument, we also have $|\tilde{\lambda}_{n - (p - k) + 1}| > \tilde{\lambda}_{k + 1}$ . Therefore, + +$$ +\{\tilde {\lambda} _ {\pi (1)}, \tilde {\lambda} _ {\pi (2)}, \dots , \tilde {\lambda} _ {\pi (p)} \} = \{\tilde {\lambda} _ {1} \geq \tilde {\lambda} _ {2} \geq \dots \geq \tilde {\lambda} _ {k} > 0 \geq \tilde {\lambda} _ {n - (p - k) + 1} \geq \tilde {\lambda} _ {n - (p - k) + 2} \geq \dots \geq \tilde {\lambda} _ {n} \}, +$$ + +as we want. Note that the gap condition of eigenvalues cannot guarantee this fact. For example, consider the following matrices + +$$ +A = \left( \begin{array}{c c} 3 0 \sqrt {n} & 0 \\ 0 & - 2 8 \sqrt {n} \end{array} \right), E = \left( \begin{array}{c c} - 2 \sqrt {n} & 0 \\ 0 & - 2 \sqrt {n} \end{array} \right), \text {t h e n} \tilde {A} = \left( \begin{array}{c c} 2 8 \sqrt {n} & 0 \\ 0 & - 3 0 \sqrt {n} \end{array} \right). +$$ + +Here, clearly, $S = \{1\}$ , $\tilde{S} = \{1\}$ and $|\lambda_1|$ is the largest singular value of $A$ , but $|\tilde{\lambda}_1|$ is not the largest singular value of $\tilde{A}$ ( $\tilde{\lambda}_1$ is still the largest eigenvalue). + +Proof of Theorem D.1 Let $1 \leq k \leq p$ be a natural number such that + +$$ +\{\lambda_ {\pi (1)}, \lambda_ {\pi (2)}, \dots , \lambda_ {\pi (p)} \} = \{\lambda_ {1}, \lambda_ {2}, \dots , \lambda_ {k} > 0 \geq \lambda_ {n - (p - k) + 1}, \lambda_ {n - (p - k) + 2}, \dots , \lambda_ {n} \}. +$$ + +Thus, we can split $A_{p}$ as $A_{k} + B_{p - k}$ , in which + +$$ +B _ {p - k} = \sum_ {n \geq i \geq n - (p - k) + 1} \lambda_ {i} u _ {i} u _ {i} ^ {\top}. +$$ + +Similarly, $\tilde{A}_p = \tilde{A}_k + \tilde{B}_{p - k}$ . Therefore, + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| = \left\| \tilde {A} _ {k} + \tilde {B} _ {p - k} - A _ {k} - B _ {p - k} \right\| \leq \left\| \tilde {A} _ {k} - A _ {k} \right\| + \left\| \tilde {B} _ {p - k} - B _ {p - k} \right\|. +$$ + +Applying the contour bootstrapping argument on $\left\| \tilde{A}_k - A_k\right\|$ with contour $\Gamma^{[1]}$ and on $\left\| \tilde{B}_{p - k} - B_{p - k}\right\|$ with another contour $\Gamma^{[2]}$ (we define these contours later), we obtain + +$$ +\begin{array}{l} \frac {\left\| \tilde {A} _ {k} - A _ {k} \right\|}{2} \leq F _ {1} ^ {[ 1 ]} := \frac {1}{2 \pi} \int_ {\Gamma^ {[ 1 ]}} \left\| z (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \right\| | d z |, \\ \frac {\left\| \tilde {B} _ {p - k} - B _ {p - k} \right\|}{2} \leq F _ {1} ^ {[ 2 ]} := \frac {1}{2 \pi} \int_ {\Gamma^ {[ 2 ]}} \left\| z (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \right\| | d z |, \tag {4} \\ \end{array} +$$ + +and hence, + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| \leq 2 \left(F _ {1} ^ {[ 1 ]} + F _ {1} ^ {[ 2 ]}\right). +$$ + +We set $\Gamma^{[1]}$ and $\Gamma^{[2]}$ to be rectangles, whose vertices are + +$$ +\Gamma^ {[ 1 ]}: (a _ {0}, T), (a _ {1}, T), (a _ {1}, - T), (a _ {0}, - T) \text {w i t h} a _ {0} := \lambda_ {k} - \delta_ {k} / 2, a _ {1} := 2 \sigma_ {1}, T := 2 \sigma_ {1}; +$$ + +and + +$$ +\Gamma^ {[ 2 ]}: (b _ {0}, T), (b _ {1}, T), (b _ {1}, - T), (b _ {0}, - T) \text {w i t h} b _ {0} := \lambda_ {n - (p - k) + 1} + \delta_ {n - (p - k)} / 2, b _ {1} := - 2 \sigma_ {1}, T := 2 \sigma_ {1}. +$$ + +Now, we are going to bound $F_1^{[1]}$ . First, we split $\Gamma^{[1]}$ into four segments: + +- $\Gamma_1 \coloneqq \{(a_0, t) | -T \leq t \leq T\}$ . +- $\Gamma_2 \coloneqq \{(x, T) | a_0 \leq x \leq a_1\}$ . +- $\Gamma_3 \coloneqq \{(a_1, t) | T \geq t \geq -T\}$ . +- $\Gamma_4 := \{(x, -T) | a_1 \geq x \geq a_0\}$ . + +![](images/e52aea10146d96868a333d4e543645cb9df451ec96edbf62b434f3172d72622a.jpg) + +Therefore, + +$$ +F _ {1} ^ {[ 1 ]} = \sum_ {l = 1} ^ {4} \int_ {\Gamma_ {l}} \left\| z (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \right\| | d z |. +$$ + +Notice that + +$$ +\left| \left| z (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \right| \right| \leq \| E \| _ {\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}}, +$$ + +we further obtain + +$$ +2 \pi F _ {1} ^ {[ 1 ]} \leq \| E \| \left(\sum_ {l = 1} ^ {4} N _ {l}\right), +$$ + +in which + +$$ +N _ {l} := \int_ {\Gamma_ {l}} \frac {| z |}{\min _ {i} | z - \lambda_ {i} | ^ {2}} | d z | \text {f o r} l = 1, 2, 3, 4. +$$ + +We use the following lemmas, whose proofs are delayed to the next section. + +Lemma D.2. Under the assumption of Theorem D.1, + +$$ +N _ {1} \leq \frac {2 \pi a _ {0}}{\delta_ {k}} + 4 \log \left| \frac {3 T}{\delta_ {k}} \right|. +$$ + +Lemma D.3. Under the assumption of Theorem D.1, + +$$ +N _ {3} \leq \frac {\pi a _ {1}}{\left| a _ {1} - \lambda_ {1} \right|} + 4 \log \left| \frac {3 T}{a _ {1} - \lambda_ {1}} \right|. +$$ + +Lemma D.4. Under the assumption of Theorem D.1, + +$$ +N _ {2}, N _ {4} \leq \frac {\sqrt {2} (a _ {1} - a _ {0})}{T}, +$$ + +Since $p < n$ , then $k + 1 > n - (p - k) + 1$ and hence $k + 1 \notin \{\pi(1), \ldots, \pi(p)\}$ . It means $|\lambda_{k+1}| \leq \lambda_k$ . Thus $0 \leq a_0 \leq \lambda_k$ , and hence + +$$ +N _ {1} \leq \frac {2 \pi \lambda_ {k}}{\delta_ {k}} + 4 \log \left| \frac {6 \sigma_ {1}}{\delta_ {k}} \right|. +$$ + +By the setting that $a_1 = T = 2\sigma_1$ + +$$ +N _ {2}, N _ {4} \leq \frac {\sqrt {2} a _ {1}}{T} = \sqrt {2}, +$$ + +$$ +N _ {3} \leq \frac {2 \pi \sigma_ {1}}{2 \sigma_ {1} - \lambda_ {1}} + 4 \log \left| \frac {3 T}{a _ {1} - \lambda_ {1}} \right| \leq \frac {2 \pi \sigma_ {1}}{\sigma_ {1}} + 4 \log \left| \frac {6 \sigma_ {1}}{\sigma_ {1}} \right| = 2 \pi + 4 \log 6. +$$ + +Thus, using above estimates, we obtain + +$$ +\begin{array}{l} F _ {1} ^ {[ 1 ]} \leq \frac {\| E \|}{2 \pi} \left(2 \pi + 4 \log 6 + 2 \sqrt {2} + \frac {2 \pi \lambda_ {k}}{\delta_ {k}} + 4 \log \left| \frac {6 \sigma_ {1}}{\delta_ {k}} \right|\right) \\ \leq \frac {\| E \|}{2 \pi} \left(1 5 \log \left| \frac {6 \sigma_ {1}}{\delta_ {k}} \right| + \frac {2 \pi \lambda_ {k}}{\delta_ {k}}\right) \tag {5} \\ \leq 3 \| E \| \left(\log \left| \frac {6 \sigma_ {1}}{\delta_ {k}} \right| + \frac {\lambda_ {k}}{\delta_ {k}}\right). \\ \end{array} +$$ + +Applying a similar argument on contour $\Gamma^{[2]}$ , we obtain + +$$ +F _ {1} ^ {[ 2 ]} \leq 3 \| E \| \left(\log \left| \frac {6 \sigma_ {1}}{\delta_ {n - (p - k)}} \right| + \frac {\left| \lambda_ {n - (p - k) + 1} \right|}{\delta_ {n - (p - k)}}\right). \tag {6} +$$ + +Combining (4), (5) and (6), we complete our proof. + +# D.2 Extension of Theorem 2.2 to the symmetric matrices + +Let $A$ be a symmetric matrix with eigenvalues $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n$ , in which $\lambda_n$ is not necessarily positive. Recall the setting from the previous section that $1 \leq k \leq p$ is the positive integer such that the set of the top $p$ singular values is $\{\lambda_{\pi(1)}, \ldots, \lambda_{\pi(p)}\} = \{\lambda_1, \ldots, \lambda_k > 0 \geq \lambda_{n - (p - k) + 1}, \ldots, \lambda_n\}$ . To extend Theorem 2.2, we first generalize the definition of the halving distance $r$ and interaction term $x$ as follows. Let $r_1, r_2$ respectively be the smallest positive integer satisfying $\frac{\lambda_k}{2} \leq \lambda_k - \lambda_{r_1 + 1}$ , and $\frac{|\lambda_{n - (p - k) + 1}|}{2} \leq \lambda_{n - r_2 + 1} - \lambda_{n - (p - k) + 1}$ . Define the "halving distance" $r := \max\{r_1, r_2\}$ . Next, let $x_1 := \max_{1 \leq i,j \leq r_1} |u_i^\top Eu_j|$ and $x_2 := \max_{1 \leq i,j \leq r_2} |u_{n - i + 1}^\top Eu_{n - j + 1}|$ . Define the interaction parameter $\bar{x} := \max\{x_1, x_2\}$ . + +Theorem D.5 (Extension of Theorem 2.2 to the symmetric matrices). Assume that $4\| E\| \leq \min \{\delta_k,\delta_{n - (p - k)}\}$ and $2\| E\| < \sigma_p - \sigma_{p + 1}$ , then + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| \leq 1 2 \left(\| E \| + r ^ {2} \bar {x}\right) \left(\log \left(\frac {6 \sigma_ {1}}{\delta_ {k}}\right) + \log \left(\frac {6 \sigma_ {1}}{\delta_ {n - (p - k)}}\right)\right) + 3 0 r ^ {2} \bar {x} \left(\frac {\lambda_ {k}}{\delta_ {k}} + \frac {\left| \lambda_ {n - (p - k) + 1} \right|}{\delta_ {n - (p - k)}}\right). +$$ + +Proof of Theorem D.5 First, we still split $(\tilde{A}_p,A_p)$ into $(A_{k},B_{p - k},\tilde{A}_{k},\tilde{B}_{p - k})$ and apply the contour bootstrapping argument on $\left\| \tilde{A}_k - A_k\right\|$ , $\left\| \tilde{B}_{p - k} - B_{p - k}\right\|$ . We also obtain + +$$ +\left| \left| \tilde {A} _ {p} - A _ {p} \right| \right| \leq 2 \left(F _ {1} ^ {[ 1 ]} + F _ {1} ^ {[ 2 ]}\right). +$$ + +However, we will treat $F_1^{[1]}$ , $F_1^{[2]}$ a bit differently. Indeed, + +$$ +2 \pi F _ {1} ^ {[ 1 ]} \leq M _ {1} + \| E \| (N _ {2} + N _ {3} + N _ {4}), +$$ + +in which + +$$ +M _ {1} := \int_ {\Gamma_ {1}} \left\| z (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \right\| | d z | = \int_ {\Gamma_ {1}} \left\| \sum_ {1 \leq i, j \leq n} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} E u _ {j} u _ {j} ^ {\top} \right\| | d z |, +$$ + +and + +$$ +N _ {l} := \int_ {\Gamma_ {l}} \frac {| z |}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} | d z | \text {f o r} l \in \{2, 3, 4 \}. +$$ + +We additionally use the following lemma (its proof will be delayed in the next section). + +Lemma D.6. Under the assumption of Theorem D.5, + +$$ +M _ {1} \leq r ^ {2} \bar {x} \left(\frac {2 \pi a _ {0}}{\delta_ {k}} + 2 \log \left(\frac {6 \sigma_ {1}}{\delta_ {k}}\right)\right) + (2 0 + 4 \pi / \log (1 0) \| E \| \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right). +$$ + +Together with the estimates for $N_{2}, N_{3}, N_{4}$ from the previous section, we obtain + +$$ +\begin{array}{l} 2 \pi F _ {1} ^ {[ 1 ]} \leq r ^ {2} \bar {x} \left(\frac {2 \pi \lambda_ {k}}{\delta_ {k}} + 2 \log \left(\frac {3 T}{\delta_ {k}}\right)\right) + (2 0 + \frac {4 \pi}{\log 1 0}) \| E \| \log \left(\frac {5 T}{\delta_ {k}}\right) + \| E \| (2 \sqrt {2} + 2 \pi + 4 \log 6) \\ \leq r ^ {2} \bar {x} \left(\frac {2 \pi \lambda_ {k}}{\delta_ {k}} + 2 \log \left(\frac {6 \sigma_ {1}}{\delta_ {k}}\right)\right) + (2 0 + \frac {4 \pi}{\log 1 0}) \| E \| \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right) + \frac {2 \sqrt {2} + 2 \pi + 4 \log 6}{\log 1 0} \| E \| \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right). \\ \end{array} +$$ + +Thus, + +$$ +F _ {1} ^ {[ 1 ]} \leq 6 \left(\| E \| \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right) + r ^ {2} \bar {x} \frac {\lambda_ {k}}{\delta_ {k}} + r ^ {2} \bar {x} \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right)\right). \tag {7} +$$ + +Similarly, + +$$ +F _ {1} ^ {[ 2 ]} \leq 6 \left(\| E \| \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {n - (p - k)}}\right) + r ^ {2} \bar {x} \frac {\left| \lambda_ {n - (p - k) + 1} \right|}{\delta_ {n - (p - k)}} + r ^ {2} \bar {x} \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {n - (p - k)}}\right)\right). \tag {8} +$$ + +Therefore, combining (4), (7), and (8), we finally obtain + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| \leq 1 2 \left(\| E \| \log \left(\frac {3 6 \sigma_ {1} ^ {2}}{\delta_ {k} \delta_ {n - (p - k)}}\right) + r ^ {2} \bar {x} \frac {\lambda_ {k}}{\delta_ {k}} + r ^ {2} \bar {x} \frac {\left| \lambda_ {n - (p - k) + 1} \right|}{\delta_ {n - (p - k)}} + r ^ {2} \bar {x} \log \left(\frac {3 6 \sigma_ {1} ^ {2}}{\delta_ {k} \delta_ {n - (p - k)}}\right)\right). +$$ + +# E Estimating integrals over segments + +In this section, we present in detail the integral estimations mentioned in the previous section: Lemma D.2, Lemma D.3, Lemma D.6 (integration over vertical segments); and Lemma D.4 (integration over horizontal segments). We first present a technical lemma, which is used several times in the upcoming sections. + +Lemma E.1. Let $a, T$ be positive numbers such that $a \leq T$ . Then, + +$$ +\int_ {- T} ^ {T} \frac {1}{t ^ {2} + a ^ {2}} d t \leq \frac {\pi}{a}. +$$ + +Proof of Lemma E.1 We have + +$$ +\begin{array}{l} \int_ {- T} ^ {T} \frac {1}{t ^ {2} + a ^ {2}} d t = 2 \int_ {0} ^ {T} \frac {1}{t ^ {2} + a ^ {2}} \\ = \frac {2}{a} \arctan (T / a) \\ \leq \frac {2}{a} \cdot \frac {\pi}{2} = \frac {\pi}{a}. \\ \end{array} +$$ + +# E.1 Estimating integrals over vertical segments for interaction-dependent bound + +In this Section, we now estimate $M_1$ - integral over the left vertical segment (prove Lemma D.6) and estimate $N_3$ - the integral over the right vertical segment (prove Lemma D.3). First, we estimate $M_1$ as follows. + +Using the spectral decomposition $(zI - A)^{-1} = \sum_{i=1}^{n} \frac{u_i u_i^\top}{(z - \lambda_i)}$ , we can rewrite $M_1$ as + +$$ +M _ {1} = \int_ {\Gamma_ {1}} \left\| \sum_ {n \geq i, j \geq 1} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z |. +$$ + +Define $x_{1} \coloneqq \max_{1 \leq i,j \leq r_{1}} \left| u_{i}^{\top} E u_{j} \right|$ . By the triangle inequality, $M_{1}$ is at most + +$$ +\begin{array}{l} \int_ {\Gamma_ {1}} \left\| \sum_ {1 \leq i, j \leq r _ {1}} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z | + \int_ {\Gamma_ {1}} \left\| \sum_ {n \geq i, j > r _ {1}} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z | \\ + \int_ {\Gamma_ {1}} \left\| \sum_ {i < r _ {1} < j \atop i > r _ {1} \geq j} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z |. \\ \end{array} +$$ + +Consider the first term, by the triangle inequality, we have + +$$ +\begin{array}{l} \int_ {\Gamma_ {1}} \left\| \sum_ {1 \leq i, j \leq r _ {1}} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z | \leq \sum_ {1 \leq i, j \leq r _ {1}} \int_ {\Gamma_ {1}} \left\| \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z | \\ = \sum_ {1 \leq i, j \leq r _ {1}} \int_ {\Gamma_ {1}} \frac {\left| u _ {i} ^ {\top} E u _ {j} \right| \cdot \left\| u _ {i} u _ {j} ^ {\top} \right\| \cdot | z |}{\left| (z - \lambda_ {i}) (z - \lambda_ {j}) \right|} | d z | \\ \leq \sum_ {i, j \leq r _ {1}} x _ {1} \int_ {- T} ^ {T} \frac {\sqrt {a _ {0} ^ {2} + t ^ {2}}}{\sqrt {\left((a _ {0} - \lambda_ {i}) ^ {2} + t ^ {2}\right) \left((a _ {0} - \lambda_ {j}) ^ {2} + t ^ {2}\right)}} d t \\ \end{array} +$$ + +$$ +\begin{array}{l} (s i n c e \max _ {1 \leq i, j \leq r _ {1}} | u _ {i} ^ {\top} E u _ {j} | \leq x _ {1}, \| u _ {i} u _ {j} ^ {\top} \| = 1, \\ \text {a n d} \Gamma_ {1} := \left\{z \mid z = a _ {0} + \mathrm {i} t, - T \leq t \leq T \right\}) \\ \leq \sum_ {i, j \leq r _ {1}} x _ {1} \int_ {- T} ^ {T} \frac {\left| a _ {0} \right| + \left| t \right|}{\sqrt {\left((a _ {0} - \lambda_ {i}) ^ {2} + t ^ {2}\right) \left((a _ {0} - \lambda_ {j}) ^ {2} + t ^ {2}\right)}} d t. \\ \end{array} +$$ + +By the construction of $\Gamma_{1}$ , we have + +$$ +\left| a _ {0} - \lambda_ {i} \right| \geq \frac {\delta_ {k}}{2} \text {f o r a l l} 1 \leq i \leq n. \tag {9} +$$ + +Thus, the r.h.s. is at most + +$$ +r _ {1} ^ {2} x _ {1} \int_ {- T} ^ {T} \frac {\left| a _ {0} \right| + | t |}{t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}} d t = r _ {1} ^ {2} x _ {1} \left(\int_ {- T} ^ {T} \frac {\left| a _ {0} \right|}{t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}} d t + \int_ {0} ^ {T} \frac {2 t}{t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}} d t\right). \tag {10} +$$ + +By Lemma E.1, we have + +$$ +\int_ {- T} ^ {T} \frac {\left| a _ {0} \right|}{t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}} d t \leq \frac {2 \pi \left| a _ {0} \right|}{\delta_ {k}} = \frac {2 \pi a _ {0}}{\delta_ {k}} (\text {s i n c e} a _ {0} \geq 0). +$$ + +The second integral is estimated by what follows. + +$$ +\begin{array}{l} \int_ {0} ^ {T} \frac {2 t}{t ^ {2} + (\delta_ {k} / 2) ^ {2}} d t = \int_ {(\delta_ {k} / 2) ^ {2}} ^ {T ^ {2} + (\delta_ {k} / 2) ^ {2}} \frac {1}{u} d u (u = t ^ {2} + (\delta_ {k} / 2) ^ {2}) \\ = \log \left(\frac {T ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}}{\left(\delta_ {k} / 2\right) ^ {2}}\right) \tag {11} \\ = \log \left(\frac {4 T ^ {2} + \delta_ {k} ^ {2}}{\delta_ {k} ^ {2}}\right) \leq 2 \log \left(\frac {3 T}{\delta_ {k}}\right). \\ \end{array} +$$ + +Therefore, + +$$ +\int_ {\Gamma_ {1}} \left\| \sum_ {1 \leq i, j \leq r _ {1}} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z | \leq r _ {1} ^ {2} x _ {1} \left(\frac {2 \pi a _ {0}}{\delta_ {k}} + 2 \log \left(\frac {3 T}{\delta_ {k}}\right)\right). \tag {12} +$$ + +Next, we bound the second term as follows + +$$ +\begin{array}{l} \int_ {\Gamma_ {1}} \left\| \sum_ {n \geq i, j > r _ {1}} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z | \\ = \int_ {\Gamma_ {1}} \left\| z \left(\sum_ {n \geq i > r _ {1}} \frac {u _ {i} u _ {i} ^ {\top}}{z - \lambda_ {i}}\right) E \left(\sum_ {n \geq i > r _ {1}} \frac {u _ {i} u _ {i} ^ {\top}}{z - \lambda_ {i}}\right) \right\| | d z | \\ \leq \int_ {\Gamma_ {1}} | z | \cdot \left\| \sum_ {n \geq i > r} \frac {u _ {i} u _ {i} ^ {\top}}{z - \lambda_ {i}} \right\| \times \| E \| \times \left\| \sum_ {n \geq i > r} \frac {u _ {i} u _ {i} ^ {\top}}{z - \lambda_ {i}} \right\| | d z | \\ \leq \| E \| \int_ {\Gamma_ {1}} \frac {| z |}{\min _ {n \geq i > r _ {1}} | z - \lambda_ {i} | ^ {2}} | d z | \\ = \| E \| \int_ {- T} ^ {T} \frac {\sqrt {a _ {0} ^ {2} + t ^ {2}}}{\min _ {n \geq i > r _ {1}} [ (a _ {0} - \lambda_ {i}) ^ {2} + t ^ {2} ]} d t. \\ \end{array} +$$ + +Moreover, by the construction of $\Gamma_{1}$ and the definition of $r_1$ + +$$ +\left| a _ {0} - \lambda_ {i} \right| = \left| \frac {\lambda_ {k} + \lambda_ {k + 1}}{2} - \lambda_ {i} \right| \geq \left| \frac {\lambda_ {k} + \lambda_ {k + 1} - 2 \lambda_ {r + 1}}{2} \right| \geq \frac {\lambda_ {k} - \lambda_ {r + 1}}{2} \geq \frac {\lambda_ {k}}{4}, \tag {13} +$$ + +where the second inequality follows the fact $i > r_1$ . Thus, the r.h.s. is at most + +$$ +\| E \| _ {- T} ^ {T} \frac {\sqrt {a _ {0} ^ {2} + t ^ {2}}}{t ^ {2} + (\lambda_ {k} / 4) ^ {2}} d t \leq \| E \| _ {- T} ^ {T} \frac {a _ {0} + | t |}{t ^ {2} + (\lambda_ {k} / 4) ^ {2}} d t. +$$ + +Similar to (10) and (11), we also have + +$$ +\begin{array}{l} \int_ {- T} ^ {T} \frac {a _ {0} + | t |}{t ^ {2} + \left(\lambda_ {k} / 4\right) ^ {2}} d t \leq \frac {4 \pi a _ {0}}{\lambda_ {k}} + \log \left(\frac {T ^ {2} + \left(\lambda_ {k} / 4\right) ^ {2}}{\left(\lambda_ {k} / 4\right) ^ {2}}\right) \\ \leq \frac {4 \pi a _ {0}}{\lambda_ {k}} + \log \left(\frac {2 T ^ {2}}{\delta_ {k} ^ {2}}\right) \tag {14} \\ \leq 4 \pi + 2 \log \left(\frac {2 T}{\delta_ {k}}\right) (\text {s i n c e} a _ {0} \leq \lambda_ {k}). \\ \end{array} +$$ + +It follows that + +$$ +\int_ {\Gamma_ {1}} \left\| \sum_ {n \geq i, j > r _ {1}} \frac {z}{(z - \lambda_ {i}) (z - \lambda_ {j})} u _ {i} u _ {i} ^ {\top} E u _ {j} u _ {j} ^ {\top} \right\| | d z | \leq \| E \| \left(4 \pi + 2 \log \left(\frac {2 T}{\delta_ {k}}\right)\right). \tag {15} +$$ + +Now we consider the last term: + +$$ +\int_{\Gamma_{1}}\left\| \sum_{\substack{i\leq r_{1} < j\\ i > r_{1}\geq j}}\frac{z}{(z - \lambda_{i})(z - \lambda_{j})} u_{i}u_{i}^{\top}Eu_{j}u_{j}^{\top}\right\| |dz|\leq 2\| E\| \int_{\Gamma_{1}}\frac{|z|}{\min_{i\leq r_{1} < j}|(z - \lambda_{i})(z - \lambda_{j})|} |dz|. +$$ + +By (13) and (9), the r.h.s. is at most + +$$ +\begin{array}{l} 2 \| E \| \int_ {- T} ^ {T} \frac {| z |}{\sqrt {\left(t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}\right) \left(t ^ {2} + \left(a _ {0} - \lambda_ {r + 1}\right) ^ {2}\right)}} d t = 4 \| E \| \int_ {0} ^ {T} \frac {\sqrt {a _ {0} ^ {2} + t ^ {2}}}{\sqrt {\left(t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}\right) \left(t ^ {2} + \left(a _ {0} - \lambda_ {r + 1}\right) ^ {2}\right)}} d t \tag {16} \\ \leq 4 \| E \| \int_ {0} ^ {T} \frac {a _ {0} + t}{\sqrt {\left(t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}\right) \left(t ^ {2} + \left(a _ {0} - \lambda_ {r + 1}\right) ^ {2}\right)}} d t. \\ \end{array} +$$ + +Moreover, $\int_0^T\frac{a_0 + t}{\sqrt{(t^2 + (\delta_k / 2)^2)(t^2 + (a_0 - \lambda_{r + 1})^2)}} dt$ equals + +$$ +\begin{array}{l} \int_ {0} ^ {T} \frac {a _ {0} d t}{\sqrt {(t ^ {2} + (\delta_ {k} / 2) ^ {2}) (t ^ {2} + (a _ {0} - \lambda_ {r + 1}) ^ {2})}} + \int_ {0} ^ {T} \frac {t d t}{\sqrt {(t ^ {2} + (\delta_ {k} / 2) ^ {2}) (t ^ {2} + (a _ {0} - \lambda_ {r + 1}) ^ {2})}} d t \\ = \int_ {0} ^ {T} \frac {a _ {0} d t}{\sqrt {(t ^ {2} + (\delta_ {k} / 2) ^ {2}) (t ^ {2} + (a _ {0} - \lambda_ {r + 1}) ^ {2})}} + \frac {1}{2} \log \left(\frac {(T ^ {2} + (\delta_ {k} / 2) ^ {2}) (a _ {0} - \lambda_ {r + 1}) ^ {2}}{(T ^ {2} + (a _ {0} - \lambda_ {r + 1}) ^ {2}) \delta_ {k} / 2}\right) \\ \leq \frac {a _ {0}}{\max \{\delta_ {k} / 2 , a _ {0} - \lambda_ {r + 1} \}} \times \log \left(\frac {T + \sqrt {T ^ {2} + \min \{\delta_ {k} / 2 , a _ {0} - \lambda_ {r + 1} \} ^ {2}}}{\min \{\delta_ {k} / 2 , a _ {0} - \lambda_ {r + 1} \}}\right) + \frac {1}{2} \log \left(\frac {(T ^ {2} + (\delta_ {k} / 2) ^ {2}) (a _ {0} - \lambda_ {r + 1})}{(T ^ {2} + (a _ {0} - \lambda_ {r + 1}) ^ {2}) \delta_ {k} / 2}\right). \\ \end{array} +$$ + +Note that $a_0 - \lambda_{r+1} \geq \delta_k / 2$ and $a_0 - \lambda_{r+1} + \delta_k / 2 = \lambda_k - \lambda_{r+1} \geq \frac{\lambda_k}{2}$ . Therefore, $a_0 - \lambda_{r+1} = \max\{\delta_k / 2, a_0 - \lambda_{r+1}\} \geq \frac{\lambda_k}{4}$ . We further obtain that $\int_0^T \frac{a_0 + t}{\sqrt{(t^2 + (\delta_k / 2)^2)(t^2 + (a_0 - \lambda_{r+1})^2)}} dt$ is at most + +$$ +\frac {a _ {0}}{\lambda_ {k} / 4} \cdot \log \left(\frac {5 T}{\delta_ {k}}\right) + \frac {1}{2} \log \left(\frac {2 T}{\delta_ {k}}\right) \leq 4. 5 \log \left(\frac {5 T}{\delta_ {k}}\right). \tag {17} +$$ + +The estimates (16) and (17) together imply that the last term is at most + +$$ +1 8 \| E \| \left(\frac {5 T}{\delta_ {k}}\right). \tag {18} +$$ + +Combining (12), (15) and (18), we finally obtain that $M_{1}$ is at most + +$$ +r _ {1} ^ {2} x _ {1} \left(\frac {2 \pi a _ {0}}{\delta_ {k}} + 2 \log \left(\frac {3 T}{\delta_ {k}}\right)\right) + \| E \| \left(4 \pi + 2 \log \left(\frac {2 T}{\delta_ {k}}\right)\right) + 1 8 \| E \| \log \left(\frac {5 T}{\delta_ {k}}\right) +$$ + +$$ +\leq r _ {1} ^ {2} x _ {1} \left(\frac {2 \pi a _ {0}}{\delta_ {k}} + 2 \log \left(\frac {6 \sigma_ {1}}{\delta_ {k}}\right)\right) + (2 0 + 4 \pi / \log (1 0) \| E \| \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right) (\text {s i n c e} \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right) \geq \log 1 0) +$$ + +$$ +\leq r ^ {2} \bar {x} \left(\frac {2 \pi a _ {0}}{\delta_ {k}} + 2 \log \left(\frac {6 \sigma_ {1}}{\delta_ {k}}\right)\right) + (2 0 + 4 \pi / \log (1 0) \| E \| \log \left(\frac {1 0 \sigma_ {1}}{\delta_ {k}}\right). \tag {19} +$$ + +This proves Lemma D.6. + +Next, we estimate $N_{3}$ . Notice that + +$$ +\begin{array}{l} N _ {3} = \int_ {\Gamma_ {3}} \frac {| z |}{\min _ {i} | z - \lambda_ {i} | ^ {2}} | d z | \\ = \int_ {- T} ^ {T} \frac {\sqrt {a _ {1} ^ {2} + t ^ {2}}}{\min _ {i \in [ n ]} \left[ \left(a _ {1} - \lambda_ {i}\right) ^ {2} + t ^ {2} \right]} d t (\text {s i n c e} \Gamma_ {3} := \{z \mid z = a _ {1} + \mathbf {i} t, - T \leq t \leq T \}) \\ \leq \int_ {- T} ^ {T} \frac {\sqrt {a _ {1} ^ {2} + t ^ {2}}}{t ^ {2} + (a _ {1} - \lambda_ {1}) ^ {2}} d t \\ \leq \int_ {- T} ^ {T} \frac {\left| a _ {1} \right|}{t ^ {2} + \left(a _ {1} - \lambda_ {1}\right) ^ {2}} d t + \int_ {- T} ^ {T} \frac {| t |}{t ^ {2} + \left(a _ {1} - \lambda_ {1}\right) ^ {2}} d t \\ \leq \left| \frac {\pi a _ {1}}{a _ {1} - \lambda_ {1}} \right| + 2 \log \left(\left| \frac {T}{a _ {1} - \lambda_ {1}} \right| ^ {2} + 1\right) (\text {b y}) \\ \leq \frac {\pi a _ {1}}{a _ {1} - \lambda_ {1}} + 4 \log \left| \frac {3 T}{a _ {1} - \lambda_ {1}} \right|. \\ \end{array} +$$ + +This proves Lemma D.3. + +# E.2 Estimating integrals over horizontal segments + +We are going to bound $N_{2}, N_{4}$ - integral over top horizontal segment (prove Lemma D.4). We have + +$$ +\begin{array}{l} N _ {2} = \int_ {\Gamma_ {2}} \frac {| z |}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} | d z | \\ = \int_ {a _ {0}} ^ {a _ {1}} \frac {\sqrt {x ^ {2} + T ^ {2}}}{\min _ {i \in \{n \}} \left(\left(x - \lambda_ {i}\right) ^ {2} + T ^ {2}\right)} d x (\text {s i n c e} \Gamma_ {2} := \left\{z \mid z = x + \mathbf {i} T, a _ {0} \leq x \leq a _ {1} \right\}) \\ \leq \int_ {a _ {0}} ^ {a _ {1}} \frac {\sqrt {2} T}{T ^ {2}} d x \text {(s i n c e} x \leq a _ {1} \leq T) \\ = \frac {\sqrt {2} \left| a _ {1} - a _ {0} \right|}{T}. \\ \end{array} +$$ + +By similar arguments, we also obtain + +$$ +N _ {4} \leq \frac {\sqrt {2} | a _ {1} - a _ {0} |}{T}. +$$ + +These estimates on $N_{2}, N_{4}$ prove Lemma D.4. + +# E.3 Estimating integrals over vertical segments for non-interaction bound + +In this Section, we estimate $N_{1}$ , proving Lemma D.2. The estimation of $N_{3}$ follows the case of the interaction-dependent bound at the end of Section E.1. + +$$ +\begin{array}{l} N _ {1} = \int_ {\Gamma_ {1}} \frac {| z |}{\min _ {i} | z - \lambda_ {i} | ^ {2}} | d z | \\ = \int_ {- T} ^ {T} \frac {\sqrt {a _ {0} ^ {2} + t ^ {2}}}{\min _ {i \in [ n ]} \left[ (a _ {0} - \lambda_ {i}) ^ {2} + t ^ {2} \right]} d t (\text {s i n c e} \Gamma_ {1} := \{z \mid z = a _ {0} + \mathbf {i} t, - T \leq t \leq T \}) \\ \leq \int_ {- T} ^ {T} \frac {\sqrt {a _ {0} ^ {2} + t ^ {2}}}{t ^ {2} + (\delta_ {k} / 2) ^ {2}} d t \left(\text {b y (9)}\right) \\ \leq \int_ {- T} ^ {T} \frac {\left| a _ {0} \right|}{t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}} d t + \int_ {- T} ^ {T} \frac {\left| t \right|}{t ^ {2} + \left(\delta_ {k} / 2\right) ^ {2}} d t \\ \leq \left| \frac {2 \pi a _ {0}}{\delta_ {k}} \right| + 2 \log \left(\left| \frac {2 T}{\delta_ {k}} \right| ^ {2} + 1\right) (\text {b y}) \\ \leq \left| \frac {2 \pi a _ {0}}{\delta_ {k}} \right| + 4 \log \left| \frac {3 T}{\delta_ {k}} \right|. \\ \end{array} +$$ + +This proves Lemma D.2. + +# F Perturbation of matrix functionals - Theorem 2.3 + +In this section, we complete the delayed proof of Theorem 2.3. By Remark 3.2, to prove Theorem 2.3, we need to show that + +$$ +2 \pi F _ {1} (1) := \int_ {\Gamma} \| (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \| | d z | = 4 \pi \frac {\| E \|}{\delta_ {p}}, +$$ + +in which the contour $\Gamma$ is set to be a rectangle with vertices + +$$ +(x _ {0}, T), (x _ {1}, T), (x _ {1}, - T), (x _ {0}, - T), \text {w h e r e} x _ {0} := \lambda_ {p} - \delta_ {p} / 2, x _ {1} := 2 \lambda_ {1}, T := 2 \lambda_ {1}. +$$ + +We split $\Gamma$ into four segments: + +$$ +\Gamma_ {1} := \left\{\left(x _ {0}, t\right) \mid - T \leq t \leq T \right\}; \Gamma_ {2} := \left\{\left(x, T\right) \mid x _ {0} \leq x \leq x _ {1} \right\}; +$$ + +$$ +\Gamma_ {3} := \left\{\left(x _ {1}, t\right) \mid T \geq t \geq - T \right\}; \Gamma_ {4} := \left\{\left(x, - T\right) \mid x _ {1} \geq x \geq x _ {0} \right\}. +$$ + +Therefore, + +$$ +\int_ {\Gamma} \| (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \| | d z | = \sum_ {i = 1} ^ {4} M _ {k}, \tag {20} +$$ + +in which + +$$ +M _ {i} := \int_ {\Gamma_ {i}} \| (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \| | d z | \text {f o r} i \in \{1, 2, 3, 4 \}. +$$ + +By a similar strategy from previous section, we bound $M_{1}$ as follows. Notice that + +$$ +\left| \left| (z - A) ^ {- 1} E (z - A) ^ {- 1} \right| \right| \leq \frac {\| E \|}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}}. +$$ + +Therefore, $M_{1}$ is at most + +$$ +\begin{array}{l} \| E \| \cdot \int_ {\Gamma_ {1}} \frac {1}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} | d z | \\ \leq \| E \| \cdot \int_ {\Gamma_ {1}} \frac {1}{| z - \lambda_ {p} | ^ {2}} | d z | \text {(s i n c e} \lambda_ {p} \text {i s c l o s e s t t o} \Gamma_ {1} \text {a m o n g a l l e i g e n v a l u e s o f} A) \\ = \| E \| \cdot \int_ {- T} ^ {T} \frac {1}{\left(\delta_ {p} / 2\right) ^ {2} + t ^ {2}} d t (\text {b y d e f i n i t i o n} \Gamma_ {1} := \left\{\left(x _ {0}, t\right) \mid - T \leq t \leq T \right\} \text {a n d} \left| x _ {0} - \lambda_ {p} \right| = \delta_ {p} / 2) \\ \leq \frac {2 \pi \| E \|}{\delta_ {p}} (\text {b y} \\ \end{array} +$$ + +Next, we bound $M_3$ as what follows. + +$$ +\begin{array}{l} M _ {3} \leq \| E \| \int_ {\Gamma_ {3}} \frac {1}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} | d z | \\ = \| E \| \int_ {- T} ^ {T} \frac {1}{\min _ {i \in [ n ]} \left(\left(x _ {1} - \lambda_ {i}\right) ^ {2} + t ^ {2}\right)} d t (\text {s i n c e} \Gamma_ {3} := \{z \mid z = x _ {1} + \mathrm {i} t, - T \leq t \leq T) \\ = \| E \| \int_ {- T} ^ {T} \frac {1}{t ^ {2} + (x _ {1} - \lambda_ {1}) ^ {2}} d t \\ \leq \frac {\pi \| E \|}{\left| x _ {1} - \lambda_ {1} \right|} (\text {b y L e m m a E . 1}) \\ = \frac {\pi \| E \|}{\lambda_ {1}} (\text {s i n c e} x _ {1} = 2 \lambda_ {1}). \\ \end{array} +$$ + +Next, we estimate $M_2$ as + +$$ +M _ {2} \leq \int_ {\Gamma_ {2}} \frac {1}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} \| E \| | d z | = \| E \| \int_ {\Gamma_ {2}} \frac {1}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} | d z |. +$$ + +Moreover, since $\Gamma_2\coloneqq \{z|z = x + \mathbf{i}T,x_0\leq x\leq x_1\}$ + +$$ +\int_ {\Gamma_ {2}} \frac {1}{\min _ {i \in [ n ]} | z - \lambda_ {i} | ^ {2}} | d z | = \int_ {x _ {0}} ^ {x _ {1}} \frac {1}{\min _ {i \in [ n ]} ((x - \lambda_ {i}) ^ {2} + T ^ {2})} d x \leq \int_ {x _ {0}} ^ {x _ {1}} \frac {1}{T ^ {2}} d x = \frac {| x _ {1} - x _ {0} |}{T ^ {2}}. +$$ + +Therefore, $M_2 \leq \frac{\|E\|\cdot|x_1 - x_0|}{T^2} \leq \frac{\|E\|\lambda_1}{4\lambda_1^2} = \frac{\|E\|}{4\lambda_1}$ . Similarly, we also obtain that $M_4 = \frac{\|E\|}{4\lambda_1}$ . These estimates on $M_1, M_2, M_3, M_4$ and Equation 20 imply + +$$ +\int_ {\Gamma} \| (z I - A) ^ {- 1} E (z I - A) ^ {- 1} \| | d z | = \frac {2 \pi \| E \|}{\delta_ {p}} + (\pi + \frac {1}{4} + \frac {1}{4}) \frac {\| E \|}{\lambda_ {1}} \leq 4 \pi \frac {\| E \|}{\delta_ {p}}. +$$ + +The last inequality follows the trivial fact that $\lambda_1 > \delta_p$ for any PSD matrix $A$ . We complete the proof. + +# G Some classical perturbation bounds + +This section recalls standard results referenced in Section 2, Section 3, and Section A. + +Theorem G.1 (Eckart-Young-Mirsky bound [16]). Let $A, \tilde{A} \in \mathbb{R}^{n \times n}$ , and let $A_p, \tilde{A}_p$ denote their respective best rank- $p$ approximations. Set $E := \tilde{A} - A$ . Then, + +$$ +\left\| \tilde {A} _ {p} - A _ {p} \right\| \leq 2 \left(\sigma_ {p + 1} + \| E \|\right), +$$ + +where $\sigma_{p + 1}$ is the $(p + 1)$ st singular value of $A$ . + +Theorem G.2 (Weyl's inequality [46]). Let $A, E \in \mathbb{R}^{n \times n}$ be symmetric, and define $\tilde{A} \coloneqq A + E$ . Then, for any $1 \leq i \leq n$ , + +$$ +\left| \tilde {\lambda} _ {i} - \lambda_ {i} \right| \leq \| E \| \quad a n d \quad \left| \tilde {\sigma} _ {i} - \sigma_ {i} \right| \leq \| E \|, +$$ + +where $\lambda_{i},\tilde{\lambda}_{i}$ are the ith eigenvalues of $A$ and $\tilde{A}$ , and $\sigma_{i},\tilde{\sigma}_{i}$ are the corresponding singular values. + +# H Notation + +This section summarizes the key notations used throughout the paper. Let $A, E$ be symmetric $n \times n$ matrices, and define the perturbed matrix $\tilde{A} \coloneqq A + E$ . Let $f$ be an entire function, and let $s \in \mathbb{N}$ . + +Table 3: Summary of notation + +
SymbolDefinition
nDimension of A, A
pTarget rank parameter
ApBest rank-p approximation of A
AipBest rank-p approximation of A
λ1≥···≥λnEigenvalues of A in descending order
λ1≥···≥λnEigenvalues of A in descending order
σ1≥···≥σnSingular values of A in descending order
δi for i∈[n-1]ith eigengap: δi := λi - λi+1
ui for i∈[n]Eigenvector of A corresponding to λi
ūi for i∈[n]Eigenvector of A corresponding to ŷi
sr(A)Stable rank: sr(A) := ||A||2/F/||A||2(p. 22)
Halving distance rSmallest integer such that λp/2 ≥ λr+1 (p. 3, Thm. 2.2)
Interaction term xx := max1≤i,j≤r|uiT Euj| (p. 3, Thm. 2.2)
fp(A)fp(A) := ∑i=1p f(λi)uiuT (p. 4, Thm. 2.3)
fp(Å)fp(Å) := ∑i=1p f(Åi)ūiūT (p. 4, Thm. 2.3)
ΓContour enclosing {λ1, ..., λp} (p. 5)
F(f)1/2π ∫f(z)[(zi-Å)-1-(zi-A)-1]| |dz| (p. 5, Eq. (2))
Fs(f)1/2π ∫f(z)(zi-A)-1[E(zi-A)-1]s || |dz| (p. 6)
F1(f)1/2π ∫f(z)(zi-A)-1E(zi-A)-1|| |dz| (p. 6, Lem. 3.1)
F(z)1/2π ∫|z[(zi-Å)-1-(zi-A)-1]| || |dz| (p. 6)
F1(z)1/2π ∫|z(zi-A)-1E(zi-A)-1|| |dz| (p. 6)
||·||Spectral norm
||·||FFrobenius norm
EYM boundEckart-Young-Mirsky bound
M-V boundMangoubi-Vishnoi bound
PSDPositive semi-definite
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Lossless and Training-Free Acceleration for Offloaded LLMs via Substitute Speculative Decoding/full.md @@ -0,0 +1,619 @@ +# Speculate Deep and Accurate: Lossless and Training-Free Acceleration for Offloaded LLMs via Substitute Speculative Decoding + +Pei-Shuo Wang $^{1}$ , Jian-Jia Chen $^{*1}$ , Chun-Che Yang $^{*1}$ , Chi-Chih Chang $^{2}$ , Ning-Chi Huang $^{1}$ , Mohamed S. Abdelfattah $^{2}$ , and Kai-Chiang Wu $^{1}$ + +$^{1}$ National Yang Ming Chiao Tung University $^{2}$ Cornell University + +# Abstract + +The immense model sizes of large language models (LLMs) challenge deployment on memory-limited consumer GPUs. Although model compression and parameter offloading are common strategies to address memory limitations, compression can degrade quality, and offloading maintains quality but suffers from slow inference. Speculative decoding presents a promising avenue to accelerate parameter offloading, utilizing a fast draft model to propose multiple draft tokens, which are then verified by the target LLM in parallel with a single forward pass. This method reduces the time-consuming data transfers in forward passes that involve offloaded weight transfers. Existing methods often rely on pretrained weights of the same family, but require additional training to align with custom-trained models. Moreover, approaches that involve draft model training usually yield only modest speedups. This limitation arises from insufficient alignment with the target model, preventing higher token acceptance lengths. To address these challenges and achieve greater speedups, we propose SUBSPEC, a plug-and-play method to accelerate parameter offloading that is lossless and training-free. SubSpec constructs a highly aligned draft model by generating low-bit quantized substitute layers from offloaded target LLM portions. Additionally, our method shares the remaining GPU-resident layers and the KV-Cache, further reducing memory overhead and enhance alignment. SubSpec achieves a high average acceptance length, delivering $9.1 \times$ speedup for Qwen2.5 7B on MT-Bench (8GB VRAM limit) and an average of $12.5 \times$ speedup for Qwen2.5 32B on popular generation benchmarks (24GB VRAM limit). The code is available at https://github.com/NYCU-EDgeAi/subspec. + +# 1 Introduction + +Large language models (LLMs) [1, 4, 37] have achieved widespread popularity in tasks ranging from chat models to code generation. Local deployment of these models on consumer hardware offers compelling advantages: data privacy, potential cost reductions compared to API access, freedom for model customization, and direct control over the inference process [47]. + +The substantial memory requirement is the primary barrier to such local deployment. Popular open source model families like Llama [15, 39], Qwen [42, 43], and DeepSeek [26] often exceed the memory constraint in common consumer-level GPUs (typically ranging from 8GB to 24GB). For + +![](images/b0ffb7b30720db7236cabba6d6d19dc18f4e9db0e0c8f6698d64d89702811527.jpg) +Qwen2.5 7B with Offloading (8GB VRAM Limit) + +![](images/d5df38da26b3ba552f1815e010936a2e2c1144fb24b7195cda29f8703043ce84.jpg) +Figure 1: Impact of draft model characteristics on speculative decoding performance, tested under the MT-Bench [46] benchmark. Left: Maximal speedup achieved by different draft models varies with draft depth in tree-based speculative decoding. Right: Average inference latency per token for Qwen2.5 7B (8GB GPU memory constraint) of different methods with optimal draft depths. SubSpec utilizes a higher draft model computation to minimize costly memory transfers of target model parameters. + +instance, Llama3.1 8B requires 16GB VRAM, surpassing the capacity of a consumer card such as an RTX 3060 (12GB). + +A common strategy to accommodate LLMs within limited memory is model compression, primarily via techniques like quantization [13, 25]. Quantization reduces model memory demands by encoding weights in low-precision formats (e.g., INT8, INT4); however, this lossy compression inevitably alters model outputs and impairs model quality. + +Parameter offloading presents a lossless alternative, storing inactive layers in host memory and streaming them to the GPU when needed [20]. Unfortunately, frequent data transfers over the PCIe bus severely throttle throughput. This results in prohibitively long latencies for token generation, often only one to two tokens per second on consumer cards like the RTX 4090 supporting PCIe 4.0. Such latency undermines practical usability and fails to satisfy demands for real-time interaction. + +Speculative decoding (SD) has emerged as a promising acceleration technique to mitigate the excessive latency due to parameter offloading. SD utilizes a smaller, faster draft model to rapidly propose multiple tokens, which are then verified by the larger target model in parallel using a single forward pass. Potentially accepting multiple tokens, SD can substantially reduce the number of expensive forward passes involving target model weight transfers. However, some existing SD approaches [10, 35] achieve notable speedups by relying on smaller pretrained models from the same family that align well with the target LLM. These strategies cannot be directly applied to custom-trained models and require additional training for alignment. Other techniques involving draft model training [24, 45] often yield relatively modest speedups when applied on offloading scenarios. This limitation stems from insufficient alignment, leading to average acceptance lengths generally below seven tokens. + +Hence, we introduce Substitute Speculative Decoding (SUBSPEC), a plug-and-play and training-free approach designed to maximize inference throughput for offloaded LLMs on consumer-grade devices. SubSpec constructs a highly aligned draft model by sharing GPU-resident weights and KV-Cache with the target model, while maintaining low-bit substitute weights for the offloaded layers. This innovative design ensures exceptionally close alignment between the draft and target models and enables deployment of large models locally with competitive inference speeds. The effectiveness of this approach in reducing data transfer latency is demonstrated in Figure 1. + +In this work, our key contributions are as follows: + +- Our analysis identifies the predominance of the alignment and draft depth of the draft model under parameter offloading scenarios, compared to its speed. + +- We propose a plug-and-play and training-free method that constructs highly aligned draft models using shared components and low-bit substitutes, minimizing VRAM overhead. +- We introduce refinements to draft tree construction, such as probability sharpening, to effectively leverage deeper trees and boost acceptance length. +- We developed an efficient system to accelerate offloading. Using this system, popular models like Qwen2.5 7B achieve 25 tokens per second on a single consumer GPU across diverse benchmarks (within an 8GB VRAM limit), demonstrating over $10 \times$ speedups compared to baseline offloading inference. + +# 2 Background + +# 2.1 Parameter Offloading + +Parameter offloading is a technique commonly employed by inference frameworks [14, 18, 21] to manage models whose memory requirements exceed GPU capacity. This strategy primarily stores model parameters in CPU memory, transferring them to the GPU only when required for computation. After the GPU completes computations for a specific layer, its parameters may be either discarded or overwritten by those required for subsequent layers. Typically, frameworks aim to retain as many model layers as possible directly within GPU memory, offloading only the remainder to minimize data transfer overhead. + +However, these frequent parameter transfers between CPU and GPU memory introduce considerable latency, as GPU operations often stall while awaiting data. This latency significantly impacts the total inference time, making performance highly sensitive to PCIe bandwidth limitations. + +While techniques like Deepspeed-inference [3] and FlexGen [34] improve throughput via large batches, such approaches are not suitable for latency-critical online inference, where small batch processing is standard. Each forward pass remains constrained by the PCIe bus data transfer bottleneck in such scenarios. + +# 2.2 Speculative Decoding + +Speculative decoding [7, 22] accelerates the target autoregressive LLM by generating multiple tokens per iteration, rather than just one. In each iteration, a smaller, faster draft model quickly produces a set of draft tokens. These draft tokens are then evaluated in parallel by the target model (the original LLM being accelerated) with a single forward pass. Tokens confirmed by the target model are then accepted, reducing the total forward passes required to generate the full context. The average acceptance length $(\tau)$ is the mean number of accepted draft tokens per iteration. + +Miao et al. further advanced this approach with tree decoding. This method improves the number of tokens accepted per iteration while conserving computational efficiency. Tree decoding maintains a hierarchical token structure instead of parallel beams. Multiple branching token paths are flattened and evaluated in one forward pass. Positional encodings and attention masks are modified to preserve tree structure dependency. Subsequent works [6, 10, 23, 24] have built on this foundation. These works developed advanced tree-based speculative decoding strategies, reporting $2 \times$ to $4 \times$ faster than standard autoregressive decoding when no offloading is required. + +Recent methods illustrate the benefits of using SD to accelerate offloading scenarios. SD significantly reduces data transfer overhead by reducing the total number of forward passes required by the target model, without losing quality. For example, SpecExec [35] demonstrates that speculating and verifying with a larger budget (from 128 to 2048 draft tokens) per iteration can achieve higher average acceptance length and additional speedup. In contrast, Dovetail [45] focuses on accelerating smaller LLMs on heterogeneous setups, running Llama-2 7B by offloading partial computation to the CPU. The authors of Dovetail also trained a draft model larger than EAGLE [23] to achieve better alignment. These techniques have showcased notable speedups. Such speedups result from choosing mid-sized, accurate draft models and speculating deeper token trees. These approaches either assume access to a compatible smaller model in the same family or require fine-tuning to align draft and target distributions. + +# 3 Analysis of Speculative Decoding Speedup in Offloading Scenarios + +# 3.1 Theoretical Speedup Analysis + +This subsection derives the theoretical speedup of speculative decoding (SD) over autoregressive decoding (AR). The goal is to clarify the factors governing SD performance and highlight the distinct optimization challenges posed by standard (model fully GPU-resident) versus parameter-offloading inference scenarios. + +First, we establish the original time required for the token generation of the AR-based target LLM. The total time $T_{AR}^{\mathcal{N}}$ required to generate $\mathcal{N}$ tokens using autoregressive decoding is directly proportional to the number of tokens, as each token necessitates one forward pass of the target model: + +$$ +T _ {A R} ^ {\mathcal {N}} = \mathcal {N} \cdot t _ {\text {T a r g e t}}, \tag {1} +$$ + +where $t_{Target}$ represents the latency of a single forward pass of the target model. + +In contrast, the total time $T_{SD}^{\mathcal{N}}$ to generate $\mathcal{N}$ tokens using speculative decoding (SD) is given by: + +$$ +T _ {S D} ^ {\mathcal {N}} = \mathcal {N} \cdot \frac {\mathcal {D} \cdot t _ {D r a f t} + \gamma \cdot t _ {T a r g e t}}{\tau}, \quad 1 \leq \tau \leq \mathcal {D} + 1, \tag {2} +$$ + +$t_{Draft}$ is the latency of a single draft model forward pass, and $\mathcal{D}$ is the draft depth. For each iteration, the draft model runs $\mathcal{D}$ forward passes to generate a draft token sequence or tree of depth $\mathcal{D}$ (as to "speculate"). The target model then performs a single forward pass over all draft tokens, checking their correctness (as to "verify"). The factor $\gamma$ reflects the relative cost of this parallel verification compared to a normal AR forward pass $t_{Target}$ (typically ranging from $1 \leq \gamma \leq 2$ ). The term $\tau$ , known as the average acceptance length, denotes the mean number of tokens accepted per iteration (potentially including a bonus token derived from the final accepted token, thus $1 \leq \tau \leq \mathcal{D} + 1$ ). + +The theoretical speedup is then the ratio $T_{AR}^{\mathcal{N}} / T_{SD}^{\mathcal{N}}$ . Combining Equations (1) and (2) yields: + +$$ +\frac {T _ {A R} ^ {\mathcal {N}}}{T _ {S D} ^ {\mathcal {N}}} = \frac {\tau \cdot t _ {T a r g e t}}{\mathcal {D} \cdot t _ {D r a f t} + \gamma \cdot t _ {T a r g e t}} = \frac {\tau}{\frac {\mathcal {D} \cdot t _ {D r a f t}}{t _ {T a r g e t}} + \gamma}, \quad 1 \leq \tau \leq \mathcal {D} + 1. \tag {3} +$$ + +Equation (3) reveals a key trade-off when selecting the draft model and its draft depth $(\mathcal{D})$ . In standard settings, both the target latency $t_{Target}$ and the speculation overhead $\mathcal{D} \cdot t_{Draft}$ impact the denominator. Increasing draft depth $(\mathcal{D})$ to improve $\tau$ potentially must therefore be balanced against this linear rise in speculation overhead. This balance typically favors smaller, faster draft models $(t_{Draft} < t_{Target})$ with moderate $\mathcal{D}$ , though such models often offer lower alignment with the target, potentially capping the achievable $\tau$ . + +Conversely, this optimization landscape changes dramatically in parameter offloading scenarios where data transfers significantly increase $t_{Target}$ . Here, the relative impact of the speculation latency $(\mathcal{D} \cdot t_{Draft})$ diminishes against the target verification cost $(\gamma \cdot t_{Target})$ . Maximizing the average acceptance length $(\tau)$ thus becomes crucial to minimize the frequency of expensive target model forward passes. This priority favors draft models with superior target alignment and contextual quality, even if their $t_{Draft}$ is larger, as reducing calls to the costly target model results in greater overall speedup. + +# 3.2 Empirical Validation and Motivation for Efficient Draft Model Generation + +To empirically demonstrate these contrasting dynamics of speculative decoding (SD) in standard versus parameter-offloading scenarios, we present an illustrative evaluation with Qwen2.5 7B as the target model. Figure 2 showcases the performance of two representative existing draft model types: EAGLE-Qwen2.5* [23], a smaller, generally faster draft model, and Qwen2.5 1.5B [43], a moderately-sized model from the same family that might offer better intrinsic alignment. Figure 2 also provides an early glimpse of SubSpec, our proposed method, which will be detailed in the next section. + +![](images/d33513425c7973e3bda9442199fe2fb7ee926fd7480bee7e673faea58954d0c2.jpg) +(a) Average acceptance length $(\tau)$ vs. draft depth (D) + +![](images/fbd412d03290a15c37b3f68e682eb05e816dfb0ecda55fefaac864e3b2cde576.jpg) +(b) Standard (w/o offloading) + +![](images/43423971b918c291459a3f7aba76503f97840930e7e7eee42b7ef56f16e0dc6b.jpg) +(c) With offloading (8GB limit) +Figure 2: Comparative performance of Qwen2.5 7B using SD with different draft approaches across varying draft depths, tested under the MT-Bench benchmark. + +The results presented in Figure 2 confirm our theoretical speedup analysis (Equation 3): the faster EAGLE-Qwen2.5 draft model performed best in standard settings due to its low overhead, while the better-aligned Qwen2.5 1.5B model achieved superior speedup in the offloading setting, where large $t_{Target}$ dominates due to obtaining a higher average acceptance length $(\tau)$ . These empirical findings highlight a critical challenge: maximizing SD benefits in offloading scenarios demands highly aligned draft models. This then raises a practical question for us: How can we efficiently obtain such highly aligned draft models, particularly for custom or fine-tuned LLMs? + +While some open-source model families offer pretrained models of various sizes, allowing smaller versions to serve as potentially efficient and aligned draft models, this option is often unavailable for custom-trained or fine-tuned LLMs. Creating a sufficiently aligned draft model for such custom models typically necessitates additional training or distillation. These processes introduce further costs (computation, memory, data, and time), preventing widespread SD adoption in many real-world deployments. + +To bridge this gap, we propose a practical and efficient alternative: utilizing a low-bit quantized version of the target LLM itself as the draft model, which remains fully GPU-resident. This approach capitalizes on efficient data-free quantization techniques, eliminating the need for training datasets and resource-intensive training runs. Detailed later in Section 4, our method also incorporates weight and KV-Cache sharing, significantly reducing VRAM overhead while enhancing draft model alignment. + +# 4 Substitute Speculative Decoding (SubSpec) + +We introduce Substitute Speculative Decoding (SubSpec), a novel method enabling efficient LLM inference on consumer-grade GPUs, particularly when model weights exceed available GPU memory. SubSpec achieves this by constructing highly aligned, fully GPU-resident draft models and performing tree-based speculative decoding, constructing deep draft trees with optimizations. + +SubSpec constructs the draft model with quantized 'substitute' layers for the offloaded portions of the target model, while GPU-resident layers and the KV-Cache are shared between the draft and target. This design creates a draft model that is highly aligned with the target model. By constructing deep draft token trees with such a model, it obtains extremely high average acceptance lengths $(\tau)$ , significantly reducing the expensive data transfers, boosting overall inference throughput. The subsequent sections detail the draft model construction (Section 4.1), adaptations for draft tree construction (Section 4.2), and complementary performance optimizations (Section 4.3). + +# 4.1 Draft Model Construction + +SubSpec employs three synergistic strategies to ensure that the draft model remains entirely on the GPU. These strategies involve using substitute weights, sharing GPU-resident layers, and employing a shared KV-Cache, as illustrated in Figure 3. + +![](images/c81e25f9cca6650d03facd5528f2ecc77d2813631bda3fabac41728ea2f41cef.jpg) +Figure 3: Left: Draft model architecture of SubSpec. SubSpec maintains additional low-bit substitute weights to keep the full draft model on the GPU. Right: Shared KV-Cache generation pipeline of SubSpec. The draft model reuses the KV-Cache of the target model to achieve better alignment and memory efficiency. This illustration serves as a simple sequential demonstration, while in practice, we maintain a flattened token tree for tree decoding. + +![](images/55195de405b46becc1fcb253fdf52df9b699283a939229d193c2b09ecd174d2c.jpg) + +Quantized Substitute Weights for Offloaded Layers. A core principle of SubSpec involves replacing the offloaded layers of the target model with lightweight, low-bit 'substitute' layers within the draft model. These substitutes reside entirely on the GPU and approximate the functionality of their corresponding target layers. For example, layers 2 and 3 on the left side of Figure 3 of the target model are offloaded and the corresponding low-bit substitutes of the target layers are utilized for the GPU-resident draft model. We generate these substitutes using fast, data-free quantization methods (e.g., HQQ [5], HIGGS [28]), which require minimal processing time (under minutes for 7B to 70B parameter models on a single consumer GPU). During inference, these quantized layers enable rapid execution through highly optimized low-bit GEMM kernels [5, 16]. + +GPU-Resident Layer Sharing. The draft model reuses target model layers that remain in GPU memory. For example, the weight of layer one on the left side of Figure 3 is GPU-resident and shared by both the target model and our draft model. This sharing strategy maximizes GPU resource utilization and inherently improves the alignment between the draft and target models, given that identical weights are employed for these shared layers. + +Shared KV-Cache. The structural similarity between the draft model of SubSpec and the target model allows for a unified KV-Cache. This sharing approach yields significant advantages: it halves the KV-Cache memory footprint compared to using separate caches and enhances alignment by ensuring that both models operate on an identical contextual history. Furthermore, sharing the KV-Cache eliminates the need for a distinct prefilling phase for the draft model, directly contributing to faster overall inference. The demonstration of this pipeline is illustrated on the right side of Figure 3. The draft model extends new KV-Cache values on speculation, which the target model then overwrites during verification to ensure identical results. + +# 4.2 Optimized Draft Tree Construction + +Constructing Deep Context-Aware Dynamic Draft Tree. The high alignment of the SubSpec draft model (detailed in Section 4.1) enables the exploration of deeper draft trees to achieve higher average acceptance lengths $(\tau)$ . Our sampling approach thus extends established context-aware dynamic draft tree methods like EAGLE-2 [24] and SpecExec [35] to support these greater depths. + +A context-aware dynamic draft tree is built by iteratively generating draft tokens with the draft model over $\mathcal{D}$ future time steps. In each of these $\mathcal{D}$ forward passes, all leaf nodes are input to the draft model, each yielding probability distributions for the potential next tokens. The score for each potential next token is the cumulative product of its conditional generation probability (from the draft model) and its parent path score. The top-k tokens with the highest scores are selected to form the new $k$ leaf nodes for the subsequent time step. This iterative procedure produces a draft tree of depth $\mathcal{D}$ , presenting $k \times \mathcal{D}$ draft tokens for target model validation (not including the root token). + +![](images/8968935e9cc1fbf29f2c5cb5b0d629865818a9c907de84da1b22cfe4a1bc57ba.jpg) +Figure 4: Demonstration of the false positive path issue (with $k = 2$ for simplification). Correct draft tokens may be dropped due to a lower cumulative probability score. The number in the parentheses in each token denotes the cumulative probability score. + +Addressing Cumulative Probability Divergence. While exploring substantially deeper draft trees than typically used in prior work (often $\mathcal{D} \leq 7$ ) showed promise for increasing $\tau$ , construction for greedy decoding (target temperature $= 0$ ) revealed a subtle issue. We observed that paths initiated by less probable tokens could accumulate an erroneously high overall likelihood through high-probability subsequent selections. This phenomenon can lead to 'false-positive' paths that achieve a higher cumulative probability than the genuinely optimal path (illustrated in Figure 4). Such divergence is problematic in greedy decoding, as it can misguide path selection, potentially causing early termination of the correct sequence and thereby limiting the achievable $\tau$ . + +Draft Probability Sharpening. To counteract this divergence under greedy decoding, we employ draft probability sharpening. This technique involves applying a low temperature $(= 0.2)$ to the output distribution of the draft model before calculating cumulative probabilities for tree sampling. Such sharpening makes the probability distribution more peaked, reducing the probability mass allocated to tokens with lower initial probabilities. Further analysis is shown in Appendix D. + +# 4.3 Complementary Performance Optimizations + +Beyond the core SubSpec framework for draft construction and sampling, two complementary techniques are integrated to further enhance performance and efficiency: + +Asynchronous Data Transfer. We mitigate the computation cost of the target layers by overlapping computation with data movement. While the GPU processes the current layer, the parameters for the next required offloaded layer are concurrently prefetched from CPU memory. Unlike some prefetching strategies limited to adjacent layers within the same decoder block, our implementation operates across decoder layers, maximizing potential computation hiding. Furthermore, these prefetched layers are loaded into the same reused memory regions to avoid increasing peak memory usage. This asynchronous data transfer technique effectively conceals the computation time of the SD verification step during forward passes of large draft trees. + +Chunked Prefill for Long Contexts. Prefilling long input prompts can lead to substantial peak GPU memory usage because intermediate activation sizes scale with context length, restricting the number of target model layers that can remain GPU-resident. To address this, we employ chunked prefetch, where the input is segmented into fixed-length chunks (e.g., 256 tokens), and the KV-Cache is built incrementally. This method significantly reduces peak memory during the prefetching phase. While Sarathi-Serve [2] introduced this approach primarily to minimize pipeline bubbles in token serving, our adaptation specifically focuses on curtailing peak memory of the prefetching phase for long contexts to maximize the GPU residency of target model layers, further enhancing overall inference efficiency. + +Table 1: Throughputs and average acceptance lengths $\tau$ by using different draft models. L31 represents Llama-3.1-Instruct, L32 represents Llama-3.2-Instruct, Q represents Qwen-2.5-Instruct. None represents vanilla autoregressive decoding is used. The number in each parentheses below the target model name denotes the restricted VRAM limit. + +
TargetMT-BenchHumanEvalGSM8KAlpacaCNN/DMMean
Drafttokens/sτtokens/sτtokens/sτtokens/sτtokens/sτtokens/sτ
Temperature = 0
L31 8B(8GB)None2.4012.3912.4012.4012.3412.39 (1.00×)1
EAGLE-27.563.908.584.438.054.157.593.906.453.347.65 (3.20×)3.95
L32 1B15.1411.9125.7920.2320.1715.7114.9811.5410.058.6617.23 (7.22×)13.61
SubSpec24.2928.3528.0931.6327.7831.5522.1325.3822.6031.3924.98 (10.47×)29.66
Q 7B(8GB)None2.7712.7712.7712.7712.3412.68 (1.00×)1
EAGLE-28.003.738.764.088.283.857.473.476.473.027.80 (2.91×)3.63
Q 1.5B15.7812.2728.2921.8822.8617.5912.379.479.347.8817.73 (6.61×)13.82
SubSpec25.3527.0833.4834.7733.0434.1822.3023.1921.2826.3327.09 (10.10×)29.11
Q 14B(12GB)None1.2211.2211.2211.2211.1711.20 (1.00×)1
Q 1.5B8.1710.8116.2021.4512.5816.546.468.434.406.409.56 (7.92×)12.73
SubSpec12.0526.3615.7033.7615.4432.9410.3421.799.2724.0812.56 (10.4×)27.79
Q 32B(24GB)None0.5210.5210.5210.5210.5010.52 (1.00×)1
Q 7B3.6812.556.0920.735.3017.913.2010.742.158.324.08 (7.86×)14.05
Q 1.5B4.4910.878.2219.906.7416.173.618.622.396.465.09 (9.80×)12.40
SubSpec6.3327.537.5832.707.9633.665.8024.504.8026.486.50 (12.50×)28.97
Temperature = 0.6
L31 8B(8GB)None2.4012.3912.4012.4012.3412.39 (1.00×)1
EAGLE-27.383.818.304.297.603.927.493.866.193.217.39 (3.10×)9.96
L32 1B12.309.6921.3716.8215.8812.4212.809.838.166.9314.10 (5.91×)11.14
SubSpec14.6217.5822.5926.0416.5119.5413.1715.4611.6115.3715.70 (6.58×)18.80
Q 7B(8GB)None2.7712.7712.7712.7712.3412.68 (1.00×)1
EAGLE-27.423.458.493.968.073.766.473.015.692.667.23 (2.69×)3.37
Q 1.5B13.1910.4322.9618.0619.1915.0210.598.217.636.4414.71 (5.48×)11.63
SubSpec15.9217.0929.1330.5723.8925.3214.4314.9810.1512.0718.70 (6.97×)20.00
Q 14B(12GB)None1.2211.2211.2211.2211.1711.20 (1.00×)1
Q 1.5B6.538.6211.9015.739.8712.995.437.093.775.427.50 (6.21×)9.97
SubSpec6.9015.2212.1726.749.5621.026.1413.054.4010.907.83 (6.49×)17.39
Q 32B(24GB)None0.5210.5210.5210.5210.5010.52 (1.00×)1
Q 7B2.649.084.8816.673.9013.242.498.371.766.613.13 (6.03×)10.79
Q 1.5B3.719.035.7213.934.8011.612.947.061.945.263.82 (7.35×)9.38
SubSpec3.7416.406.1526.544.4119.323.3714.162.6513.334.06 (7.82×)17.95
+ +# 5 Performance Evaluation + +# 5.1 Evaluation Setup + +Evaluation Benchmarks. We evaluated performance across five diverse generative tasks, consistent with the benchmarks from EAGLE [23] and Spec-Bench [41]. These tasks included multi-turn conversation (MT-Bench [46]), code generation (HumanEval [9]), mathematical reasoning (GSM8K [11]), instruction following (Alpaca [36]), and summarization (CNN/Daily Mail [31]). + +Hardware and Simulated Environments. All experiments were run on a system with an RTX 4090 GPU, an Intel i7-13700K CPU, a PCIe-4.0 x16 bus, and 128GB of DDR5 RAM. GPU memory utilization during evaluations was programmatically restricted to 8GB, 12GB, and 24GB VRAM capacities to simulate diverse consumer device environments. + +Comparative Methodology and Parameters. We compared the end-to-end speedup of SubSpec against EAGLE-2 $^{\dagger}$ and chat models from the Qwen2.5 (7B, 14B, 32B) and Llama3.1 (8B) families. Evaluations used a batch size of 1 under both greedy (target temperature $= 0$ ) and stochastic (target temperature $= 0.6$ ) generation. For fair comparison, all methods used an identical context-aware dynamic draft tree algorithm without additional tree pruning techniques. A portion of the target model decoder layers was kept resident on the GPU within the VRAM limits. All SD methods were evaluated on 20 identical samples, randomly selected from each dataset. The baseline (offloading with no SD) used the initial five samples due to its significantly longer runtime. + +The key parameters were configured as follows: The low-bit substitute layers in SubSpec were quantized to 4 bits with a group size 64 using HQQ. EAGLE-2 used its default published parameters + +$(k = 10, \mathcal{D} = 6)$ . For SubSpec and the smaller pretrained draft models, the top-k value of tree construction was set to $k = 6$ . Their optimal draft depths $(\mathcal{D})$ , identified through the grid search reported in Section 3.2 (results shown in Figure 2), were $\mathcal{D} = 48$ for SubSpec and $\mathcal{D} = 32$ for the pretrained models. While further parameter tuning might yield additional improvements, such exhaustive optimization was beyond the scope of this research. Chunked prefetch was also applied to prevent out-of-memory (OOM) errors and maximize the number of decoder layers on the GPU. + +To better reflect typical real-world usage, all draft models were executed using torch.compile with the max-autotune configuration. A static KV-Cache with a context length of 2048 tokens was applied consistently across all methods. The comparative results are summarized in Table 1. + +# 5.2 End-to-end Performance + +The results in Table 1 demonstrate the effectiveness and robustness of SubSpec. SubSpec consistently achieved average acceptance lengths $(\tau)$ near 30 across tasks, delivering the highest average throughput. This performance translates to a speedup of $10\times$ to $12.5\times$ between different model sizes, underscoring the broad applicability and significant performance gains offered by SubSpec. We also evaluated the performance of the reasoning models on additional reasoning benchmarks listed in the Appendix G. + +Further highlighting its efficiency, SubSpec achieves an additional $30\%$ to $50\%$ speedup compared to smaller draft models from the same family as the target model, without any additional training. This advantage underscores the critical role of the enhanced draft alignment of SubSpec in accelerating offload scenarios. All SD methods showed reduced performance in stochastic settings (target temperature $= 0.6$ ). For SubSpec, this meant a decrease in speedup of approximately $60\%$ . Despite this, SubSpec maintained a considerable speedup of $5.8\times$ to $7.8\times$ , showcasing its resilience and sustained effectiveness even under less favorable generation conditions. We achieved a low standard deviation of 0.101 tokens/sec on five independent runs of SubSpec on MT-Bench benchmark. + +# 5.3 Ablation Study + +Finally, we performed an ablation study on MT-Bench under an 8GB VRAM constraint to assess the impact of individual SubSpec components. The results are detailed in Table 2. A baseline configuration of only implementing the core concept of SubSpec (a quantized, GPU-resident draft model using only 'substitute' layers for offloaded portions and shared GPU-resident target model layers), achieved a $7.05 \times$ speedup (19.54 tokens/s). + +Table 2: Ablation study of SubSpec component contributions for accelerating Qwen2.5 7B target model on MT-Bench under greedy decoding (8GB VRAM limit). Performance is shown as the components were added cumulatively. The final row represents the complete SubSpec system. + +
Methodtokens/sτ
Substitute and layer sharing19.54 (7.05×)23.07
+ Shared KV-Cache21.99 (7.94×)25.14
+ Draft prob. sharpening23.66 (8.54×)27.08
+ Async data transfer25.35 (9.15×)27.08
+ +Sequentially integrating additional enhancements of shared KV-Cache, draft probability sharpening, and asynchronous data transfer yielded further performance gains. Each of these components contributed an approximate $7\%$ to $13\%$ increase in throughput. The complete SubSpec system, incorporating all optimizations, ultimately delivered a $9.15 \times$ speedup and a throughput of 25.35 tokens/s. These results affirm the individual and collective efficacy of the components of SubSpec. + +# 6 Conclusion + +This paper addressed the challenge of efficiently performing the inference of large language models on memory-constrained consumer GPUs using parameter offloading. Our analysis confirmed that a highly aligned draft model is crucial for speculative decoding to accelerate parameter offloading effectively. We introduced SubSpec, a novel lossless and training-free technique based on this insight. SubSpec constructs an aligned draft model by utilizing low-bit substitute layers for offloaded portions of the target LLM while sharing GPU-resident components. Evaluations demonstrate that SubSpec is robust across various model sizes and benchmarks under realistic memory limits, achieving substantial average speedups of $10 \times$ to $12.5 \times$ compared to baseline offloading inference. These results significantly advance the feasibility of deploying large, high-quality LLMs locally on widely available consumer hardware. + +# Acknowledgments + +The authors would like to thank ASUS for their generous support in providing the computing resources necessary for this work. + +# References + +[1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. 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Gptq: Accurate post-training quantization for generative pre-trained transformers. arXiv preprint arXiv:2210.17323, 2022. +[14] Georgi Gerganov. ggerganov/llama.cpp: Port of facebook's llama model in c/c++. https://github.com/ggerganov/llama.cpp, 2023. +[15] Aaron Grattafori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, et al. The llama 3 herd of models. arXiv preprint arXiv:2407.21783, 2024. + +[16] Han Guo, William Brandon, Radostin Cholakov, Jonathan Ragan-Kelley, Eric Xing, and Yoon Kim. Fast matrix multiplications for lookup table-quantized llms. In *Findings of the Association for Computational Linguistics: EMNLP* 2024, pages 12419–12433, 2024. +[17] Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring mathematical problem solving with the math dataset, 2021. 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Fast inference from transformers via speculative decoding. In International Conference on Machine Learning, pages 19274-19286. PMLR, 2023. +[23] Yuhui Li, Fangyun Wei, Chao Zhang, and Hongyang Zhang. EAGLE: Speculative sampling requires rethinking feature uncertainty. In International Conference on Machine Learning, 2024. +[24] Yuhui Li, Fangyun Wei, Chao Zhang, and Hongyang Zhang. EAGLE-2: Faster inference of language models with dynamic draft trees. In Empirical Methods in Natural Language Processing, 2024. +[25] Ji Lin, Jiaming Tang, Haotian Tang, Shang Yang, Wei-Ming Chen, Wei-Chen Wang, Guangxuan Xiao, Xingyu Dang, Chuang Gan, and Song Han. Awq: Activation-aware weight quantization for on-device llm compression and acceleration. Proceedings of Machine Learning and Systems, 6:87-100, 2024. +[26] Aixin Liu, Bei Feng, Bin Wang, Bingxuan Wang, Bo Liu, Chenggang Zhao, Chengqi Dengr, Chong Ruan, Damai Dai, Daya Guo, et al. Deepseek-v2: A strong, economical, and efficient mixture-of-experts language model. arXiv preprint arXiv:2405.04434, 2024. +[27] Fangcheng Liu, Yehui Tang, Zhenhua Liu, Yunsheng Ni, Duyu Tang, Kai Han, and Yunhe Wang. Kangaroo: Lossless self-speculative decoding for accelerating llms via double early exiting. Advances in Neural Information Processing Systems, 37:11946-11965, 2024. +[28] Vladimir Malinovskii, Andrei Panferov, Ivan Ilin, Han Guo, Peter Richtárik, and Dan Alistarh. Pushing the limits of large language model quantization via the linearity theorem. arXiv preprint arXiv:2411.17525, 2024. +[29] Michael R Metel, Peng Lu, Boxing Chen, Mehdi Rezagholizadeh, and Ivan Kobyzev. Draft on the fly: Adaptive self-speculative decoding using cosine similarity. arXiv preprint arXiv:2410.01028, 2024. +[30] Xupeng Miao, Gabriele Oliaro, Zhihao Zhang, Xinhao Cheng, Zeyu Wang, Zhengxin Zhang, Rae Ying Yee Wong, Alan Zhu, Lijie Yang, Xiaoxiang Shi, et al. Specinfer: Accelerating generative large language model serving with tree-based speculative inference and verification. arXiv preprint arXiv:2305.09781, 2023. + +[31] Ramesh Nallapati, Bowen Zhou, Caglar Gulcehre, Bing Xiang, et al. Abstractive text summarization using sequence-to-sequence rnns and beyond. arXiv preprint arXiv:1602.06023, 2016. +[32] Mathematical Association of America. Aime, February 2024. URL https://artofproblemsolving.com/wiki/index.php/AIME_Problems_and_Solutions/. +[33] David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R. Bowman. Gpqa: A graduate-level google-proof q&a benchmark, 2023. URL https://arxiv.org/abs/2311.12022. +[34] Ying Sheng, Lianmin Zheng, Binhang Yuan, Zhuohan Li, Max Ryabinin, Beidi Chen, Percy Liang, Christopher Ré, Ion Stoica, and Ce Zhang. Flexgen: High-throughput generative inference of large language models with a singlegpu. In International Conference on Machine Learning, pages 31094-31116. PMLR, 2023. +[35] Ruslan Svirschevski, Avner May, Zhuoming Chen, Beidi Chen, Zhihao Jia, and Max Ryabinin. Specexec: Massively parallel speculative decoding for interactive lvm inference on consumer devices. Advances in Neural Information Processing Systems, 37:16342-16368, 2025. +[36] Rohan Taori, Ishaan Gulrajani, Tianyi Zhang, Yann Dubois, Xuechen Li, Carlos Guestrin, Percy Liang, and Tatsunori B Hashimoto. Alpaca: A strong, replicable instruction-following model. Stanford Center for Research on Foundation Models. https://crfm.stanford.edu/2023/03/13/alpaca.html, 3(6):7, 2023. +[37] Gemini Team, Petko Georgiev, Ving Ian Lei, Ryan Burnell, Libin Bai, Anmol Gulati, Garrett Tanzer, Damien Vincent, Zhufeng Pan, Shibo Wang, et al. Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context. arXiv preprint arXiv:2403.05530, 2024. +[38] Qwen Team. Qwq-32b: Embracing the power of reinforcement learning, March 2025. URL https://qwenlm.github.io/blog/qwq-32b/. +[39] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Jasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. Llama 2: Open foundation and fine-tuned chat models. arXiv preprint arXiv:2307.09288, 2023. +[40] Heming Xia, Yongqi Li, Jun Zhang, Cunxiao Du, and Wenjie Li. Swift: On-the-fly self-speculative decoding for lIm inference acceleration. arXiv preprint arXiv:2410.06916, 2024. +[41] Heming Xia, Zhe Yang, Qingxiu Dong, Peiyi Wang, Yongqi Li, Tao Ge, Tianyu Liu, Wenjie Li, and Zhifang Sui. Unlocking efficiency in large language model inference: A comprehensive survey of speculative decoding. arXiv preprint arXiv:2401.07851, 2024. +[42] An Yang, Baosong Yang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Zhou, Chengpeng Li, Chengyuan Li, Dayiheng Liu, Fei Huang, et al. Qwen2 technical report. arXiv preprint arXiv:2407.10671, 2024. +[43] An Yang, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoyan Huang, Jiandong Jiang, Jianhong Tu, Jianwei Zhang, Jingren Zhou, Junyang Lin, Kai Dang, Kexin Yang, Le Yu, Mei Li, Minmin Sun, Qin Zhu, Rui Men, Tao He, Weijia Xu, Wenbiao Yin, Wenyuan Yu, Xiafei Qiu, Xingzhang Ren, Xinlong Yang, Yong Li, Zhiying Xu, and Zipeng Zhang. Qwen2.5-1m technical report, 2025. URL https://arxiv.org/abs/2501.15383. +[44] Hanling Yi, Feng Lin, Hongbin Li, Peiyang Ning, Xiaotian Yu, and Rong Xiao. Generation meets verification: Accelerating large language model inference with smart parallel auto-correct decoding. arXiv preprint arXiv:2402.11809, 2024. +[45] Libo Zhang, Zhaoning Zhang, Baizhou Xu, Songzhu Mei, and Dongsheng Li. Dovetail: A cpu/gpu heterogeneous speculative decoding for lIm inference. arXiv preprint arXiv:2412.18934, 2024. +[46] Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, Hao Zhang, Joseph E. Gonzalez, and Ion Stoica. Judging LLM-as-a-judge with MT-Bench and chatbot Arena. arXiv preprint arXiv:2306.05685, 2023. + +[47] Yue Zheng, Yuhao Chen, Bin Qian, Xiufang Shi, Yuanchao Shu, and Jiming Chen. A review on edge large language models: Design, execution, and applications, 2025. URL https://arxiv.org/abs/2410.11845. + +# A Limitations + +Minimum GPU Memory Requirement. SubSpec has a higher minimum GPU memory requirement. A general prerequisite for speculative decoding to work effectively is that the entire draft model must be GPU-resident. SubSpec architecture necessitates maintaining low-bit substitutes for offloaded target model layers. An approximate 7.1GB minimum GPU memory is required for Qwen2.5 7B, thus fewer layers of the target model can remain GPU-resident compared to some alternative speculative decoding methods. Despite this, SubSpec demonstrates superior throughput in all benchmarks. + +Quantization Granularity. Our research only experimented with 4-bit quantization for substitute layers. Although this might affect draft model alignment, more aggressive methods (e.g., 2-bit or 3-bit quantization) could further reduce VRAM demands. Such VRAM savings could permit strategic memory reallocation, for instance, retaining critical draft or target model layers at full GPU precision. A thorough exploration of these trade-offs between draft quality, VRAM usage, and performance represents an important direction for future research. + +Applicability to Model Architectures. SubSpec mainly suits dense LLM architectures. Applying SubSpec to alternative architectures, such as Mixture-of-Experts (MoE) models, requires further adaptation and research. + +# B Extended Discussion + +# B.1 Comparison with Standalone Quantized Models + +While a standalone 4-bit quantized model may fully fit on a GPU and eliminate the offloading bottleneck, this approach inevitably alters outputs and degrades model quality. In contrast, SubSpec avoids this accuracy trade-off entirely. + +SubSpec makes parameter offloading practical. As shown in Table 3, this method accelerates naive offloading from an unusable 2.4 tokens/sec to an acceptable 24 tokens/sec for interactive use on consumer-grade hardware. We therefore position SubSpec as a simple, training-free solution for users who refuse to compromise on the model quality. + +Table 3: Throughput comparison of various quantization methods for the Llama3.1 8B model on the MT-Bench (8GB VRAM). + +
Model Configurationtokens/s
Original (fp16)2.40
GPTQ [13] (int4)58.87
AWQ [25] (int4)52.32
HQQ [5] (int4)135.84
SubSpec24.29
+ +# B.2 Why Tree-Based Speculative Decoding + +This research focuses on tree-based speculative decoding (SD) because tree-based SD generally achieves higher average acceptance lengths $(\tau)$ than sequential or self-speculation methods [23, 30, 41]. Besides, high $\tau$ values are crucial to minimize expensive forward passes of the target model, especially in offloading scenarios that involve significant data transfer overhead. + +# B.3 Future Outlook and Technological Advancements. + +Forthcoming interconnect advancements (e.g., PCIe 5.0 and 6.0), along with the continuous progress on model compression methods and kernel optimizations, are anticipated to further enhance inference performance. Each PCIe generation roughly doubles raw bandwidth, halving expensive target model data transfers. While faster PCIe reduces target model latency, potentially increasing the relative cost of speculation iterations, concurrent GPU computation and kernel efficiency improvements are expected to accelerate speculation proportionally. Consequently, SubSpec is projected to retain its relative speedup advantage over standard autoregressive decoding. These combined technological trends promise a progressive reduction in the performance gap between offloaded and non-offloaded LLM inference. + +# C Related Works + +SpecExec [35]. SpecExec is a speculative decoding method that introduces a pruning strategy during tree construction and an early-exit mechanism to reduce speculation time. However, these features increase computational complexity and hinder the application of torch.compile due to dynamic shapes from pruning. + +We performed a parameter sweep for SpecExec on the MT-Bench benchmark, with results in Table 4. While SpecExec exceeds the 8GB VRAM constraint, SubSpec achieves $76\%$ speedup without requiring an additional draft model. + +Table 4: Performance of SpecExec using Llama3.1 8B as target model and Llama3.2 1B as draft model on MT-Bench (greedy decoding). "Budget" denotes the number of tokens concurrently verified, "VRAM" denotes the peak GPU memory. + +
BudgetVRAM (GB)tokens/sτ
648.3811.758.64
1288.4312.219.42
2568.5513.0210.79
5128.7913.7712.40
10249.2712.9512.93
+ +Dovetail [45]. Dovetail is a speculative decoding + +method to accelerate LLM inference on consumer-grade devices by offloading portions of the target computation to the CPU, with the draft model on the GPU. Unlike GPU processing, CPU computation for verification scales linearly with the number of draft tokens. This trait typically restricts the amount of verifiable draft tokens per iteration to maintain optimal speedup. For instance, Dovetail used only 16 draft tokens for verification, whereas our proposed SubSpec verified 288 draft tokens in parallel. + +Self-Speculative Decoding. Self-speculative decoding builds a draft model by reusing layers from a target model to reduce parameter overhead and improve alignment. While many approaches add new parameters and require extra training [6, 24, 27, 44], recent works propose training-free layer skipping methods [8, 29, 40]. This distinction highlights an opportunity for hybrid solutions. Future work could explore the combination of SubSpec with layer skipping to produce a draft model with a lower VRAM requirement and faster inference. + +# D Analysis of Draft Probability Sharpening + +Table 7 discusses the results of varying draft temperatures among draft model types. The first result (denoted Self in the second row) performs speculative decoding using the target model itself, i.e., Qwen2.5 7B, as the draft model. Although this setup does not make sense from both memory and speedup perspectives, the experiment clearly demonstrates the cumulative probability divergence problem mentioned in Section 4.2, which + +shows that even the same model cannot correctly predict all the tokens when using the default draft tree method. + +Table 5: Average acceptance length $(\tau)$ for various draft models and temperatures with Qwen2.5 7B target model on MT-Bench (greedy decoding), where a draft temperature of 1.0 represents the baseline without draft probability sharpening. + +
Draft Temp.0.20.40.60.81.01.2
Qwen2.5 7B (Self)34.5033.0331.9529.6427.7926.57
EAGLE-23.423.623.703.713.763.77
Qwen2.5 1.5B11.8612.2012.4812.3612.4212.51
SubSpec27.0827.1927.1626.0925.2923.92
+ +By lowering the draft temperature, the average acceptance length $(\tau)$ increases from 27.79 to 34.50, showing that this aggressive sharpening effectively counters the "false positive" path issue. For SubSpec, $\tau$ also increases from 23.92 to 27.08. This method does not yield a better $\tau$ due to the lower alignment of the pretrained models and EAGLE-2. + +# E Verification Pipeline with Asynchronous Data Transfer + +Figure 5 illustrates the execution timelines of the verification step in vanilla autoregressive decoding, speculative decoding, and speculative decoding with asynchronous data transfer. The bottom diagram (c) illustrates how asynchronous data transfer works. This method overlaps GPU computation (Q, K) with the data transfer of the subsequent layer (Loading K, V weights to the GPU) to hide the higher verification computation time for speculative decoding. + +![](images/c8495074fe9802a61af104b92ca7686eaa1b8b15c253adc622443717d4a8a74e.jpg) +(a) Vanilla Autoregressive Decoding + +![](images/0726626ce9d54f867c6aa00a58d4e23e5d228439de9c93367593e4ab9854c07e.jpg) +(b) Speculative Decoding + +![](images/cc41f13b9bb83552ee0361c1d1435ddc56c470ba290b6f892ba3a17329fcae9a.jpg) +(c) Speculative Decoding $^+$ Asynchronous Data Transfer +Figure 5: Execution timelines of the verification step for different decoding strategies + +# F Temporal Analysis: Speculation vs. Verification + +For each iteration in speculative decoding, the draft model runs $\mathcal{D}$ forward passes to generate a draft tree of depth $\mathcal{D}$ (speculate). The target model then performs a single forward pass over all draft tokens, checking their correctness (verify). We compare the average execution time of these two phases, along with the actual obtained throughput, as shown in Figure 6. The optimal throughput for SubSpec occurs at $\mathcal{D} = 48$ , where the speculation and verification times are nearly equal. + +![](images/cfc56088570f344fef0ee5c84c8aecaca8dc90edd13aedf031689a933e7169fa.jpg) +Figure 6: Execution time breakdown (speculation vs. verification) and average acceptance lengths $(\tau)$ of SubSpec accelerating Qwen2.5 7B under an 8GB VRAM limit. + +# G Performance Evaluation on Reasoning Models + +To further demonstrate the robustness of the lossless and training-free method, SubSpec, we also evaluated its performance on reasoning models (DeepSeek-R1 distilled variants [12], QWQ [38]). This evaluation utilized benchmarks widely used in the field: AIME 2024 [32], MATH 500 [17], GPQA Diamond [33], and LiveCodeBench [19]. As shown in Table 6, SubSpec demonstrated promising results of over $10 \times$ speedup on all benchmarks. + +Table 6: Speedup ratios and average acceptance lengths $(\tau)$ for SubSpec compared to baseline (None) on reasoning models under greedy decoding. DSL represents DeepSeek-R1-Distill-Llama, and DSQ represents DeepSeek-R1-Distill-Qwen. The number in each parentheses below the target model name denotes the restricted VRAM limit. + +
AIME 2024Math 500GPQA DiamondLiveCodeBenchMean
TargetDrafttokens/sτtokens/sτtokens/sτtokens/sτtokens/sτ
DSL 8B (8GB)None2.4012.4012.4012.3912.40 (1.00×)1
SubSpec31.3038.6432.3939.8927.5434.1828.4435.6929.92 (12.49×)37.10
DSQ 7B (8GB)None2.7712.7712.7712.7612.77 (1.00×)1
SubSpec33.4537.7635.2539.5527.7431.4129.9434.4531.60 (11.42×)34.72
DSQ 14B (12GB)None1.2211.2211.2111.2111.21 (1.00×)1
SubSpec17.3840.5717.6040.9614.9335.0715.7337.4216.41 (13.51×)38.51
DSQ 32B (24GB)None0.5210.5210.5210.5210.52 (1.00×)1
SubSpec9.5643.599.7044.008.5739.458.4739.709.07 (13.76×)41.68
OWQ 32B (24GB)None0.5210.5210.5210.5210.52 (1.00×)1
SubSpec7.9536.298.1136.936.8631.545.8927.477.20 (13.76×)33.06
+ +# H Quantized Target Model Scenarios + +We analyzed the practical trade-off between model precision and performance for an 8-bit quantized target model. As detailed in Table 7, this quantization nearly halves the memory footprint and offloading time relative to the 16-bit version. While the process introduces an acceptable loss in accuracy, this configuration increases throughput by $36\%$ with all other configurations held constant. This outcome demonstrates a compelling trade-off, where a minor reduction in precision yields a substantial performance gain. + +Table 7: Performance of Llama3.1 8B across different SubSpec settings on MT-Bench (greedy decoding, 8GB VRAM). + +
Config (Target, Draft).tokens/sτ
SubSpec (fp16, int4)24.2929.66
SubSpec (int8, int4)33.0329.76
+ +# I Supplementary Evaluation Configurations + +We retain as many target model layers as possible directly within GPU memory to reduce the expensive data transfer overhead. The draft models and KV-Cache are default GPU-resident. + +The following details the number of GPU-resident layers of the target model for various methods within VRAM limits. Embedding and head layers are default to be GPU-resident: + +- Vanilla (Baseline Offloading): 11 layers for 7B/8B target models, 15 layers for 14B target models, and 20 layers for 32B target models. +EAGLE-2: 7 layers for 7B/8B target models. +- SD with Mid-Size Pretrained Draft Model (1B/1.5B): 3 layers for 7B/8B target models, 7 layers for 14B target models, and 16 layers for 32B target models. +- SD with Large Pretrained Draft Model (7B): 3 layers for the 32B target model. +- SubSpec: All decoder layers of the target model were offloaded, with their 4-bit quantized substitutes retained on the GPU. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The main claims in the abstract and Section 1 accurately reflect the paper's contributions and scope. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. 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For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. +- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. 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+1,759 @@ +# Speculative Jacobi-Denoising Decoding for Accelerating Autoregressive Text-to-image Generation + +Yao Teng + +Fuyun Wang + +Xian Liu2 + +Zhekai Chen + +Han Shi3 + +Yu Wang + +Zhengu + +Weiyang Liu + +Difan Zou + +Xihui Liu* + +1The University of Hong Kong 2CUHK 3Huawei Noah's Ark Lab 4Tsinghua University + +# Abstract + +As a new paradigm of visual content generation, autoregressive text-to-image models suffer from slow inference due to their sequential token-by-token decoding process, often requiring thousands of model forward passes to generate a single image. To address this inefficiency, we propose Speculative Jacobi-Denoising Decoding (SJD2), a framework that incorporates the denoising process into Jacobi iterations to enable parallel token generation in autoregressive models. Our method introduces a next-clean-token prediction paradigm that enables the pre-trained autoregressive models to accept noise-perturbed token embeddings and predict the next clean tokens through low-cost fine-tuning. This denoising paradigm guides the model towards more stable Jacobi trajectories. During inference, our method initializes token sequences with Gaussian noise and performs iterative next-clean-token-prediction in the embedding space. We employ a probabilistic criterion to verify and accept multiple tokens in parallel, and refine the unaccepted tokens for the next iteration with the denoising trajectory. Experiments show that our method can accelerate generation by reducing model forward passes while maintaining the visual quality of generated images. + +# 1 Introduction + +Autoregressive models have emerged as a cornerstone of visual generative tasks through next-token prediction [1-4]. However, the autoregressive paradigm suffers from significant inference latency due to its sequential, token-by-token decoding process. For instance, generating a single high-resolution image often requires thousands of sequential forward passes. To address this challenge, we focus on accelerating autoregressive text-to-image generation models via parallel token decoding. + +Jacobi decoding [5] is an iterative method that accelerates the inference of autoregressive models through parallel token decoding without any training. This method operates on a sequence of randomly initialized tokens and iteratively calls the neural network to refine the tokens until the convergence (i.e., tokens are correctly decoded). Its variant, Speculative Jacobi Decoding (SJD) [6], improves Jacobi decoding with a probabilistic criterion tailored for accelerating text-to-image generation using discrete tokens. The core of SJD is a verification-refinement process. Specifically, given a sequence of tokens, in each Jacobi iteration, SJD first predicts the probability for each input token, and then these probabilities enable the criterion to determine the acceptance of a prefix of the tokens (i.e., verification) while also guiding the resampling of unaccepted tokens for the next iteration (i.e., refinement). The above verification-refinement process operates within a fixed-length sliding Jacobi window where the accepted tokens are removed and newly initialized tokens are appended. By accepting multiple tokens (at least one token) per iteration, SJD reduces model forward passes, speeding up generation compared to token-by-token decoding. + +![](images/7e8371ef8e235a07af68a83af167b0db87d9353a88301e7710a8ab07ff107bdc.jpg) +Steps: 2357 + +![](images/b34831b64e9bf9dd9ec523fb26cfd66453278dc71611f368e52f6acce5299f4c.jpg) +Steps: 529 (4.5 × Fewer) + +![](images/0e791a770d028a5a58b616142c828ba0a273d5bf4e575e5b0f7b0b5ef3a84051.jpg) +Steps: 2357 + +![](images/92e90592746e34e8044a9bc83bfd73b7702c8573bdf456089d4cb1ba631d1fa7.jpg) +Steps: 571 (4.1 × Fewer) + +![](images/7aea62e3ef5afacf39520f17e7383632d1ef6ef0a3a914f2fc1cb0723484b7b0.jpg) +Prompt: A huge black bear wearing a red cassock, in front of a blazing fire +Steps: 2357 + +![](images/65f8c8a10fea3de755614e12f8024d3e33975a2cb424b3f465e066712d8f65ce.jpg) +Steps: 578 (4.1 × Fewer) + +![](images/64c77d64703e97cb638b0ab7e4ff92d4d54f246f06a065499a9f8d34c159b0d8.jpg) +Prompt: An image of a hamster with one paw grasping a trident and pointing the other finger forward in the desert +Steps: 2357 + +![](images/d361852aad341a8976dd4fde196e66b73b2f4ad5f88a62f5a97d3b30708b08f3.jpg) +Steps: 549 (4.3 × Fewer) + +![](images/5b01226ff7a1947e85ba58e144fb00f304b44a76a4327fe8d38483a2cd4647f5.jpg) +Prompt: Jars in the laboratory with yellow crystals growing inside, film grained +Steps: 2357 +Prompt: An imaginary picture of cyborg German shepherd, standing in the ruins of a modern city under a red sunset sky, film grained +Figure 1: We propose Speculative Jacobi-Denoising Decoding to accelerate autoregressive text-to-image generation via multi-token prediction. On Lumina-mGPT, the number of model forward passes for inference (denoted as steps) is reduced. The inference step for our decoding is marked in green. + +![](images/4cc49aa776e590083a89e58ad758c8f6528f8d9c76c1e9aea04cb4842e3ce9df.jpg) +Steps: 600 $(3.9\times$ Fewer) + +![](images/157c5920ad940a1c6f4c94d824e8ec6ffbdecdff352e59dd9589647d45a5e650.jpg) +Prompt: An oil painting of a beautiful young lady with crimson eyes, short white hair, a sly smile on her face, and an violet outfit +Steps: 2357 +Prompt: A muscular space warrior wearing dark brown armor with a helmet, a mask over his face (black mask covering his eyes), and a huge oxygen tube plugged into the mask + +![](images/acd656065ff088c03cfb8b3c0ccfcd3204585749bbfdaf3da7f2c3474a542419.jpg) +Steps: 564 (4.2 × Fewer) + +Although SJD accelerates autoregressive text-to-image generation by a non-negligible margin, the token refinement process in SJD is inherently unconstrained, making it difficult to control the refinement to achieve the correct token predictions. Consequently, some tokens undergo many refinement iterations before being accepted, resulting in a low speedup ratio. In contrast, diffusion models explicitly define a trajectory for the iterative input refinement, known as the denoising process, which is governed by the principles of stochastic differential equations [7]. More importantly, previous works [8-10] have demonstrated that this trajectory can be remarkably short (as few as tens of iterations) for any length of inputs (i.e., various image resolutions). Inspired by this intuition, to further accelerate the autoregressive text-to-image generation, we leverage the denoising process from the diffusion models to assist the Jacobi decoding and fine-tune the autoregressive model to adapt to the noise perturbation. + +In this paper, we propose Speculative Jacobi-Denoising Decoding (SJD2), along with a fine-tuning strategy, to enable pre-trained autoregressive text-to-image generation models to perform parallel token-level denoising decoding. To achieve this, we introduce a task called next-clean-token prediction, where an autoregressive model accepts noisy input tokens and predicts the clean tokens at a one-position offset (i.e., noise-free next tokens). Our noise-augmented fine-tuning strategy + +equips pre-trained autoregressive models with the denoising ability, i.e., we randomly select segments of input tokens and add Gaussian noise to their token embeddings during training. The training supervision remains consistent with conventional autoregressive models, i.e., the one position-offset discrete tokens are taken as ground-truth supervision and the model is trained with cross-entropy loss. After fine-tuning, the model has the ability to process noisy inputs and thus perform Jacobi-Denoising decoding during inference. The decoding process begins with a sequence (Jacobi window) of token embeddings randomly initialized with Gaussian noise. In each Jacobi-Denoising iteration, this sequence is fed into the autoregressive model to predict the probability distribution of clean tokens at each token position. The noisy tokens are refined for the next iteration through the denoising process using the probability predicted in the current iteration. If a token is sufficiently denoised, it becomes a clean token and undergoes the standard Jacobi refinement. The iterative process is repeated until all tokens are accepted. + +To validate our method, we conduct experiments on two open-source large autoregressive models, Lumina-mGPT [11] and Emu3 [4]. Experimental results demonstrate that our method reduces the number of forward passes by about $4 \times$ on Lumina-mGPT and more than $5 \times$ on Emu3 and thus achieves latency speedup by more than $2 \times$ . We further verify image quality to show that our method accelerates autoregressive text-to-image generation without compromising image quality. + +# 2 Related Work + +Integration of Autoregression and Continuous Diffusion The AR-diffusion language model [12] integrates the concept of autoregression into an embedding diffusion language model [13-18]. The embedding diffusion performs the standard denoising process on (normalized) token embeddings instead of next-token prediction, and use the regression loss [12] or cross-entropy loss [14] for training. The recent diffusion-forcing model and its variants [19-25] share a similar pipeline as the AR-diffusion model and introduce the autoregressive sampling into the temporal dimension of the continuous latent video diffusion models. They employ causal masks to support history-conditioned video generation, but have independent noise levels across input tokens. Moreover, for the diffusion process on the temporal dimension, works like the rolling diffusion model [26] further introduce the sliding window mechanism on the time series. Transfusion and its variants [27-29] utilize a single backbone model to jointly perform diffusion-based image generation in a continuous latent space and autoregressive language generation in a discrete space, using separate loss, different lightweight decoders, and specific attention masks for training. MAR and its variants [30-32] propose the use of an efficient MLP diffusion head to decode features from autoregressive backbones, enabling autoregression in continuous latent space. Several multimodal large language models (MLLMs) [33-38] generate images via pre-trained diffusion models such as SDXL [39] which take the output features of autoregressive models as conditions. + +Parallel Token Decoding in Autoregressive Models Blockwise Parallel Decoding [40] and Medusa [41] employ auxiliary modules to predict multiple next tokens simultaneously, thereby accelerating language models. Speculative decoding [42, 43, 41, 44-47] enhances inference efficiency by using a smaller model to generate candidate tokens, which are then verified and accepted by the larger model in parallel. DiffuLLaMA [48] fine-tunes the large autoregressive language models into discrete diffusion models [49] for parallel decoding. Jacobi Decoding [5], initially applied to pixel-level autoregressive generation models, iteratively decodes tokens in parallel until their values converge, often in a training-free manner. Lookahead Decoding [50] employs the token trajectories of Jacobi decoding to form a pool of $n$ -grams generated using greedy sampling to accelerate language models. CLLMs [51] collect the Jacobi trajectories into a dataset and then distill the language models with it. Speculative Jacobi Decoding [6] revisits the original Jacobi Decoding in image generation and adapts it for modern autoregressive text-to-image generation based on discrete tokens by simply introducing a probabilistic criterion. Spatially parallel image autoregressive decoding [52, 53] decodes multiple tokens simultaneously by leveraging spatial dependencies. Distilled-Decoding [17] distills an autoregressive model into a consistency model [54] via an embedding-prediction head with millions of prepared noise-token pairs and hundreds of training epochs. + +Discussion In contrast to the above works, this paper integrates the denoising process into discrete autoregressive models while preserving the properties like next-token prediction. Our method enables flexible modulation between the autoregression, the speculative decoding, and the denoising process. + +We fine-tune pre-trained autoregressive text-to-image generation models to achieve our goal with a few epochs and off-the-shelf image data while avoiding adding additional modules. + +# 3 Preliminaries + +Autoregressive Generation Let $\{x_1, \ldots, x_N\}$ denote a sequence of discrete tokens, where $N$ is the sequence length and $x_i \in V$ is an integer from a vocabulary of size $|V|$ . In autoregressive models, the joint probability of the sequence is factorized as: $\mathcal{P}(x_1, x_2, \ldots, x_N) = \mathcal{P}(x_1) \prod_{i=2}^{N} \mathcal{P}(x_i | x_1, \ldots, x_{i-1})$ , which assumes each discrete token $x_i$ depends only on its preceding tokens, following a causal structure. Autoregressive models parameterize the conditional probability as $\mathcal{P}_{\theta}(x_i | x_1, \ldots, x_{i-1})$ and sequentially decode (i.e., sampling a token based on the predicted probability) each discrete token $x_i$ based on the preceding outputs. + +Jacobi Decoding and Speculative Jacobi Decoding Jacobi Decoding treats the autoregressive decoding as solving a non-linear equation in a triangular system via fixed-point iteration [5]. Instead of sequentially generating tokens by the rule $x_{i} \sim \mathcal{P}_{\theta}(x|x_{1},\dots ,x_{i - 1})$ , Jacobi Decoding introduces an iteration index $j$ to enable parallel updates across all the sequential positions: $x_{i}^{(j + 1)} \sim \mathcal{P}_{\theta}(x|x_{1}^{(j)},\dots ,x_{i - 1}^{(j)})$ where $x_{i}^{(j)}$ denotes a token at position $i$ at iteration $j$ . Jacobi Decoding starts with randomly initialized tokens and iterates across the dimension $j$ until convergence, i.e., the tokens remain unchanged between two consecutive iterations. Since the number of iterations is proven to be not greater than the number of tokens [5], the acceleration can be achieved. + +To adapt Jacobi Decoding for the modern autoregressive text-to-image generation which is based on a large range of discrete tokens, Speculative Jacobi Decoding (SJD) [6] improves it by introducing the probabilistic criterion from speculative sampling [42, 43] to determine the convergence of tokens: + +$$ +x _ {i} ^ {(j)} \text {c o n v e r g e d i f} r < \min \left(1, \frac {\mathcal {P} _ {\theta} \left(x _ {i} ^ {(j)} \mid x _ {1} ^ {(j)} , \cdots , x _ {i - 1} ^ {(j)}\right)}{\mathcal {P} _ {\theta} \left(x _ {i} ^ {(j)} \mid x _ {1} ^ {(j ^ {\prime})}, \cdots , x _ {i - 1} ^ {(j ^ {\prime})}\right)}\right), r \sim \mathcal {U} [ 0, 1 ], \tag {1} +$$ + +where $\mathcal{U}[0,1]$ denotes a uniform distribution between 0 and 1, and $j^{\prime}$ is set to $j - 1$ by default, i.e., the tokens generated in previous Jacobi iterations serve as draft tokens for the current Jacobi iteration. SJD also maintains a fixed-length Jacobi window within which Jacobi iterations are performed and token convergence is determined. In each iteration, a prefix of the tokens is determined to be converged and removed from the window (i.e., the tokens are accepted), while the remaining tokens are resampled and newly initialized tokens are appended to the window. + +Continuous Diffusion Models Continuous diffusion models [55-59, 8, 60] generate data by learning to reverse a noise-corruption process. In this process, a clean continuous input $x_0$ is gradually corrupted into pure Gaussian noise $\epsilon$ . This can be formulated as: at any timestep $t$ (normalized to the range of [0, 1]) the noise perturbation can be written as $x_{t} = \alpha_{t}x_{0} + \sigma_{t}\epsilon$ where $\alpha_{t}$ and $\sigma_t$ are manually defined functions. As $t$ increases, $\alpha_{t}$ monotonically decreases but $\sigma_t$ increases. The reverse denoising trajectory between two timesteps can be solved by: $x_{t} = \frac{\sigma_{t}}{\sigma_{s}} x_{s} + \alpha_{s}\left(\frac{\alpha_{t}}{\alpha_{s}} -\frac{\sigma_{t}}{\sigma_{s}}\right)D_{\theta}(x_{s},s)$ [61]. Here, $D_{\theta}$ is a neural network trained via regression loss to predict $x_0$ given a noisy input and a function of timestep. In this paper, we incorporate the denoising process into the Jacobi decoding. + +# 4 Method + +In Jacobi Decoding, the refinement of tokens is unconstrained and difficult to control, with no guarantee that tokens will follow the fastest path to reach the correct values. In contrast, diffusion models have been demonstrated to generate high-resolution, high-quality images through short trajectories (as few as dozens of iterations) [8-10]. Motivated by this, we propose integrating the denoising process from the diffusion models into Jacobi Decoding. + +Overview Our decoding process, illustrated in Figure 2, comprises the following steps: (1) Token initialization: Given a sequence of noisy normalized token embeddings (illustrated as blue-bordered patches in the first row) and prefilling/already-accepted tokens (depicted as green circles in the first row), the noise levels of the token embeddings are configured to be non-strictly monotonically + +![](images/ddb23cc6baa640fd2ed7d66b7c6a8bdc95e7850280b94953a6bb8b5652ffa9a3.jpg) +Figure 2: Overview of our decoding process. The noisy token embeddings with increasing noise levels undergo a parallel forward pass with a causal attention mask, predicting conditional probabilities and then sampling clean tokens. A probabilistic criterion selects a prefix of tokens for acceptance (green area). For unaccepted tokens, if clean, the next-token prediction performs on them (green solid arrows marked with AR). If noisy, they are denoised with one-position offset (blue solid $\hat{x}^0$ arrows and dash arrows). + +![](images/6b39de1b96592da9c59aae8bfc9dcfdef213d8c11700195e413af23c49b4b5a8.jpg) +Figure 3: Overview of our training strategy and model process. The input token embedding sequence is randomly divided into segments and the adjacent segments are perturbed with the noise of consecutive levels. The noisy embeddings with timestep tokens are fed into transformer blocks and a prediction head. During training, the predicted probability is used to compute the cross-entropy loss for next-token prediction. During inference, the probability is for token sampling and then generating embeddings. + +increasing. Before the iterations start, the sequence is initialized with pure Gaussian noise. (2) Parallel forward: After initialization, these token embeddings undergo a Jacobi-denoising iteration, where they are fed into the neural network along with timestep encodings for a single parallel forward pass using a causal attention mask. The network predicts conditional probability and performs token sampling for the next clean token at each position. Sampled tokens from noisy embedding inputs are denoted as the one-position-offset $\hat{x}^0$ -predictions (marked by the down-right blue solid arrows) while those from prefilling or accepted token inputs are denoted as autoregressive (AR) predictions, i.e., the standard next-token prediction (marked by green solid arrows). (3) Verification: After the parallel forward pass, a prefix of sampled tokens is accepted based on the probabilistic criterion outlined in Equation (1). For example, in the second row of Figure 2, the first two sampled tokens are accepted and marked with green borders. (4) Refinement for unaccepted tokens: After the verification, the refinement is performed for unaccepted tokens. For each unaccepted noisy token, a denoising step, outlined in Equation (3), is performed on its embedding. Specifically, a linear combination is performed on the token embedding from the previous iteration at the same spatial position (indicated by vertical blue dashed lines) and the embedding of the predicted clean tokens from the one-position offset (the down-right blue solid arrows). If an unaccepted token has been sufficiently denoised to be clean in previous iterations, there is no further denoising needed and the refinement follows the standard next-token prediction. This iterative process repeats until all required tokens are accepted, serving as the final outputs. + +# 4.1 Model Parameterization + +To seamlessly integrate the denoising process with next-token prediction, we propose a paradigm of next-clean-token prediction based on noisy normalized token embeddings. In this paradigm, the autoregressive neural network learns to accept noisy input tokens and predict the clean tokens at a one-position offset (i.e., noise-free next tokens). Let $e$ denote a normalized token embedding and $\theta$ + +denote the autoregressive model, our paradigm can be formulated as follows: + +$$ +\hat {x} _ {i + 1} ^ {0} \sim \mathcal {P} _ {\theta} \left(x \mid e _ {1} ^ {0}, \dots , e _ {i ^ {\prime} - 1} ^ {0}, e _ {i ^ {\prime}} ^ {t _ {0}}, \dots , e _ {i} ^ {t _ {k}}\right), +$$ + +$$ +\hat {\boldsymbol {m}} _ {i + 1} ^ {0} = \boldsymbol {W} _ {e} \cdot \text {O n e - h o t} \left(\hat {x} _ {i + 1} ^ {0}\right), \tag {2} +$$ + +$$ +\hat {\boldsymbol {e}} _ {i + 1} ^ {0} = \operatorname {N o r m a l i z e} \left(\hat {\boldsymbol {m}} _ {i + 1} ^ {0}\right), +$$ + +where $e_i^t$ is the normalized token embedding at spatial position $i$ in the sequence and at timestep $t$ , and $\hat{e}_{i+1}^0$ represents the predicted clean normalized token embedding at position $i+1$ , $\mathcal{P}_{\theta}(x|e_1^0, \dots, e_{i'-1}^0, e_{i'}^{t_0}, \dots, e_i^{t_k})$ denotes the predicted conditional probability of a discrete token $x$ with the embedding sequence $\{e_1^0, \dots, e_{i'-1}^0, e_{i'}^{t_0}, \dots, e_i^{t_k}\}$ as conditions, One-hot(·) denotes transforming a token category into a one-hot vector, $W_e \in \mathbb{R}^{D \times |V|}$ denotes the learned embedding weight matrix of the model, and $\tilde{m}_{i+1}^0 \in \mathbb{R}^D$ denotes the $D$ -dimensional predicted token embedding at position $i+1$ . Normalize(·) is normalizing embeddings with their statistics (details in Section 4.3). The timesteps $\{t_0, \dots, t_k\}$ , i.e., noise levels, are configured to be non-strictly monotonically increasing, and we discuss the specific selection of timesteps for decoding in Section 4.2. In this paradigm, the pre-trained autoregressive models still predict the categorical probabilities of tokens instead of the continuous token embeddings, and we employ these probabilities to generate the embeddings. + +# 4.2 Speculative Jacobi-Denoising Decoding + +Jacobi Window and token initialization In practice, our method employs a sliding Jacobi window during the decoding phase, rather than directly performing the decoding process at fixed token positions. The Jacobi window is a fixed-length sliding window containing noisy normalized token embeddings as the draft. Initially, the window is filled with pure Gaussian noise. In each iteration, a prefix of accepted tokens is removed, and an equal number of new tokens, sampled from pure Gaussian noise, is appended. Consequently, the noise levels of the tokens within the window are non-strictly monotonically increasing across iterations. + +Parallel Forward and Verification The model takes draft token embeddings as the inputs and predicts the conditional probability of their next clean tokens in parallel. Then, Equation (1) is employed to determine whether to accept or reject each draft token which is transformed from the input token embedding. This transformation is finding the nearest discrete tokens in the vocabulary through cosine similarity [47]. + +Refinement with Denoising After verification, the unaccepted tokens will go through a refinement process to refine the token embeddings for the next iteration. The refinement of noisy tokens is realized by the denoising process. For denoising, we first define a fixed monotonically decreasing timestep sequence $\{t_{K},\dots ,t_{k}\dots ,t_{0},0\}$ , where $t_0$ is set very close to zero [62]. The values of these timesteps can follow the Karras timestep scheduler [62]. Then, we show the specific denoising formula based on these timesteps when $k > 0$ : + +$$ +\boldsymbol {e} _ {i} ^ {t _ {k - 1}} = \frac {\sigma_ {t _ {k - 1}}}{\sigma_ {t _ {k}}} \boldsymbol {e} _ {i} ^ {t _ {k}} + \alpha_ {t _ {k}} \left(\frac {\alpha_ {t _ {k - 1}}}{\alpha_ {t _ {k}}} - \frac {\sigma_ {t _ {k - 1}}}{\sigma_ {t _ {k}}}\right) \hat {\boldsymbol {e}} _ {i} ^ {0}, \tag {3} +$$ + +where $\hat{e}_i^0$ is the token embedding predicted according to Equation (2). In the above equation, we perform a denoising step like the diffusion models to obtain the embedding with a smaller noise level at position $i$ with timestep $t_k$ . If $k = 0$ , i.e., the denoising process is just complete, Equation (3) reduces into the standard Jacobi iteration: $e_i^0 \coloneqq \mathbf{0} + 1 \cdot \hat{e}_i^0$ . Additionally, for the unaccepted sufficiently denoised tokens, as no further denoising is required, we resample the tokens whose threshold from Equation (1) is below 0.5 but retain the others for the next Jacobi iteration. + +By introducing the denoising trajectories into the decoding process of autoregressive models, our method stabilizes the token trajectories in the decoding process, accelerating the token convergence. + +# 4.3 Fine-tuning Strategy + +In this section, we introduce the strategy of fine-tuning a pretrained autoregressive text-to-image generation model to accept noisy input tokens for Speculative Jacobi-Denoising Decoding. Figure 3 illustrates an overview of our training strategy and the specific process of the neural network: During + +training, the normalized embeddings (initially clean) are first transformed into noisy embeddings. For example, in the right-side image of Figure 3, noise levels increase non-monotonically across patches (i.e., token positions) in a raster scan order (left to right, top to bottom). When reaching a randomly determined position, the noise level stops increasing and the noise level of the next position resets to zero, forming segments with non-monotonically increasing noise levels in the token sequence. Next, these noisy embeddings are appended with timestep encodings, which indicate the noise level at each position. Together, the embeddings and encodings are denormalized and are fed into transformer blocks and a prediction head to produce logits for each position. The cross-entropy loss is then applied to each position, using the clean token indices as labels, with one-position offset (shown by the dotted frame in Figure 3) for next-clean-token prediction. During inference, the input normalized embeddings are already noisy and are not further perturbed. These embeddings are also appended with timestep encodings and processed to generate logits through the denormalization, the transformer blocks and the prediction head. As described in Equation (2), these logits are used for token sampling, and these sampled clean tokens are then transformed into normalized token embeddings. + +Noise Perturbation We add noise to the embeddings of these discrete input tokens, bypassing the discrete values. Although we can directly perform a linear combination of the embedding and a Gaussian random variable, the distribution of the pre-trained embeddings may deviate from the scale of the standard Gaussian distribution. For instance, if the variance of the embedding is small, the pre-trained transformer blocks may only handle small values, making the values from a standard Gaussian distribution unsuitable for these blocks. To address this issue, we add noise to the normalized token embeddings. Subsequently, these noisy normalized embeddings are de-normalized and then fed into the transformer blocks. The detailed procedure is as follows: + +$$ +\boldsymbol {e} ^ {*} = \text {N o r m a l i z e} (\boldsymbol {m} ^ {*}) = \frac {1}{\sigma_ {e}} \odot (\boldsymbol {m} ^ {*} - \boldsymbol {\mu} _ {e}), +$$ + +$$ +\boldsymbol {e} ^ {t} = \alpha_ {t} \boldsymbol {e} ^ {*} + \sigma_ {t} \boldsymbol {\epsilon}, \tag {4} +$$ + +$$ +\boldsymbol {m} ^ {t} = \text {D e - n o r m a l i z e} (\boldsymbol {e} ^ {t}) = \sigma_ {e} \odot \boldsymbol {e} ^ {t} + \boldsymbol {\mu} _ {e}, +$$ + +where $\pmb{m}^{*} \in \mathbb{R}^{D}$ denotes the $D$ -dimensional embedding of a ground-truth token. $\pmb{\mu}_{e} \in \mathbb{R}^{D}$ and $\pmb{\sigma}_{e} \in \mathbb{R}^{D}$ denote the mean and standard deviation of the learned embedding weight $W_{e} \in \mathbb{R}^{|V| \times D}$ , respectively. In practice, we directly average this weight across its first dimension to compute the mean, and the standard deviation is also obtained across this dimension. $\frac{1}{\pmb{\sigma}_{e}} \in \mathbb{R}^{D}$ represents the element-wise reciprocal of $\pmb{\sigma}_{e}$ , and $\odot$ is the element-wise product. $\alpha_{t} \in \mathbb{R}$ and $\sigma_{t} \in \mathbb{R}$ are the hyper-parameters for the denoising timestep $t$ , and $\epsilon \in \mathbb{R}^{D}$ denotes the standard Gaussian noise. With Equation (4), we can transform a clean embedding $m^{*}$ into a noisy embedding $m^{t}$ . For fine-tuning, the input sequence is divided into randomly sized segments, and we add the identical level of noise to the tokens from the same segment. + +Finetuning Objective and Loss Function We fine-tune a pre-trained autoregressive text-to-image model to predict the next clean token from the inputs of noisy token embeddings. The model accepts noisy token embeddings and outputs logits representing the categorical probability distribution of the next clean token. The cross-entropy loss is computed between these logits and the ground truth token categories, optimizing the model to denoise inputs while maintaining autoregressive prediction. Specifically, analogous to standard next-token prediction, the cross-entropy loss is performed as follows: $\mathcal{L} = \sum_{i=0}^{N} \text{Cross-Entropy}\left(x_{i+1}^{*}, \hat{p}_{i+1}^{0}\right)$ , where $x_{i}^{*}$ denotes the one-hot label of the ground-truth token at position $i$ , $N$ is the total number of tokens, and $\hat{p}_{i+1}^{0}$ denotes the conditional probability in Equation (2). Here, we assume that only the first token is prefilled with text conditioning. With this training objective and parameterization, the model learns to decode or verify clean tokens even when noisy tokens are present in the input conditions. + +Timestep Injection Injecting the information of timesteps into noisy inputs is a common design for the denoising process [56, 63, 64]. To avoid introducing additional modules like AdaLN [56], we take the sinusoidal encodings of timesteps as a sequence of special token embeddings and append them to the sequence of input token embeddings during fine-tuning and decoding. Then, the sequence, which comprises input token embeddings and timestep encodings, is fed into the transformer blocks. Within the attention modules of these blocks, we use the attention mask to force each noisy token embedding to attend to the corresponding timestep encoding which indicates its noise level. To ensure + +![](images/c31d2e9bd81ba24379fce6dbee0e774a13b6f9ba111d9dd8e5153607d7fa25b6.jpg) +Figure 4: Correlation between denoising iterations and Jacobi window length on the latency. Circle areas represent absolute latency values, and the lowest latency (a difference within 3 seconds is allowed) is in orange. + +![](images/3270f93ff2ec3d9f07212c1bfe241eceda9d5649fdf420ef83e239e0cbe16d01.jpg) +Figure 5: Study on embedding normalization for denoising process. Left: Denoising output without embedding normalization, failing to generate a coherent image. Right: Denoising output with embedding normalization, generating a semantically meaningful image. + +Table 1: Evaluation on the validation sets of MSCOCO [65]. + +
ConfigurationCOCO2017 (5k)COCO2014 (30k)
Average Steps (↓)Step Comp- resection (↑)FIDCLIP- ScoreAverage Steps (↓)Step Comp- resection (↑)FIDCLIP- Score
Lumina-mGPT [11]23571.00×30.831.323571.00×21.031.3
SJD [6]10602.23×31.131.310572.23×20.831.3
SJD25924.02×31.431.85993.93×21.131.9
Emu3 [4]81931.00×31.131.081931.00×19.331.2
SJD [6]35282.32×30.630.935352.32×21.431.0
SJD214615.62×31.530.414615.63×21.830.8
+ +that the distribution of the timestep encodings aligns with that of the token embeddings, we apply a normalization-then-denormalization process to these encodings similar to Equation (4). + +# 5 Experiments + +# 5.1 Implementation Details + +We perform experiments on two baselines, Lumina-mGPT [11] and Emu3 [4]. When generating an image at least $720 \times 720$ , Lumina-mGPT only needs about 2k tokens while Emu3 requires more than 8k tokens because of the difference between their tokenizers. For each fine-tuning, 8 GPUs with 80G memory are required for each model. Since all model parameters are used for fine-tuning, we leverage DeepSpeed ZeRO-3 or FSDP with gradient checkpointing to save GPU memory at the cost of increased training time. The global batch size is set to 64, and the learning rate is set to $2 \times 10^{-5}$ with the AdamW optimizer. We tune each model only within 6 epochs, costing approximately $14 \times 8$ A100 hours for Lumina-mGPT and $26 \times 8$ H100 hours for Emu3. By default, we set classifier-free guidance to 3.0 and use top-2000 for the quantitative results of our method. + +Evaluation metrics. To assess visual quality, we employ two key metrics: FID [66] and CLIP-Score [67] on COCO benchmark [65] and GenEval benchmark [68]. To quantify the efficiency of the decoding process, we introduce the step compression ratio [50]: $S = \frac{\# \text{generated tokens}}{\# \text{decoding steps}}$ , which serves as a theoretical measure of parallelization and thus reflects the acceleration. For each benchmark, we compute the average step compression ratio across all generated images. Additionally, this ratio is included alongside individual image samples in qualitative comparisons to highlight the performance differences between our method and other approaches. More importantly, we provide the latency of model forward passes on a single GPU to evaluate the practical speedup achieved. + +Table 2: Visual quality on GenEval benchmark [68]. + +
MethodColorsPositionCountingTwoColor AttriSingleOverall
Lumina-mGPT [11]0.830.090.260.600.150.960.48
Lumina-mGPT + SJD [6]0.820.080.230.630.120.960.47
Lumina-mGPT (tuned)0.810.130.270.650.270.980.52
Lumina-mGPT (tuned) + SJD [6]0.810.120.310.680.240.970.52
Lumina-mGPT (tuned) + SJD20.790.110.310.640.230.960.51
Emu3 [4]0.780.150.330.690.160.980.52
Emu3 + SJD [6]0.790.120.280.610.130.970.48
Emu3 (tuned) + SJD20.730.140.280.610.240.960.49
+ +# 5.2 Qualitative Results + +In Figure 1 and Figure 6, we compare the standard autoregressive decoding and our method on Lumina-mGPT [11]. The results show that our method can achieve about $4\times$ fewer steps for autoregressive text-to-image generation and the visual quality is preserved. We also compare different decoding methods on Emu3 [4] in Figure 7. + +# 5.3 Quantitative Results + +Our SJD2 significantly reduces the steps for autoregressive text-to-image generation while maintaining visual quality, as demonstrated by our evaluation on MS-COCO [65] validation sets in Table 1. We also find that our method achieves a higher step compression ratio on Emu3 [4] than that on LuminamGPT [11]. Specifically, SJD2 can achieve a step compression about $4.0 \times$ on Lumina-mGPT and about $5.6 \times$ on Emu3. For visual quality, we further compare our method to autoregressive decoding and SJD [6] on the GenEval benchmark [68] with Lumina-mGPT [11] as the baseline in Table 2. Our method achieves an overall score of 0.51, nearly matching the 0.52 of tuned AR and SJD, demonstrating preserved visual quality. More importantly, following [47], we select 100 COCO prompts on the identical A100 server to compare the practical average speedup and visual quality among these decoding methods. According to the latency reported in Table 3, our method is still faster than other decoding methods on the real server by more than $2 \times$ . We also provide the GPU memory usage during inference. While our parallel decoding method achieves significant latency reductions, we acknowledge it incurs an additional memory overhead of about 3GB compared to autoregressive decoding, because of the variables for denoising like the timestep tokens. + +# 5.4 Ablation Studies + +We perform experiments for our method with Lumina-mGPT as the baseline and on one RTX 4090 by default. The selected 100 prompts used in Table 3 are for the evaluation in the ablation studies. We also include a specific and detailed analysis for our denoising process in Appendix B. + +Study on the sampling timesteps and Jacobi window length. While contemporary diffusion models typically require dozens of denoising iterations to achieve satisfactory results, extending Jacobi window lengths introduces computational overhead. This establishes a trade-off between denoising iteration counts and Jacobi window length for the minimization of latency. According to the results in Figure 4, when constraining denoising steps to 20 while maintaining Jacobi window lengths beyond 80, the latency reduction converges to a minimum. + +Study on the embedding normalization. When verifying the usefulness of embedding normalization, we focus on our denoising process by enforcing the denoised tokens to be immediately accepted (as detailed in Appendix B). As shown in Figure 5, deactivating embedding normalization results in a complete failure of the denoising process, yielding pure noise instead of coherent images. Conversely, activating normalization enables the denoising process to generate reasonable outputs, demonstrating the critical role of embedding normalization in our denoising process. + +Table 3: The computational cost of decoding methods on a subset of COCO prompts. + +
MethodsStepsLatencyCLIP-ScoreGPU Memory
Lumina-mGPT [11]235788.55s32.017G
SJD [6]105841.99s31.517G
SJD259633.64s32.220G
Emu3 [4]8193375.29s30.920G
SJD [6]3537207.60s31.220G
SJD21470147.65s30.723G
+ +![](images/cd01f6a2d2d18c827c4a131e08ac92221e72c95d3e555169f388bd31809c42a5.jpg) +Prompt: A photo of a spring with purple gas coming out in a jungle, film grained, high-quality, 8K + +Latency: 102.63s + +Steps:2357 + +Latency: 58,11s (1.77 × Faster) + +Steps:1238 $(1,90\times$ Fewer) + +Latency:38,41s (2.67 x Faster) + +Steps:672 (3.51 $\times$ Fewer) + +![](images/ee193d5aa2336c74c2bb931a2525db2a9ab28732a335c4ad6724798c7e94d87b.jpg) +Prompt: An imaginary picture of a dark blue-haired minotaur wearing black armor, surrounded by groups of purple + +flames,filmgrained,high-quality,8K + +![](images/a415836e77ba3db3dafe06ea7efb1a2e1f4b6f2aeb70e3c645ae6eca65cf94bc.jpg) +Prompt: An oil painting of a cool man wearing a suit in village + +Latency: 391.68 + +Steps:8193 + +Latency: 217.45s (1.80 × Faster) + +Steps: 3490 (2.34 × Fewer) + +ency:181.24s (2.16 $\times$ Faster) + +Steps: 1581(5.18 $\times$ Fewer) + +![](images/0defbe65f7cd546a0415585841de119eeaf4f8c0c8a8a40d75ee51f5e2425877.jpg) +Prompt: A cool man with a blond beard and a white hood stands on the deck with the sea behind him. + +Latency:392.98s + +Steps: 8193 + +Latency: 224.13s (1.75 × Faster) + +Steps:3623(2.26 $\times$ Fewer) + +Latency: 182.04s (2.16 × Faster) + +Steps: 1579 (5.19 × Fewer) + +![](images/dce8668a2b6f2d48c268cd4f5dbaf2603e6ab6ff7103ec0150ff8197bd008a41.jpg) +Figure 6: The comparison of the original autoregressive decoding, SJD [6], and our method with Lumina-mGPT [11] as the baseline and on one RTX 4090. +Prompt: A strikingly beautiful girl with deep red eyes, short white hair, and a sly smile, dressed in elegant purple attire +Figure 7: The comparison of the original autoregressive decoding, SJD [6], and our method with Emu3 [4] as the baseline and on one RTX 4090. + +with a hood. The image features perfect eye-symmetry and facial-symmetry. + +# 6 Conclusion + +This paper introduces Speculative Jacobi-Denoising Decoding, a new algorithm integrating the continuous denoising process into Jacobi decoding to accelerate autoregressive text-to-image generation. By extending next-token prediction into next-clean-token prediction with noisy inputs, we enable pre-trained autoregressive models to learn to denoise noise-perturbed token embeddings through a fine-tuning strategy. The proposed Jacobi-Denoising decoding initializes token sequences with Gaussian noise. It then iteratively refines them using a process that combines denoising steps with Jacobi decoding and an improved probabilistic prefix acceptance criterion. Experiments show that our method can reduce the number of model forward passes for acceleration while keeping the visual quality of generated images. + +# Acknowledgment + +This work is supported by the National Nature Science Foundation of China (No. 62402406). + +# References + +[1] Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yinfei Yang, Burcu Karagol Ayan, et al. Scaling autoregressive models for content-rich text-to-image generation. arXiv preprint arXiv:2206.10789, 2022. 1 +[2] Dan Kondratyuk, Lijun Yu, Xiuye Gu, José Lezama, Jonathan Huang, Rachel Hornung, Hartwig Adam, Hassan Akbari, Yair Alon, Vighnesh Birodkar, et al. Videopoet: A large language model for zero-shot video generation. arXiv preprint arXiv:2312.14125, 2023. +[3] Chameleon Team. Chameleon: Mixed-modal early-fusion foundation models. arXiv preprint arXiv:2405.09818, 2024. +[4] Emu3 Team BAAI. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [Yes] + +Justification: The LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigourousness, or originality of the research, declaration is not required. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# A More Implementation Details + +Experimental settings. We perform experiments on two baselines, Lumina-mGPT [11] and Emu3 [4]. We have collected approximately 80k high-resolution (at least $720 \times 720$ ) images from the Internet. For images lacking text descriptions, we caption them with Qwen2-VL [69]. We follow the advanced flow matching setting [70, 71] with $\alpha_{t} = 1 - t$ and $\sigma_{t} = t$ for our denoising process. By default, the length of the Jacobi window of our method is set to 96 for Lumina-mGPT and 128 for Emu3. The number of denoising steps in SJD2 is set to 25. + +# B Further Analysis + +The pure denoising process also generates reasonable images. In our approach, the speculative Jacobi decoding is applied after the denoising steps, so we need to recognize that the acceleration benefits stem not only from the Jacobi iterations but also from the effectiveness of the denoising process. To systematically verify that the denoising stage alone can produce valid tokens, we conduct the following experiment: when a token is successfully denoised and its distance from the first point of the Jacobi window is smaller than $\frac{L}{T}$ (where $L$ represents the length of the Jacobi window and $T$ denotes the number of denoising iterations), we immediately accept it. The ratio $\frac{L}{T}$ guarantees that the last token in the Jacobi window completes exactly $T$ denoising iterations when it reaches the left part of the window. The results of this experiment are presented in Figure 9a. According to these results, we observe that our denoising process can generate reasonable images in certain cases, particularly those featuring characters. Additionally, the immediate acceptance of denoised tokens significantly reduces the number of steps and overall latency. In comparison, as shown in Figure 9b, we reintroduce Jacobi iterations into the decoding process. These iterations serve as a refinement mechanism, enhancing the image generation with more intricate details and reducing the artifacts. However, this improvement comes at the cost of increased computational steps. Therefore, to preserve image quality, we opt to keep the Jacobi iterations in our method. + +Analysis of unifying noise perturbation for discrete and continuous inputs. We demonstrate the feasibility of unifying noise perturbation for both the discrete and continuous inputs in the state-of-the-art transformer-based models. Since the noise perturbation is commonly used in diffusion models, we first analyze the behavior of noisy inputs in diffusion transformer (DiT) architecture. Although noise perturbation appears to occur in the latent space, the learnable linear transformation $\mathbf{W} : \mathbb{R}^{dHW} \to \mathbb{R}^{DH_{S}W_{S}}$ (usually implemented by a 2D convolution without receptive field overlapping) before the transformer blocks causes the perturbation to actually happen in the feature space: $\mathbf{W}\mathbf{x}_{t} = \alpha_{t}(\mathbf{W}\mathbf{x}^{*}) + \sigma_{t}\mathbf{n}$ , where $\mathbf{n} \in \mathbb{R}^{DH_{S}W_{S}}$ is a random Gaussian variable formed by a linear weighted sum of independent Gaussian noises with the elements of $\mathbf{W}$ as weights, and $D, H_{S}, W_{S}, d$ denote feature dimension, latent height, latent width and latent dimension, respectively. Therefore, based on the above equation, the noise perturbation can be interpreted as a linear combination of a clean feature vector and a noise vector. Since both DiTs and autoregressive models rely on transformer blocks operating in the feature space, we aim to align their noise perturbation on the input embedding. + +Analysis of Flops in inference. We present the average Flops per output token in Table 4. The results reveal that autoregressive decoding requires fewer Flops than SJD2. Although more Flops are used for decoding, the practical latency becomes lower. Actually, this Flops overload stems from the paradigm of all the speculative decoding methods. Their drafting-and-verification mechanism, which inherently introduces computational overhead: the number of accepted tokens per sampling step is substantially lower than the number of input draft tokens. + +Smaller baseline. We also implement SJD2 on Janus-pro-1B [72], an advanced autoregressive model much smaller than Lumina-mGPT [11]. The results are in Table 5 and Table 6. From the results, we observe that SJD2 still can accelerate Janus-pro without sacrificing on generated image quality, as evidenced by the following Geneval metrics [68]. + +Comparison to other accelerating methods. We compare our method with the recent and classic speculative/parallel decoding methods, including Lantern [47], ZipAR [53], Eagle [46] and Jacobi + +Table 4: The inference Flops of autoregressive models with different decoding methods. + +
MethodGFlops (↓)Latency (↓)
Lumina-mGPT [11]18.7288.55s
Lumina-mGPT + SJD2219.6033.64s
Emu3 [4]18.15375.29s
Emu3 + SJD2465.92147.65s
+ +Table 5: Inference performance comparison on Janus-Pro-1B [72]. + +
MethodLatency (↓)Steps (↓)
Janus-Pro-1B [72]9.1s576
Janus-Pro-1B + SJD22.5s144
+ +Decoding [5], on COCO2017 validation set [65] with Lumina-mGPT [11] as baseline. As shown in Table 7, our approach achieves superior acceleration while maintaining comparable visual quality. + +Comparison to diffusion models. While many autoregressive models currently underperform state-of-the-art diffusion models in image quality and face acceleration challenges, our SJD2 narrows the speed gap. In Table 8, we evaluate inference latency for several commonly-used diffusion models (smaller than 3B) and Janus-Pro [72] at the same resolution $(384 \times 384)$ . We set the number of sampling steps for diffusion models as 50. Results demonstrate that our SJD2 reduces latency of Janus-Pro-1B from 9.1s to 2.5s, narrowing the gap between Janus-pro and the advanced diffusion models like SD3. Moreover, this result means Janus-Pro-1B with SJD2 already outperforms SDXL in speed (2.5s vs. 4.3s). + +Investigation on the refinement. In this paragraph, we demonstrate that SJD2 stabilizes the refinement trajectory, as illustrated in Figure 8. Specifically, we apply SJD2 and SJD [6] on Emu3 [4] respectively, and then we examine the first five tokens from the Jacobi window in one iteration, computing the times of the token change between adjacent steps and the cumulative changes. As shown in the first five figures, SJD2 yields identical initial predictions across the 25 sampling steps. As the noise level decreases, token predictions diversify but then become unchanged at several steps, indicating the stabilization of the token trajectory. In contrast, for SJD, the tokens consistently change, appearing to oscillate and remain unstable. The last figure of Figure 8 also shows that the SJD causes more times of token change than SJD2. + +# C Limitations and Future Work + +Although SJD2 can achieve similar step compression in various models, the improvements on actual latency are not consistent, shown by Figure 6 and Figure 7. We speculate that this is caused by the different sizes of KV cache in different autoregressive models (the model whose tokenizer has a low image compression ratio leads to a large KV cache). A promising direction is to stabilize the latency acceleration of Jacobi-based acceleration methods across the models with different KV caches. + +Table 6: Visual quality on Geneval benchmark [68] with Janus-pro-1B [72] as the baseline. + +
MethodColorsPositionCountingTwoColor AttriSingleOverall
Janus-Pro-1B (tuned) [72]0.820.390.390.550.420.940.59
Janus-Pro-1B (tuned) + SJD20.830.450.370.590.380.960.60
+ +Table 7: Comparison to other accelerating methods with Lumina-mGPT [11] as baseline. + +
ConfigurationAcceleration Latency (↑)Acceleration Step (↑)CLIP-Score (↑)
Autoregressive Decoding1.00×1.00×31.3
Jacobi Decoding [5]1.02×1.04×31.4
SJD [6]2.05×2.23×31.3
EAGLE [46]2.10×2.94×33.3
LANTERN [47]2.56×3.63×32.7
ZipAR [53]1.82×4.00×31.2
SJD22.63×4.02×31.8
+ +Table 8: Efficiency comparison with the diffusion model. + +
MethodLatency (↓)Steps (↓)
Janus-Pro-1B [72]9.1s576
SD1.5 [55]1.7s50
SDXL [39]4.3s50
SD3-Medium [70]1.7s50
Janus-Pro-1B + SJD22.5s144
+ +![](images/4071c723c67b1f5c88585675da3da621beb25bfdef0d758010de39c97775be9f.jpg) + +![](images/939e2889e2f7cf1e1733c3a624beb276c0e315e446a0d5c710ccb84f3f9b2d9b.jpg) + +![](images/f69c987858475d17d271e8f773eef75df6ead428706d142aac806090733a3ac2.jpg) + +![](images/a5279e77ce68c5a96255ad3b6848235004d3e5640955d1e6518c4f8574e21fcf.jpg) +Figure 8: The trajectories of token difference. + +![](images/6c8cc194036b6c9ab15db3d2257eb617d27baf694344c96e9d4689b7302133dd.jpg) + +![](images/7c6aa02fbae0dc5b90a28d5114198acdfa65367b10f8c0ee75aea986a1986745.jpg) + +# D Impact Statement + +Our paper proposes a new method of autoregressive text-to-image generation for research purposes. Real-world deployment would require additional safeguards beyond technical implementation. We recognize that this open technical accessibility could lead to potential societal risks, including misuse for generating misleading content or harmful biases. Fortunately, this problem can be alleviated by strict dataset filtering to exclude harmful content. + +![](images/1649e3a0db716471c9f01d57e9c16c94bd1fabede0249d134bdd462e49c4e794.jpg) +Latency: 19.49s +Steps: 306 +Prompt: Miss Mexico portrait of the most beautiful mexican woman, Exquisite detail + +![](images/c0b85f3a9738a257f0e528aece64cdcc31b7494149592d8ff8b4635bccf168da.jpg) +Latency: 19.35s Steps: 305 +Prompt: An old elephant wearing Chinese armor, Exquisite detail, 30-megapixel, 4k + +![](images/d235f1268aaa160f746a552dc326fa140e17fe316f55c7eb39a2421be526fa6e.jpg) +Latency: 19.40s +Steps: 306 +Prompt: Atlantis, the most Fantasy high-quality photos + +![](images/0d102a0ac1f88087298a9792310a3a1bb43c0ef9fec3640016c114ead3305d99.jpg) +Latency: 32.80s +Steps: 538 +Prompt: Miss Mexico portrait of the most beautiful mexican woman, Exquisite detail +(b) The images generated with the combination of Jacobi iterations and denoising. +Figure 9: Analysis of our denoising process. (a) Our denoising process without Jacobi iterations can generate reasonable images with few model forward passes and small latency. (b) The further Jacobi iterations refine the results of our denoising process, resulting in more details and fewer artifacts. + +![](images/6b43887b0a1e1715e256267ef9d30d6fdd390991a7c150cc9e26b196f7fd4552.jpg) +Latency: 34.83s +Steps: 575 +Prompt: An old elephant wearing Chinese armor. Exquisite detail, 30-megapixel, 4k + +![](images/edd44a493f9935c567b281c4cd33b276f27cffab20cbbe5534ab1e68cc215446.jpg) +(a) The images generated with the immediate acceptance of denoised tokens (without Jacobi iterations). +Latency: 35.18s +Steps: 587 +Prompt: Atlantis, the most Fantasy high-quality photos \ No newline at end of file diff --git a/NeurIPS/2025/Speculative Jacobi-Denoising Decoding for Accelerating Autoregressive Text-to-image Generation/images.zip b/NeurIPS/2025/Speculative Jacobi-Denoising Decoding for Accelerating Autoregressive Text-to-image Generation/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..c605f0eb64f816f5b2a142892c5fb34cb450b4d4 --- /dev/null +++ b/NeurIPS/2025/Speculative Jacobi-Denoising Decoding for Accelerating Autoregressive Text-to-image Generation/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:49d64a16c9c9c118a5c58dc2f26e386879af9c5be16f3519d697fe41a6892fcb +size 951842 diff --git a/NeurIPS/2025/Speculative Jacobi-Denoising Decoding for Accelerating Autoregressive Text-to-image Generation/layout.json b/NeurIPS/2025/Speculative Jacobi-Denoising Decoding for Accelerating Autoregressive Text-to-image Generation/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..caf431cbff2cf5201d87737090746de9f9640b30 --- /dev/null +++ b/NeurIPS/2025/Speculative Jacobi-Denoising Decoding for Accelerating Autoregressive Text-to-image Generation/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2dc3730525b88ec7b761dc1b7a58b284ca5cead1a1c130e47626f01136027240 +size 835067 diff --git a/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_content_list.json b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..8aaee2c89ce46193e8548d9129025888570e0abc --- /dev/null +++ b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fb1f0f2ba46bb43ad06c5b504df7e8be8cb15fa4469ac9e35225e6ea0c5d24da +size 247524 diff --git a/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_model.json b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_model.json new file mode 100644 index 0000000000000000000000000000000000000000..8f6885f6eefa6853da429277b697e464180c3bbf --- /dev/null +++ b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:70f35def6cf80f0fbe5639ed8adf0f68ca3b2de8802c15ef4afbe32d5e2247b5 +size 304057 diff --git a/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_origin.pdf b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..898b1772cb7f47e263558b7e4ae29aafd87018c9 --- /dev/null +++ b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/cf72017a-65d1-40de-9723-91fb71e5374c_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f6d5ea722f7063b2c0926b7de6bf85d01e4ce80d9ddd49a95d0494417c80cb19 +size 1083575 diff --git a/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/full.md b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/full.md new file mode 100644 index 0000000000000000000000000000000000000000..deab1d0e77808d7568017411b4629e853e1020d3 --- /dev/null +++ b/NeurIPS/2025/Spend Wisely_ Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping/full.md @@ -0,0 +1,1474 @@ +# Spend Wisely: Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping + +Pu Yang* Peking University + +Yunzhen Feng* New York University + +Ziyuan Chen* Peking University + +Yuhang Wu +UC Berkeley + +Zhuoyuan Li† +National University of Singapore + +# Abstract + +Modern foundation models often undergo iterative "bootstrapping" in their posttraining phase: a model generates synthetic data, an external verifier filters out low-quality samples, and the high-quality subset is used for further fine-tuning. Over multiple iterations, the model performance improves, raising a crucial question: How should the total budget for generation and training be allocated across iterations to maximize final performance? In this work, we develop a theoretical framework for analyzing budget allocation strategies. Specifically, we show that constant policies fail to converge with high probability, while increasing policies—particularly exponential growth policies—exhibit significant theoretical advantages. Experiments on image denoising with diffusion probabilistic models and math reasoning with large language models show that both exponential and polynomial growth policies consistently outperform constant policies, with exponential policies often providing more stable performance. + +# 1 Introduction + +Supervised fine-tuning is a critical stage in training large language models (LLM). Pre-training supplies them with broad linguistic and factual knowledge, but task-specific skills—such as tool use (Schick et al., 2023), reasoning pattern (Gandhi et al., 2025), and agentic behavior (Shao et al., 2023)—emerge when the model is further refined on carefully curated, supervised examples. Securing these high-quality human-annotated data, however, remains a major bottleneck, as it requires domain expertise and substantial resources. + +To address this limitation, synthetic data have emerged as a promising alternative that offers scalability and cost-effectiveness. Despite concerns about potential risks of model collapse (Shumailov et al., 2024; Dohmatob et al., 2024b), synthetic data is typically selected and verified before use in posttraining (Feng et al., 2024; Setlur et al., 2024), ensuring its quality. A common paradigm for leveraging synthetic data employs an iterative bootstrapping process (Zelikman et al., 2022; Trung et al., 2024): the model generates synthetic data, rewards or verifiers are used to filter and select high-quality data, and the model is then fine-tuned on selected data. This process is repeated iteratively to fully improve performance. An illustration of this approach is provided in Figure 1. + +However, for practitioners who implement this approach, an important question arises: How should a fixed computational budget be allocated to decide the amount of synthetic data to generate and select at each iteration to maximize performance? + +![](images/2bc04b86f1d4e986601b5ef869cf2b72fe1df46a48f868c484a2ff5cc6cb8f54.jpg) +Figure 1: Iterative learning with synthetic data. In this framework, synthetic data is generated, filtered using a reward model, and the selected data is used to further train the generator. The budget policy is defined as the quantity of data retained after selection, $n_t$ . Our goal is to identify the optimal policy across iterations to achieve the best final performance, given a fixed budget. + +In this paper, we establish some foundational principles for crafting optimal strategies for synthetic data generation across iterations. To the best of our knowledge, this is the first attempt to address this problem. We begin with a theoretical analysis of policies that control the amount of selected synthetic data used in each iteration, as illustrated in Figure 1. In a simplified setting with Gaussian data and exponential reward functions, we could identify the optimal policy. Contrary to common strategies that maintain a fixed number of synthetic data across iterations (constant policies), our analysis shows that the optimal strategy requires exponentially increasing the amount of data at each iteration. + +Furthermore, in more general settings with only mild assumptions about regularity, we demonstrate that constant policies fail to converge with high probability. In contrast, policies that increase the amount of synthetic data across iterations can converge to optimal performance. Among such increasing policies, we prove that exponential growth policies guarantee exponential convergence, and in the worst-case scenarios, they achieve no larger computational costs compared to polynomial growth policies when attaining similar near-optimal performance. + +Building on these theoretical insights, we validate our findings with two experiments: an image-denoising task using diffusion probabilistic models (DPMs), and a math-reasoning task with large language models (LLMs). Across these experiments, exponential and linear (polynomial) growth policies outperform constant policies. Moreover, the best exponential policies match or exceed the performance of linear policies in every case, confirming their theoretical robustness. + +We summarize our contribution as follows. + +- We formulate the problem of optimizing post-training gains under a limited budget for iterative learning in Section 3.1. +- In a solvable Gaussian setting, we prove the optimality of an exponential growth policy in Section 3.2 and validate it through numerical simulations. +- In a more general setting, we demonstrate in Section 4 that constant policies fail to converge to the optimal reward with high probability, while increasing policies ensure convergence. Among these, the exponential growth policy achieves an exponential convergence rate and outperforms polynomial growth policies in the worst case. +- Experiments on image denoising (diffusion probabilistic model) and math reasoning (large language model) in Section 5 confirm that exponential and polynomial growth policies outperform constant policies, with exponential policies exhibiting greater stability. + +# 2 Related Work + +Synthetic Data and Iterative Bootstrapping. The iterative bootstrapping framework, illustrated in Figure 1, is widely used for training foundational models in both text and vision domains. In this approach, synthetic data are generated and then iteratively refined or filtered to produce new + +training sets that more closely align with the target objectives. For large language models (LLMs) and vision language models (VLMs), this approach has been applied to instruction following (Touvron et al., 2023) and alignment tasks (Dong et al., 2023), and vision question answering (Yang & Dong, 2025), wherein reward models filter synthetic data to retain only high-quality samples. Recently, it has been particularly effective in improving reasoning abilities, as first demonstrated in Zelikman et al. (2022). Subsequent works (Lehnert et al., 2024; Su et al., 2024; Trung et al., 2024; Guan et al., 2025; Singh et al., 2025) have extended this approach to generate diverse reasoning paths with search algorithms. Beyond text, iterative bootstrapping has also been utilized to generate training data for image segmentation (Kirillov et al., 2023), create image-text pairs (Fan et al., 2023, 2024), and synthesize images to address distribution shifts (Hemmat et al., 2024; Azizi et al., 2023). However, in these empirical works, a principled method for controlling generation in each iteration is still missing. + +To our knowledge, Ferbach et al. (2024) provides the only theoretical analysis of this paradigm, proving convergence to the optimal policy under the infinite-sample assumption. Crucially, their framework fails to characterize finite sample requirements, offering little actionable guidance to practitioners implementing this framework. We address this gap by establishing optimal computational budget allocation strategies across iterations to achieve maximum performance gains. + +Model Collapse. A series of papers has shown that the incorporation of synthetic data into the training corpus can lead to performance degradation, either in a single training instance or iteratively over time (Shumailov et al., 2023; Alemohammad et al., 2024; Dohmatob et al., 2024b,a, 2025). Feng et al. (2024) has explored the use of data selection to overcome model collapse by employing synthetic data in a single generation, combined with verification. In our work, we also leverage verification, but focus on addressing the question of how to optimally generate synthetic data during learning. + +# 3 Problem Formulation and Preliminary Analysis + +In this section, we formalize the iterative learning setting in Section 3.1 and then analyze optimal budget allocation strategies with a simple Gaussian case in Section 3.2. + +# 3.1 Problem Formulation + +The setup is illustrated in Figure 1. Our goal is to train a parameterized generative model $f(\cdot ;\theta)$ , for example, given by a DPM or an LLM, that generates data $x\sim \mathcal{P}_{\theta}$ . A reward model $\mathcal{R}$ evaluates the quality of the generated data, serving as a measure of the model performance. We aim to train the generative model to maximize the expected reward + +$$ +r (\theta) = \mathbb {E} _ {x \sim \mathcal {P} _ {\theta}} [ \mathcal {R} (x) ], +$$ + +where we assume $\mathcal{R}(x)\in [0,1]$ + +The iterative training process leverages the generative nature of the model to perform self-generation and refinement based on reward signals. Specifically, we start with an initial model $\theta^{(0)}$ . At the $(t + 1)$ -th iteration, the model undergoes improvement through the following three steps: + +- Generation step: Generate $N_{t}$ data by $f(\cdot ;\theta^{(t)})$ +- Selection step: For each synthetic data point $x$ , include it into dataset $D_{t}$ with probability $\mathcal{R}(x)$ . Suppose $n_t$ of the $N_{t}$ data are selected to form the dataset $D_{t}$ , formally, $|D_{t}| = n_{t}$ . +- Updating step: Update $\theta^{(t)}$ to $\theta^{(t + 1)}$ with the dataset $D_{t}$ . + +This iterative approach generates additional training data using synthetic samples, with the reward-guided selection process ensuring its quality. The selected samples are then used to further enhance the model's generative performance (Feng et al., 2024). This process, carried out iteratively to maximize performance gains, is particularly effective in scenarios where high-quality specialized data is scarce, for example, in mathematical reasoning (Zelikman et al., 2022; Guan et al., 2025; Singh et al., 2025). The algorithm framework is formalized in Algorithm 1. In the selection step, the data may be selected with noise. We present a simple form here for ease of understanding, while our results extend to the noisy case in Appendix B.5. + +When practitioners use this algorithm, a key challenge is determining how to set $N_{t}$ and $n_t$ across iterations. This decision is particularly important, as the generation process of foundation models usually involves multiple forward propagations, and consequently, the generation cost often exceeds the training cost. Given a limited budget, it is crucial to design the process in a way that maximizes post-training gains. + +Specifically, we focus on identifying the optimal policy of $N_{t}$ and $n_t$ correlated with the generation cost and the training cost, respectively. Given the reward model and generator, $N_{t}$ can be viewed as a random variable dependent on $n_t$ and the selection rate. Therefore, we define the policy over $n_t$ as $\pi : \{n_t\}_{t=0,1,\dots}$ , and our objective is to identify the best scheme for $\pi$ . + +Algorithm 1 Iterative learning with synthetic data +1: Input: An initial generative model $f(\cdot; \theta^{(0)})$ , a reward model $\mathcal{R}$ , and a policy $\pi: \{n_t\}_t$ with a termination time $T$ . +2: Output: A fine-tuned generative model $f(\cdot; \theta^{(T)})$ . +3: for $t \gets 0$ to $T - 1$ do +4: Initialize selected synthetic dataset $D_t = \emptyset$ . +5: while $|D_t| < n_t$ do +6: Generate a sample $x \sim \mathcal{P}_{\theta^{(t)}}$ . +7: Add $x$ to $D_t$ with probability $\mathcal{R}(x)$ . +8: end while +9: Update $\theta^{(t)}$ to $\theta^{(t + 1)}$ using the dataset $D_t$ . +10: end for + +# 3.2 Gaussian Case Study + +To begin, we present a warm-up example with Gaussian data, where the optimal policy can be derived analytically. This example provides theoretical insight into the general iterative learning framework. The setup includes Gaussian data, an exponential reward function, and maximum likelihood estimation (MLE) as the updating algorithm. + +Gaussian Generator and Exponential Reward Model. To simplify the setting, we consider the generative model as a one-dimensional Gaussian distribution $\mathcal{P}_{\theta} = \mathcal{N}(\theta ,\sigma^2)$ , where the mean $\theta$ is the parameter to learn, and the variance $\sigma^2$ is fixed. The reward model is an exponential function $\mathcal{R}(x) = \exp \left(-x^{2} / (2\kappa^{2})\right)$ , where $\kappa$ is also fixed. + +Updating via MLE. In Algorithm 1, we employ MLE as the updating algorithm at line 9. Specifically, the parameter $\theta^{(t + 1)}$ updated in the $t$ -th iteration is obtained by + +$$ +\theta^ {(t + 1)} = \arg \max _ {\theta} \prod_ {x \in D _ {t}} p _ {\theta} (x), +$$ + +where $p_{\theta}(x)$ is the density function of the distribution $\mathcal{P}_{\theta}$ . + +**Optimization Problem.** We aim to find the optimal iterative learning policy within this setup. We assume that the cost of sampling the Gaussian distribution is negligible, while the training cost for MLE is proportional to the size of the dataset, $n_t$ . The objective is to maximize the expected performance of the generator in iteration $T$ , as measured by the reward. Given a fixed budget $C$ , the optimal policy will be the solution to + +$$ +\max _ {\left\{n _ {t} \right\} _ {t = 0} ^ {T - 1}} \mathbb {E} _ {\theta^ {(T)}} \left[ r \left(\theta^ {(T)}\right) \right] \quad \text {s . t .} \quad \sum_ {t = 0} ^ {T - 1} n _ {t} \leq C. \tag {1} +$$ + +For the above setup, we can theoretically identify the optimal policy in the following theorem. + +![](images/f00bc21521eefa669c202b8efe05382a49d2d1a7316834406cafa2ad6aed6833.jpg) +Figure 2: Empirical results of the toy example. We compare the exponential, constant, and linear policy, and show the gap to the optimal expected reward as a function of the computational cost $(\sum_{t} n_{t})$ . All the results are averaged over 1,000 runs. + +Theorem 3.1. If the initial parameter satisfies $\theta^{(0)}\leq (1 + \sigma^2 /\kappa^2)^T (\sigma^2 +\kappa^2)^{1 / 2}$ , then the optimal iterative policy, i.e., the solution of the optimization problem in Equation (1), is given by + +$$ +n _ {t} \propto \left(1 + \frac {\sigma^ {2}}{\kappa^ {2}}\right) ^ {t}. +$$ + +The proof is deferred to Appendix A and extends naturally to the high-dimensional case, as detailed in Appendix E.1. Theorem 3.1 demonstrates that the optimal policy in this setting follows an exponential scheme. The optimal strategy follows this rationale: In an iterative optimization algorithm, the contribution of an early update to the overall error decays exponentially with the total number of iterations $T$ . As a result, achieving a low final error increasingly depends on the accuracy of later updates. To keep those updates precise enough, each iteration must therefore draw on an exponentially larger number of samples. We will further analyze this exponential scheme in the general analysis in Section 4. + +Experiments. We include an empirical experiment to confirm Theorem 3.1 with data in two-dimensional space. The generator is set to $\mathcal{N}(\theta, I_2)$ . The reward function is $\mathcal{R}(\pmb{x}) = \exp\left(-\|\pmb{x}\|^2/(2\kappa^2)\right)$ , which favors data concentrated near the origin. $\kappa$ controls the flatness of the reward. Full details are deferred to Appendix E.2. Figure 2 plots the gap to the optimal reward as a function of the cumulative cost, defined as the total computational cost incurred up to that point. The results demonstrate that the exponential policy consistently approaches the optimal reward faster than the other schemes. While the linear policy converges slightly more slowly, the constant policy, as shown in the left figure, clearly fails to converge. These findings validate our theoretical proof. Additional results with various parameters are left in Appendix F.1. + +# 4 Main Theoretical Results + +We now proceed to analyze the general setting. We will show that constant policies fail to converge with high probability, whereas there exists an exponential growth policy that achieves an exponential convergence rate and outperforms polynomial alternatives in the worst case. The assumptions and setup are outlined below, while detailed proofs for the subsequent theorems are provided in Appendix B.4. + +Assumptions. We consider a general function class for the generative model, a general loss with regularity assumptions (Assumptions B.1 and B.2), and a reward model with mild assumptions about its relationship with the loss (Assumption B.3). The formal descriptions are provided in Appendix B. The warm-up example in the previous section serves as a special case that satisfies all these assumptions. Our results also naturally extend to noisy rewards, as discussed in Appendix B.5. + +Updating via Gradient Descent. In Algorithm 1, we use gradient descent with a learning rate $\eta >0$ as the update algorithm at line 9. Specifically, the update for the $t$ -th iteration writes + +$$ +\theta^ {(t + 1)} = \theta^ {(t)} - \frac {\eta}{n _ {t}} \sum_ {x \in D _ {t}} \nabla_ {\theta} l (x; \theta^ {(t)}), \quad t = 0, 1, \dots , +$$ + +where $l(x; \theta)$ is the loss function. For any policy $\pi$ , let $\left\{\theta_{\pi}^{(t)}\right\}_{t}$ denote the iterative trajectory of the parameter. + +Reward and Cost. Similarly to the warm-up example, we evaluate the generative model using the expected reward $r(\theta)$ . The optimal performance is the supremum of the expected reward over the whole parameter space + +$$ +r ^ {*} = \sup _ {\theta} r (\theta). +$$ + +We consider a general computational cost that accounts for both generation and training, with constant coefficients $c_{\mathrm{g}}$ and $c_{\mathrm{t}}$ for each data point, respectively. Then, the total cost is given by + +$$ +C (\pi , T) = \sum_ {t = 0} ^ {T - 1} c _ {\mathrm {g}} N _ {t} + c _ {\mathrm {t}} n _ {t}. +$$ + +The first policy class that we consider is + +- constant policy: $\pi_{\mathrm{const}}: n_t = n_0, \quad t = 0,1,\dots$ + +This is the most standard policy, where $n_0$ can be set either as the total number of prompts in the dataset (Zelikman et al., 2022), or as a multiple of the batch size in an online setting (Guo et al., 2024). However, in the following theorem we show that constant policies fail to converge to the optimal reward with high probability. + +Theorem 4.1 (Bounded Reward for Constant policy). Under Assumption B.3, there exists a constant $c > 0$ such that, for any $T$ and any constant policy $\pi_{\mathrm{const}}$ , with probability at least $1/4$ , + +$$ +r ^ {*} - r \left(\theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right) \geq c n _ {0} ^ {- 1 / 2}. +$$ + +The theory establishes the non-convergence of constant policies, which stems from persistent random noise in the gradient. Specifically, the noise term is proportional to $n_t^{-1}$ and remains non-decaying under any constant policy, resulting in suboptimal performance. + +To address this limitation, we introduce increasing policies, where $n_t$ grows monotonically over iterations. The following theorem shows that increasing policies ensure convergence to the optimal reward with high probability. + +Theorem 4.2 (Optimal Reward for Increasing Policies). Under Assumption B.1 to Assumption B.3, for any increasing policy $\pi$ and $\varepsilon >0$ , there exists a sufficiently large $T$ such that, with probability greater than $\left(1 - \sum_{t = 0}^{T - 1}n_t^{-4}\right)$ + +$$ +r ^ {*} - r \left(\theta_ {\pi} ^ {(T)}\right) \leq \varepsilon . +$$ + +This theorem suggests that increasing policies should be used in practice to maximize performance. Hence, we consider the following two classes, namely, + +- polynomial growth policy: $\pi_{\mathrm{poly}}: n_t = n_0(1 + t)^\alpha, \quad \alpha > 0, t = 0, 1, \dots$ ; +- exponential growth policy: $\pi_{\mathrm{exp}}: n_t = n_0(1 + u)^t$ , $u > 0$ , $t = 0, 1, \dots$ . + +For the exponential policy, we can identify a specific policy that achieves an exponential convergence rate as follows. + +Table 1: A summarization of all experimental setups and implementations. + +
ExperimentSetup TypeGeneratorRewardDataset
Image DenoisingTheoretical ValidationDPMPSNRMNIST (Deng, 2012)
Math ReasoningPractical ScenarioLLMAccuracyGSM-Symbolic (Mirzadeh et al., 2025)
+ +Theorem 4.3 (Convergence Rate for the Exponential Policy). Under Assumptions B.1 to B.3, there exists an exponential growth policy $\pi_{\mathrm{exp}}^*$ such that with probability at least $\left(1 - \sum_{t=0}^{T-1} n_t^{-4}\right)$ , + +$$ +r ^ {*} - r \left(\theta_ {\pi_ {\exp} ^ {*}} ^ {(T)}\right) = \mathcal {O} \left((1 + \zeta) ^ {- 2 T}\right), +$$ + +where $\zeta > 0$ is a constant defined in Theorem B.6. + +This theorem also applies to noisy rewards, as discussed in Appendix B.5. + +Notice that the above three theorems analyze the convergence of the respective policies (constant, polynomial, and exponential) as the number of iterations $T$ grows without imposing any limit on the total computational budget. We now introduce the budget constraint and compare the exponential policy with the polynomial policy. Specifically, for any policy $\pi$ , we introduce + +$$ +T ^ {*} (\pi , \varepsilon) = \min _ {T} \left\{T \mid r ^ {*} - r \left(\theta_ {\pi} ^ {(T)}\right) \leq \varepsilon \right\}, \tag {2} +$$ + +denoting the minimum number of iterations needed for the policy $\pi$ to achieve a reward that is within $\varepsilon$ of the optimal $r^*$ . The cost $C(\pi, T^*(\pi, \varepsilon))$ is then the minimum cost to attain the target performance of the policy $\pi$ . The following theorem compares the total cost incurred by the exponential growth policy and polynomial policies in a worst-case scenario. + +Theorem 4.4 (Worst-Case Optimality of the Exponential Policy, informal). For any constant or polynomial growth policy $\pi$ and a sufficiently small $\varepsilon$ , we have + +$$ +\sup C \left(\pi_ {\exp} ^ {*}, T ^ {*} \left(\pi_ {\exp} ^ {*}, \varepsilon\right)\right) < \sup C (\pi , T ^ {*} (\pi , \varepsilon)) +$$ + +with probability at least $\left(1 - \sum_{t=0}^{T^*(\pi_{\exp}^*), \varepsilon) - 1} n_t^{-4}\right)$ . Here, the supremum is taken over all feasible problem settings, including parameter spaces, loss functions, and reward functions under mild assumptions. + +For the formal version, please refer to Theorem B.7. This theorem shows that, in a worst-case analysis, the exponential policy $\pi_{\mathrm{exp}}^{*}$ from Theorem B.6 incurs a total cost no greater than that of any polynomial growth or constant policy to reach a specified performance threshold, establishing its efficiency and robustness. This analysis theoretically highlights the exponential policy as the preferred choice for our iterative learning tasks. + +In summary, we draw the following conclusion: + +- Constant policies lead to rewards that are consistently below optimal by a constant margin. +- Increasing policies such as polynomial and exponential ones can achieve optimal performance. +- In the worst case, there exists an exponential growth policy ensuring the same performance at equal or lower computational cost compared to any constant or polynomial growth policy. + +# 5 Experiments + +Our theoretical results show that, in iterative bootstrapping, synthetic data curation with an exponential scheme outperforms both linear and polynomial schemes, while linear and polynomial schemes outperform the constant scheme. In our experiments, we evaluate constant, linear (as a simplified polynomial scheme), and exponential schemes. + +We conduct two experiments to evaluate our schemes: image denoising with DPMs, and mathematical reasoning with LLMs, covering both image and text domains. The setups and implementation details are summarized in Table 1. + +![](images/626b2f5f6107f524508523d6c240a19dbf4880a6d37d60c40c4f48a6e7241457.jpg) + +![](images/8bc825456457726ee85436241007dc37891f63ac93a16ec6ca738adf21392ac8.jpg) +Figure 3: Empirical results of image denoising. The figure shows the average PSNR of the generated denoised images as a function of computational cost. Computational cost is measured in floating-point operations (FLOPs) during iterative learning, including both generation and training. The training batch size is set to $B = 640$ . We adopt $n_t = 1.1^t \cdot B$ and $n_t = 1.05^t \cdot B$ as the exponential growth policies for $s = 10$ and $s = 20$ , respectively. + +![](images/d1d960cc62a4b953db026a1b219a27d948e491e0709fb86bb591b2fce4f6de01.jpg) + +In these settings, we emphasize that our goal is not to propose a novel task-solving method or generation technique. Rather, we focus on optimizing the allocation of generation and training costs in iterative learning with synthetic data—an objective that is method-agnostic and can be applied to any generation approach. Here, we validate our theoretical results from Section 4 and conduct a performance comparison among exponential, linear, and constant schemes on image data (using DPMs) and text data (with LLMs). + +# 5.1 Image Denoising with Diffusion Models + +We consider applying iterative learning to the image denoising task, a representative problem in image processing, where the goal is to recover $\pmb{x}$ from a noisy observation $\pmb{y} = \pmb{x} + \pmb{\eta}$ with an unknown Gaussian noise $\pmb{\eta}$ . + +Setup. We choose diffusion probabilistic models (DPMs) (Ho et al., 2020; Song et al., 2021) as denoisers due to their intrinsic design, which aligns closely with the denoising process (Xie et al., 2023) since the reverse process of a DPM gradually estimates and removes this noise using a trained network, effectively performing denoising. Specifically, we fine-tune a pre-trained diffusion model3 for denoising on the MNIST (Deng, 2012) dataset, and readers may refer to Appendix C for more details. + +We follow the iterative learning framework to annotate synthetic "clean" images to improve the denoising process. In each iteration, the diffusion model first generates synthetic denoised images $\hat{x}$ from the noisy observations. These are then filtered with a reward model linear to the Peak Signal-to-Noise Ratio (PSNR) values compared with the real images. The filtered synthetic data are then used to fine-tune the model through a standard noise-prediction training procedure, iteratively improving its denoising capability. Although an oracle reward model may not be practical in many real-world scenarios, its use here helps to validate the generality of our theoretical results. Furthermore, existing models or correction functions with prior knowledge could be utilized as the reward, as demonstrated in Gillman et al. (2024). + +**Implementations.** We perform denoising for two noise levels corresponding to the diffusion step $s = 10$ and 20, where the PSNR of noisy images are 33.25 and 28.40, respectively. A summary of the algorithm is provided in Algorithm 2. We consider three constant and three linear policies, each with small, medium, and large $n_0$ . For the exponential policy, we have performed a small hyperparameter search over $n_0$ and $u$ , selecting good configurations for two settings. The complete results of the hyperparameter search are presented as an ablation study on different exponential schemes in Appendix F.2, demonstrating its robustness. The configurations for all schemes are presented in the caption and legend of Figure 3. + +Table 2: Comparison of each policy on the image denoising and math reasoning tasks. The final accuracy are determined using a held-out validation set. We use boldface for the best result while the underline for the second best. For the exponential policies, we only exhibit a single set of configurations for the comparison. We plot for all the policies in Figure 3 and Figure 4, respectively. +(a) Image denoising + +
POLICYnts=10s=20
PRE-TRAINEDN/A40.9236.96
CONSTANT1·B42.7538.91
10·B43.1539.45
100·B44.1839.91
LINEARt·B43.9039.90
3t·B44.2340.08
10t·B44.6440.26
EXPONENTIAL—44.8440.14
+ +(b) Math reasoning + +
ntSYMBOLICP1P2
N/A55.6034.9815.72
10·B60.2838.5016.93
30·B60.1639.6817.73
100·B64.0441.6618.81
3t·B62.5438.8817.01
10t·B61.9639.9417.29
30t·B64.2440.8419.17
—65.6647.2620.65
+ +Results. We summarize the final performance of different schemes and baselines in Table 2 and show the trade-off between performance and computational cost in Figure 3. We observe that the exponential policy yields the best performance for $s = 10$ , and performs comparable to the best linear policy for $s = 20$ . This observation demonstrates both the benefits of an increasing policy (Theorem 4.2) and the robustness of the exponential growth policy (Theorem 4.4). In contrast, the constant policy only matches the exponential policy in the early stages of training, but soon reaches a performance plateau and eventually decays due to overfitting, corroborating Theorem 4.1. + +# 5.2 Math Reasoning with LLMs + +Now we move to natural language processing. Iterative learning with synthetic data is also an emerging method for LLMs. Previous research (Zelikman et al., 2022) leverages the model to generate synthetic data, which is then filtered using a reward model or accuracy metrics and subsequently utilized to fine-tune the model. This framework is employed in various settings, including instruction tuning (Touvron et al., 2023), alignment (Dong et al., 2023), and recent efforts to enhance reasoning capabilities through chain-of-thought (CoT) approaches (Hosseini et al., 2024; Lehnert et al., 2024; Singh et al., 2025). + +Setup. We select mathematical reasoning to evaluate the effectiveness of our strategy in iterative learning. Let $q$ and $a$ represent the question and its corresponding answer, respectively. We use the LLM to generate synthetic solutions with CoT and in-context examples. These synthetic data are filtered according to the correctness of the final generated answer $\hat{a}$ , defined as the reward $\mathcal{R}(\hat{a}) = 1(\hat{a} = a)$ . The filtered data are used to further train the model in an iterative process, without filtering based on the correctness of CoT itself. In summary, synthetic data with the correct final solutions are generated and used for training in each iteration, closely mimicking the STaR method (Zelikman et al., 2022). + +**Implementations.** Many contemporary LLMs are known to overfit to GSM8k (Cobbe et al., 2021) or exhibit a certain degree of data leakage (Zhang et al., 2024; Mirzadeh et al., 2025), where even minor changes to numbers or names can result in significant performance degradation. This issue renders the evaluation results on GSM8k unreliable. To address this, we regenerate three math datasets—GSM_SYMBOLic, GSM_p1, and GSM_p2—using symbolic math templates from Mirzadeh et al. (2025), one of the most outstanding papers on evaluating math reasoning. Each dataset consists of questions paired with their correct answers. GSM_SYMBOLic retains a similar difficulty level to GSM8k, while GSM_p1 and GSM_p2 introduce one and two additional clauses per question, respectively, making them progressively more challenging. Together, these datasets complete a robust evaluation set. + +In our iterative learning framework, we generate synthetic data using a mixing ratio of 7:2:1 across these three datasets, filter the data based on the correctness of the final answers, and fine-tune the model on the selected data. We use the Llama-3-8B-Base model (Dubey et al., 2024) with full + +
constant, nt=10·Blinear, nt=3t·B
constant, nt=30·Blinear, nt=10t·B
constant, nt=100·Blinear, nt=30t·B
+ +![](images/93e973625fe362edff45a69bb709712b5570a6c886cac2f351ae2a4a86ccbd53.jpg) +Figure 4: Empirical results of math reasoning. The figure shows the accuracies with respect to the computational cost, measured in FLOPs as well. The training batch size is set to $B = 256$ . We adopt $n_t = 10 \cdot 2^t \cdot B$ as the exponential policy. We set the temperature to 0.3 during the generation step and include the results for a temperature of 0.7 in Appendix F.4, which lead to the same conclusions. + +![](images/c53ea3e7e9f08a6bc15d771899754e9a4ec2aa06d7c524e7c5a32ec3c8189646.jpg) + +![](images/278a894e30903efd4a4cac28fd9bbd02abf965140d1748d847881b191b2889eb.jpg) + +weight fine-tuning. For evaluation, the model is tested on held-out datasets from all three difficulty levels. During data generation and evaluation, in-context examples and CoT reasoning are used, but in-context examples are excluded during fine-tuning. + +A summary of the algorithm is provided in Algorithm 3. Similarly to image denoising, we consider three constant and three linear policies, each with small, medium, and large $n_0$ . For the exponential scheme, we have selected one configuration and left the full ablation results in Appendix F.3. The configurations for all schemes are detailed in Figure 4 and its caption. We also include the performance of the pre-trained model as a baseline. + +Results. Figure 4 shows the performance versus computational cost across the three datasets under iterative training, while Table 2 summarizes the final performance for all policies. Across all three difficulty levels, the exponential growth policy demonstrates the steadiest improvement compared to other schemes. Although the linear policy ranks second overall, it fails to match the exponential growth policy when the training budget is large. In particular, with the exponential growth policy, iterative learning increases accuracy on GSM_p1 from $35\%$ to $47\%$ and on GSM_p2 from $16\%$ to $21\%$ . Based on these results, we recommend the exponential scheme for iterative learning. + +# 6 Conclusion + +In this work, we take the first step toward understanding how to design optimal budget allocation policies in iterative bootstrapping for supervised fine-tuning. Through a combination of theoretical analysis and empirical validations, we demonstrate that constant policies are insufficient for achieving convergence, whereas increasing policies offer a more effective alternative. Among increasing policies, the exponential growth policy emerges as a robust and efficient method. + +Although our focus here is on SFT in an iterative learning framework, recent research has highlighted the potential of Reinforcement Learning with Human Feedback (RLHF) (Yuan et al., 2024; Pang et al., 2024; Setlur et al., 2024) for similar iterative setups. Extending our framework to incorporate RLHF is a promising direction for future work. By answering these open questions, we aim to lay a foundation for more efficient iterative training approaches that can improve the post-training of foundation models. + +# References + +Alemohammad, S., Casco-Rodriguez, J., Luzi, L., Humayun, A. I., Babaei, H., LeJeune, D., Siahkoohi, A., and Baraniuk, R. Self-consuming generative models go MAD. In The Twelfth International Conference on Learning Representations, 2024. +Azizi, S., Kornblith, S., Sahara, C., Norouzi, M., and Fleet, D. J. Synthetic data from diffusion models improves imagenet classification. Transactions on Machine Learning Research, 2023. +Cobbe, K., Kosaraju, V., Bavarian, M., Chen, M., Jun, H., Kaiser, L., Plappert, M., Tworek, J., Hilton, J., Nakano, R., et al. Training verifiers to solve math word problems. arXiv preprint arXiv:2110.14168, 2021. +Contributors, O. OpenCompass: A universal evaluation platform for foundation models. https://github.com/open-compass/opencompass, 2023. +Deng, L. 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Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: We only leverage LLMs for writing and editing on sentence level. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Theoretic Intuition + +# A.1 Discussion on the Gaussian Setting + +In the Gaussian case, we use a simplified toy example that omits the generation cost only to provide a clear, introductory setup for the reader. In our complete framework (see Section 4), however, we assume a generation cost for standard estimators and incorporate this cost into the objective using the hyperparameter $c_{g}$ . This hyperparameter offers flexibility: setting $c_{g} = 0$ corresponds to free generation, while a large $c_{g}$ reflects expensive generation. We also provide general theorems that hold for all values of $c_{g}$ , and explain how the Gaussian case satisfies all assumptions of the general theorems in Appendix B.2. Thus, the Gaussian case can be viewed as a special instance, and the theoretical results still remain valid regardless of whether the sampling is free or costly. + +# A.2 Proof of Theorem 3.1 + +We first calculate the distribution of $\theta^{(T)}$ given an initial parameter $\theta^{(0)}$ and a specific policy $\pi : \{n_t\}_t$ with the terminal time $T$ in the following lemma. + +Lemma A.1. With the MLE updation and Gaussian distributions, given an initial parameter $\theta^{(0)}$ and a specific policy $\pi : \{n_t\}_t$ with the terminal time $T$ , we have + +$$ +p (\theta^ {(T)} \mid \theta^ {(0)}, \pi) = \mathcal {N} \left(\frac {\theta^ {(0)}}{(1 + \sigma^ {2} / \kappa^ {2}) ^ {T}}, \sigma^ {2} \sum_ {t = 0} ^ {T - 1} \frac {1}{n _ {t} (1 + \sigma^ {2} / \kappa^ {2}) ^ {2 (T - t) - 1}}\right). +$$ + +Proof. Suppose that there are $N_{t}$ data $\{x_{i}\}_{i = 1}^{N_{t}}$ generated from the $t$ -th step generator such that the selected sample size is $n_t$ . Let $b_{i}\sim \mathrm{Bernoulli}(\mathcal{R}(x_i))$ as a selecting sign where $b_{i} = 1$ indicates that $x_{i}$ is selected and placed into $D_{t}$ , otherwise $b_{i} = 0$ indicates that $x_{i}\notin D_{t}$ . + +We first consider one-step updating. We have the distribution of a data $x_{i}$ conditioned on $x_{i} \in D_{t}$ , + +$$ +\begin{array}{l} p \left(x _ {i} \mid x _ {i} \in D _ {t}\right) = p \left(x _ {i} \mid b _ {i} = 1\right) \\ \propto \mathcal {R} (x _ {i}) \cdot \mathcal {N} \left(x _ {i}; \theta^ {(t)}, \sigma^ {2}\right) \\ \propto \exp \left(- \frac {x _ {i} ^ {2}}{2 \kappa^ {2}} - \frac {\left(x _ {i} - \theta^ {(t)}\right) ^ {2}}{2 \sigma^ {2}}\right) \\ \sim \mathcal {N} \left(\frac {\theta^ {(t)}}{1 + \sigma^ {2} / \kappa^ {2}}, \frac {\sigma^ {2}}{1 + \sigma^ {2} / \kappa^ {2}}\right). \\ \end{array} +$$ + +Since $\theta^{(t + 1)}$ is the solution of the MLE satisfying + +$$ +\theta^ {(t + 1)} = \arg \max _ {\theta} \prod_ {x \in D _ {t}} \mathcal {N} (x; \theta , \sigma^ {2}) = \frac {1}{n} \sum_ {x \in D _ {t}} x _ {i}, +$$ + +we have the distribution of $\theta^{(t + 1)}$ conditioned on the parameter $\theta^{(t)}$ and the sample size $n_t$ of the previous iteration, written as + +$$ +\theta^ {(t + 1)} \mid \theta^ {(t)}, n _ {t} \sim \mathcal {N} \left(\frac {\theta^ {(t)}}{1 + \sigma^ {2} / \kappa^ {2}}, \frac {\sigma^ {2}}{n _ {t} (1 + \sigma^ {2} / \kappa^ {2})}\right). +$$ + +Considering the decomposition + +$$ +\theta^ {(t + 1)} = \frac {\theta^ {(t)}}{1 + \sigma^ {2} / \kappa^ {2}} + \frac {\sigma}{n _ {t} ^ {1 / 2} (1 + \sigma^ {2} / \kappa^ {2}) ^ {1 / 2}} \xi_ {t}, +$$ + +with an independent auxiliary random $\xi_t \sim \mathcal{N}(0,1)$ recursively, we have + +$$ +\theta^ {(T)} = \frac {\theta^ {(0)}}{(1 + \sigma^ {2} / \kappa^ {2}) ^ {T}} + \sum_ {t = 0} ^ {T - 1} \frac {\sigma}{n _ {t} ^ {1 / 2} (1 + \sigma^ {2} / \kappa^ {2}) ^ {1 / 2 + (T - 1 - k)}} \xi_ {t}. +$$ + +Therefore, + +$$ +\theta^ {(T)} \mid \theta^ {(0)}, \pi \sim \mathcal {N} \left(\frac {\theta^ {(0)}}{(1 + \sigma^ {2} / \kappa^ {2}) ^ {T}}, \sum_ {t = 0} ^ {T - 1} \frac {\sigma^ {2}}{n _ {t} (1 + \sigma^ {2} / \kappa^ {2}) ^ {2 (T - t) - 1}}\right). +$$ + +![](images/0d907aa2d9f81af05ae6765f00bc1aa16b1a0a4964b28e58f3fe76323f2cca8b.jpg) + +Back to the proof of the theorem, we define the mean and variance as + +$$ +\mu_ {T} = \frac {\theta^ {(0)}}{\left(1 + \sigma^ {2} / \kappa^ {2}\right) ^ {T}}, +$$ + +and + +$$ +\sigma_ {T} ^ {2} = \sigma^ {2} \sum_ {t = 0} ^ {T - 1} \frac {1}{n _ {t} (1 + \sigma^ {2} / \kappa^ {2}) ^ {2 (T - t) - 1}}. +$$ + +According to Lemma A.1, we have that + +$$ +\begin{array}{l} \mathbb {E} _ {\theta^ {(T)}} \mathbb {E} _ {x} [ \mathcal {R} (x) ] = \mathbb {E} _ {\theta^ {(T)}} \mathbb {E} _ {x} \left[ \exp \left(- \frac {x ^ {2}}{2 \kappa^ {2}}\right) \right] \\ = \mathbb {E} _ {\theta (T)} \left[ \int \frac {1}{\sqrt {2 \pi \sigma^ {2}}} \exp \left(- \frac {x ^ {2}}{2 \kappa^ {2}} - \frac {(x - \theta^ {(T)}) ^ {2}}{2 \sigma^ {2}}\right) \mathrm {d} x \right] \\ = \frac {1}{\sqrt {1 + \sigma^ {2} / \kappa^ {2}}} \mathbb {E} _ {\theta^ {(T)}} \left[ \exp \left(- \frac {(\theta^ {(T)}) ^ {2}}{2 (\sigma^ {2} + \kappa^ {2})}\right) \right] \\ = \frac {1}{\sqrt {1 + \sigma^ {2} / \kappa^ {2}}} \frac {1}{\sqrt {1 + \sigma_ {T} ^ {2} / (\sigma^ {2} + \kappa^ {2})}} \exp \left(- \frac {\mu_ {T} ^ {2}}{2 (\sigma^ {2} + \kappa^ {2} + \sigma_ {T} ^ {2})}\right) \\ = \frac {\kappa}{\sqrt {\sigma^ {2} + \kappa^ {2} + \sigma_ {T} ^ {2}}} \exp \left(- \frac {\mu_ {T} ^ {2}}{2 (\sigma^ {2} + \kappa^ {2} + \sigma_ {T} ^ {2})}\right). \\ \end{array} +$$ + +With the assumption that $\mu_T \leq \sqrt{\sigma^2 + \kappa^2}$ , the right term of the above equation is an increasing function with respect to $\frac{1}{\sqrt{\sigma^2 + \kappa^2 + \sigma_T^2}}$ . Therefore, the objective function gets its maximum value when we minimize $\sigma_T^2$ . Furthermore, we have the optimal iterative learning policy from Cauchy-Schwarz inequality, + +$$ +\begin{array}{l} \sigma_ {T} ^ {2} \cdot \sum_ {t = 0} ^ {T - 1} n _ {t} = \sigma^ {2} \left(\sum_ {t = 0} ^ {T - 1} \frac {1}{n _ {t} (1 + \sigma^ {2} / \kappa^ {2}) ^ {2 (T - t) - 1}}\right) \cdot \left(\sum_ {t = 0} ^ {T - 1} n _ {t}\right) \\ \geq \sigma^ {2} (1 + \sigma^ {2} / \kappa^ {2}) \left(\sum_ {t = 0} ^ {T - 1} \frac {1}{(1 + \sigma^ {2} / \kappa^ {2}) ^ {T - t}}\right) ^ {2}, \\ \end{array} +$$ + +where the equal sign is established if and only if + +$$ +n _ {t} \propto \left(1 + \frac {\sigma^ {2}}{\kappa^ {2}}\right) ^ {t - T}. +$$ + +Reindexing or shifting $t$ will complete the proof. + +# B Main Theoretic Results + +# B.1 Notations + +Consider a probability space $S = (\Omega, \mathcal{B}(\Omega), \mathcal{P})$ , where $\Omega$ is the sample space, $\mathcal{B}(\Omega)$ is the Borel $\sigma$ -algebra on $\Omega$ , and $\mathcal{P}$ is the probability measure. Let $\Theta \subseteq \mathbb{R}^p$ be a parameter space, and define + +$$ +\mathcal {A} _ {\theta} = \{f (\omega ; \theta) \mid \theta \in \Theta \} +$$ + +as a parametric family of random variables mapping from $\Omega$ to a measurable outcome space $\mathcal{T}$ . Each $\theta$ induces a distribution $\mathcal{P}_{\theta}$ on $\mathcal{T}$ , where + +$$ +\mathcal {P} _ {\theta} (X) = \mathcal {P} \left(f (\omega ; \theta) \in X\right) \quad \text {f o r a n y m e a s u r a b l e s e t} X \subseteq \mathcal {T}. +$$ + +Any function $\mathcal{R}:\mathcal{T}\to [0,1]$ induces an additional distribution $\mathcal{P}_{r;\theta}$ on $[0,1]$ , where + +$$ +\mathcal {P} _ {\mathcal {R}; \theta} (\mathcal {B}) = \mathcal {P} _ {\theta} \left(\mathcal {R} (x) \in \mathcal {B}\right) \quad \text {f o r a n y B o r e l s e t} \mathcal {B} \subseteq [ 0, 1 ]. +$$ + +Given a reward model $\mathcal{R}(x)$ , the reward $r(\theta)$ for $\theta \in \Theta$ is defined as the expectation of the distribution $\mathcal{P}_{\mathcal{R};\theta}$ , i.e., + +$$ +r (\theta) := \int_ {[ 0, 1 ]} r \mathrm {d} \mathcal {P} _ {\mathcal {R}; \theta} (r) = \mathbb {E} _ {x \sim \mathcal {P} (\theta)} [ \mathcal {R} (x) ]. +$$ + +Define the loss function $l(x; \theta) : \mathcal{T} \times \Theta \to [0, \infty)$ . Now we consider the optimizing algorithm that starts with an initial parameter $\theta^{(0)}$ and updates the parameter $\theta^{(t)}$ at the $t$ -th iteration using gradient descent with a learning rate $\eta > 0$ and data $D_t \in \mathcal{T}^{\otimes |D_t|}$ , formulated as + +$$ +\theta^ {(t + 1)} = \theta^ {(t)} - \frac {\eta}{n _ {t}} \sum_ {x \in D _ {t}} \nabla_ {\theta} l (x; \theta^ {(t)}), \quad t = 0, 1, \dots . +$$ + +# B.2 Assumptions + +We always consider the general case where the initial value $\theta^{(0)}$ is sufficiently accurate. The distribution $\mathcal{P}_{\theta^{(0)}}$ is assumed as non-degenerate and there exists a constant $c_{r}$ such that + +$$ +\mathrm {d} \mathcal {P} _ {r; \theta^ {(0)}} \geq c _ {r} > 0. +$$ + +Assumption B.1 (Basic properties of the optimization space and loss function). The parameter space $\Theta$ for $\theta$ and the loss function $l(x;\theta)$ indexed by $\theta \in \Theta$ satisfy the following requirements. + +1. The covering number $\mathcal{N}(\epsilon, \Theta, \| \cdot \|_2)$ for the parameter space $\Theta$ is bounded, which means the inequality + +$$ +\mathcal {N} (\epsilon , \Theta , \| \cdot \| _ {2}) \leq c _ {\Theta} \varepsilon^ {- C _ {\Theta}} \vee 1, \forall \epsilon > 0 \tag {3} +$$ + +holds for some positive constants $c_{\Theta}$ and $C_{\Theta}$ . + +2. The loss function $l(x; \theta)$ is Lipschitz continuous. Formally, there exists a constant $L > 0$ independent of $x$ such that + +$$ +\left| l (x; \theta) - l (x; \theta_ {1}) \right| \leq C _ {L} \| \theta - \theta_ {1} \| _ {2}, \forall \theta , \theta_ {1} \in \Theta . \tag {4} +$$ + +3. For any $\theta \in \Theta$ , there exist positive constants $c$ and $\beta$ satisfying + +$$ +\left\| \nabla_ {\theta} l (x; \theta) \right\| _ {2} ^ {2} \geq c l (x; \theta), \quad \left\| \nabla_ {\theta \theta^ {\prime}} ^ {2} l (x; \theta) \right\| _ {\mathrm {o p}} \leq \beta . \tag {5} +$$ + +Remark. This assumption governs both the complexity of the parameter space $\Theta$ and the smoothness of the likelihood function $l(x;\theta)$ . Here, we provide two examples to show that these assumptions are reasonable. Firstly, if $\Theta$ is a compact finite-dimensional space or a functional space sufficiently smooth, then the coverage number requirement (Equation (3)) is satisfied. Secondly, for a special case when $l(x;\theta)$ is the negative log-likelihood for a Gaussian distribution, the derivative condition (Equation (5)) holds with $c = 2$ and $\beta = 1$ . + +We define the post-selection distribution $\mathcal{Q}_{\theta}$ via its Radon-Nikodym derivative relative to $\mathcal{P}_{\theta}$ as + +$$ +\frac {\mathrm {d} \mathcal {Q} _ {\theta} (x)}{\mathrm {d} \mathcal {P} _ {\theta} (x)} = \frac {\mathcal {R} (x)}{\mathbb {E} _ {x \sim \mathcal {P} _ {\theta}} [ \mathcal {R} (x) ]}, +$$ + +which indicates that $\mathcal{Q}_{\theta}$ is just the original distribution $\mathcal{P}_{\theta}$ weighted by the reward function $\mathcal{R}(x)$ . Equivalently, we can think of $\mathcal{Q}_{\theta}$ as the conditional distribution of $x$ given that $x$ is "selected" in proportion to its reward. For convenience, let $p_{\theta}, q_{\theta}$ , and $p_{r;\theta}$ be the density functions (with respect to some appropriate base measure) of $\mathcal{P}_{\theta}, \mathcal{Q}_{\theta}$ , and $\mathcal{P}_{r;\theta}$ , respectively. + +Next, we define the expected likelihood function $\mathbb{P}l_{\theta_1}(\theta_2)$ as + +$$ +\mathbb {P} l _ {\theta_ {1}} (\theta_ {2}) = \mathbb {E} _ {x \sim \mathcal {Q} _ {\theta_ {1}}} \big [ l (x; \theta_ {2}) \big ] = \mathbb {E} _ {x \sim q _ {\theta_ {1}}} \big [ - \log p _ {\theta_ {2}} (x) \big ]. +$$ + +Intuitively, $\mathbb{P}l_{\theta_1}(\theta_2)$ measures how well the model $p_{\theta_2}$ "fits" data drawn from the post-selection distribution $\mathcal{Q}_{\theta_1}$ . + +We define + +$$ +\Theta^{*} = \arg \min_{\theta \in \Theta}\mathbb{P}l_{\theta}(\theta) +$$ + +as the optimal parameter set that minimize all possible $\mathbb{P}l_{\theta}(\theta)$ + +Finally, to introduce the sampling error, we define + +$$ +\mathbb {P} _ {n} l (X; \theta) = \frac {1}{n} \sum_ {i = 1} ^ {n} l \left(x _ {i}; \theta\right) +$$ + +as an empirical evaluation for the negative log-likelihood based on samples $X = (x_{1},\ldots ,x_{n})$ + +Assumption B.2 (Local reducibility of the expected loss function). There exists a constant $\gamma \in (0,1]$ such that for any $\theta ,\theta_1\in \Theta$ with $\| \theta -\theta_1\| _2\leq 2C_L / \beta$ and $\mathbb{P}l_{\theta_1}(\theta)\leq \mathbb{P}l_{\theta_1}(\theta_1)$ , we have + +$$ +\mathbb {P} l _ {\theta} (\theta) - \mathbb {P} l _ {\theta_ {1}} (\theta_ {1}) \leq \gamma \left(\mathbb {P} l _ {\theta_ {1}} (\theta) - \mathbb {P} l _ {\theta_ {1}} (\theta_ {1})\right). +$$ + +Remark. This assumption ensures that, within a local neighborhood of $\theta$ , any parameter update that reduces the expected likelihood under $\theta_{1}$ 's distribution also decreases the expected likelihood under $\theta$ ’s distribution, relative to $\mathbb{P}l_{\theta_1}(\theta_1)$ . Concretely, each iteration of the algorithm samples from $\theta_{1}$ but then updates the parameter $\theta$ . Since the data remain the same while $\theta$ changes, this condition guarantees that a reduction in the loss is preserved across iterations. For instance, if $\mathcal{P}_{\theta}$ is a Gaussian distribution with mean $\theta$ and standard deviation 1, and $\mathcal{R}(x)$ is the normalized standard Gaussian density, one can verify that Assumption B.2 holds with $\gamma = 1/2$ . + +Assumption B.3 (Relationship between the loss and the reward function). For any $\theta \in \Theta$ , it holds for some $c_{2}, c_{3} \leq 1$ that + +$$ +c _ {3} \mathbb {P} l _ {\theta} (\theta) \leq r ^ {*} - r (\theta) \leq c _ {2} \mathbb {P} l _ {\theta} (\theta). +$$ + +Remark. This assumption establishes a direct relationship between the performance gap $(r^{*} - r(\theta))$ and the expected loss $\mathbb{P}l_{\theta}(\theta)$ . Specifically, the upper bound shows that if $\mathbb{P}l_{\theta}(\theta)$ is small, then $(r^{*} - r(\theta))$ must also be small, which is crucial for proving convergence of our proposed policy. For example, in the Gaussian setting discussed in the remark of Assumption B.2, one can verify that $c_{2} = 1 / 12$ . Conversely, the lower bound indicates that a small performance gap necessarily implies a correspondingly small expected loss, thereby capturing the sensitivity of the loss to improvements in the expected reward. Since both $(r^{*} - r(\theta))$ and $\mathbb{P}l_{\theta}(\theta)$ quantify how well the policy performs, it is natural that each can bound the other. This relationship serves as a key underpinning for the theoretical results presented in subsequent sections. + +# B.3 Main Theorems + +Our theoretical results start from the defect of constant policies by showing that the final reward remains bounded away from the optimum with non-negligible probability when using constant policies. + +Theorem B.4 (Bounded Reward for Constant policy). Under Assumption B.3, there exists a constant $c > 0$ such that, for any $T$ and any constant policy $\pi_{\mathrm{const}}$ , with probability at least $1/4$ , + +$$ +r ^ {*} - r \left(\theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right) \geq c n _ {0} ^ {- 1 / 2}. +$$ + +On the contrary, any increasing policy eventually closes the optimization gap within arbitrarily small range as long as the training iteration goes to infinity. + +Theorem B.5 (Optimal Reward for Increasing Policies). Under Assumptions B.1 to B.3, for any increasing policy $\pi$ and $\varepsilon > 0$ , there exists a sufficiently large $T$ such that, with probability greater than $\left(1 - \sum_{t=0}^{T-1} n_t^{-4}\right)$ , + +$$ +r ^ {*} - r \left(\theta_ {\pi} ^ {(T)}\right) \leq \varepsilon . +$$ + +Next, we demonstrate that the exponential-growth policy from Theorem B.6 achieves this convergence more effectively than polynomial growth policies, including the constant policies. + +Theorem B.6 (Convergence Rate for Exponential Policy). Under Assumptions B.1 to B.3, there exists an exponential growth policy $\pi_{\mathrm{exp}}^{*}$ such that with probability at least $\left(1 - \sum_{t=0}^{T-1} n_t^{-4}\right)$ , + +$$ +r ^ {*} - r \left(\theta_ {\pi_ {\mathrm {e x p}} ^ {*}} ^ {(T)}\right) = \mathcal {O} \left((1 + \zeta) ^ {- 2 T}\right) +$$ + +with a constant $\zeta = \gamma c / (8\beta -\gamma c) > 0$ + +Consider the minimum number of iterations $T^{*}(\pi, \varepsilon)$ needed for a policy $\pi$ to achieve a reward that is within $\varepsilon$ of the optimal $r^{*}$ as defined in Equation (2), then the cost $C(\pi, T^{*}(\pi, \varepsilon))$ is the minimum cost to attain the target performance for policy $\pi$ . The following theorem compares the total cost incurred by exponential growth policies and polynomial growth policies (including constant policies) in a worst-case scenario. + +Theorem B.7 (Worst-Case Optimality of Exponential Policy). Under the same conditions as in Theorem B.6, for any constant or polynomial growth policy $\pi$ and a sufficiently small $\varepsilon$ , we have + +$$ +\sup _ {\Theta , \Theta^ {*}, l (x; \theta), \mathcal {R} (x)} C \left(\pi_ {\exp} ^ {*}, T ^ {*} \left(\pi_ {\exp} ^ {*}, \varepsilon\right)\right) \leq \sup _ {\Theta , \Theta^ {*}, l (x; \theta), \mathcal {R} (x)} C \left(\pi , T ^ {*} (\pi , \varepsilon)\right) +$$ + +with probability at least $\left(1 - \sum_{t=0}^{T^*(\pi_{\mathrm{exp}}, \varepsilon)-1} n_t^{-4}\right)$ . The supremum is taken over all possible parameter spaces $\Theta, \Theta^*$ , loss functions $l(x; \theta)$ , and reward models $\mathcal{R}(x)$ satisfying the same conditions as in Theorem B.6. + +Such a worst-case perspective shows that the exponential growth policy remains robust and outperforms (or at least matches) any constant or polynomial policy. (Although the policy $\pi$ is fixed in advance, the iterative sequence $\left\{\theta_{\pi}^{(t)}\right\}_{t = 0}^{T - 1}$ is still random because data are sampled and selected probabilistically at each iteration. The stated probability bound reflects this inherent randomness and guarantees high-probability performance under the specified assumptions. + +# B.4 Proofs + +In this subsection, we present proofs for all statements in our theoretical results. We begin with the proof of Theorem B.6, proceed to prove Theorems B.4 and B.5, and conclude with the proof of Theorem B.7. + +Proof of Theorem B.6. + +We begin by noting the following decomposition + +$$ +\begin{array}{l} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(1)}\right) - \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) = \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(1)}\right) - \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(1)}\right) \\ + \mathbb {P} _ {n _ {0}} l (X; \theta^ {(1)}) - \mathbb {P} _ {n _ {0}} l (X; \theta^ {(0)}) \\ + \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(0)}\right) - \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right), \\ \end{array} +$$ + +which gives + +$$ +\begin{array}{l} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(1)}\right) - \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \leq \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(1)}\right) - \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(0)}\right) \tag {6} \\ + \left| \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(1)}\right) - \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(1)}\right) \right| + \left| \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) - \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(0)}\right) \right|. \\ \end{array} +$$ + +By Assumption B.1 and a Taylor expansion, choosing the step size $\eta = 1 / (2\beta)$ gives + +$$ +\begin{array}{l} \mathbb {P} _ {n _ {0}} l (X; \theta^ {(1)}) - \mathbb {P} _ {n _ {0}} l (X; \theta^ {(0)}) \leq (\nabla_ {\theta} \mathbb {P} _ {n _ {0}} l (X; \theta^ {(0)})) ^ {\top} (\theta^ {(1)} - \theta^ {(0)}) + \beta \| \theta^ {(1)} - \theta^ {(0)} \| _ {2} ^ {2} \\ \leq - \eta \left(\nabla_ {\theta} \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(0)}\right)\right) ^ {\top} \nabla_ {\theta} \mathbb {P} _ {n _ {0}} l \left(X; \theta^ {(0)}\right) \\ + \beta \eta^ {2} \left\| \nabla_ {\theta} \mathbb {P} _ {n _ {0}} l (X; \theta^ {(0)}) \right\| _ {2} ^ {2} \\ \leq - \frac {1}{4 \beta} \left\| \nabla_ {\theta} \mathbb {P} _ {n _ {0}} l (X; \theta^ {(0)}) \right\| _ {2} ^ {2} \\ \leq - \frac {c}{4 \beta} \mathbb {P} _ {n _ {0}} l (X; \theta^ {(0)}). \tag {7} \\ \end{array} +$$ + +Define + +$$ +\mathcal {R} _ {\theta} = \mathbb {E} _ {\varepsilon} \left[ \sup _ {\theta \in \Theta} \left| \frac {1}{n} \sum_ {i = 1} ^ {n} \varepsilon_ {i} l \left(x _ {i}; \theta\right) \right| \right], +$$ + +where $\varepsilon_{i}$ $(1\leq i\leq n)$ are i.i.d. Rademacher random variables. By Dudley's entropy integral and Assumption B.1, we then get + +$$ +\begin{array}{l} \sup _ {\theta \in \Theta} | \mathbb {P} l _ {\theta^ {(0)}} (\theta) - \mathbb {P} _ {n} l (X; \theta) | \leq 2 \mathcal {R} _ {\theta} \\ \leq 2 L n ^ {- 1 / 2} \int_ {0} ^ {\infty} \sqrt {\log \mathcal {N} (\epsilon , \Theta , \| \cdot \| _ {2})} d \epsilon \tag {8} \\ \leq 2 L C _ {\Theta} \log c _ {\Theta} n ^ {- 1 / 2}. \\ \end{array} +$$ + +By Chernoff-Hoeffding theorem, + +$$ +P \left(\left| n _ {0} - N _ {0} r \left(\theta^ {(0)}\right) \right| \geq t _ {1}\right) \leq \exp \left(\frac {- 2 t _ {1} ^ {2}}{N _ {0} r \left(\theta^ {(0)}\right)}\right). +$$ + +Combining Equations (7) and (8) with Equation (6) and choosing + +$$ +t _ {1} = 2 \left(N _ {0} r \left(\theta^ {(0)}\right)\right) ^ {1 / 2} \log N _ {0}, +$$ + +we see that, with probability at least $\left(1 - N_0^{-4}\right)$ + +$$ +\begin{array}{l} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(1)}\right) - \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \\ \leq - \frac {c}{4 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) + \left(2 - \frac {c}{4 \beta}\right) \sup _ {\theta \in \Theta} | \mathbb {P} l _ {\theta^ {(0)}} (\theta) - \mathbb {P} _ {n _ {0}} l (X; \theta) | \\ \leq - \frac {c}{4 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) + \left(2 - \frac {c}{4 \beta}\right) 2 L C _ {\Theta} \log c _ {\Theta} n _ {0} ^ {- 1 / 2} \\ \leq - \frac {c}{4 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) + \left(2 - \frac {c}{4 \beta}\right) 2 L C _ {\Theta} \log c _ {\Theta} \left(N _ {0} r \left(\theta^ {(0)}\right) - 2 \left(N _ {0} r \left(\theta^ {(0)}\right)\right) ^ {1 / 2} \log N _ {0}\right) ^ {- 1 / 2}. \\ \end{array} +$$ + +We choose $N_0$ to satisfy + +$$ +\left(2 - \frac {c}{4 \beta}\right) 2 L C _ {\Theta} \log c _ {\Theta} \left(N _ {0} r \left(\theta^ {(0)}\right) - 2 \left(N _ {0} r \left(\theta^ {(0)}\right)\right) ^ {1 / 2} \log N _ {0}\right) ^ {- 1 / 2} \leq \frac {c}{8 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right). +$$ + +By Assumption B.2, we have + +$$ +\mathbb {P} l _ {\theta^ {(1)}} \left(\theta^ {(1)}\right) - \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \leq \gamma \left(\mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(1)}\right) - \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right)\right) \leq - \frac {\gamma c}{8 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right), +$$ + +which implies + +$$ +\mathbb {P} l _ {\theta^ {(1)}} \left(\theta^ {(1)}\right) \leq \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {1}. +$$ + +Define $\tilde{\mathbb{P}} l_{\theta} = \mathbb{P}l_{\theta}(\theta)$ . This can be rewritten as + +$$ +\tilde {\mathbb {P}} l _ {\theta^ {(1)}} - \tilde {\mathbb {P}} l _ {\theta^ {(0)}} \leq - \frac {\gamma c}{8 \beta} \tilde {\mathbb {P}} l _ {\theta^ {(0)}}, +$$ + +and by Assumption B.3 we have the inequality + +$$ +r \left(\theta^ {(1)}\right) \geq r ^ {*} - c _ {2} \tilde {\mathbb {P}} l _ {\theta^ {(1)}} \geq r ^ {*} - c _ {2} \tilde {\mathbb {P}} l _ {\theta^ {(0)}} \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {1}. +$$ + +By induction, we see that + +$$ +\mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t)}\right) \leq \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {t}. +$$ + +For each $t$ , with probability at least $1 - N_t^{-4}$ + +$$ +\begin{array}{l} \mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t + 1)}\right) - \mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t)}\right) \\ \leq - \frac {c}{4 \beta} \mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t)}\right) + \left(2 - \frac {c}{4 \beta}\right) \sup _ {\theta \in \Theta} | \mathbb {P} l _ {\theta^ {(t)}} (\theta) - \mathbb {P} _ {n _ {t}} l (X; \theta) | \\ \leq - \frac {c}{4 \beta} \mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t)}\right) + \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} n _ {t} ^ {- 1 / 2} \\ \leq - \frac {c}{4 \beta} \mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t)}\right) + \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} \left(N _ {t} r \left(\theta^ {(t)}\right) - 2 \left(N _ {t} r \left(\theta^ {(t)}\right)\right) ^ {1 / 2} \log N _ {t}\right) ^ {- 1 / 2}. \\ \end{array} +$$ + +We now choose $N_{t}$ so that + +$$ +\left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} \left(N _ {t} r \left(\theta^ {(t)}\right) - 2 \left(N _ {t} r \left(\theta^ {(t)}\right)\right) ^ {1 / 2} \log N _ {t}\right) ^ {- 1 / 2} \leq \frac {c}{8 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {t}. +$$ + +Then + +$$ +\begin{array}{l} \mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t + 1)}\right) \leq \left(1 - \frac {c}{4 \beta}\right) \mathbb {P} l _ {\theta^ {(t)}} \left(\theta^ {(t)}\right) + \frac {c}{8 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {t} \\ \leq \left(1 - \frac {c}{4 \beta}\right) \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \left(1 - \frac {c}{8 \beta}\right) ^ {t} + \frac {c}{8 \beta} \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {t} \\ \leq \mathbb {P} l _ {\theta^ {(0)}} \left(\theta^ {(0)}\right) \left(1 - \frac {c}{8 \beta}\right) ^ {t + 1}. \\ \end{array} +$$ + +Similarly, + +$$ +r \left(\theta^ {(t + 1)}\right) \geq r ^ {*} - c _ {2} \tilde {\mathbb {P}} l _ {\theta^ {(t + 1)}} \geq r ^ {*} - c _ {2} \tilde {\mathbb {P}} l _ {\theta^ {(0)}} \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {t + 1}. +$$ + +Consequently, we choose + +$$ +N _ {t} = r \left(\theta^ {(t)}\right) ^ {- 1} \left(2 (1 6 \beta - 2 c) c ^ {- 1} C _ {L} C _ {\Theta} \log c _ {\Theta}\right) ^ {2} \left(\tilde {\mathbb {P}} l _ {\theta} ^ {(0)}\right) ^ {- 2} \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {- 2 t}, +$$ + +then for the initial value $\theta^{(0)}$ which is sufficiently accurate, we have + +$$ +r \left(\theta^ {(t)}\right) ^ {1 / 2} \geq r \left(\theta^ {(0)}\right) ^ {1 / 2} \geq 4 N _ {t} ^ {- 1 / 2} \log N _ {t}, +$$ + +ensuring that our choice of $N_{t}$ meets the needed condition. + +Finally, we define $\zeta = \gamma c / (8\beta -\gamma c)$ , then + +$$ +n _ {t} = N _ {t} r \left(\theta^ {(t)}\right) = \left(2 \left(1 6 \beta - 2 c\right) c ^ {- 1} C _ {L} C _ {\Theta} \log c _ {\Theta}\right) ^ {2} \left(\tilde {\mathbb {P}} l _ {\theta} ^ {(0)}\right) ^ {- 2} (1 + \zeta) ^ {- 2 t}. +$$ + +By the inequality + +$$ +1 - \sum_ {t = 0} ^ {T - 1} N _ {t} ^ {- 4} \geq 1 - \sum_ {t = 0} ^ {T - 1} n _ {t} ^ {- 4} +$$ + +we obtain the results stated in Theorem B.6. + +![](images/4c6d122e48aabc5e6d7d5d8ed850deedc88c17ec5f1654b0fca4352e0e2afce0.jpg) + +Proof of Theorem B.4. By Assumption B.3, we have + +$$ +\begin{array}{l} \sup _ {\theta \in \Theta} \mathbb {E} _ {x \sim \mathcal {P} _ {\theta}} [ \mathcal {R} (x) ] - \mathbb {E} _ {x \sim \mathcal {P} _ {\theta_ {\pi_ {\mathrm {c o n s t}} ^ {(T)}}}} [ \mathcal {R} (x) ] \geq c _ {3} \mathbb {P} l _ {\theta_ {\pi_ {\mathrm {c o n s t}} ^ {(T)}}} \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)} \\ \geq c _ {3} \left(\mathbb {P} l _ {\theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}} \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)} - \mathbb {P} _ {n _ {0}} l \left(X; \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right)\right) + \mathbb {P} _ {n _ {0}} l \left(X; \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right) \\ \geq c _ {3} \frac {1}{n _ {0}} \sum_ {i = 1} ^ {n _ {0}} \left(l \left(x _ {i}; \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right) - \mathbb {E} _ {X \sim \mathcal {P}} _ {l \left(x _ {i}; \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right)} \left[ l \left(X; \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right) \right]\right) \\ \geq c _ {3} n _ {0} ^ {- 1 / 2} \Phi (0. 7 5) \operatorname {V a r} _ {X \sim \mathcal {P} _ {l \left(x _ {i}; \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right)}} \left[ l \left(X; \theta_ {\pi_ {\mathrm {c o n s t}}} ^ {(T)}\right) \right] ^ {2} \\ \end{array} +$$ + +with probability larger than $1 - 0.75 = 0.25$ , where we invoke the central limit theorem for the last inequality. + +Proof of Theorem B.5. Let $\varepsilon > 0$ be given. We first show that there exists some $t$ satisfying + +$$ +r \left(\theta_ {\pi} ^ {(t)}\right) \geq r ^ {*} - \frac {1}{2} \varepsilon . +$$ + +Following the proof of Theorem B.6, we have + +$$ +\tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t + 1)}} - \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} \leq - \frac {\gamma c}{4 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} + \gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} n _ {t} ^ {- 1 / 2}. +$$ + +There exists a sufficiently large $T'$ such that + +$$ +\gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} n _ {T} ^ {- 1 / 2} \leq \min \left\{\frac {\gamma c}{1 6 \beta c _ {2}} \varepsilon , \frac {1}{2 c _ {2}} \varepsilon \right\}. +$$ + +For any $t \geq T'$ , consider the following two cases: + +(a) $\tilde{\mathbb{P}} l_{\theta_{\tau}^{(t)}} \leq \varepsilon / (2c_2)$ . + +By Assumption B.3, we have + +$$ +r \left(\theta_ {\pi} ^ {(t)}\right) \geq r ^ {*} - c _ {2} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} \geq r ^ {*} - \frac {1}{2} \varepsilon . +$$ + +(b) $\tilde{\mathbb{P}} l_{\theta_{\pi}^{(t)}} > \varepsilon /(2c_2)$ + +Since $n_t \to \infty$ as $t \to \infty$ , for all $t \geq T'$ we have + +$$ +\gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} n _ {t} ^ {- 1 / 2} \leq \frac {\gamma c}{1 6 \beta c _ {2}} \varepsilon \leq \frac {\gamma c}{8 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}}. +$$ + +Therefore, + +$$ +\tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t + 1)}} - \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} \leq - \frac {\gamma c}{8 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}}. +$$ + +Repeating this argument for $t = T', T' + 1, \dots$ whenever $\tilde{\mathbb{P}}l_{\theta_{\pi}^{(t)}} > \varepsilon/(2c_2)$ shows that there must exist a finite $K$ (so $T = T' + K$ ) where either + +$$ +\begin{array}{l} r \left(\theta_ {\pi} ^ {(t)}\right) \geq r ^ {*} - c _ {2} \frac {\gamma c}{4 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(T)}} \\ \geq r ^ {*} - c _ {2} \frac {\gamma c}{4 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(T ^ {\prime})}} \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {T - T ^ {\prime}} \geq r ^ {*} - \frac {1}{2} \varepsilon , \\ \end{array} +$$ + +or else $\tilde{\mathbb{P}} l_{\theta_{\pi}^{(T)}} \leq \varepsilon / (2c_2)$ , in which case (a) implies + +$$ +r \left(\theta_ {\pi} ^ {(t)}\right) \geq r ^ {*} - \frac {1}{2} \varepsilon . +$$ + +Thus, there exists some $T$ such that + +$$ +r \left(\theta_ {\pi} ^ {(T)}\right) \geq r ^ {*} - \frac {1}{2} \varepsilon . +$$ + +Next, we prove that for any $t \geq T$ , + +$$ +r \left(\theta_ {\pi} ^ {(t)}\right) \geq r ^ {*} - \varepsilon . +$$ + +For $t\geq T$ , we have + +$$ +\begin{array}{l} \sup _ {t \geq T} \left[ \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t + 1)}} - \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} \right] \leq \sup _ {t \geq T} \left[ - \frac {\gamma c}{4 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} + \gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} n _ {t} ^ {- 1 / 2} \right] \\ \leq \sup _ {t \geq T} \left[ \gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} n _ {t} ^ {- 1 / 2} \right] \\ \leq \frac {\varepsilon}{2 c _ {2}}. \\ \end{array} +$$ + +From part (b) above, whenever $\tilde{\mathbb{P}} l_{\theta_{\pi}^{(t)}} \geq \varepsilon/(2c_2)$ , we have + +$$ +\tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t + 1)}} - \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} \leq - \frac {\gamma c}{8 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}}. +$$ + +Hence, + +$$ +\sup _ {t \geq T} \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} \leq \frac {\varepsilon}{2 c _ {2}} + \sup _ {t \geq T ^ {\prime}} \left[ \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t + 1)}} - \tilde {\mathbb {P}} l _ {\theta_ {\pi} ^ {(t)}} \right] \leq \frac {\varepsilon}{c _ {2}}. +$$ + +It follows that for all $t \geq T$ , + +$$ +r \left(\theta_ {\pi} ^ {(t)}\right) \geq r ^ {*} - \varepsilon , +$$ + +which completes the proof. + +![](images/ef981bfa11f6dac6672dc55244c8353099e811de5f969c7eaeca7268486e160b.jpg) + +Proof of Theorem B.7. We prove this theorem by showing that for a given policy $\pi'$ , there exists a specified parameter space $\Theta$ and $\Theta^*$ , a loss function $l(x; \theta)$ , and a reward model $\mathcal{R}(x)$ , under Assumptions B.1 to B.3 such that + +$$ +\begin{array}{l} C \left(\pi^ {\prime}, T ^ {*} \left(\pi^ {\prime}, \varepsilon\right)\right) = \sum_ {t = 0} ^ {T ^ {*} \left(\pi^ {\prime}, \varepsilon\right) - 1} \left(c _ {g} N _ {t} ^ {\prime} + c _ {\mathrm {t}} n _ {t} ^ {\prime}\right) \tag {9} \\ > C _ {\mathrm {e x p}} \varepsilon^ {- 2} - c _ {\mathrm {e x p}} \\ \geq C \left(\pi_ {\exp}, T ^ {*} \left(\pi_ {\exp}, \varepsilon\right)\right) \\ \end{array} +$$ + +with probability at least $\left(1 - \sum_{t=0}^{T^*(\pi_{\exp}, \varepsilon) - 1} n_t^{-4}\right)$ . + +To begin, we use Theorem B.6 to obtain + +$$ +T ^ {*} \left(\pi_ {\exp}, \varepsilon\right) \leq \log^ {- 1} \left(1 - \frac {\gamma c}{8 \beta}\right) \left(\log \varepsilon - \log c _ {2} \tilde {\mathbb {P}} l _ {\theta^ {(0)}}\right) := T _ {1} +$$ + +with probability at least $\left(1 - \sum_{t=0}^{T^*(\pi_{\mathrm{exp}}, \varepsilon) - 1} n_t^{-4}\right)$ . Then the equality + +$$ +\begin{array}{l} C \left(\pi_ {\exp}, T ^ {*} \left(\pi_ {\exp}, \varepsilon\right)\right) \leq C \left(\pi_ {\exp}, T _ {1}\right) \\ = \sum_ {t = 0} ^ {T _ {1} - 1} \left(c _ {\mathrm {g}} r _ {t} ^ {- 1} + c _ {\mathrm {t}}\right) \left(2 (1 6 \beta - 2 c) c ^ {- 1} C _ {L} C _ {\Theta} \log c _ {\Theta}\right) ^ {2} \left(\tilde {\mathbb {P}} l _ {\theta} ^ {(0)}\right) ^ {- 2} \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {- 2 t} \\ \leq \left(c _ {\mathrm {g}} r _ {0} ^ {- 1} + c _ {\mathrm {t}}\right) \left(2 \left(1 6 \beta - 2 c\right) c ^ {- 1} C _ {L} C _ {\Theta} \log c _ {\Theta}\right) ^ {2} \left(\tilde {\mathbb {P}} l _ {\theta} ^ {(0)}\right) ^ {- 2} \sum_ {t = 0} ^ {T _ {1} - 1} \left(1 - \frac {\gamma c}{8 \beta}\right) ^ {- 2 t} \\ \leq \left(c _ {\mathrm {g}} r _ {0} ^ {- 1} + c _ {\mathrm {t}}\right) \left(2 (1 6 \beta - 2 c) c ^ {- 1} C _ {L} C _ {\Theta} \log c _ {\Theta}\right) ^ {2} \left(\tilde {\mathbb {P}} l _ {\theta} ^ {(0)}\right) ^ {- 2} \frac {(1 + \zeta) ^ {2 (T _ {1} - 1) - 1}}{2 \zeta + \zeta^ {2}} \\ \leq C _ {\exp} \varepsilon^ {- 2} - c _ {\exp}. \\ \end{array} +$$ + +holds for some positive constants $C_{\mathrm{exp}}$ and $c_{\mathrm{exp}}$ that do not depend on $\varepsilon$ . So we have proven the last inequality in Equation (9). + +Next, we consider the second inequality in Equation (9). When $\pi'$ is a constant policy, Theorem B.4 implies that for sufficiently small $\varepsilon$ , $T^{*}(\pi', \varepsilon) = \infty$ and the inequality holds vacuously. Otherwise, for a polynomial growth policy $\pi'$ under Assumptions B.1 to B.3, there exist $\Theta$ , $\Theta^{*}$ , $l(x; \theta)$ , and $\mathcal{R}(x)$ such that for $\theta_{\pi'}^{(t)}$ , + +$$ +\tilde {\mathbb {P}} l _ {\theta_ {\pi^ {\prime}} ^ {(t + 1)}} - \tilde {\mathbb {P}} l _ {\theta_ {\pi^ {\prime}} ^ {(t)}} = \min \left\{- \frac {\gamma c}{4 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi^ {\prime}} ^ {(t)}} + \rho \gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} \frac {1}{\sqrt {n _ {t} ^ {\prime}}}, 0 \right\}, +$$ + +for some $0 < \rho \leq 1$ , and by Assumption B.3, + +$$ +r ^ {*} - r \left(\theta_ {\pi^ {\prime}} ^ {(t)}\right) \geq c _ {3} \tilde {\mathbb {P}} l _ {\theta_ {\pi^ {\prime}} ^ {(t)}}. +$$ + +Since $\pi'$ is a polynomial growth policy, let $\pi' = (n_0', n_1', n_2', \dots)$ with $n_t' = n_0'(1 + t)^\alpha$ , $\alpha > 0$ by definition. A decrease in expected loss occurs only if + +$$ +\rho \gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} \frac {1}{\sqrt {n _ {0} ^ {\prime} (1 + t) ^ {\alpha}}} \leq \frac {\gamma c}{4 \beta} \tilde {\mathbb {P}} l _ {\theta_ {\pi^ {\prime}} ^ {(t)}}. +$$ + +Besides, + +$$ +\tilde {\mathbb {P}} _ {\theta_ {\pi^ {\prime}} ^ {(T ^ {*} (\pi^ {\prime}, \varepsilon))}} \leq c _ {3} ^ {- 1} \left(r ^ {*} - \mathbb {E} _ {x \sim \mathcal {P} _ {\theta_ {\pi^ {\prime}} ^ {(T ^ {*} (\pi^ {\prime}, \varepsilon))}}} [ \mathcal {R} (x) ]\right) \leq c _ {3} ^ {- 1} \varepsilon , +$$ + +it follows that + +$$ +\rho \gamma \left(2 - \frac {c}{4 \beta}\right) 2 C _ {L} C _ {\Theta} \log c _ {\Theta} \frac {1}{\sqrt {n _ {0} ^ {\prime} (1 + T ^ {*} (\pi^ {\prime} , \varepsilon)) ^ {\alpha}}} \leq \frac {\gamma c}{4 \beta c _ {3}} \varepsilon , +$$ + +so + +$$ +T ^ {*} (\pi^ {\prime}, \varepsilon) \geq \left[ \left(\frac {c}{2 c _ {3} \rho (8 \beta - c) C _ {L} C _ {\Theta} \log c _ {\Theta}}\right) ^ {- 2} n _ {0} ^ {' - 1} \right] ^ {1 / \alpha} \varepsilon^ {- 2 / \alpha} - 1 := T _ {2}. +$$ + +Thus, + +$$ +\begin{array}{l} \sum_ {t = 0} ^ {T ^ {*} \left(\pi^ {\prime}, \varepsilon\right) - 1} \left(c _ {\mathrm {g}} N _ {t} ^ {\prime} + c _ {\mathrm {t}} n _ {t} ^ {\prime}\right) \geq \sum_ {t = 0} ^ {T _ {2} - 1} \left(c _ {\mathrm {g}} N _ {t} ^ {\prime} + c _ {\mathrm {t}} n _ {t} ^ {\prime}\right) \\ \geq \sum_ {t = 0} ^ {T _ {2} - 1} \left(c _ {\mathrm {g}} + c _ {\mathrm {t}}\right) n _ {0} ^ {\prime} (1 + t) ^ {\alpha} \\ \geq \left(c _ {\mathrm {g}} + c _ {\mathrm {t}}\right) n _ {0} ^ {\prime} \left(\frac {T _ {2}}{2}\right) ^ {\alpha} \frac {T _ {2}}{2} := C _ {\mathrm {p o l y}} \varepsilon^ {- 2 (\alpha + 1) / \alpha} - c _ {\mathrm {p o l y}} \\ \end{array} +$$ + +for some positive constants $C_{\mathrm{poly}}$ and $c_{\mathrm{poly}}$ independent of $\varepsilon$ . If $\varepsilon$ is small enough, it follows that + +$$ +\sum_ {t = 0} ^ {T ^ {*} (\pi^ {\prime}, \varepsilon) - 1} (c _ {\mathrm {g}} N _ {t} ^ {\prime} + c _ {\mathrm {t}} n _ {t} ^ {\prime}) \geq C _ {\mathrm {p o l y}} \varepsilon^ {- 2 (\alpha + 1) / \alpha} - c _ {\mathrm {p o l y}} > C _ {\exp} \varepsilon^ {- 2} - c _ {\exp}, +$$ + +so the second inequality in Equation (9) also holds. + +![](images/1b377739477ad8afa1dcc851b92eeea2571ce5a01ff3eb6ed25c69c2ba616c9a.jpg) + +# B.5 Extension to Rewards with Random Noise + +Our theoretical conclusions directly generalize to cases with noisy rewards. Consider a perturbed reward function $\hat{\mathcal{R}}(x) = \mathcal{R}(x) + \varepsilon$ , where $\varepsilon$ is zero-mean noise independent of $\mathcal{R}(x)$ . Crucially, the expected reward $r(\theta)$ remains unchanged, preserving all key theoretical results. Although noisy rewards affect the sample selection process, the concentration bounds for selected sample sizes (Appendix B.4) and the expected reward under the selected distribution remain valid asymptotically. This robustness stems from the exponential policy's inherent adaptivity by progressively increasing sample sizes (Theorem 4.3), it naturally compensates for reward noise variance. + +# C Diffusion Probabilistic Models + +A diffusion probabilistic model (DPM) progressively adds noise to clean data sampled from the target distribution until the data is fully corrupted. The generative process then employs a noise prediction model to transform noisy samples into progressively cleaner ones auto-regressively. This reverse-time evolution naturally resembles a denoising process, making DPMs suitable for image denoising, especially when the noise modeling is known (Xie et al., 2023). + +Formally, a DPM defines a forward stochastic process with conditional probability + +$$ +p _ {\tau | 0} \left(\boldsymbol {x} _ {\tau} \mid \boldsymbol {x} _ {0}\right) = \mathcal {N} \left(\boldsymbol {x} _ {\tau}; \alpha_ {\tau} \boldsymbol {x} _ {0}, \beta_ {\tau} ^ {2} \boldsymbol {I}\right), \quad \tau \in [ 0, 1 ], +$$ + +where $\pmb{x}_0$ is drawn from the target distribution that the model aims to learn. The parameters $\alpha_{\tau}$ and $\beta_{\tau}$ are typically chosen such that $\alpha_0 = \beta_1 = 1$ and $\alpha_{1} = \beta_{0} = 0$ . This process progressively adds noise to $\pmb{x}_0$ , resulting in a fully corrupted sample $\pmb{x}_1 \sim \mathcal{N}(\pmb{0},\pmb{I})$ . + +To model the reverse process, the time interval $[0,1]$ is discretized into steps $0 = \tau_0 < \tau_1 < \dots < \tau_K = 1$ . Starting at $\tau_K$ , the reverse diffusion process gradually removes noise step by step until reaching $\tau_0$ . At each step $k$ , the noise added during the forward process is estimated and removed using a trained noise prediction model $\varepsilon_{\theta}(\cdot ,\tau_k)$ . This model learns to predict the noise $\varepsilon$ based on the noisy sample $\pmb{x}_{\tau_k}$ , enabling the reconstruction of cleaner samples at each step. + +The standard training procedure, which is also adapted in our experiments, attempts to solve + +$$ +\min _ {\boldsymbol {\theta}} \mathbb {E} _ {\boldsymbol {x} \sim \mathcal {D}} \mathbb {E} _ {\substack {\boldsymbol {\varepsilon} \sim \mathcal {N} (\boldsymbol {0}, \boldsymbol {I}) \\ \tau \in \mathcal {U} (0, s)}} \left\| \varepsilon_ {\boldsymbol {\theta}} \left(\alpha_ {\tau} \boldsymbol {x} + \beta_ {\tau} \varepsilon , \tau\right) - \varepsilon \right\| ^ {2}, \tag{10} +$$ + +where $\mathcal{D}$ denotes the distribution of training data. + +# D Algorithm for Realistic Experiments + +We follow the algorithm in Algorithm 1. Empirically, in the two realistic settings (image denoising and math reasoning), our iterative learning policy satisfies that $n_t$ is an integer multiple of the batch size. Specifically, + +- For the constant scheme, + +$$ +n _ {t} = \lfloor n \rfloor B; +$$ + +- For the linear scheme, + +$$ +n _ {t} = \lfloor n (t + 1) \rfloor B; +$$ + +- For the exponential scheme, + +$$ +n _ {t} = \lfloor n (1 + u) ^ {t} \rfloor B, +$$ + +where $n$ and $u$ serve as hyperparameters, and $B$ is the generation batch size. + +# E Implementation Details + +We provide the code for all the experiments in the GitHub repository: https://github.com/zylipku/spend-wisely. + +# E.1 Toy Example + +In this subsection, we extend the theoretical intuition from Theorem 3.1 to high-dimensional settings and then derive the expected reward of each iteration, as well as the corresponding optimal expected reward. + +We begin with the following theorem, which shows that in a high-dimensional Gaussian setting, the same exponential growth strategy emerges as in the one-dimensional case. + +Theorem E.1. Under the MLE update, suppose that the generator is a $d$ -dimensional Gaussian distribution $\mathcal{N}(\pmb{\theta}, \sigma^2 \pmb{I}_d)$ , the reward model is an exponential function $\mathcal{R}(\pmb{x}) = \exp \left(-\|\pmb{x}\|^2 / (2\kappa^2)\right)$ and we start from + +$$ +\left\| \boldsymbol {\theta} ^ {(0)} \right\| \leq \left(1 + \sigma^ {2} / \kappa^ {2}\right) ^ {T} \left(d \left(\sigma^ {2} + \kappa^ {2}\right)\right) ^ {1 / 2}. +$$ + +Considering the optimization problem + +$$ +\max _ {\pi} \mathbb {E} _ {\boldsymbol {\theta} ^ {(T)}} \left[ \mathbb {E} _ {\boldsymbol {x} \sim \mathcal {P} _ {\boldsymbol {\theta} ^ {(T)}}} \mathcal {R} (\boldsymbol {x}) \right] \quad s. t. \quad \sum_ {k = 0} ^ {T - 1} n _ {t} \leq C, +$$ + +then the optimal policy satisfies + +$$ +n _ {t} \propto \left(1 + \frac {\sigma^ {2}}{\kappa^ {2}}\right) ^ {t}. +$$ + +Proof. Since each component of $\pmb{\theta}^{(t)}$ is i.i.d. and its distribution conditioned on the initial parameter $\theta^{(0)}$ and the policy $\pi$ is given by Lemma A.1, it follows that + +$$ +p \left(\pmb {\theta} ^ {(T)} \mid \pmb {\theta} ^ {(0)}, \pi\right) = \mathcal {N} \left(\frac {\pmb {\theta} ^ {(0)}}{(1 + \sigma^ {2} / \kappa^ {2}) ^ {T}}, \sigma^ {2} \sum_ {t = 0} ^ {T - 1} \frac {1}{n _ {t} (1 + \sigma^ {2} / \kappa^ {2}) ^ {2 (T - t) - 1}} \pmb {I} _ {d}\right) +$$ + +for a specific policy $\pi : \{n_t\}_t$ . By the definition of the reward model, $\mathcal{R}(\pmb{x})$ factors as + +$$ +\mathcal {R} (\boldsymbol {x}) = \prod_ {i = 1} ^ {d} \mathcal {R} (x _ {i}), \quad \boldsymbol {x} = (x _ {1}, x _ {2}, \dots , x _ {d}). +$$ + +Repeating the argument in the proof of Theorem 3.1 shows that + +$$ +\begin{array}{l} \mathbb {E} _ {\boldsymbol {\theta} ^ {(T)}} \mathbb {E} _ {\boldsymbol {x}} \mathcal {R} (\boldsymbol {x}) = \mathbb {E} _ {\boldsymbol {\theta} ^ {(T)}} \mathbb {E} _ {\boldsymbol {x}} \prod_ {i} \mathcal {R} (x _ {i}) \\ = \prod_ {i} \mathbb {E} _ {\boldsymbol {\theta} ^ {(T)}} \mathbb {E} _ {\boldsymbol {x}} \mathcal {R} (x _ {i}) \\ = \left(\frac {\kappa}{\sqrt {\sigma^ {2} + \kappa^ {2} + \sigma_ {T} ^ {2}}}\right) ^ {d} \exp \left(- \sum_ {i = 1} ^ {d} \frac {\left(\theta_ {i} ^ {(0)}\right) ^ {2}}{2 (\sigma^ {2} + \kappa^ {2} + \sigma_ {T} ^ {2}) (1 + \sigma^ {2} / \kappa^ {2}) ^ {2 T}}\right) \\ = \left(\frac {\kappa}{\sqrt {\sigma^ {2} + \kappa^ {2} + \sigma_ {T} ^ {2}}}\right) ^ {d} \exp \left(- \frac {\left\| \pmb {\theta} ^ {(0)} \right\| ^ {2}}{2 (\sigma^ {2} + \kappa^ {2} + \sigma_ {T} ^ {2}) (1 + \sigma^ {2} / \kappa^ {2}) ^ {2 T}}\right). \\ \end{array} +$$ + +With assumption $\| \pmb{\mu}_T\| \leq \left(d(\sigma^2 +\kappa^2)\right)^{1 / 2}$ , the right side increases as $\left(\sigma^2 +\kappa^2 +\sigma_T^2\right)^{-1 / 2}$ increases. Hence, figuring with the proof of Theorem 3.1 in Appendix A, one obtains + +$$ +n _ {t} \propto \left(1 + \frac {\sigma^ {2}}{\kappa^ {2}}\right) ^ {t - T}. +$$ + +Reindexing or shifting $t$ completes the proof. + +![](images/403fbf8698ecec8f66516c57cf88c507bcc19d882e1df3b635cd99487372326d.jpg) + +Theorem E.1 therefore shows that the optimal iterative policy in the high-dimensional case mirrors the one-dimensional scenario. Next, we derive the expected reward in the high-dimensional case. Specifically, if $\pmb{x} \sim \mathcal{N}(\pmb{\theta}, \sigma^2 \pmb{I}_d)$ , then + +$$ +\begin{array}{l} \mathbb {E} _ {\boldsymbol {x} \sim \mathcal {N} (\boldsymbol {\theta}, \sigma^ {2} I _ {d})} [ \mathcal {R} (\boldsymbol {x}) ] = \int \frac {1}{(2 \pi \sigma^ {2}) ^ {d / 2}} \exp \left(- \frac {\| \boldsymbol {x} \| ^ {2}}{2 \kappa^ {2}} - \frac {\| \boldsymbol {x} - \boldsymbol {\theta} \| ^ {2}}{2 \sigma^ {2}}\right) \mathrm {d} \boldsymbol {x} \\ = \frac {1}{(1 + \sigma^ {2} / \kappa^ {2}) ^ {d / 2}} \cdot \exp \left(- \frac {\| \pmb {\theta} \| ^ {2}}{2 (\sigma^ {2} + \kappa^ {2})}\right). \\ \end{array} +$$ + +Clearly, the maximum of this expression is attained at $\theta = 0$ , and the optimal expected reward is $(1 + \sigma^2 /\kappa^2)^{-d / 2}$ . + +# E.2 Synthetic Experiment with Gaussian Distributions + +Setup. We begin with the synthetic setting introduced in our warm-up (Section 3.2), which involves optimizing a generative distribution parameterized as a Gaussian to maximize the expected reward. Specifically, we focus on a two-dimensional space using MLE updates at each iteration. The generator is set to $\mathcal{N}(\pmb{\theta},\pmb{I}_2)$ , with fixed variance. The reward model is defined as an exponential function $\mathcal{R}(\pmb{x}) = \exp \left(-\|\pmb{x}\|^2/(2\kappa^2)\right)$ , favoring data concentrated near the origin. $\kappa$ controls the flatness of the reward. + +All theoretical results from the warm-up extend naturally to the high-dimensional case, as detailed in Appendix E.1. We now validate that with equal computational budgets and iterations, the exponential scheme outperforms the linear scheme, which outperforms the constant scheme. + +Implementation. We consider two settings with $\kappa^2 \in \{2, 4\}$ . Based on theoretical analysis, the exponential scheme is set as $n_t = \lfloor n_0 \cdot (1 + 1 / \kappa^2)^t \rfloor$ , where $n_0 = 10$ . For a fixed total budget and number of iterations, the constant scheme is then + +$$ +n _ {t} = \left\lfloor T ^ {- 1} \sum_ {k = 0} ^ {T - 1} n _ {0} \cdot (1 + 1 / \kappa^ {2}) ^ {k} \right\rfloor , +$$ + +while the linear scheme is + +$$ +n _ {t} = \left\lfloor \frac {2 (t + 1)}{T (T - 1)} \sum_ {k = 0} ^ {T - 1} n _ {0} \left(1 + 1 / \kappa^ {2}\right) ^ {k} \right\rfloor . +$$ + +The initial parameter is set to $\pmb{\theta}^{(0)} = (1,1)^{\top}$ , and the optimal solution for the reward function is $\pmb{\theta}^{*} = (0,0)^{\top}$ . The optimal reward value is then + +$$ +\mathbb {E} _ {\boldsymbol {x} \sim \mathcal {N} (\boldsymbol {0}, I)} [ \mathcal {R} (\boldsymbol {x}) ] = \left(1 + 1 / \kappa^ {2}\right) ^ {- 1}. +$$ + +We evaluate how fast the three schemes approach this optimal reward value. + +# E.3 Image Denoising + +The pre-trained model we choose is a public checkpoint that can be accessed via https://huggingface.co/google/ddpm-cifar10-32. For all denoising experiments, we fix batch size $B = 640$ and learning rate $5 \times 10^{-5}$ . + +At each iteration, synthetic samples are generated on the fly, scored by a reward model + +$$ +\mathcal {R} (\hat {\boldsymbol {x}}, \boldsymbol {x}) = \frac {\operatorname {P S N R} (\hat {\boldsymbol {x}} , \boldsymbol {x}) - r _ {\min}}{r _ {\max} - r _ {\min}}, +$$ + +and accumulated into a chunk until it reaches size $n_t$ . The full training procedure is given in Algorithm 2. Each configuration requires around 100 GPU-hours on an NVIDIA A800 cluster. Note that our setup is much more computationally intensive than the usual training of a generative model in that it takes $s$ times of network inferences to generate synthetic data for network training. + +To compare increasing-schedule policies (particularly exponential growth), we report the computational costs in terms of floating-point operations (FLOPs), which scale linearly with the number of forward passes. For diffusion models, generating one sample uses $s$ denoising steps (i.e., $s$ forward passes), and each optimization step introduces an additional forward pass. + +# E.4 Math Reasoning + +We summarize the iterative math-reasoning algorithm in Algorithm 3. Each configuration requires approximately 400 GPU-hours on an NVIDIA A800 cluster. + +Hyperparameters are as follows: one epoch per iteration on the selected data; a constant learning rate of $10^{-7}$ ; batch size $B = 256$ ; and roughly 1,000 total generator update steps. During generation we use temperature 0.3 for diversity, and temperature 0 at evaluation for accuracy. The maximum generation length is 512. Prompts follow the few-shot setting of the GSM8k dataset in OpenCompass (Contributors, 2023). Our implementation builds on the public OpenRLHF framework (Hu et al., 2024). + +Algorithm 2 Iterative learning for image denoising +1: Input: a pre-trained generative model $f(\cdot ;\pmb{\theta}^{(0)})$ , a policy $\pi :\{n_t\}_t$ with a terminal time $T$ , a reward model $\mathcal{R}$ , and a dataset $\mathcal{D} = \{\pmb{x}_i\}_i$ . +2: Output: a generative model $f(\cdot ;\pmb{\theta}^{(T)})$ fine-tuned with synthetic data. +3: for $t\gets 0$ to $T - 1$ do +4: Initialize the set of selected synthetic data $D_{t} = \emptyset$ . +5: while $|D_{t}| < n_{t}$ do +6: Sample a mini-batch $\{\pmb{x}_i\}_i\subseteq \mathcal{D}$ randomly. +7: Create noisy $y_{i}\sim p_{s|0}(\cdot \mid x_{i})$ for each $x_{i}$ . +8: Generate synthetic $\hat{x}_i$ as a denoised version for each $y_{i}$ based on the generative model $f(\cdot ;\pmb{\theta}^{(t)})$ ; +9: For each $\hat{x}_i$ , add it into $D_{t}$ with probability $\mathcal{R}(\hat{x}_i,\pmb{x}_i)$ clipped within the interval [0, 1]. +10: end while +11: Truncate $D_{t}$ so that it contains exactly $n_t$ samples. +12: for each batch $S = \{\hat{x}_i\}_i$ in $D_{t}$ do +13: Take a single optimization step with the loss + $L\left(\pmb{\theta}^{(t)}\right) = |S|^{-1}\sum_{\pmb{x}\in S}\|\pmb{\varepsilon}_{\pmb{\theta}^{(t)}}(\alpha_{\tau_x}\pmb{x} + \beta_{\tau_x}\pmb{\varepsilon}_x,\tau_x) - \pmb{\varepsilon}_x\|^2$ +to update $\pmb{\theta}^{(t)}$ , where $\varepsilon_{x}\sim \mathcal{N}(0,I)$ and $\tau_{x}\sim \mathcal{U}(0,s)$ . +14: end for +15: $\pmb{\theta}^{(t + 1)}\gets \pmb{\theta}^{(t)}$ . +16: end for + +# F Supplementary Empirical Results + +# F.1 Toy Example + +Additional results for the toy example, obtained under various parameter settings, are shown in Figure 5. In all cases, the behavior is consistent with the conclusion in Section 3.2. + +# F.2 Ablation on Exponential Growth Schemes in Image Denoising + +We conduct an ablation study for exponential growth policies to understand the impact of the $n_t$ on the performance of the iterative learning framework, as measured by PSNR in Figure 6. The results reveal that a smaller $n_t$ during the early stages of training leads to a rapid increase in PSNR and earlier convergence. In contrast, a larger $n_t$ results in a slower initial increase in PSNR but achieves a higher final PSNR when trained long enough. These findings underscore the trade-off between rapid early improvements and the potential for superior overall performance with prolonged training. + +# F.3 Ablation on Exponential Growth Schemes in Math Reasoning + +We further explore the effects of varying the parameter $n_t$ within exponential growth policies on the performance of mathematical reasoning tasks. As shown in Figure 7, configurations with higher $n_t$ demonstrate the potential for superior performance, improving the model's reasoning capabilities and ability to solve mathematical problems. However, these configurations also exhibit some instability, characterized by abrupt changes in accuracy as $n_t$ increases. This highlights the need to find a balance between achieving rapid learning and mitigating the risk of unstable training dynamics. Based on these findings, we adopt $n_t = 10 \cdot 2^t \cdot B$ in the main text. + +# F.4 Additional Empirical Results in Math Reasoning + +In the main paper, we use a temperature of 0.3 for the math reasoning experiment shown in Figure 4. Here, we present additional empirical results with a temperature of 0.7 in Figure 8. The general + +Algorithm 3 Iterative learning on math reasoning + +1: Input: a pre-trained LLM $f(\cdot; \theta^{(0)})$ , a policy $\pi: \{n_t\}_t$ with a terminal time $T$ , and a dataset $\mathcal{D} = \{(q_i, a_i)\}_i$ . +2: Output: $f(\cdot; \theta^{(T)})$ fine-tuned with synthetic data. +3: for $t\gets 0$ to $T - 1$ do +4: Initialize the set of selected synthetic data $D_{t} = \varnothing$ + +// Generate answers and select out correct ones. + +5: while $|D_t| < n_t$ do +6: Sample a mini-batch $\mathcal{B} = \{(q_j,a_j)\}_{j}\subseteq \mathcal{D}$ +7: Generate answers $\{\hat{a}_j\}$ based on $f(\cdot \mid \{q_j\}; \theta^{(t)})$ with few-shot examples and CoT strategy. +8: Add $(q_{j},a_{j})$ into $D_{t}$ if $\hat{a}_j = a_j$ for each $j$ . +9: end while +10: $D_{t}\gets$ a subset of $D_{t}$ containing exactly $n_t$ data. +// Update $\theta^{(t)}$ to $\theta^{(t + 1)}$ by the selected dataset $D_{t}$ with auto-regressive loss via Adam optimizer. +11: for each $\mathcal{B}$ as a mini-batch of $D_{t}$ do +12: Take a single optimization step with the auto-regressive loss + +$$ +L (\theta) = \frac {1}{| \mathcal {B} |} \sum_ {(\hat {a}, q) \in \mathcal {B}} \log p (\hat {a} \mid q; f (\cdot , \theta)) +$$ + +to update $\theta^{(t)}$ + +13: end for +14: $\theta^{(t + 1)}\gets \theta^{(t)}$ +15: end for + +Table 3: Additional empirical results of mixed policies in the math reasoning task. For the mixed policies, we use a linear policy initially and switching to an exponential policy when the linear phase ceased to grow. The results show that mixed policies outperform linear policies and are comparable to the exponential policies. + +
POLICYSYMBOLICP1P2
LINEAR64.2440.8419.17
MIXED65.9845.5620.16
EXPONENTIAL65.6647.2620.65
+ +trends across different curves and policy classes remain consistent when comparing these two figures. This ablation study further demonstrates the superiority of the proposed approach. + +# F.5 Mixed Policies + +We also explore mixed policies empirically. Specifically, we use a linear policy initially, switching to an exponential policy when the linear phase ceases to yield growth. We present the results in Table 3, where we directly combine the linear and exponential schemes. The results show that mixed policies do not outperform the pure exponential policy. While this approach could be seen as a more general strategy, the increased degrees of freedom in parameter tuning make hyperparameter selection considerably more challenging. We leave further exploration of this direction to future work. + +# G Discussion + +# G.1 Impact Statement + +This research advances the field of machine learning by addressing the strategic allocation of computational resources in the post-training phase using synthetic data. We introduce a novel theoretical + +![](images/aedad3c827307bc42bd25e318e16d82d68d0a04f436381f2364508db6bcde1c3.jpg) +(a) $n = 10,T = 15,\theta = (1,1)^{\top}$ + +![](images/33fc51d1710c2d11f20bdeec9b263748a76a243506071ebbbce487e006c8eeb5.jpg) +constant linear exponential + +![](images/41f238476f48bf92f8ca202ae4b3de40a5f1886bde405649d06f4e89465eddf1.jpg) +(b) $n = 15,T = 10,\theta = (1,1)^{\top}$ + +![](images/40f3c718df566069d3d0fe5267ab4303ed7217701ed49ec46b0fac3f659fac99.jpg) +constant linear exponential + +![](images/052f613851ae8cf3587c2de7c3d27baf7ba4d1b5ad8b8f154c6bd77410fcaab8.jpg) +(c) $n = 10,T = 10,\theta = (0.5,0.5)^{\top}$ + +![](images/152682b793a610002c35ca7856527bdbfa1e49b0f7445dd75712a5a659adc4c7.jpg) +constant linear exponential + +Figure 5: Additional experimental results of the toy example with diverse parameters. +![](images/f05429e5559215326c736c108f7942d5ff433fb471a56ef757327f847d5b42af.jpg) +(d) $n = 10,T = 10,\theta = (2,2)^{\top}$ + +![](images/28b0f329c0cae73bc8ef4d97c689e1dcf1a79e3795cab56523b18126d48afc1e.jpg) +constant linear exponential + +framework to optimize the distribution of a fixed budget across multiple training iterations, an approach that demonstrates significant improvement in model performance. Our findings challenge conventional constant allocation strategies, revealing that dynamically adjusted policies, particularly those with exponential growth, not only converge more stably but also do so at a reduced computational cost. This work has broad implications for the development of AI models, especially in domains where the generation of synthetic data is critical for enhancing model capabilities without the extensive need for expensive or hard-to-obtain real-world data. The practical applications of this research extend to fields such as autonomous driving, medical image analysis, and any other area where AI must perform with high stability and efficiency under constraints of limited labeled data. This study thus provides a foundation for more sustainable AI development practices that prioritize both performance and computational efficiency. + +![](images/6804520290634ad30abf9b63b5aba6d94fd8e5a074013b80bc1aecacacf27692.jpg) +Figure 6: Ablation study for the image denoising task. We show the performance of exponential scheme under the $s = 10$ setting with different parameters. + +![](images/a3a255dc8d8edb14c9aafe8063df2cc266fe0690690b498208e5d65b62633b74.jpg) + +![](images/df494845ceae698859295d6d0ad7aed64458b4d8575c1abcf50cba4d87e6ed4a.jpg) + +![](images/c0e5824be9ae2373710ee052e3d263f1dee6a7bd40c2c0565a97bc400145245d.jpg) + +![](images/540d0e938a0af6a2b61468ff3ceeb7db2f72473e4afbc00c1c1c4ba5c88486ba.jpg) + +![](images/bee24b8e15a87dbca8f136bd17208194b2772c872f764f0abef7548daa3adef3.jpg) + +![](images/5cdccfdef5035727edd25973beabaecb521246cfb58be9a85da0a3d61eca390e.jpg) + +![](images/32b7e751787a90dc2c0b7d1e31277eeb1f1120637d00dc00275f1ff64170d764.jpg) +Figure 7: Ablation study for the math reasoning task. We show the performance of exponential scheme with different parameters. + +![](images/d72ccd197740ca5ff1a6cbc2360260d46ab0c6aec104207d7e052bb20012c1db.jpg) + +![](images/b0a5ce3b5d1d3ce1019a84b3382c44d4ffc3cb53e423fdc15e6ff202dea1cb5c.jpg) + +# G.2 Connection with Model Collapse + +Prior work has introduced verification (Feng et al., 2024; El Firdoussi et al., 2025) and correction (Gillman et al., 2024) mechanisms for synthetic data to mitigate model collapse. They have theoretically derived one-step conditions under which collapse is avoided or triggered. Our methodology crucially relies on the reward model-based data verification process, using reward signals to filter data to prevent model collapse. + +Empirically, we observe that increasing policies are generally more resistant to collapse. For constant policies, even with reward-based curation, the model's performance initially improves but then degrades over successive iterations. Increasing polices yield smoother, generally upward trajectories, though some decline under very extensive finetuning. These observations extend beyond the scope of the analyses in (Feng et al., 2024; El Firdoussi et al., 2025) and suggest the need for a deeper theoretical analysis. We will add this discussion on model collapse to the paper and leave it for future work. + +
constant, nt=10·Blinear, nt=3t·B
constant, nt=30·Blinear, nt=10t·B
constant, nt=100·Blinear, nt=30t·B
+ +![](images/057709400a57c68a8ccd529b23d58ccc6fb4c06ded0d46b9edcd72bd04ccc63e.jpg) +Figure 8: Additional empirical results of math reasoning where the temperature is 0.7. Other hyperparameters remain the same them in Figure 4, i.e., the training batch size is still set to $B = 256$ and the exponential policy is still $n_t = 10 \cdot 2^t \cdot B$ . + +![](images/2d7f30c1e98bdb413c6029bc2bf30ecdffa0c11d3e2fba15c0de895e3f4efed6.jpg) + +![](images/9b1aa38cc25c7c1a156b5714629f42e5f979606878a1d5bf32557ad7f6ff947d.jpg) + +# G.3 Limitation + +In this paper, we focus on achieving efficient SFT improvement within an iterative learning framework. Recent trends (Guo et al., 2025) incorporate reinforcement learning into the entire post-training phase, while SFT remains an important component for enabling the model to exhibit certain behaviors and skills necessary for successful RL training later (Gandhi et al., 2025). We aim to extend the theory to develop an efficient training scheme for RLHF and RL from verifiable rewards. + +We also examine learning with access to a reward model with (Appendix B.5) or without independent random noises. An empirical reward, such as using an LLM as a judge, could produce biased results and lead to reward hacking. We leave the exploration of biased rewards to future work. + +We conduct two experiments to validate our theoretical results. However, incorporating the SFT scheme design into production scale is beyond our resources at academic institutions. 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However, extending them to solve PDEs on complex geometries remains a challenge. In this work, we propose SpiderSolver, a geometry-aware transformer that introduces spiderweb tokenization for handling complex domain geometry and irregularly discretized points. Our method partitions the irregular spatial domain into spiderweb-like patches, guided by the domain boundary geometry. SpiderSolver leverages a coarse-grained attention mechanism to capture global interactions across spiderweb tokens and a fine-grained attention mechanism to refine feature interactions between the domain boundary and its neighboring interior points. We evaluate SpiderSolver on PDEs with diverse domain geometries across seven datasets, including cars, airfoils, blood flow in the human thoracic aorta, as well as canonical cases governed by the Navier-Stokes, Darcy flow, elasticity, and plasticity equations. Experimental results demonstrate that SpiderSolver consistently achieves state-of-the-art performance across different datasets and metrics, with better generalization ability in the OOD setting. The code is available at https://github.com/Kai-Qi/SpiderSolver. + +# 1 Introduction + +Solving partial differential equations (PDEs) is fundamental to many computational problems in science and engineering. Classical numerical methods involve discretizing computational domains and solving the resulting algebraic systems. However, for domains with complex boundary geometries and irregular discretization, these methods require complex mesh generation and incur high computational costs. For example, computing the drag force on a car (Figure 1) requires solving the Navier-Stokes equations with a car-shaped boundary. Classical methods discretize the computational domain into irregular meshes or points (e.g., over 30,000 points for the Shape-Net Car dataset), leading to high computational complexity and cost. + +In recent years, there has been growing interest in applying deep learning methods to solve PDEs, such as PINNs [1], neural operators [2, 3, 4], etc. However, most of the current methods deal with regular computational domains. For example, the well-established FNO [3] solves PDE in rectangular domains with uniform grids due to its implementation via the Fast Fourier Transform. To efficiently solve PDEs with irregularly shaped domains, Geometry-Informed Neural Operator (GINO) [5] and Geo-FNO [6] are proposed based on deforming the irregularly shaped domain into a uniform latent mesh. Such approaches encounter difficulties with mesh transformations for complex geometries of the computational domains. The Graph Neural Operator (GNO) [2] adopts the Graph Neural Network [7] as a backbone by formulating the kernel integral operator as message passing on graphs, + +![](images/ec52cb0f15156385d15811c156ad95ff5e46068f11453e99e422e7d655de5936.jpg) +Figure 1: Visualization of datasets. For Shape-Net Car and AirfRANS, we estimate the surface pressure on the car/airfoil and the surrounding air velocity based on their shapes, aiming to predict the drag and lift forces for a driving car or a flying airplane. The object surfaces of cars, airfoils, and blood vessels serve as PDE domain boundaries, with the interior regions surrounding the car/airfoil surfaces or lying inside the blood vessel. + +which can be directly applied to irregular meshes. 3D-GenCA [8] introduces a pre-trained 3D vision model [9] for GNO, to encode the complex boundary of the domain. However, it does not fully consider the complex physical relationships between boundary and interior mesh points. + +Recently, Transformers [10], a widely used backbone in deep learning, has been applied to solve PDEs, which can also be directly applied on irregular meshes. FactFormer [11], MINO [12], and Galerkin Transformer [13] are based on the idea that the dot-product attention can be considered as an approximation of an integral transform with a non-symmetric learnable kernel function, which relates Transformers to the FNO. These transformers have demonstrated promising performance for solving PDEs with complex geometry. However, massive mesh points lead to the huge computational overhead of Transformers because the canonical attention in Transformers has quadratic complexity, and brings challenges for Transformers to capture the complex physical relations between irregular mesh points. Transolver [14] proposes a Physics-Attention mechanism that decomposes the discretized domain into a series of learnable slices, with mesh points exhibiting similar physical states assigned to the same slice. Notably, the slices are learned without explicitly utilizing the geometric information of the domain. + +In this paper, we focus on designing a transformer-based PDE solver for domains with complex boundary geometries and irregularly discretized points. The main challenges include efficient tokenization over irregular computational domains and the integration of domain and boundary geometry into the network design. Following this motivation, we propose SpiderSolver, a geometry-aware Transformer. SpiderSolver uses spiderweb tokenization to partition the domain into spiderweb-like patches, guided by spectral clustering of the boundary and the distance of interior points from the boundary. Spiderweb tokenization partitions physical space in a more fundamental and physically intuitive way, achieving a trade-off between computational efficiency and capturing the physical interactions of spatial points. Based on spiderweb tokenization, as shown in Figure 2, SpiderSolver integrates coarse-grained and fine-grained attention to capture the physical relationship between points of interior domain and boundary surface. We evaluate our SpiderSolver on two industrial-level design tasks and one blood simulation task, as well as canonical cases governed by the Navier-Stokes and Darcy flow equations. These tasks are challenging since they require the model to handle various complex boundary geometries. The experiments show that our SpiderSolver achieved state-of-the-art results compared to other neural operators and Transformer-based PDE solvers for these tasks. + +Overall, our contributions are as follows. First, we introduce a novel complex computational domain partitioning method, Spiderweb Tokenization, which divides the domain into spiderweb-like patches. Second, we propose SpiderSolver, a geometry-aware transformer for solving PDEs with complex boundary geometries and discretization. SpiderSolver integrates coarse-grained and fine-grained attention to capture the physical relationship between inner space and boundary surfaces. Third, SpiderSolver surpasses the state-of-the-art methods in the PDE solving tasks for car and airfoil design, the blood flow dynamics in the human thoracic aorta, as well as two fundamental PDE tasks. + +# 2 Related Work + +The deep learning methods have been widely applied to solve PDEs, which can be roughly categorized into two paradigms. The first paradigm, such as PINNs [1], Deep Ritz [15], etc, is to approximate the solutions of PDEs by neural networks and formalize the physical constraints (including equations, + +initial and boundary conditions) as objective functions to optimize network parameters. This kind of method requires the exact formalization of PDEs and needs to retrain the network for new PDEs. Another paradigm is operator learning, e.g., DeepONet [4] and FNO [3]. It is to learn the nonlinear mapping (i.e., operator) between the function space of parameters and the function space of the PDEs' solutions. FNO [3] approximates the solution operator by the kernel integral operator and calculates it in the frequency domain by the fast Fourier transform (FFT). Afterward, a series of variants of FNO are proposed. F-FNO [16] enhances model efficiency by employing factorization in the Fourier domain. U-NO [17] and U-FNO [18] combine FNO and U-Net for multiscale problems. However, most of these methods solve PDEs with regular computational domains. + +For solving PDEs with irregularly shaped domains, Geo-FNO [6] and GINO [5] are based on the idea of deforming the irregular input domain into a uniform latent mesh on which the FFT used in FNO can be applied. However, such mesh transformations are difficult for complex geometries of the computational domains, such as a car-shaped domain, resulting in performance degradation. Graph Neural Operator (GNO) [2] formulates the FNO's kernel integral operator as message passing on graphs, leveraging Graph Neural Networks [7] to construct neural operators. However, it should be noted that graph kernels are insufficient in their capacity to capture global information. + +Transformers [10] have also been introduced as a backbone to solve PDEs with irregularly shaped domains. MINO [12], FactFormer [11], and Galerkin Transformer [13] are based on the idea that the dot-product attention can be considered as an approximation of an integral transform with a non-symmetric learnable kernel function, which relates Transformer to the FNO. Vito [19] combines the vision transformer [20] and the U-net [21] to construct the neural operator. To overcome the quadratic complexity of attention, GNOT [22] and ONO [23] utilize the well-established linear Transformers, such as Reformer [24], Performer [25]. These methods directly apply attention to mesh points, which is computationally prohibited when the number of points is large. Transolver [14] decomposes the discretized domain into a series of learnable slices, in which mesh points under similar physical states are assigned to the same slice. Then Transolver applies attention to these learnable slices to learning intrinsic physical relations. + +In contrast, SpiderSolver is a geometry-aware Transformer-based PDE solver, which is designed to partition the domain into spiderweb-like patches utilizing physical and geometric knowledge to reduce the computation cost of attention. Besides, SpiderSolver adopts the multi-grained attention mechanism to capture the intricate physical correlation of complex boundaries and interior points. + +# 3 Our Proposed SpiderSolver + +Problem setup. We consider the partial differential equations (PDEs) defined over a domain $\Omega \in \mathbb{R}^d$ , where $d$ represents the dimensionality of the space. In this work, the domain $\Omega$ has general irregular geometry and boundary, typically discretized into $N$ mesh points, represented as $G \in \mathbb{R}^{N \times d}$ . Specifically, $G = \{g_i\}_{i=1}^{N_I} \cup \{s_j\}_{j=1}^{N_B}$ , where $\mathcal{I}_G = \{g_i\}_{i=1}^{N_I}$ represents the $N_I$ interior points of $\Omega$ (off-boundary points), and $\mathcal{B}_G = \{s_j\}_{j=1}^{N_B}$ denotes the $N_B$ boundary points of $\Omega$ , and $N_B + N_I = N$ . Our goal is to learn a non-linear operator that outputs the physical field $u$ to approximate physical quantities over the geometry $G$ . + +For PDEs with domain boundaries in general irregular geometries (e.g., the boundaries shown in Figure 1), the physical quantities at different locations within the domain are influenced by their proximity to the boundary and the geometry of the boundary itself. In automotive aerodynamics, the flow field exhibits stratification around the vehicle. Near the car surface, airflow adheres to the surface geometry, while at greater distances, the velocity gradually aligns with the freestream. The aerodynamic effect is influenced by the car's surface curvature and angle of attack. Horizontal surfaces, such as the roof, have less impact on the flow, whereas regions with larger angles, such as the front windshield, significantly alter both the flow direction and speed. + +Based on the above observations, we propose SpiderSolver, a geometry-aware transformer, specifically defined as a fast PDE solver for solving PDEs over the domains with irregular geometries. As shown in Figure 2, SpiderSolver integrates spiderweb tokenization (Figure 2 (a)) into a transformer architecture. As will be presented in Section 3.1, spiderweb tokenization partitions the domain of PDE into spiderweb-like patches considering the geometry of the domain boundary, and each patch is used to define a token. Built upon the spiderweb tokenization, we design a transformer (detailed in Section + +![](images/a85633f2b40c58047e74f3d63640de3c8b601b25df558fe6776d69a6f81c44d9.jpg) +Figure 2: Overall architecture of SpiderSolver. Spiderweb Tokenization partitions the domain into Spiderweb-like patches. Coarse-grained Attention interacts the features over the spiderweb tokens. Weighted Token Aggregation updates the point-wise features by the weighted combination of token features. Fine-grained Attention enables interactions between points of boundary and near boundary. + +3.2) consisting of cascaded attention blocks, and each block is composed of a coarse-grained attention (Figure 2 (b)) over the spiderweb tokens and a fine-grained attention (Figure 2 (c)) to interact features between points of domain boundary and their near points in the interior of domain. This transformer is the basis of our SpiderSolver, which is learned to output the PDE solutions. + +# 3.1 Spiderweb Tokenization over PDE Domain + +Overview. To solve PDEs with complex boundary geometries, we introduce a geometry-aware tokenization method as a foundational step in constructing our transformer. The aim is to quantize the computational domain $\Omega$ with interior point set $\mathcal{I}_G$ and boundary point set $\mathcal{B}_G$ into non-overlapping sub-regions adaptive to the geometry of domain boundary (e.g., the object surfaces in Figure 1). This process begins with spectral clustering for the domain boundary $\partial \Omega$ to quantize the domain boundary to sub-regions. Then the inner space of the domain $\Omega$ is partitioned based on the boundary clustering and the distance of interior points to the boundary. We term this approach "spiderweb tokenization" due to the spiderweb-like structure of the resulting divided sub-regions, as illustrated in Figure 3. + +![](images/f1befc92fbd990914785ad2451dcd7de5efc2db675a973eb7ed896074555599f.jpg) +Figure 3: Spiderweb-like sub-region +Figure 4: Optimal transport-based alignment for spectral structure from spiderweb tokenization. clustering. (a) Aligning surface points of a shape to the Colored curves on the car surface illustrate template shape for clustering. (b) Comparison of individual spectral clustering-based partition and aligned clustering results. + +![](images/82907a4254aad2c28bc82b28916f74cdb9f00e21435398101bb9ea0f92675afb.jpg) + +Boundary spectral clustering. Spectral clustering over the boundary leverages the eigenvalue problem of the Laplace operator $\Delta$ , which can be defined on non-Euclidean geometry for capturing its geometric and topological properties [26]. For the continuous domain of boundary surface $\partial \Omega \subset \mathbb{R}^h$ , the eigenvalue problem is: + +$$ +- \Delta u = \lambda u, \quad \text {i n} \partial \Omega , \tag {1} +$$ + +subject to appropriate boundary conditions. The eigenvalues $\lambda$ and eigenfunctions $u$ characterize the intrinsic geometric structure of the boundary $\partial \Omega$ , enabling its decomposition into geometric decomposed sub-regions. These eigenvectors $u$ are used to embed the data into a low-dimensional space. Specifically, the Laplacian operator can be approximated by constructing the affinity matrix + +over nearest neighbors (10 neighbors) graph using points or mesh vertices on the domain boundary. Then we use k-means clustering on the eigenvectors of the normalized Laplacian matrix associated with the $m_B$ smallest eigenvalues, to group $\partial \Omega$ into $m_B$ clusters. Spectral clustering leverages the graph structure of surface points to capture global, shape-aware connectivity beyond local geometry. See Appendix A for implementation details of spectral clustering. + +Aligned boundary clustering for different instances of object surface. As shown in Figure 4 (b) and Figure 10 (Appendix B), individual clustering of different instances of an object results in unaligned clustering. We devise an aligned clustering approach by transferring the clustering of a template shape to the remaining instances of the boundary shape. As shown in Figure 4 (a), we first align all other shapes to a randomly selected reference shape using the entropic regularization optimal transport problem solved by the Sinkhorn algorithm [27]. We then average these aligned shapes to obtain a template shape, on which spectral clustering is performed. Finally, the clustering labels from the template shape are transferred to the other instances of the object. Here, the reference shape is only used for alignment, while the template shape serves as the basis for spectral clustering. Therefore, given a new instance of object, we only need to align its points to template shape points to derive its point clustering. This above strategy to align PDE boundaries to the boundary template shape can accomplish the varying shapes of boundary geometry for a given type of geometry, e.g., car, airfoil. Please refer to Figure 9 in Appendix B for the example of template shape. + +Spiderweb tokenization. Based on aligned boundary clustering, we further divide the inner space of $\Omega$ into sub-regions $\mathcal{T}_{pq}$ with $p = 1,2,\ldots ,m_B$ and $q = 1,2,\ldots ,m_I$ , as shown in Figures 3 and 2 (a). Spectral clustering applied to the boundary $\mathcal{B}_G$ yields $m_B$ clusters, therefore $\mathcal{B}_G = \bigcup_{p = 1}^{m_B}\mathcal{C}_p$ with $\mathcal{C}_p$ denoting the $p$ -th cluster of the boundary. As shown in Figure 3, interior PDE domain is divided into sub-regions, and each sub-region $\mathcal{T}_{pq}$ includes the interior point with its closest point on domain boundary belonging to $p$ -th boundary cluster, and its signed distance function (SDF) values to domain boundary within a range $(d_{q - 1},d_q]$ , with $d_0 = 0$ . This region is expressed as: + +$$ +\mathcal {T} _ {p q} = \left\{g \in \mathcal {I} _ {G} \mid \operatorname {S D F} (g) \in \left(d _ {q - 1}, d _ {q} \right], s _ {j} ^ {*} \in \mathcal {C} _ {p}, s _ {j} ^ {*} = \underset {s _ {j} \in \mathcal {B} _ {G}} {\arg \min } \| g - s _ {j} \| \right\}. \tag {2} +$$ + +We determine the range of $(d_{q - 1},d_q]$ by ensuring an equal number of points falling within each interval $(d_{q - 1},d_q]$ . In this way, we partition the whole PDE domain into $M = m_Bm_I + m_B$ sub-regions with $m_Bm_I$ sub-regions in the interior domain and $m_B$ sub-regions on the boundary. Each sub-region is taken as the domain of a token. + +Notes: The spiderweb-tokenization method is inherently geometry-aware, as explained below. Firstly, it utilizes the geometric information of boundary to partition the domain effectively. Secondly, by optimal transport-based point alignment, it ensures that the clustering of object surfaces, i.e., domain boundaries, remains consistent across different instances. This alignment enables a well-structured tokenization of PDE domain, even when boundary shapes vary. As a result, the geometrically aligned tokens allow transformers to function as PDE solvers that inherently account for boundary geometry. + +# 3.2 Transformer Design for SpiderSolver + +Based on spiderweb tokenization (Figure 2 (a)), we define SpiderSolver (Figure 2) as a transformer defined over spiderweb tokenization, with SpiderAtten layers integrating coarse-grained (Figure 2 (b)) and fine-grained attentions (Figure 2 (c)). The coarse-grained attention facilitates interactions among spiderweb-like tokens. The fine-grained attention serves as a finer-level complement, capturing interactions between boundary points and their near interior points adjacent to the boundary. + +As shown in Figure 2, we summarize the overall process of SpiderSolver. First, a linear embedding maps the input $G \in \mathbb{R}^{N \times d}$ or with the observed physical quantity, to initial feature $\mathbf{x}^0 \in \mathbb{R}^{N \times C}$ . Next, $\mathbf{x}^0$ passes sequentially through the $L$ SpiderAtten blocks (Figure 2 (d)). The $l$ -th $(l \in [0, L - 1])$ SpiderAtten block is defined as: + +$$ +\mathbf {f} ^ {l} = \text {S p i d e r w e b - T o k e n i z a t i o n} \left(\mathbf {x} ^ {l}\right), +$$ + +$$ +\tilde {\mathbf {x}} ^ {l + 1} = \mathbf {x} ^ {l} + \operatorname {T 2 P} \left(\text {C o a r s e - A T} \left(\mathbf {f} ^ {l}\right), \mathbf {x} ^ {l}\right) + \text {F i n e - A T} \left(\operatorname {L N} \left(\mathbf {x} ^ {l}\right)\right), \tag {3} +$$ + +$$ +\mathbf {x} ^ {l + 1} = \text {F e e d F o r w a r d} \left(\mathrm {L N} \left(\tilde {\mathbf {x}} ^ {l + 1}\right)\right) + \tilde {\mathbf {x}} ^ {l + 1}, +$$ + +where the "Coarse-AT" refers to coarse-grained attention, "Fine-AT" refers to fine-grained attention, "LN" denotes layerNorm, and $\mathbf{x}^l,\tilde{\mathbf{x}}^{l + 1}\in \mathbb{R}^{N\times C},\mathbf{f}^l\in \mathbb{R}^{M\times C}$ . The T2P operator transforms the + +token features back to update the point-wise features. The "FeedForward" operator consists of two linear layers with a non-linear activation in between. The operation Spiderweb-Tokenization $(\mathbf{x}^l)$ is defined as the concatenation of average pooled features of $\mathbf{x}^l$ over spiderweb-token domains $\mathcal{T}_{pq}$ , for $p \in [1, m_B]$ , $q \in [1, m_I]$ , together with $m_B$ additional features by average feature pooling over $m_B$ boundary clusters. This coarse-level tokenization facilitates the fast self-attention computation from the interactions of $N$ points to $M$ tokens. + +After $L$ SpiderAtten blocks, a linear embedding layer over $\mathbf{x}^L$ is applied to obtain the output physical quantities, i.e., the PDE solution. We next introduce the operators of Coarse-AT(·), T2P(·,·), Fine-AT(·) of Equation (3). + +# 3.2.1 Coarse-grained Attention (Coarse-AT) + +The coarse-grained attention (Figure 2 (b)), denoted as Coarse-AT in Equation (3), is defined over the $M$ tokens of the spiderweb tokenization of the PDE domain. For the $l$ -th SpiderAtten block, with the token-level feature $\mathbf{f}^l \in \mathbb{R}^{M \times C}$ , we apply the self attention, denoted as $\mathcal{A}(\cdot)$ : + +$$ +\hat {\mathbf {f}} ^ {l} = \mathcal {A} \left(\mathbf {f} ^ {l}\right) = \left[ \mathcal {A} _ {1} \left(\mathbf {f} ^ {l}\right); \dots ; \mathcal {A} _ {h} \left(\mathbf {f} ^ {l}\right) \right] W _ {0} ^ {l}, \quad \text {w h e r e} \mathcal {A} _ {i} \left(\mathbf {f} ^ {l}\right) = \operatorname {S o f t m a x} \left(Q _ {i} K _ {i} ^ {\top} / \sqrt {d _ {k}}\right) V _ {i}, \tag {4} +$$ + +to conduct multi-head attention, where $d_{k}$ denotes the dimension of key vectors, $[\cdot]$ denotes concatenation, $T_{i} = \mathrm{Linear}(\mathbf{f}^{l},W_{T,i}^{l})$ for $T_{i}\in \{Q_{i},K_{i},V_{i}\} ,i = 1,\dots ,h$ , and $h$ denotes the number of heads. "Linear" refers to a linear layer and generates $Q,K,V\in \mathbb{R}^{M\times C}$ in attention. Note that, the self-attention is conducted over the spiderweb tokens instead of all the interior points of PDE domain. Thus, the output of the operator Coarse-AT is $\hat{\mathbf{f}}^l$ , i.e., the coarse-grained attention feature. + +# 3.2.2 From Token to Point-Wise Features (T2P) + +After coarse-grained attention, the attended token features $\hat{\mathbf{f}}^l = \mathrm{Coarse - AT}(\mathbf{f}^l)$ will be transformed into point-wise features and added to the corresponding point-wise features $\mathbf{x}^l$ . Let $\mathbf{x}_I^l\in \mathbb{R}^{N_I\times C}$ , $\mathbf{x}_B^l\in \mathbb{R}^{N_B\times C}$ be the point-wise features $\mathbf{x}^l$ in the interior and on the domain boundary respectively. Let $\hat{\mathbf{f}}_I^l\in \mathbb{R}^{m_Bm_I\times C}$ and $\hat{\mathbf{f}}_B^l\in \mathbb{R}^{m_B\times C}$ be the spiderweb token features $\hat{\mathbf{f}}^l$ of the interior and boundary tokens respectively. The updated point-wise features are computed based on the weighted token aggregation, which globally aggregates all the token features to compute the point-wise update, formulated as: + +$$ +\hat {\mathbf {x}} _ {I} ^ {l} = \eta^ {l} \hat {\mathbf {f}} _ {I} ^ {l}, \hat {\mathbf {x}} _ {B} ^ {l} = \xi_ {B} ^ {l} \hat {\mathbf {f}} _ {B} ^ {l} + \xi_ {I} ^ {l} \hat {\mathbf {f}} _ {I} ^ {l}, \text {w h e r e} \eta^ {l} = \operatorname {L i n e a r} \left(\mathbf {x} _ {I} ^ {l}\right), \xi_ {B} ^ {l} = \operatorname {L i n e a r} \left(\mathbf {x} _ {B} ^ {l}\right), \xi_ {I} ^ {l} = \operatorname {L i n e a r} \left(\mathbf {x} _ {I} ^ {l}\right), \tag {5} +$$ + +where the three "Linear" layers are respectively with learnable parameters $W_{I}^{l}, W_{B}^{l}$ and $W_{I2B}^{l}$ , and generates $\eta^l \in \mathbb{R}^{N_I \times m_B m_I}, \xi_B^l \in \mathbb{R}^{N_B \times m_B}, \xi_I^l \in \mathbb{R}^{N_B \times m_B m_I}$ as weighting matrix to combine spiderweb token features. Therefore, the operator T2P outputs updated features $[\hat{\mathbf{x}}_I^l; \hat{\mathbf{x}}_B^l]$ . It facilitates the point-wise feature updating by the coarse-grained attention features over spiderweb tokens. + +# 3.2.3 Fine-grained attention (Fine-AT) + +The fine-grained attention (Figure 2 (c)) is designed to enhance the interaction between features of points located on boundary and points near boundary. This design is motivated by the observation that as points get closer to boundary, their physical characteristics are increasingly influenced by the boundary geometry. In practical applications, crucial physical quantities, such as the drag coefficient of a car and the lift coefficient of an airfoil are derived from the features on or near the boundary. + +To effectively capture these boundary-related interactions, fine-grained attention is introduced to facilitate feature interactions between the domain boundary points $\mathcal{B}_G$ and their nearest interior domain neighboring points in the set of $\mathcal{T}_B = \{g\in \mathcal{I}_G|\mathrm{SDF}(g)\in (d_0,d_1]\}$ . The corresponding features of these boundary and near-boundary interior points in $\mathcal{T}_B$ are concatenated to form a combined feature representation $\mathbf{x}_F$ . The attention is then applied to this representation to capture the underlying physical dependencies by $\mathcal{A}(\mathbf{x}_F)$ , which is the output of the operator Fine-AT. + +# 3.2.4 Network Training and Testing + +The proposed transformer contains the learnable parameters in the two embedding layers and $L$ SpiderAtten blocks in Equation (3). The network is trained based on the relative $L_{2}$ -norm loss + +between the network output and the ground-truth PDE solutions over the training set [14]. In the training phase, spiderweb tokenization is performed over the training dataset as a data pre-processing procedure. In the testing phase, given a new instance of the PDE geometric domain and/or boundary condition, the PDE domain is divided into token sub-regions by aligned spectral clustering, and then the SpiderSolver outputs the PDE solution based on the spiderweb tokenization. + +# 4 Experiments + +We evaluate SpiderSolver on five datasets spanning industrial, biomedical, and fundamental PDE tasks. The template shapes of Shape-Net Car and AirfRANS are visualized in Figure 9 in Appendix B. As the Blood Flow, Bounded Navier-Stokes and Darcy Flow datasets have fixed geometries, optimal transport-based alignment is not required. See Appendix C for more details on five datasets. + +Shape-Net Car [33] consists of 889 simulated samples based on Reynolds-Average Navier-Stokes equations, with car shapes from the ShapeNet "car" category [34]. Each sample includes velocity and pressure fields solved via a finite element method over 32,186 mesh points. Using irregularly discretized car and surrounding space as input, the model is trained to predict these fields, from which the drag coefficient is subsequently derived. The shapes of cars randomly selected from Shape-Net Car are visualized in Figure 7 in Appendix B. + +Table 1: Results on Shape-Net Car and AirfRANS datasets. Vol: error of surrounding physics field; Surf: error of surface physics field. $C_D$ , $C_L$ : error of drag and lift coefficients; $\rho_D$ , $\rho_L$ : Spearman's rank correlation of drag and lift coefficients. + +
MethodsShape-Net CarAirfRANS
Vol ↓Surf ↓CD ↓ρD ↑Vol ↓Surf ↓CL ↓ρL ↑
Simple MLP0.05120.13040.03070.94960.00810.02000.21080.9932
G-SAGE [28]0.04610.10500.02700.96950.00870.01840.14760.9964
PointNet [29]0.04940.11040.02980.95830.02530.09960.19730.9949
G-U-Net [30]0.04710.11020.02260.97250.00760.01460.16770.9944
MG-Net [31]0.03540.07810.01680.98400.02140.03870.22520.9945
GNO [2]0.03830.08150.01720.98340.02690.04050.20160.9934
Galerkin [13]0.03390.08780.01790.97640.00740.01590.23360.9957
Geo-FNO [6]0.16700.23780.06640.82800.03610.08200.66140.9257
GNOT [32]0.03290.07980.01780.98330.00490.01520.19920.9942
GINO [5]0.03860.08100.01840.98260.02970.04820.18210.9958
3D-GeoCA [8]0.03190.07790.01590.9842////
Transolver [14]0.02280.07930.01290.99160.00250.00800.10260.9977
SpiderSolver0.02100.07380.01000.99280.00170.00430.07410.9988
+ +# AirfRANS [35] comprises 1,000 + +high-fidelity simulations over 32,000-point meshes of 4- and 5-digit NACA airfoils, with variations in shape, Reynolds number, and angle of attack. Using irregularly discretized airfoil and surrounding space as input, the model is trained to predict velocity, pressure, and viscosity fields, from which lift coefficient is computed. The shapes of airfoils are visualized in Figure 8 in Appendix B. + +Blood Flow dataset [36] consists of 500 simulations of blood flow in a fixed human thoracic aorta geometry, by the Navier-Stokes equations. Each sample varies in inlet and outlet conditions, and is discretized over a tetrahedral mesh with 1,656 spatial nodes and 121 time steps. The model is trained to predict the velocity field of the flow, using pressure at the inlets and outlets. + +![](images/b5f3c0296c3f73b51bae1126892050c9ba8b7a4827d3919d2399efce135e173f.jpg) +Figure 5: Examples from three datasets. The first column shows the ground truth. In (a), (b), and (c), the first row presents the predictions by Transolver and our model, while the second row displays the point-wise $L_{2}$ norm of the difference between ground truth and predictions. + +Bounded Navier-Stokes dataset with multiple separate boundaries [37] simulates 2D fluid flow through a pipe containing several fixed pillar-like obstacles, resulting in multiple disconnected boundaries. Despite the presence of separate boundaries, the SDF at each point is uniquely defined as the minimum distance to all boundary components. See Appendix C for more details on datasets. + +Darcy Flow dataset [3] considers the steady-state of the 2-d Darcy Flow equation on the unit box. In the Darcy Flow dataset, we take the spatial outer rectangle boundary as the boundary to compute clustering and SDF for tokenization. See Appendix C for more details on datasets. + +Elasticity and plasticity datasets. Elasticity [6] models a unit cell with an arbitrary void, governed by the incompressible Rivlin-Saunders constitutive law. Plasticity [6] considers a block impacted by a rigid die, governed by an elasto-plastic constitutive model with time-dependent deformation. + +Implementation details. To ensure that our model parameters are comparable to other + +Transformer-based models, such as Trans solver, we set the number of layers as 8 and the channel of hidden features as 256 or 512, depending on the number of observed quantities of input data. All experiments are performed on a GeForce RTX 4090 GPU. See Appendix C for more details. + +Compared methods and metrics. We compare SpiderSolver with more than 18 baselines. For the Shape-Net Car and AirfRANS, we evaluate the estimation error of physical fields in the PDE domain (Vol), domain boundary surface (Surf), drag coefficient $(C_D)$ and lift coefficient $(C_L)$ using the relative $L_{2}$ except that we follow [35, 14] to use MSE for the Vol and Surf on the AirfRANS dataset. We additionally use Spearman's rank correlations of drag and lift coefficients, respectively in the Shape-Net Car and AirfRANS datasets. For the other datasets, we use the relative $L_{2}$ error as the metric. See Appendix E for details. + +Table 2: Results on Blood Flow dataset. + +
MethodsVelocity ↓
Simple MLP0.3080
DeepONet [4]0.8926
POD-D [38]0.3742
Geo-FNO [6]0.1209
GNOT [32]0.0411
NORM [36]0.0453
Geo-FNO [6]0.1209
3D-GeoCA [8]0.2863
GINO [5]0.1864
Transolver [14]0.0438
SpiderSolver0.0322
+ +# 4.1 Main Results + +Results on Shape-Net Car. As shown in Table 1, SpiderSolver outperforms various methods. The Spearman's rank correlation indicates that our predicted drag coefficients better match the true ranking. Figure 11 in Appendix B shows an example of spiderweb tokenization by SpiderSolver. + +Results on AirfRANS. As shown in Table 1, SpiderSolver reduces the MSE of the volume and surface physics fields by $32.0\%$ and $46.3\%$ , compared to Transolver. Figures 12 and 13 in Appendix B show an example of spiderweb tokenization by SpiderSolver and the patches learned by Transolver. + +Results on Blood Flow. As shown in Table 2, SpiderSolver achieves state-of-the-art performance. Figure 5 (c) visualizes the predicted velocity fields. + +Results on Bounded Navier-Stokes and Darcy Flow. SpiderSolver outperforms various methods on both Bounded Navier-Stokes and Darcy flow datasets. See Tables 13 and 14 in Appendix F for results. Darcy Flow dataset is defined on a regular quadrilateral domain, and both the Bounded Navier-Stokes and Darcy Flow datasets use uniform grids. Although SpiderSolver is designed for complex boundaries and irregular grids, it remains applicable to such settings. Figure 14 in Appendix B shows two examples of spiderweb tokenization on Bounded Navier-Stokes dataset. For the Bounded Navier-Stokes dataset, SpiderSolver applies spectral clustering to all obstacle surfaces, where coarse-grained attention directly captures in + +Table 3: Generalization in OOD on AirfRANS. + +
MethodsReynolds OODAngles OOD
CL↓ρL↑CL↓ρL↑
Simple MLP0.62050.95780.41280.9572
G-SAGE [28]0.43330.97070.25380.9894
PointNet [29]0.38360.98060.44250.9784
G-U-Net [30]0.46640.96450.37560.9186
GNO [2]0.44080.98780.30380.9836
Galerkin [13]0.46150.98620.38140.9821
GNOT [32]0.32680.98650.34970.9863
GINO [5]0.41800.96450.25830.9923
Transolver [14]0.38890.99110.24900.9940
SpiderSolver0.22910.99220.10620.9941
+ +teractions among tokens. As shown in Figure 14 (Appendix B), the resulting interior partitions align well with obstacle geometry, enabling attention maps to encode mutual influences between obstacles. Ablation results further show that merging obstacle surface tokens reduces accuracy (relative $L_{2}$ error 0.0432), while explicit obstacle token modeling in SpiderSolver achieves superior performance (relative $L_{2}$ error 0.0376). + +Results on elasticity and plasticity datasets As shown in Table 15 in Appendix F, SpiderSolver consistently outperforms all baselines, achieving the lowest relative $L_{2}$ error on both datasets. + +Generalization to Reynolds number and airfoil angle variations on AirfRANS. We assess the generalizability of SpiderSolver in two scenarios using AirfRANS [35]. (1) Reynolds extrapolation (Reynolds OOD): the training set includes samples with Reynolds between 3 and 5 million, while the Reynolds of the test set spans 2 to 3 and 5 to 6 million. (2) Angle of attack extrapolation (Angles OOD): the training set covers angles from $-2.5^{\circ}$ to $12.5^{\circ}$ , and the test set includes angles from $-5^{\circ}$ to $-2.5^{\circ}$ and $12.5^{\circ}$ to $15^{\circ}$ . Table 3 shows the metrics for the OOD experiments on AirfRANS dataset. The results demonstrate that our proposed SpiderSolver achieves consistently better out-of-distribution generalization on the AirfRANS dataset. + +Generalization to shape variations of cars on ShapeNet-Car. We compute the average point-wise Euclidean distance from each car shape to the template shape, and then select the 200 nearest shapes for training and the 100 farthest shapes for testing. The results are reported in Table 4. SpiderSolver achieves good out-of-distribution generalization to shape variations of cars on ShapeNet-Car. In Appendix D, we analyze the diversity of car geometries, + +Table 4: Generalization to shape variations of cars on Shape-Net Car dataset. + +
MethodsShape-Net Car
Vol ↓Surf ↓CD ↓ρD ↑
Transolver [14]0.06600.1910.07350.9142
SpiderSolver0.05100.1610.05500.9222
+ +and we further visualize how prediction error varies with different distances to the template shape, but observe no clear correlation between model accuracy and shape distance to template shape. + +# 4.2 Ablation Study and Model Analysis + +Effects of key components of the network. We conducted the ablation study on the key components in Table 5. Referring to Equation (3), "w/o Coarse-A" denotes our SpiderSolver without the Coarse-AT and the T2P operator, and "w/o Fine-AT" denotes SpiderSolver without Fine-AT operator. "T2P-local" refers to T2P operator in Equation (5) that is computed locally in each spiderweb token. Specifically, in "T2P-local", Equation (5) is computed locally for each spiderweb token to generate the updated feature for the points belonging to the token. As shown in Table 5, removing Coarse-AT and Fine-AT considerably degrades the model's performance. Compared with removing Fine-AT, the performance drop when Coarse-AT is removed is more remarkable, highlighting its crucial role. Table 12 in Appendix E compares the runtime of SpiderSolver with and without Fine-AT. + +Table 5: Ablation study of key components of SpiderSolver. + +
Ablation methodsShape-Net CarAirfRANSBlood Flow
Vol ↓Surf ↓CD ↓ρD ↑Vol ↓Surf ↓CL ↓ρL ↑Velo ↓
T2P-local0.02180.07410.01140.98810.00320.00550.07170.99890.0345
w/o Fine-AT0.02360.08960.01360.98910.00220.00990.10130.99860.0347
w/o Coarse-AT0.03240.09620.02040.97650.12580.01010.15760.99620.0634
w/o Coarse-AT and Fine-AT0.06450.23900.06290.85850.19170.53140.17930.99150.0874
SpiderSolver0.02140.07250.00960.99570.00160.00390.06650.99840.0322
+ +Ablation study on reference shape selection. The reference shape serves only to provide a consistent indexing, and it does not affect the geometry of the resulting template shape. As shown in Table 6, our model exhibits strong robustness to the choice of reference shape on Shape-Net Car, consistently outperforming Transolver across different reference settings. Furthermore, spectral clustering results across five reference shapes demonstrate high consistency, with Dice of 0.9975 and IoU of 0.9951. Dice and IoU are computed based on the intersection and union of the five clusterings. + +Analysis of hyper-parameters on validation set. We compare the performance and number of parameters of models with varying feature channels $C$ and network layers $L$ in Figure 6, and the varying $m_I$ and $m_B$ in Table 8. Ours consistently outperforms Trans solver across different $C$ and $L$ . + +Robustness to noisy boundary points. To evaluate the robustness of spiderweb tokenization to noisy boundaries, we introduced Gaussian perturbations with varying noise levels to the car surface point cloud. As shown in Table 7, the spectral clustering and SDF-based partitioning preserve domain structure under perturbations. Furthermore, model evaluation on Shape-Net Car confirms that the robustness of patch partitioning directly contributes to stable performance against noisy boundary conditions. + +Table 6: Ablation study on selection of the reference shape for Shape-Net Car. + +
ShapesVol ↓Surf ↓CD ↓ρD ↑
Reference shape 10.02150.07300.00950.9934
Reference shape 20.02120.07210.01030.9935
Reference shape 30.02150.07250.01050.9921
Reference shape 40.02100.07270.01030.9934
Shape used in Table 10.02100.07380.01000.9928
Mean0.02120.07280.01010.9930
Standard Deviation0.00020.00060.00030.0005
+ +Table 7: Robustness of spiderweb tokenization and model performance under noisy boundary perturbations on Shape-Net Car ( $\sigma$ is the noise standard deviation). + +
σDice ↑IoU ↑Vol ↓Surf ↓CD ↓ρD ↑
0.00011.00001.0000.02100.07380.01000.9928
0.0010.99940.99870.02170.07390.01030.9924
0.010.98680.97430.02300.07900.01190.9919
0.030.95340.91350.03100.08290.01350.9901
+ +![](images/f044cd9d320f50a02bb6f083dd7330933c082794f99dd806400e4d7d8c703d30.jpg) +Figure 6: Performance of our model and Transolver on the Blood Flow dataset with varying channels $C$ and layers $L$ (point size $\propto$ model size). + +![](images/6c0f92e59ab46fce6300c16b11562e9dfb3538f14b0fdf6c81ce22e57c23bfe8.jpg) + +Table 8: Ablation study of SpiderSolver with varying $m_I$ and $m_B$ on the validation set. + +
(mI,mB)AirfRANSBlood Flow
Vol ↓Surf ↓CL ↓ρL ↑Velo ↓
(2, 2)0.00260.01300.08850.99870.0372
(4, 2)0.00210.01010.08180.99880.0366
(2, 4)0.00190.00640.07510.99890.0359
(6, 2)0.00200.00790.06240.99900.0352
(2, 6)0.00190.00750.05830.99880.0350
+ +Computational cost. On the Shape-Net Car dataset, SpiderSolver has 4.05M parameters and the network inference time for one PDE is 0.058s, compared with the 3.86M and 0.022s respectively by previous state-of-the-art method Transolver [14]. Note that it takes around 50 minutes [33] required by the traditional $k$ -epsilon turbulence simulations. See Appendix E for details on inference time. + +# 5 Conclusion, Limitations and Future Work + +Conclusion and impact statement. This paper proposed a novel transformer SpiderSolver, to solve PDEs with complex domain geometry. SpiderSolver is based on the proposed spiderweb tokenization and integrates coarse-grained and fine-grained attention. Experiments on five datasets demonstrate its superiority. This work has made a fundamental contribution to the transformer architecture design in scientific computation, and may have an impact on the fast computation of numeric solutions to PDEs with complex boundaries, in the applications of science and engineering problems. + +Limitations and future work. SpiderSolver depends on the boundary and is currently not applicable to PDEs with open-boundary domains. Moreover, it is currently applied to PDE with boundary shapes of one type of objects. In future work, we plan to extend the framework to multiple classes of shape boundaries, possibly by making the network parameters aware to different geometric classes. 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In International Conference on Machine Learning, 2023. + +# A Implementation Details of Spectral Clustering + +The process of spectral clustering [39] can be summarized in three main stages (clusters $= k$ ): + +Step 1: Constructing the Affinity Matrix. Given the input data points $(x_{1}, x_{2}, x_{3}, \ldots, x_{n})$ , the algorithm begins by treating all points as nodes in a graph. The similarity between these nodes is then quantified and stored in an affinity matrix $W$ by K-Nearest Neighbors (KNN) method. The process is as follows: + +$$ +w _ {i j} = w _ {j i} = \left\{ \begin{array}{l l} \exp \left(- \frac {\| x _ {i} - x _ {j} \| ^ {2}}{2 \sigma^ {2}}\right), & \text {i f} x _ {i} \in \operatorname {K N N} (x _ {j}) \text {o r} x _ {j} \in \operatorname {K N N} (x _ {i}) \\ 0, & \text {o t h e r w i s e} \end{array} \right. +$$ + +Step 2: Constructing the Laplacian Matrix. We use the normalized Laplacian matrix, defined as follows: + +$$ +L = I - D ^ {- 1 / 2} W D ^ {- 1 / 2} +$$ + +where $I$ is the identity matrix, $W$ is the affinity matrix, and $D = \mathrm{diag}(d_1,\dots,d_n)$ , $d_i = \sum_{j=1}^n w_{i,j}$ . + +Step 3: Computation of Eigenvectors and Clustering Completion. First, compute the first $k$ eigenvectors $(\mathbf{u}_1,\dots ,\mathbf{u}_k)$ of the normalized Laplacian matrix $L$ , forming the eigenvector matrix $U\in \mathbb{R}^{n\times k}$ . Second, denote $U = (\mathbf{y}_1,\dots ,\mathbf{y}_n)^\top$ , where each $\mathbf{y}_i\in \mathbb{R}^k$ corresponds to the $i$ th row of $U$ . Third, cluster the points $\mathbf{y}_i$ $(i = 1,\dots ,n)$ in $\mathbb{R}^k$ with the k-means algorithm into clusters $C_1,\dots ,C_k$ . + +We employ standard SpectralClustering function from the scikit-learn library, configured as follows: SpectralClustering(n_clusters=clusters, affinity='nearest_neighbors', random_state=42). + +# B Visualization of Template Shapes, Boundary Clustering and Spiderweb Tokenization + +In this section, we present visualizations of the template shapes of Shape-Net Car and AirfRANS, together with examples of surface clustering and spiderweb tokenization. + +![](images/c9a60ecf5232187f4f085ba3dc54507a56bc9a0eaf725ffb7c4e7e1325be3547.jpg) +Random Shape 1 + +![](images/07fdd1a3fe463c65ecd207b77143e6480ff0880534dce3977e67f79aa2b384e5.jpg) +Random Shape 2 + +![](images/305f5f1762e33c02cbb23b1ec96bf5ac1f90942a8f12f1c31038d6b0c9d58aa6.jpg) +Random Shape 3 + +![](images/33d201cfd641775a02873953130869cacda7e0350975fe1a81c0bb813f216226.jpg) +Random Shape 4 + +![](images/ff106253e3b2d6321af778a9d231e1e39136994aada95b8d57ac078e9753f858.jpg) +Random Shape 5 + +![](images/c2e30dcf29a61ddb2ddc3a44800e6c9ee1bbca325afd569330dfa8dd17362a71.jpg) +Random Shape 6 + +![](images/44d1ec64237ff8305c4a683767eba1e7944fa24e5244575d384e7c8581c13442.jpg) +Random Shape 7 +Figure 7: Eight randomly selected shapes from the Shape-Net Car dataset. + +![](images/7de6eacd60f454b147d34e00e312aa4f06e0e1b80db5ba1fbfc9425e35a2ca5e.jpg) +Random Shape 8 + +Figures 7 and 8 show the eight randomly selected shapes from the Shape-Net Car and AirfRANS, respectively. Figure 9 shows the template shapes of the Shape-Net Car and AirfRANS. Figure 10 shows the clustering of instances of cars in the Shape-Net Car, with individual clustering (i.e., separately clustering each car surface) and our aligned clustering method. The results demonstrated that our proposed aligned clustering can cluster different instances of cars with aligned clusters including the cluster indexes highlighted by different colors. + +Figure 11 shows the point sets of different spiderweb tokens for an instance of a car in the Shape-Net Car dataset. Each row of Figure 11 shows the tokens corresponding to the same cluster of car surface, with increasing SDF values to the domain boundary, i.e., the car surface, from left to right. We can observe that the different tokens contain point sets with geometric structures resembling the shape of segments of a car surface. + +![](images/9cdf3463844a47c8af2ada0fdcc871e2fa8e88b3a3ba2479bd6110c493fa8950.jpg) +Random Shape 1 + +![](images/2871160bc2c8a478b53d0d30bcc4be5ca0465e467770f7888bdc543468bc1e60.jpg) +Random Shape 2 + +![](images/32629d9807a29f57da1c3de87f1435fee708af50fc4c8171fc5d9033ad246aa5.jpg) +Random Shape 3 + +![](images/e15a3069c80088d5e4a39795389349a19dc5fc400c4d3266b390bbd6bd8eb75b.jpg) +Random Shape 4 + +![](images/322d58f4ef7fdfca336c3f37dcb6b643852efb74d6b82976db984796102ce75e.jpg) +Random Shape 5 + +![](images/cb50e0ad6b61a3fd86ddaea032535567209149c892b91e430d49fb8ff46134a2.jpg) +Random Shape 6 + +![](images/0cc44b37246ab010d7376b10aac8f379e8ab64f29551ac89d61a1d6cd8f776f3.jpg) +Random Shape 7 + +![](images/3f550dc7a8224ea894362c98ee3f075f262a599dec257edb7bc797d589b301be.jpg) +Random Shape 8 + +![](images/f31742d44dcc4bc6d6db0db120559d314afcf61a232ff0ddf05c9ba3413beba8.jpg) +Figure 8: Eight randomly selected shapes from the AirfRANS dataset. +Figure 9: Template shapes of the Shape-Net Car dataset (left) and AirfRANS dataset (right). + +![](images/c1cd50f228a4975fe29ad8dd4b9363eb7cc43c5876c7664429ecab0aa6c54ea6.jpg) + +![](images/41c299d1699bcdc280b52d2cacf959d2437f9188c00f2f8fda6605887bdffe3c.jpg) + +![](images/d65d7a35a806db36af08ddf20b9d96a26f02f2b7b43a0a50f10bf9345ba06c48.jpg) + +![](images/feef74c064e304bcd059c029494e4d2949c8e50018425faa3640ab740205897c.jpg) + +![](images/54292b95cbf8a4e64c84119e2c90481a23f72e6b62c63655e73e40f4795bc48e.jpg) + +![](images/16a6e9370743fd9087a0b6906f76fbff0cb882c25b8f464236ab493425ed1db4.jpg) + +![](images/b7b0c471eeaef034150c0a0b0bad9cf1d0e51de5b0341cbb56ef0f91b2158086.jpg) + +![](images/7f44c16b8eefef6260bf9c190d7655dccca9fbd003b7692d7e63dc157b6a08d4.jpg) +Figure 10: Clustering of car surface (i.e., the PDE domain boundary) by conducting spectral clustering on separate cars (denoted as individual clustering), and our proposed aligned clustering. +Figures 12 and 14 show an example of spiderweb tokenization of an instance of the AirfRANS and the Bounded Navier-Stokes dataset, respectively. Different colors highlight different tokens, which + +![](images/1a320851543f8138de7a6fa620434308c250a64d2ca7fff72ca3d010eeb632eb.jpg) + +![](images/f395fa3eae128a813505a86196b41f173f6d2dfcc5123dd4565fa31f6f3f50ac.jpg) +Figure 11: Point sets of different spiderweb tokens on an instance of car. The $p$ -th row visualizes the point sets belonging to tokens $T_{p,q}, q = 1, \dots, m_q$ for the $p$ -th $(p = 1, \dots, m_p)$ cluster of the car surface. + +![](images/0f71ea5961e727319cf4d9452de11584367e7036bfbda061bed87c90329c3d68.jpg) +Figure 12: An example of spiderweb tokenization for an airfoil with different tokens highlighted by different colors. + +well split the point sets surrounding the airfoil into sub-regions. As presented in Section 3.1.2, the determination of width, i.e., $[d_{q - 1}, d_q]$ of each token, is to ensure that the regions with $[d_{q - 1}, d_q]$ for $q = 1, \dots, m_q$ contain the same number of points, thus adaptive to the different density of point clouds. For comparison, Figure 13 shows representative patches learned by Transolver on the AirfRANS dataset. + +![](images/19f3c07066a3b1b3304723c34b836e83323eff0ec9348cd562f8e0e8cee0d9e6.jpg) +Figure 13: Representative patches learned by Transolver on the AirfRANS dataset. Colors indicate the magnitude of the weights of patches. + +![](images/e81d390fff8f9c4ab66d4688965032c7ae9d59d22f2a6d32b87f813d05359f27.jpg) +Figure 14: Spiderweb tokenization for Bounded Navier-Stokes dataset highlighted by different colors. Left: $m_B = 2$ , $m_I = 5$ ; Right: $m_B = 4$ , $m_I = 5$ . + +![](images/4dc24c2ea6fe0213f57f69036221cd0a6a652e5e00fd7aac0b9fb80cef590045.jpg) + +# C Detailed Descriptions of Five Datasets and More Implementation Details of Experiments + +Shape-Net Car dataset. This dataset is designed to simulate the behavior of a car traveling at a speed of $72\mathrm{km / h}$ , using realistic physical parameters and applying the Reynolds-averaged Navier-Stokes (RANS) equations for airflow modeling. The input models are derived from the "car" category in ShapeNet [34], with modifications to remove elements like side mirrors, spoilers, and tires, focusing more on the vehicle's aerodynamic properties. High-resolution spatial grids are used to solve the Navier-Stokes equations, simulating airflow for a 10-second duration. The time-averaged velocity and pressure fields are obtained by averaging the results from the last 4 seconds of the simulation. Each simulation involves degrees of freedom ranging from 600k to 700k, and each run takes around 50 minutes to complete. A finite element Navier-Stokes solver resolves the fluid dynamics, using the k-epsilon turbulence model and SUPG stabilization technique [40]. This setup accurately represents the flow around the vehicle, capturing complex phenomena such as boundary layers and vortex shedding at the surface. + +For the Shape-Net Car dataset, the space is discretized into 32186 mesh points. Overall, a total of 889 samples with different car shapes are generated to simulate. 789 samples are used for training, and the other 100 samples are for testing, which follows that of the Transolver [14]. The model is trained to predict the velocity and pressure value for each point, then the drag coefficient can be calculated based on these estimated physical fields. + +For the Shape-Net Car dataset, the input dimension of our model is $32186 \times 7$ , where 32186 represents the number of discretized points in the PDE domain, and 7 corresponds to the point coordinates, SDF values, normal vectors of boundary points and direction vectors of interior points. The output of our model is $32186 \times 4$ , where 32186 represents the number of discretized points in the PDE domain, and 4 corresponds to the wind field of interior points and the pressure of boundary points (car surface). The drag coefficient can be calculated based on these predicted physics quantities. + +AirfRANS dataset. AirrRANS [35] contains high-fidelity simulation data for RANS equations on airfoils that are the 4 and 5-digit series of the National Advisory Committee for Aeronautics. Each case is discretized into 32,000 mesh points. By changing the airfoil shape, Reynolds number, and angle of attack, AirrRANS provides 1000 samples, where 720 samples are used for training, 80 samples are used for validation, and 200 for the test set. The model is trained to predict the air velocity, pressure, and viscosity for each point, and calculate the lift coefficient. AirrRANS includes high-fidelity simulation data for RANS equations applied to both the NACA 4-digit and 5-digit series airfoils. Specifically, 1000 simulations were run, each defined by the airfoil, Reynolds number, and angle of attack. The goal of the simulations is to reflect realistic conditions, with the Mach number capped at 0.3 and the Reynolds number ranging from 2 million to approximately 6 million. The angle of attack varies from $-5^{\circ}$ to $15^{\circ}$ . The simulations were performed using the steady-state RANS solver simpleFOAM with the SIMPLEC algorithm [41], combined with the $k-\omega$ SST turbulence model [42], until the lift and drag coefficients converged. + +For the AirfRANS dataset, the input dimension of our model is $32000 \times 7$ , where 7 corresponds to the point coordinates, inlet velocity, SDF values, normal vectors of boundary points, and direction vectors of interior points. The output of our model is in size of $32000 \times 4$ , where 32000 represents the number of discretized points in the PDE domain, and 4 corresponds to the wind field of interior points, the pressure of boundary points (airfoil surface), and turbulent viscosity of interior points. We aim to estimate the lift coefficient of airfoils, as well as the pressure on both interior and boundary points. + +Blood Flow dataset. Blood Flow [36] contains the simulation data of the Navier-Stokes equations for the blood flow dynamics in the human thoracic aorta. In this dataset, the computational domains are the same, i.e., the human thoracic aorta, but the pressure and velocity at the inlet/outlets are given 500 different samples as boundary conditions. The spatial domain is represented by a tetrahedral mesh with 1656 nodes and the temporal domain is discrete with 121 temporal time points. 400 samples are used as training data, 50 samples are used as validation data and the remaining 50 as test data. The model is trained to output the velocity field of the blood flow. This dataset contains simulation data for the Navier-Stokes equations modeling blood flow dynamics in the human thoracic aorta. The aorta has one inlet (ascending aorta) and five outlets (descending aorta, left/right subclavian arteries, and left/right common carotid arteries), with velocity boundary conditions at the inlet and pressure boundary conditions at the outlets. The dataset describes the time-varying velocity and pressure over one cardiac cycle (1.2 s). Blood is modeled as a homogeneous Newtonian fluid with a density of $1060\mathrm{kg / m^3}$ and a viscosity of $0.0035\mathrm{N\cdot s / m^2}$ , assuming laminar flow in the aorta and rigid vessel walls with no-slip conditions. A total of 500 velocity/pressure curves are used as inputs, and the velocity field is simulated as the output using COMSOL Multiphysics [36]. + +For the Blood Flow dataset, we aim to predict the velocity field at the interior points of the aorta using pressure values at the inlets and outlets of the aorta. The geometry of the aorta remains consistent across samples, eliminating the need for aligned boundary clustering for different instances of the object surface. For the Blood Flow dataset, the spatial domain is represented by a tetrahedral mesh with 1,656 nodes, and the velocity field is characterized by a dimension of $1656 \times 1 \times 3$ . Pressure values at the two inlets and four outlets are recorded across 121 time steps, with a variable dimension of $1 \times 121 \times 6$ . The first two dimensions of the above two variables are replicated to match each other, and they are concatenated along the third dimension, yielding a variable dimension of $1656 \times 121 \times 9$ . This is then resized to $1656 \times 1089$ and used as input to our model. The output of the model has a + +dimension of $1656 \times 363$ , and it is resized to $1656 \times 121 \times 3$ , representing the predicted velocity field at each point across the 121 time steps. + +Bounded Navier-Stokes dataset. The Bounded Navier-Stokes dataset simulates incompressible dye flow governed by the Navier-Stokes equations, with a Reynolds number of 256. At this regime, flow instability leads to periodic vortex shedding and the formation of a Kármán vortex street [43] behind cylindrical obstacles. The presence of multiple cylinders and downstream obstructions induces complex interactions that challenge predictive models. To generate the data, [37] run simulations over $10^{5}$ steps with fixed cylinder positions but varying initial conditions. From each trajectory, multiple frames are extracted and randomly partitioned into training and test sets. The numerical solver employs a finite difference scheme (MAC method) with CIP-based advection, enabling high-resolution flow capture while minimizing numerical dissipation. + +The model predicts the physical field at the 10th time step based on the initial conditions. A total of 1,000 training sequences, 200 validation sequences and 200 test sequences are used at a spatial resolution of $64 \times 64$ . Despite the presence of separate boundaries, the signed distance function (SDF) at each point is uniquely defined as the minimum distance to all boundary components. Consequently, our SDF-based spiderweb tokenization produces non-overlapping partitions, even in such complex domains. + +Darcy Flow dataset. The Darcy Flow equation is a second-order elliptic PDE, which is defined as + +$$ +- \nabla \cdot (a (x) \nabla u (x)) = f (x), \quad x \in (0, 1) ^ {2}, +$$ + +$$ +u (x) = 0, \quad x \in \partial (0, 1) ^ {2}, \tag {6} +$$ + +where $f(x)$ is the forcing function, $a(x)$ is the diffusion coefficient, and $u(x)$ is the density of fluid. We use the same parameter settings for the Darcy Flow equation as in FNO [3]. The diffusion coefficient $a(x)$ is generated from $\Psi_{\sharp}\mathcal{N}(0,(-\Delta +9I)^{-2})$ with zero Neumann boundary conditions on the Laplacian operator $\Delta$ . The $\Psi_{\sharp}$ is the point-wise push-forward operator that takes the value of 12 if $x > 0$ , and 3 elsewhere. We learn the mapping from $a(x)$ to the steady state $u(x)$ , fixing the forcing term $f(x) = 1$ . We use 1000, 200, and 200 pairs of $a(x)$ and $u(x)$ in the train, validation, and test sets, respectively. The Darcy Flow equation dataset is set to four resolutions, i.e., $s = 85$ , where $s$ represents that both the $a(x)$ and $u(x)$ are discretized into $s\times s$ grid. + +For the Darcy Flow Dataset, the coefficient $a(x)$ results in disconnected multi-regions. In such a dataset, we can take the spatial outer rectangle boundary as the boundary to compute clustering and SDF for tokenization. + +Table 9 shows more implementation details for the network structure and network training. + +Table 9: Settings of hyper-parameters in the SpiderSolver network and SpiderSolver training. + +
DatasetmImBLhCdkLossEpochsLROptimizerBatch Size
Shape-Net Car548825632Relative L22000.001Adam1
AirfRANS548825632Relative L24000.001Adam1
Blood Flow568851264Relative L25000.001Adam10
Bounded Navier-Stokes448825632Relative L22000.0005Adam10
Darcy Flow888825632Relative L210000.001Adam4
+ +Elasticity dataset. The Elasticity dataset consists of simulations on unit cells with arbitrary voids at the center, where the bottom edge is clamped and tension is applied on the top. Each case takes as input a tensor of shape $972 \times 2$ representing the 2D positions of discretized points, and outputs the corresponding stress field in the shape of $972 \times 1$ . A total of 1000 training samples and 200 test samples are provided, obtained using a finite element solver with about 100 quadratic quadrilateral elements. The inputs are represented as point clouds of size around 1000, and the outputs correspond to the stress fields. This dataset features an unstructured input format, which is designed to evaluate the capability of models to handle irregular geometries. + +Plasticity dataset. The Plasticity dataset is constructed from plastic forging problems, where a block of material is impacted by a rigid die with random shapes sampled from spline interpolations. The simulations are performed on a $101 \times 31$ structured mesh with 20 time steps, resulting in 900 training samples and 80 test samples. Each case takes as input the die shape discretized into a structured mesh + +of size $101 \times 31$ , and outputs the deformation of each mesh point over 20 time steps in a tensor of shape $20 \times 101 \times 31 \times 4$ , containing the deformation in four directions. The outputs correspond to the deformation fields induced by time-dependent die motions. This dataset emphasizes temporal dynamics and structured mesh inputs, providing a challenging benchmark for models that combine spatial and temporal learning. + +# D Shape Distribution of Shape-Net Car and Error Analysis across Shape Distance + +In this section, we quantify shape variation using the distance between each car and the template shape, and analyze the distribution of car geometries. We further visualize how prediction error varies with shape differences, but observe no clear correlation between shape distance and model accuracy. + +Distance from each car shape to the template shape. We align each shape to the template shape and compute the average point-wise Euclidean distance from all 789 training car surfaces to the template shape, as summarized in Table 10. + +Table 10: Distance to template shape + +
CasesDistance (meter)
Min0.03254
Max0.6296
Mean0.1802
Standard Deviation0.09741
Top 200 closest distances[0.03254, 0.1031]
Top 100 farthest distances[0.3005, 0.6296]
+ +The template shape serves as an intermediate reference for aligning and clustering different car geometries. Figure 15 presents the template shape, along with the four nearest and four farthest shapes to the template shape, respectively. The distance to the template is computed as the average point-wise Euclidean distance following optimal transport (OT) alignment. As shown in Figure 1, this distance metric captures shape variation across the Shape-Net Car dataset. + +![](images/7fd086346aa9d8b18bfde84e357c489682ebef8f18f3fe148c1ad322f2249869.jpg) +Figure 15: Template shape and closest and farthest four shapes to the template shape, with distances (meters) in parentheses. + +Shape distribution of cars on Shape-Net Car dataset. To characterize the shape distribution, we visualize the distances from training shapes to the template shape using a histogram and a Q-Q plot in Figure 16. The histogram reveals a right-skewed distribution with a gradually decaying tail, while the Q-Q plot indicates significant deviation from normality in the upper quantiles, suggesting heavier-than-Gaussian tails. Note: A Q-Q plot assesses whether a dataset follows a reference distribution by comparing their quantiles. In the context of heavy-tailed detection, a pronounced upward deviation of the upper quantiles from the reference line indicates that the data exhibits heavier tails than the reference (e.g., normal) distribution. + +Analyzing error across shape distance of cars. We further analyze how prediction error varies with the distance to the template shape on the test set, as shown in Figure 17 (ours) and Figure 18 (Transolver). For the prediction of our model, the two shapes farthest from the template $(0.5\mathrm{m})$ do + +![](images/3ee4589be8d4cf3277fb529cae87b3c250159ffaf8271900ab886cda28490e11.jpg) +Figure 16: The histogram and Q-Q plot of different distances to template shape of Shape-Net Car Dataset. + +![](images/dec150387ad440fdbbd53c925ffa4fb1e62d6ebd14c14dcbd3b33e137ba03f58.jpg) + +not yield the highest prediction errors. No clear correlation is observed between shape distance and prediction accuracy. + +![](images/1091ad2b2d79328c21639e9f4f72f820b5b6e92346a4904ad63672cdb2e36bac.jpg) +Figure 17: Our model's testing error with different car distance to template shape on Shape-Net Car Dataset. + +![](images/f07b7e2e2be4aef45b72467a09f36f26e125066847282c9631d461643e310efd.jpg) + +![](images/d14ac5adc587d2bf87ab942bf7fc5e6ea3a6427b3fa1cc2cb2e691cfa192e89d.jpg) + +![](images/ccb42f760b9a4f86b9ec28aa97fe2c6476d0478fb59b4e9be5be2e350d059818.jpg) +Figure 18: Transolver's testing error with different car distance to template shape on Shape-Net Car Dataset. + +![](images/e0f0343d932a57d4bf6ac2ac69b8e04af948cf185a0b24af85363af64fe70966.jpg) + +![](images/e63e5048d1d2bb12a276ba5bd08520d1ec1381b2be8d1fb74008dc3499b21e3c.jpg) + +# E Inference Time and Evaluation Metrics + +Inference time. The inference time of our model comprises the spiderweb tokenization (includes optimal transport-based matching) and the network forward pass. Spectral clustering is only performed once for each shape during training; during inference, cluster labels are propagated from the template shape to the test shape via optimal transport-based boundary points correspondences. For the Shape-Net Car dataset, spectral clustering is performed on 3,586 surface points using only the top $m_B$ eigenvectors of the Laplacian matrix (e.g., 6 eigenvectors for $m_B = 6$ ), which keeps the computational cost manageable. We report the model inference time in Table 11. Note that the prediction commonly takes around 50 minutes, which is required by the traditional $k$ -epsilon turbulence simulations. + +Comparison of models with and without Fine-AT. We also compare the inference time of the model with and without the Fine-AT on the Shape-Net Car dataset, as shown in Table 12. + +Table 11: Inference time and number of parameters on the Shape-Net Car dataset. + +
Different termsSpiderSolver (Ours)Transolver
Time of spectral clustering (only model training)0.26sN/A
Number of surface points of car35863586
Time of point matching via optimal transport0.012sN/A
Time of interior domain partition before model forward0.0032sN/A
Time of model forward0.043s0.022s
Total inference time0.058s0.022s
Number of parameters40503403860804
+ +Table 12: Comparison of Inference time of Fine-AT on the Shape-Net Car dataset. + +
SettingModel Forward time (s)Total Inference time (s)
SpiderSolver (Ours) w/o Fine-AT0.0320.047
SpiderSolver (Ours) w/ Fine-AT0.0430.058
+ +Since our evaluation is based on practical tasks of different datasets, we include several design-oriented metrics for physics fields, which will be introduced as follows. + +Relative $L_{2}$ error for physics fields. The relative $L_{2}$ error $L(u, \hat{u})$ between the true physics field $u$ and the model's predicted field $\hat{u}$ is calculated as follows: + +$$ +L (u, \hat {u}) = \frac {\| u - \hat {u} \| _ {2}}{\| u \| _ {2}}. \tag {7} +$$ + +Drag and lift coefficients. For the Shape-Net Car and AirFANS datasets, drag and lift coefficients are computed from the estimated physics fields. For unit-density fluids, the coefficients are given by: + +$$ +C = \frac {2}{v ^ {2} A} \left(\int_ {\partial \Omega} p (\omega) \left(\widehat {n} (\omega) \cdot \widehat {i} (\omega)\right) \mathrm {d} \omega + \int_ {\partial \Omega} \tau (\omega) \cdot \widehat {i} (\omega) \mathrm {d} \omega\right), \tag {8} +$$ + +where $v$ is the inlet flow velocity, $A$ is the reference area, and $\partial \Omega$ is the object surface. The pressure $p$ , the outward unit normal $\hat{n}$ , the flow direction $\hat{i}$ , and the wall shear stress $\tau$ are defined as usual. The shear stress $\tau$ is typically much smaller than the pressure term and can be approximated by the air velocity near the surface [44]. We use the setting of drag and lift coefficient from Transolver [14]. + +Spearman's rank correlation for drag and lift coefficients. Given $K$ test samples, let the true drag or lift coefficients be $D = \{D^{1},\ldots ,D^{K}\}$ and the predicted coefficients be $\widehat{D} = \{\widehat{D}^1,\dots ,\widehat{D}^K\}$ . The Spearman rank correlation is computed by ranking both sets of coefficients and then calculating the Pearson correlation between the ranks [14]: + +$$ +\rho = \frac {\operatorname {c o v} (R (D) , R (\widehat {D}))}{\sigma_ {R (D)} \sigma_ {R (\widehat {D})}}, \tag {9} +$$ + +where $R$ is the ranking function, cov denotes covariance of the ranks, and $\sigma$ is the standard deviation. Higher $\rho$ indicates better alignment between the predicted and true coefficients [45]. + +# F Results on Bounded Navier-Stokes, Darcy Flow, elasticity and plasticity datasets + +Tables 13 and 14 show the results on Bounded Navier-Stokes and Darcy Flow, respectively. Table 15 shows the results on Elasticity and Plasticity datasets. + +Table 13: The results on the Bounded Navier-Stokes dataset. + +
MethodsRelative L2 error ↓
FNO [3]0.0472
GNOT [32]0.0589
Galerkin [13]0.4908
Factformer [11]0.0452
Transolver [14]0.0555
DeepLag [37]0.0382
SpiderSolver0.0376
+ +Table 14: The results on the Darcy Flow dataset. + +
MethodsRelative L2 error ↓
FNO [3]0.0108
WMT [46]0.0082
GEO-FNO [6]0.0108
F-FNO [16]0.0077
Galerkin [13]0.0084
LSM [47]0.0065
Transolver [14]0.0068
SpiderSolver0.0064
+ +# G More Visualization Results on Shape-Net Car and Blood Flow + +Figure 19 visualizes the Shape-Net Car estimation results by Transolver and our model. The visual results demonstrate that our approach achieved less error in the estimation of physical fields, especially the press value in the car surfaces. + +Figure 20 visualizes the blood flow estimation results by different compared methods. The estimation absolution error, i.e., the point-wise $L_{2}$ norm of the difference between ground truth and prediction values of different models, is highlighted by heatmaps. The visual results demonstrate that our approach achieved less error in the estimation of physical fields. + +![](images/3a1b0f0f69fa44c2307cd56e2a5abd974c50a77f85bcbf909b527b7fe12c993b.jpg) +Figure 19: Comparisons of the Shape-Net Car estimation by Transolver and our model, with the estimation error maps, i.e., point-wise $L_{2}$ norm of the difference between ground truth and prediction. + +Table 15: The results (Relative $L_{2}$ error) on the Elasticity and Plasticity datasets. + +
MethodsElasticity ↓Plasticity ↓
LSM0.02180.0025
GALERKIN0.02400.0120
OFFORMER0.01830.0017
GNOT0.00860.0336
ONO0.01180.0048
Transolver0.00640.0012
SpiderSolver0.00610.0011
+ +![](images/383c369b89ffe149a4664653ffefb8324038a04d90f2bb01ec7ff940cc8d3386.jpg) +Figure 20: Comparisons of the blood flow estimation by different methods, with the estimation error maps, i.e., point-wise $L_{2}$ norm of the difference between ground truth and prediction. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: See abstract and introduction section for details. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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In this work, we extend the use of SNNs to neural rendering tasks and introduce Spik-NeRF (Spiking Neural Radiance Fields with Ternary Spike). We observe that the binary spike activation map of traditional SNNs lacks sufficient information capacity, leading to information loss and a subsequent decline in the performance of spiking neural rendering models. To address this limitation, we propose the use of ternary spike neurons, which enhance the information-carrying capacity in the spiking neural rendering model. With ternary spike neurons, Spik-NeRF achieves performance that is on par with, or nearly identical to, traditional ANN-based rendering models. Additionally, we present a re-parameterization technique for inference that allows Spik-NeRF with ternary spike neurons to retain the event-driven, multiplication-free advantages typical of binary spike neurons. Furthermore, to further boost the performance of Spik-NeRF, we employ a distillation method, using an ANN-based NeRF to guide the training of our Spik-NeRF model, which is more compatible with the our ternary neurons compared to the standard binary neurons and other neuron forms. We evaluate Spik-NeRF on both realistic and synthetic scenes, and the experimental results demonstrate that Spik-NeRF achieves rendering performance comparable to ANN-based NeRF models. + +# 1 Introduction + +Spiking Neural Networks (SNNs) [36, 3, 9, 10, 35, 18, 17], known for their energy efficiency compared to Artificial Neural Networks (ANNs), have garnered significant attention due to their event-driven computation mechanism and the energy-saving advantages of multiplication-free operations. SNNs have shown great potential in a wide range of applications. For instance, in [34], SNNs were applied to object detection and demonstrated substantial energy efficiency improvements, outperforming their ANN counterparts by orders of magnitude. In [16], SNNs were used to improve the image de-occlusion task. In [33], SNNs were employed for sequential learning, showing better performance and reduced energy consumption compared to ANNs with similar scale. Similarly, in [26], SNNs were utilized for Human Activity Recognition (HAR), achieving up to a $94\%$ reduction in energy consumption while maintaining comparable accuracy to ANN-based models. Additionally, SNNs have been applied to pose tracking [45], 3D recognition [37], and even autonomous driving [39], + +where LaneSNNs demonstrated low power consumption (1 W) in lane detection using event-based cameras. + +Given these successes, the question naturally arises: Can SNNs be adapted to the more complex task of neural rendering, such as rendering neural radiance fields (NeRF)? + +In this paper, we introduce Spik-NeRF, a spiking neural network approach tailored for neural rendering tasks, the first one directly-trained SNN-based NeRF model building upon the initial NeRF framework [31]. However, we found that applying SNNs to neural rendering tasks led to suboptimal performance. This is primarily due to the limited information capacity of the binary spike activation maps in SNNs. Unlike the rich activation maps of ANNs, the binary spike maps in SNNs fail to retain enough useful information during the quantization process, resulting in significant information loss and a decrease in performance. A more detailed explanation is provided in Sec. 3.3. + +![](images/2b36c17a48d6486ccb44811b40487f50a9dc9a6e9c71d4285ddff2c875568379.jpg) +Figure 1: The overall workflow of the proposed Spik-NeRF, along with the ternary spike neuron and re-parameterization technique. + +To address this challenge, we propose + +the ternary spike neuron for the Spik-NeRF, which extends the traditional binary spike representation $(\{0,1\})$ to a ternary form $(\{0,1,2\})$ . This new approach significantly increases the information capacity of SNNs, as detailed in Sec. 3.3. Furthermore, in the inference phase, we introduce a reparameterization technique that transforms the ternary spikes $(\{0,1,2\})$ into the set of $(\{-1,0,1\})$ , preserving the multiplication-free and event-driven advantages of SNNs. The overall workflow of the proposed Spik-NeRF, along with the ternary spike neuron and re-parameterization technique, is illustrated in Fig. 1. + +Additionally, to further improve the performance of Spik-NeRF, we propose using an ANN-based NeRF model for distillation. This technique, leveraging the superior capabilities of ANN-based NeRF models, is particularly well-suited for our ternary spike neuron, offering additional performance gains. + +In summary, the main contributions of this work are as follows: + +- We present Spik-NeRF, a spiking neural network for rendering neural radiance fields. To our best knowledge, this is the first directly-trained SNN model building on the original NeRF framework [31]. +- We demonstrate, with theoretical justification, that binary spike activation maps in SNN-based NeRF are insufficient in carrying information, leading to performance degradation. To solve this issue, we propose the ternary spike neuron, which effectively increases the information capacity while retaining the multiplication-free and event-driven advantages of standard SNNs in the inference, aided by our re-parameterization technique. +- We introduce a distillation method using an ANN-based NeRF teacher, which is more suitable for our ternary neuron compared to other spike neurons, to further enhances the performance of Spik-NeRF. +- We evaluate Spik-NeRF on both realistic and synthetic scenes. The experimental results demonstrate that Spik-NeRF achieves rendering performance comparable to ANN-based NeRF models. + +# 2 Related Work + +# 2.1 Spiking Neural Networks + +There are generally three primary methods for training SNNs [13]: (1) spike-timing-dependent plasticity (STDP) [1] approaches, (2) ANN-to-SNN conversion approaches [22, 21, 29, 6, 8, 2, 19, 24], and (3) direct training methods [5, 32, 42, 35, 25, 41, 40]. + +The STDP method is biologically inspired [20, 7] and updates synaptic weights using an unsupervised learning algorithm called Hebbian learning [23]. However, this approach is still limited to small-scale datasets. + +The ANN-to-SNN conversion method [6, 24, 22, 21] involves converting a well-trained ANN model to an SNN counterpart. This method is advantageous because training an ANN is faster than training an SNN. Consequently, the ANN-to-SNN conversion offers a quick way to obtain an SNN without using gradient descent. However, the converted SNN essentially mimics the original ANN and lacks learned features, thus not fully exploiting the benefits of SNNs. Additionally, this method typically requires many time steps to achieve high accuracy. + +Direct training methods, on the other hand, aim to find an alternative function to replace the firing function of spiking neurons during backpropagation. These methods can significantly reduce the number of time steps needed, sometimes even to fewer than five [12, 25, 14], and have attracted considerable attention recently. However, finding an appropriate surrogate function for SNNs with large time steps remains a challenging problem. Our work focuses on addressing this issue. + +# 2.2 Spiking Neural Networks for NeRF + +Some research has explored applying SNNs to Neural Radiance Fields, but these studies differ from our approach. For instance, hybrid ANN-SNN models were proposed in Spiking NeRF [28] and Spiking Nerfacto [11]. These works employ non-linear, non-spike functions to post-process the density-related outputs of the original ANN-based NeRF models. In contrast, our work focuses on utilizing a fully SNN-based model for neural radiance fields. In Spiking-Nerf [27], an ANN-SNN model is proposed to develop energy-efficient spiking neural rendering using the ANN-to-SNN conversion approach, which, as mentioned earlier, requires many time steps. In comparison, our work concentrates on developing a direct training SNN method for NeRF. Another relevant study, SpikingNerf [43], presents a spiking neural radiance field model based on the DVGO [38] and TensoRF [4] frameworks. These frameworks enhance the original NeRF model by integrating various innovations. Except for these, this method requires numerous time steps as each sampled point on the ray is associated with a particular time step and represented in a hybrid manner. + +Our approach, however, aims to develop a spiking neural radiance field model based on the original NeRF framework [31], which could pave the way for future advancements in this field. Additionally, we aim to achieve competitive performance with fewer time steps, which is directly related to the energy efficiency of the model. + +# 3 Preliminary and Methodology + +In this paper, we primarily apply the SNN to the NeRF framework [31], which is the first deep learning model that represents a scene as a neural radiance field and renders novel views from this representation. We then modify it to create Spik-NeRF. First, we provide a detailed introduction to NeRF and the widely used SNN neuron model, the Leaky Integrate-and-Fire (LIF) model. Subsequently, we address the information loss issue when applying SNN to NeRF. Finally, we present the Spik-NeRF model, which is based on ternary spike neurons to resolve the aforementioned problem, along with an isomorphic network knowledge distillation method to further enhance performance. + +# 3.1 NeRF + +In contrast to conventional explicit 3D reconstruction techniques that employ discrete voxel grids or point clouds, NeRF [31] introduces a novel continuous implicit representation through differentiable volumetric rendering. The core innovation lies in encoding the 3D scene as a 5D neural radiance + +field using a multi-layer perceptron (MLP), which maps spatial coordinates $\mathbf{p} = (x,y,z)\in \mathbb{R}^3$ and viewing directions $\mathbf{v} = (\theta ,\phi)\in \mathbb{S}^2$ to volume density $\sigma \in \mathbb{R}^{+}$ and view-dependent RGB color $\mathbf{c} = (r,g,b)\in [0,1]^3$ . This parametric representation is formalized through two cascaded MLP components: + +$$ +\text {G e o m e t r y n e t w o r k :} \quad F _ {\theta}: \mathbf {p} \mapsto (\mathbf {e}, \sigma) \tag {1} +$$ + +$$ +\text {A p p a r a n c e} G _ {\gamma}: (\mathbf {e}, \mathbf {v}) \mapsto \mathbf {c} \tag {2} +$$ + +where $\theta$ and $\gamma$ denote network parameters, $\mathbf{e} \in \mathbb{R}^N$ represents intermediate feature embeddings, and $\sigma$ corresponds to the differential opacity at point $\mathbf{p}$ . These networks are all composed of several fully connected layers. Each fully connected layer implements the transformation: + +$$ +\mathbf {h} = \operatorname {R e L U} (\operatorname {L i n e a r} (\mathbf {a})) \tag {3} +$$ + +where $\mathbf{a}$ is the activation from the previous layer. + +The rendering process employs classical volume rendering principles [30] with neural adaptation. For a camera ray $\mathbf{r}(g) = \mathbf{o} + g\mathbf{d}$ with near/far bounds $[g_n, g_f]$ , we sample $K$ stratified points $\{g_i\}_{i=1}^K$ and compute the pixel color via numerical integration: + +$$ +\hat {C} (\mathbf {r}) = \sum_ {i = 1} ^ {K} G _ {i} \alpha_ {i} \mathbf {c} _ {i} \quad \text {w h e r e} \quad \left\{ \begin{array}{l} G _ {i} = \exp \left(- \sum_ {j = 1} ^ {i - 1} \sigma_ {j} \delta_ {j}\right) \\ \alpha_ {i} = 1 - \exp (- \sigma_ {i} \delta_ {i}) \\ \delta_ {i} = g _ {i + 1} - g _ {i} \end{array} \right. \tag {4} +$$ + +Here $G_{i}$ represents the transmittance probability for ray segment $[g_{n}, g_{i}]$ , and $\alpha_{i}$ denotes the alpha-compositing weight for the $i$ -th sample. The hierarchical sampling strategy combines coarse and fine networks to importance-sample along rays. + +The model is optimized through photometric reconstruction over a set of rays $\mathcal{R}$ using an $L_{2}$ loss between rendered and observed pixel colors: + +$$ +\mathcal {L} _ {\mathrm {p h o t o}} = \mathbb {E} _ {\mathbf {r} \sim \mathcal {R}} \left[ \| \hat {C} _ {c} (\mathbf {r}) - C (\mathbf {r}) \| _ {2} ^ {2} + \| \hat {C} _ {f} (\mathbf {r}) - C (\mathbf {r}) \| _ {2} ^ {2} \right] \tag {5} +$$ + +where $\hat{C}_c$ and $\hat{C}_f$ denote outputs from the coarse and fine networks respectively. + +# 3.2 Vanilla SNN for NeRF (Denoted as Spiking NeRF) + +SNNs use the spiking neuron, which is inspired by the brain's natural mechanisms, to transmit information. A spiking neuron will receive input spike trains from the previous layer neuron models along times to update its membrane potential, $\mathbf{u}$ . In the paper, we adopt the widely used leaky integrate and fire (LIF) neuron model. The LIF neuron model governs membrane potential $\mathbf{u}(t)$ evolution through time $t$ : + +$$ +\tau_ {\mathrm {m}} \frac {d \mathbf {u}}{d t} = - (\mathbf {u} - u _ {\text {r e s e t}}) + R \cdot I (t), \quad \mathbf {u} < V _ {\text {t h}} \tag {6} +$$ + +$$ +\mathbf {s} ^ {t} = \Theta (\mathbf {u} ^ {t} - V _ {\mathrm {t h}}) \tag {7} +$$ + +where $\Theta (\cdot)$ denotes the Heaviside step function, $\tau_{\mathrm{m}}$ is the membrane time constant, and $V_{\mathrm{th}}$ the firing threshold. For practical implementation, we adopt the discrete-time formulation: + +$$ +\mathbf {v} ^ {t} = \mathbf {u} ^ {t - 1} + \frac {1}{\tau} \left(\mathbf {W s} ^ {t - 1} - \mathbf {u} ^ {t - 1} + u _ {\text {r e s e t}}\right) \tag {8} +$$ + +$$ +\mathbf {s} ^ {t} = \Theta (\mathbf {v} ^ {t} - V _ {\mathrm {t h}}) \tag {9} +$$ + +$$ +\mathbf {u} ^ {t} = \mathbf {v} ^ {t} \odot (1 - \mathbf {s} ^ {t}) + u _ {\text {r e s e t}} \mathbf {s} ^ {t} \tag {10} +$$ + +where $\mathbf{W}$ denotes synaptic weights and $\odot$ the Hadamard product. Parameters follow biological constraints: $\tau = 4$ , $u_{\mathrm{reset}} = 0$ , $V_{\mathrm{th}} = 0.5$ [25]. + +We transform NeRF's MLP layers to spiking domains through temporal unfolding and potential accumulation: + +$$ +\text {S p i k i n g G e o m e t r y N e t w o r k :} \quad F _ {\theta} ^ {t}: \mathbf {p} ^ {t} \mapsto \left(\mathbf {e} ^ {t}, \sigma^ {t}\right) \tag {11} +$$ + +$$ +\text {S p i k i n g A p p a r a n c e N e t w o r k :} \quad G _ {\gamma} ^ {t}: \left(\mathbf {e} ^ {t}, \mathbf {v} ^ {t}\right) \mapsto \mathbf {c} ^ {t} \tag {12} +$$ + +Each spiking MLP layer implements temporal-aware computation: + +$$ +\mathbf {h} _ {k} ^ {t} = \operatorname {L I F} \left(\mathbf {W} _ {k} \mathbf {s} _ {k - 1} ^ {t} + \mathbf {b} _ {k}\right) \tag {13} +$$ + +where $\mathbf{s}_{k - 1}^{t}$ denotes spike inputs from layer $k - 1$ at timestep $t$ . The membrane potential $\mathbf{u}_k^t$ tracks temporal dependencies across layers. + +The spiking radiance field outputs are integrated through: + +$$ +\bar {\sigma} = \frac {1}{T} \sum_ {t = 1} ^ {T} \sigma^ {t} \tag {14} +$$ + +$$ +\bar {\mathbf {c}} = \frac {1}{T} \sum_ {t = 1} ^ {T} \mathbf {c} ^ {t} \tag {15} +$$ + +where $T$ is the total timestep of the Spiking NeRF. Then volumetric rendering then follows Eq. (4) with $\sigma_{i} = \bar{\sigma}_{i}$ and $\mathbf{c}_i = \bar{\mathbf{c}}_i$ . + +# 3.3 Spik-NeRF + +# 3.3.1 Information Loss in Spiking NeRF + +While employing binary spike feature embeddings in Spiking NeRF offers substantial energy efficiency, it inherently has a limited representational capacity compared to the high-precision feature embeddings utilized in ANN-based NeRF. This limitation ultimately restricts its performance. To better illustrate this issue, we begin by providing a theoretical analysis based on the concept of entropy. Given a set $\mathbf{X}$ , its representational capability, denoted as $\mathcal{C}(\mathbf{X})$ , can be quantified by the maximum entropy of $\mathbf{X}$ , as expressed below: + +$$ +\mathcal {C} (\mathbf {X}) = \max \mathcal {H} (\mathbf {X}) = \max \left(- \sum_ {x \in \mathbf {X}} p _ {\mathbf {X}} (x) \log p _ {\mathbf {X}} (x)\right), \tag {16} +$$ + +where $p_{\mathbf{X}}(x)$ represents the probability of observing a sample $x$ from $\mathbf{X}$ . The following proposition can be easily derived: + +Proposition: Given a set $X$ , we have $\mathcal{C}(X) = \max \mathcal{H}(X) = \max \left(-\sum_{x \in X} p_X(x) \log p_X(x)\right)$ . When the probability distribution is defined as $p_X(x) = \frac{1}{M}$ , where $M$ represents the total number of samples in $X$ , the entropy $\mathcal{H}(X)$ reaches its maximum value of $\log(M)$ . Therefore, it follows that $\mathcal{C}(X) = \log(M)$ . + +Next, we calculate the representational capacities of the binary spike feature embeddings in Spiking NeRF and the real-valued feature embeddings in the ANN-based counterpart. Let $\mathbf{E}_{\mathrm{B}} \in \mathbb{B}^{C \times N}$ represent the binary feature embeddings of the Spiking NeRF, and $\mathbf{E}_{\mathrm{R}} \in \mathbb{R}^{C \times N}$ denote the real-valued feature embeddings of the ANN-based NeRF. A binary spike output $s$ can be represented by 1 bit, with two possible samples from $s$ . Therefore, the number of samples in $\mathbf{E}_{\mathrm{B}}$ is $2^{(C \times N)}$ , and the corresponding representational capacity is: + +$$ +\mathcal {C} \left(\mathbf {E} _ {\mathrm {B}}\right) = \log \left(2 ^ {(C \times N)}\right) = C \times N. \tag {17} +$$ + +In contrast, a real-valued output requires 32 bits, leading to $2^{32}$ possible samples. Hence, the representational capacity for the real-valued embeddings is: + +$$ +\mathcal {C} \left(\mathbf {E} _ {\mathrm {R}}\right) = \log \left(2 ^ {3 2 \times (C \times N)}\right) = 3 2 \times C \times N. \tag {18} +$$ + +This comparison clearly demonstrates that the representational capacity of the binary spike feature embeddings is substantially limited, which consequently results in degraded performance for Spiking NeRF. + +# 3.3.2 Ternary Spike Neuron Mechanism for Spik-NeRF + +Our theoretical analysis reveals that enhancing the information capacity of spike neuron activations directly correlates with improved task performance. To capitalize on this insight, we propose a novel + +ternary spike neuron formulation that forms the foundation of our Spik-NeRF architecture. The membrane dynamics and spike generation mechanism operate through three distinct phases: + +$$ +\mathbf {v} ^ {t} = \mathbf {u} ^ {t - 1} + \frac {1}{\tau} \left(\mathbf {W s} ^ {t - 1} - \mathbf {u} ^ {t - 1} + u _ {\text {r e s e t}}\right) \tag {19} +$$ + +$$ +\mathbf {s} ^ {t} = \left\{ \begin{array}{l l} 2, & \mathbf {v} ^ {t} > V _ {\mathrm {t h}} + \Delta_ {v} \\ 1, & V _ {\mathrm {t h}} \leq \mathbf {v} ^ {t} \leq V _ {\mathrm {t h}} + \Delta_ {v} \\ 0, & \mathbf {v} ^ {t} < V _ {\mathrm {t h}} \end{array} \right. \tag {20} +$$ + +$$ +\mathbf {u} ^ {t} = \left\{ \begin{array}{l l} \mathbf {v} ^ {t}, & \mathbf {v} ^ {t} < V _ {\text {t h}} \\ u _ {\text {r e s e t}}, & \text {o t h e r w i s e} \end{array} \right. \tag {21} +$$ + +where $\Delta_v$ represents our adaptive threshold margin (fixed at 1 in implementation). Obviously, this ternary formulation significantly enhances the representational capacity of Spik-NeRF. To quantitatively analyze the representational advantage, we also resort to the information entropy theory. Let $\mathbf{E}_T \in \mathbb{T}^{C \times N}$ denote as a ternary feature embedding in our Spik-NeRF. The ternary spike embeddings $\mathbf{E}_T$ consists of $3^{C \times N}$ samples. Hence, + +$$ +\mathcal {C} \left(\mathbf {E} _ {T}\right) = \log_ {2} 3 ^ {C \times N} = C \times N \cdot \log_ {2} 3 \approx 1. 5 8 5 \cdot C \times N \tag {22} +$$ + +The $\approx 58.5\%$ increase in theoretical information capacity directly translates to enhanced scene representation capabilities, which benefits performance improvement. + +# 3.3.3 Training-Inference Decoupling via Spike Reparameterization + +While ternary spike neurons enhance representational capacity, their direct implementation introduces computational challenges: the $\{0,1,2\}$ activation space prevents efficient conversion of weight-activation multiplications to additions, a critical advantage in SNN acceleration. To resolve this fundamental efficiency conflict, we propose a novel spike re-parameterization technique that preserves both the information richness of ternary signals and the computational benefits of binary networks. + +Our solution employs a train-infer decoupling strategy with affine transformation of spike representations. During training, we maintain the native $\{0,1,2\}$ spike formulation for gradient stability. For inference, we apply a linear transformation to the spike tensor: + +$$ +\hat {\mathbf {s}} ^ {t} = \mathbf {s} ^ {t} - \Delta \cdot \mathbf {1} \quad \text {w h e r e} \quad \Delta = 1 \tag {23} +$$ + +This shifts the spike space to $\{-1,0,1\}$ while preserving ordinal relationships. The membrane potential update equations consequently adapt as follows: + +$$ +\text {T r a i n i n g :} \quad \mathbf {v} ^ {t} = \mathbf {u} ^ {t - 1} + \frac {1}{\tau} \left(\mathbf {W s} ^ {t - 1} - \mathbf {u} ^ {t - 1} + u _ {\text {r e s e t}}\right) \tag {24} +$$ + +$$ +\text {I n f e r e n c e :} \quad \mathbf {v} ^ {t} = \mathbf {u} ^ {t - 1} + \frac {1}{\tau} \left(\mathbf {W} \hat {\mathbf {s}} ^ {t - 1} + \mathbf {w} _ {b} - \mathbf {u} ^ {t - 1} + u _ {\text {r e s e t}}\right) \tag {25} +$$ + +where $\mathbf{w}_b = \mathbf{W1}$ constitutes a pre-computable bias term. Thus in the inference, the linear layer will only consist of addition operations and keep the event-driven advantage. + +Note that we can also use the $\{-1,0,1\}$ activation spike [15] during training. However, we observe that its performance is inferior to that of the $\{0,1,2\}$ spike. Additionally, the $\{0,1,2\}$ spike activation resembles ReLU activation more closely, making it better suited for ANN-SNN distillation compared to the $\{-1,0,1\}$ spike activation. With these two reasons, we chose this form of neuron for our work. + +# 3.3.4 Isomorphic Network Knowledge Distillation + +To further increase the performance, we propose an isomorphic distillation framework that transfers knowledge from an ANN-based NeRF (teacher) to our Spik-NeRF (student). We establish direct supervision on both density and color predictions through mean squared error (MSE) distillation. For any 3D point $\mathbf{p}$ and viewing direction $\mathbf{v}$ , + +$$ +\mathcal {L} _ {\text {d e n s i t y}} = \mathbb {E} _ {\mathbf {p} \sim \mathcal {P}} \left[ \| \sigma^ {\mathrm {A N N}} (\mathbf {p}) - \bar {\sigma} ^ {\mathrm {S N N}} (\mathbf {p}) \| _ {2} ^ {2} \right] \tag {26} +$$ + +$$ +\mathcal {L} _ {\text {c o l o r}} = \mathbb {E} _ {(\mathbf {p}, \mathbf {v}) \sim \mathcal {P} \times \mathcal {V}} \left[ \| \mathbf {c} ^ {\text {A N N}} (\mathbf {p}, \mathbf {v}) - \bar {\mathbf {c}} ^ {\text {S N N}} (\mathbf {p}, \mathbf {v}) \| _ {2} ^ {2} \right] \tag {27} +$$ + +The final training objective combines photometric reconstruction loss with distillation is + +$$ +\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {p h o t o}} + \lambda_ {d} \mathcal {L} _ {\text {d e n s i t y}} + \lambda_ {c} \mathcal {L} _ {\text {c o l o r}} \tag {28} +$$ + +where $\lambda_{d}$ and $\lambda_{c}$ control the distillation strength for density and color respectively. We set them as 0.5 in the paper. + +Table 1: Per-scene quantitative results from the synthetic dataset + +
MetricMethodChairDrumsFicusHotdogLegoMaterialsMicShipAvg.
PSNR↑ANN-based NeRF34.1525.6429.1536.8531.4829.3433.1229.4231.15
Spiking-NeRF12.2411.1413.9814.859.8110.369.8113.2211.93
Spiking NeRF31.9824.5124.0034.5529.3327.7531.6527.8528.95
Spik-NeRF33.4025.2126.0535.9930.8228.8632.7828.7530.23
SSIM↑ANN-based NeRF0.9790.9290.9660.9790.9650.9580.9780.8740.953
Spiking NeRF0.9630.9070.8910.9690.9400.9370.9690.8380.927
Spik-NeRF0.9730.9210.9290.9760.9570.9500.9750.8560.942
LPIPS↓ANN-based NeRF0.0140.0530.0220.0150.0200.0240.0230.0860.032
Spiking NeRF0.0340.0860.1190.0320.0390.0470.0430.1260.066
Spik-NeRF0.0210.0650.0630.0210.0270.0330.0280.1050.045
+ +Table 2: Per-scene quantitative results from the realistic dataset + +
MetricMethodRoomFernLeavesFortressOrchidsFlowerT-RexHornsAvg.
PSNR↑ANN-based NeRF31.3826.2521.9831.3521.2027.5127.2728.1026.88
Spiking-NeRF17.0715.919.6814.519.1210.3515.0513.0513.09
Spiking NeRF30.1225.0820.7530.0720.5126.1925.2626.3425.54
Spik-NeRF30.9025.7021.4630.7920.9426.9426.1527.1626.26
SSIM↑ANN-based NeRF0.9310.8360.7900.8960.7340.8530.8960.8770.851
Spiking NeRF0.9040.7690.6910.8370.6440.7820.8250.8000.781
Spik-NeRF0.9200.8020.7440.8690.6880.8200.8620.8350.817
LPIPS↓ANN-based NeRF0.0490.1010.1190.0590.1220.0750.0620.0780.083
Spiking NeRF0.0980.2070.2030.1300.2230.1340.1350.1660.162
Spik-NeRF0.0660.1640.1570.0920.1700.1010.0950.1230.121
+ +# 4 Experiment + +We assess the rendering performance of Spik-NeRF on both synthetic and real-world datasets [31]. The synthetic dataset includes eight scenes featuring different objects. For each scene, there are 100 views used for training and 200 views for testing, with each view image having a resolution of $400 \times 400$ pixels. The real-world dataset consists of eight scenes captured with mobile phones. Each scene contains between 20 and 60 images, and the images are resized to $400 \times 400$ pixels in this paper. Additionally, one-eighth of the images are reserved for testing. + +The network architecture follows the design outlined in NeRF [31]. All models are trained using the Adam optimizer for 300,000 iterations with a batch size of 1,024 rays. We initialize the learning rate at $5 \times 10^{-4}$ , which is decayed exponentially as training progresses. For synthetic scenes, the number of sampled points is set to 64 for the coarse network and 128 for the fine network. Similarly, for real-world scenes, 64 and 128 sampled points are used for the coarse and fine networks, respectively. The total number of timesteps for both Spiking NeRF and our Spik-NeRF is set to 2, while for the Spiking-NeRF, it is 8 timesteps. Since our Spik-NeRF with 2 timesteps achieves rendering performance comparable to the ANN-based NeRF model, we did not explore larger timesteps. + +# 4.1 Rendering Performance + +We evaluate the rendering performance of Spik-NeRF both quantitatively and qualitatively. Tables 1 and 2 present per-scene quantitative results from the synthetic and realistic datasets, respectively. + +As mentioned earlier, although previous work has explored the application of SNNs to NeRF, these studies differ from our approach. Spiking-NeRF [27], an ANN-SNN hybrid model, uses the same NeRF [31] framework as ours, and is thus selected for comparison. Additionally, we implemented the original NeRF [31] and a spiking NeRF based on a vanilla binary spike neuron for comparison, referred to as ANN-based NeRF and Spiking NeRF, respectively. + +For performance evaluation, we adopt standard metrics: PSNR and SSIM (higher values are better), and LPIPS [44] (lower values are better), as used in NeRF [31]. Since Spiking-Nerf [27] only reports PSNR, we present SSIM and LPIPS results for ANN-based NeRF, Spiking NeRF, and Spik-NeRF. + +![](images/cf2de120e13a4569a6213cd17431e0e5755f261a300559215db76a24ed759345.jpg) +Figure 2: The rendering performance in the synthetic dataset. + +On the synthetic dataset, Spiking-NeRF with 8 timesteps achieved an average PSNR of 11.93. In comparison, our directly trained Spiking NeRF achieved a significant improvement with an average PSNR of 28.95. More notably, our Spik-NeRF, utilizing ternary spikes, further boosts the average PSNR to 30.23, approaching the performance of the ANN-based NeRF (31.15 PSNR). Furthermore, in every scene, Spik-NeRF consistently outperforms both Spiking-NeRF and Spiking NeRF, highlighting the effectiveness of our approach. + +On the realistic dataset, our method also surpasses Spiking-NeRF and Spiking NeRF. For instance, Spik-NeRF achieves a PSNR of 26.26, outperforming Spiking-NeRF and Spiking NeRF by 13.17 and 0.72 PSNR, respectively. In terms of SSIM and LPIPS, Spik-NeRF achieves scores of 0.817 and 0.121, while Spiking NeRF achieves 0.781 and 0.162, respectively. + +Figure 2 illustrates the rendering results from synthetic datasets, including Lego, Drums, Mic, and Ship, for ANN-based NeRF, Spiking NeRF, and Spik-NeRF. It is evident that our method recovers fine details in both geometry and appearance, comparable to the ANN-based NeRF. This includes features such as the Drums' pedal, the Microphone's mesh grille, and the Ship's rigging. In contrast, Spiking NeRF produces blurry and distorted renderings, particularly for the Microphone's mesh grille. + +We also include the rendering results for the realistic dataset in the appendix. As seen in the figures, our method consistently represents fine geometry more accurately across rendered views than Spiking NeRF. + +# 4.2 Ablation Study for Knowledge Distillation + +In this section, we evaluate the performance of our method with the proposed isomorphic network knowledge distillation on complex scenes, such as Room, Orchids, and Drums. The results are presented in Tab. 3. + +As illustrated in Tab. 3, applying knowledge distillation to Spik-NeRF results in a noticeable improvement in performance, bringing its results much closer to those of the ANN-based NeRF. The quantitative analysis reveals two important findings: (1) Knowledge distillation effectively narrows the performance gap between Spik-NeRF and ANN-based NeRF, as demonstrated by the improvement in both PSNR and SSIM metrics. (2) Even in challenging scenes that involve specular reflections, such as Orchids, our method achieves rendering quality comparable to ANN-based NeRF, suggesting that knowledge distillation is + +Table 3: Per-scene quantitative results for knowledge distillation. + +
MetricMethodRoomOrchidsDrums
PSNR↑ANN-based NeRF31.3821.2025.64
Spik-NeRF30.9020.9425.21
Spik-NeRF with KD31.1221.0925.40
SSIM↑ANN-based NeRF0.9310.7340.929
Spik-NeRF0.9200.6880.921
Spik-NeRF with KD0.9240.6920.924
LPIPS↓ANN-based NeRF0.0490.1220.053
Spik-NeRF0.0660.1700.065
Spik-NeRF with KD0.0640.1650.063
+ +beneficial in maintaining high-quality renderings in complex environments. + +These findings also highlight the compatibility of our Spik-NeRF with the ternary spike neuron model and the isomorphic network knowledge distillation, which appears to facilitate the optimization process and enhance performance significantly. + +# 5 Conclusion + +We present Spik-NeRF, achieving ANN-comparable rendering quality. Theoretical analysis reveals binary spikes' limited representational capacity for SNN-based NeRF. To address limitations of binary spike neurons based Spiking NeRF, we propose the ternary spike neuron for Spik-NeRF, which increase representational capacity by $58.5\%$ using three activation states. We also propose a train-infer decoupling via spike reparameterization technique to keep the energy efficiency of SNNs. In addition, we also propose the isomorphic distillation method, which transfers knowledge from ANN-based NeRF to compensate for information loss. Our experiments show that Spik-NeRF achieves PSNR metric within $2.9\%$ of ANN baselines with only 2 timesteps, while retaining energy efficiency through multiplication-free operations. Our work bridges the efficiency-performance gap in neural fields, enabling future energy-efficient 3D reconstruction. + +# Acknowledgements + +This work was supported by the National Key Research and Development Program of China (No. 2024YDLN0013) and the National Natural Science Foundation of China (No. 12202412). + +# References + +[1] Bi, G.-q. and Poo, M.-m. Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type. Journal of neuroscience, 18(24): 10464-10472, 1998. +[2] Bu, T., Ding, J., Yu, Z., and Huang, T. Optimized potential initialization for low-latency spiking neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 11-20, 2022. +[3] Carnevale, N. T. and Hines, M. L. The NEURON book. Cambridge University Press, 2006. +[4] Chen, A., Xu, Z., Geiger, A., Yu, J., and Su, H. Tensorf: Tensorial radiance fields. In European conference on computer vision, pp. 333-350. Springer, 2022. +[5] Cheng, X., Hao, Y., Xu, J., and Xu, B. Lisnn: Improving spiking neural networks with lateral interactions for robust object recognition. In *IJCAI*, pp. 1519–1525, 2020. +[6] Deng, S. and Gu, S. Optimal conversion of conventional artificial neural networks to spiking neural networks. arXiv preprint arXiv:2103.00476, 2021. +[7] Diehl, P. U. and Cook, M. Unsupervised learning of digit recognition using spike-timing-dependent plasticity. Frontiers in computational neuroscience, 9:99, 2015. +[8] Ding, J., Yu, Z., Tian, Y., and Huang, T. Optimal ann-snn conversion for fast and accurate inference in deep spiking neural networks. arXiv preprint arXiv:2105.11654, 2021. +[9] Fang, W., Yu, Z., Chen, Y., Masquelier, T., Huang, T., and Tian, Y. Incorporating learnable membrane time constant to enhance learning of spiking neural networks. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 2661-2671, 2021. +[10] Ghosh-Dastidar, S. and Adeli, H. Spiking neural networks. International journal of neural systems, 19(04):295-308, 2009. +[11] Gu, Y., Wang, Z., Ye, D., and Xu, R. Sharpening your density fields: Spiking neuron aided fast geometry learning. arXiv preprint arXiv:2412.09881, 2024. +[12] Guo, Y., Zhang, L., Chen, Y., Tong, X., Liu, X., Wang, Y., Huang, X., and Ma, Z. Real spike: Learning real-valued spikes for spiking neural networks. In Computer Vision-ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23-27, 2022, Proceedings, Part XII, pp. 52-68. Springer, 2022. +[13] Guo, Y., Huang, X., and Ma, Z. Direct learning-based deep spiking neural networks: a review. Frontiers in Neuroscience, 17:1209795, 2023. +[14] Guo, Y., Zhang, Y., Chen, Y., Peng, W., Liu, X., Zhang, L., Huang, X., and Ma, Z. Membrane potential batch normalization for spiking neural networks. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp. 19420-19430, October 2023. +[15] Guo, Y., Chen, Y., Liu, X., Peng, W., Zhang, Y., Huang, X., and Ma, Z. Ternary spike: Learning ternary spikes for spiking neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pp. 12244-12252, 2024. +[16] Guo, Y., Peng, W., Chen, Y., Zhou, J., and Ma, Z. Improved event-based image de-occlusion. IEEE Signal Processing Letters, 2024. +[17] Guo, Y., Peng, W., Liu, X., Chen, Y., Zhang, Y., Tong, X., Jie, Z., and Ma, Z. Enof-snn: Training accurate spiking neural networks via enhancing the output feature. Advances in Neural Information Processing Systems, 37:51708-51726, 2024. +[18] Guo, Y., Zhang, Y., Jie, Z., Liu, X., Tong, X., Chen, Y., Peng, W., and Ma, Z. Reverbssn: Reversing bit of the weight and activation for spiking neural networks. arXiv preprint arXiv:2506.07720, 2025. + +[19] Han, B., Srinivasan, G., and Roy, K. Rmp-snn: Residual membrane potential neuron for enabling deeper high-accuracy and low-latency spiking neural network. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 13558–13567, 2020. +[20] Hao, Y., Huang, X., Dong, M., and Xu, B. A biologically plausible supervised learning method for spiking neural networks using the symmetric stdp rule. Neural Networks, 121:387-395, 2020. +[21] Hao, Z., Bu, T., Ding, J., Huang, T., and Yu, Z. Reducing ann-snn conversion error through residual membrane potential. arXiv preprint arXiv:2302.02091, 2023. +[22] Hao, Z., Ding, J., Bu, T., Huang, T., and Yu, Z. Bridging the gap between anns and snns by calibrating offset spikes. arXiv preprint arXiv:2302.10685, 2023. +[23] Hebb, D. O. The organization of behavior: A neuropsychological theory. Psychology press, 2005. +[24] Li, Y., Deng, S., Dong, X., Gong, R., and Gu, S. A free lunch from ann: Towards efficient, accurate spiking neural networks calibration. In International Conference on Machine Learning, pp. 6316-6325. PMLR, 2021. +[25] Li, Y., Guo, Y., Zhang, S., Deng, S., Hai, Y., and Gu, S. Differentiable spike: Rethinking gradient-descent for training spiking neural networks. Advances in Neural Information Processing Systems, 34:23426-23439, 2021. +[26] Li, Y., Yin, R., Park, H., Kim, Y., and Panda, P. Wearable-based human activity recognition with spatio-temporal spiking neural networks. arXiv preprint arXiv:2212.02233, 2022. +[27] Li, Z., Ma, Y., Zhou, J., and Zhou, P. Spiking-nerf: Spiking neural network for energy-efficient neural rendering. ACM Journal on Emerging Technologies in Computing Systems, 20(3):1-23, 2025. +[28] Liao, Z., Liu, Y., Zheng, Q., and Pan, G. Spiking nerf: Representing the real-world geometry by a discontinuous representation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 13790-13798, 2024. +[29] Liu, F., Zhao, W., Chen, Y., Wang, Z., and Jiang, L. Spikeconverter: An efficient conversion framework zipping the gap between artificial neural networks and spiking neural networks. In Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Artificial Intelligence, EAAI 2022 Virtual Event, February 22 - March 1, 2022, pp. 1692-1701. AAAI Press, 2022. URL https://ojs.aaai.org/index.php/AAAI/article/view/20061. +[30] Max, N. Optical models for direct volume rendering. IEEE Transactions on visualization and computer graphics, 1(2):99-108, 2002. +[31] Mildenhall, B., Srinivasan, P. P., Tancik, M., Barron, J. T., Ramamoorthi, R., and Ng, R. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1):99-106, 2021. +[32] Neftci, E. O., Mostafa, H., and Zenke, F. Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks. IEEE Signal Processing Magazine, 36(6):51-63, 2019. +[33] Ponghiran, W. and Roy, K. Spiking neural networks with improved inherent recurrence dynamics for sequential learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 8001-8008, 2022. +[34] Qu, J., Gao, Z., Zhang, T., Lu, Y., Tang, H., and Qiao, H. Spiking neural network for ultra-low-latency and high-accurate object detection, 2023. +[35] Rathi, N. and Roy, K. Diet-snn: Direct input encoding with leakage and threshold optimization in deep spiking neural networks. arXiv preprint arXiv:2008.03658, 2020. + +[36] Rathi, N., Srinivasan, G., Panda, P., and Roy, K. Enabling deep spiking neural networks with hybrid conversion and spike timing dependent backpropagation. arXiv preprint arXiv:2005.01807, 2020. +[37] Ren, D., Ma, Z., Chen, Y., Peng, W., Liu, X., Zhang, Y., and Guo, Y. Spiking pointnet: Spiking neural networks for point clouds. arXiv preprint arXiv:2310.06232, 2023. +[38] Sun, C., Sun, M., and Chen, H.-T. Direct voxel grid optimization: Super-fast convergence for radiance fields reconstruction. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 5459-5469, 2022. +[39] Viale, A., Marchisio, A., Martina, M., Masera, G., and Shafique, M. Lanesnns: Spiking neural networks for lane detection on the loihi neuromorphic processor. In 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 79-86. IEEE, 2022. +[40] Wu, J., Chua, Y., Zhang, M., Li, G., Li, H., and Tan, K. C. A tandem learning rule for effective training and rapid inference of deep spiking neural networks. IEEE Transactions on Neural Networks and Learning Systems, 2021. +[41] Wu, J., Xu, C., Han, X., Zhou, D., Zhang, M., Li, H., and Tan, K. C. Progressive tandem learning for pattern recognition with deep spiking neural networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(11):7824-7840, 2021. +[42] Wu, Y., Deng, L., Li, G., Zhu, J., Xie, Y., and Shi, L. Direct training for spiking neural networks: Faster, larger, better. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 1311-1318, 2019. +[43] Yao, X., Hu, Q., Liu, T., Mo, Z., Zhu, Z., Zhuge, Z., and Cheng, J. Spikingnerf: Making bio-inspired neural networks see through the real world. arXiv preprint arXiv:2309.10987, 2023. +[44] Zhang, R., Isola, P., Efros, A. A., Shechtman, E., and Wang, O. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 586-595, 2018. +[45] Zou, S., Mu, Y., Zuo, X., Wang, S., and Cheng, L. Event-based human pose tracking by spiking spatiotemporal transformer, 2023. + +# A Technical Appendices and Supplementary Material + +# A.1 Rendering Performance in Realistic Dataset + +Figure 3 presents the rendering results from realistic datasets, including Flower, Room, T-rex, and Horns, for ANN-based NeRF, Spiking NeRF, and Spik-NeRF. As shown, our method captures fine details in both geometry and appearance, achieving results comparable to those of the ANN-based NeRF. In contrast, Spiking NeRF produces blurry and distorted renderings in certain areas. + +![](images/c1963d7b687685526a49d122fe3f477ceacc0123bc9f648519a8d2c5a8594d4a.jpg) +Flower + +![](images/2bb3196e91cb0331db9534e688f1c5f86dcc8776a6b282e3965de41b7f4c20cc.jpg) + +![](images/824ca7fd6d6f1b2a8b4e59aed8cafe170ee428a7b9728c2be3b970ae0706f045.jpg) + +![](images/64266cc79f5ed593fa94e238e8a931e0e2b8456fbdba7a7c14cc130aa5cc3ab3.jpg) + +![](images/1b1e4f36ed08e54e36579a9613d7a9c9a2d2d798f5019ea532b7a63589196ccb.jpg) + +![](images/065ce2e714969cac6f4d26f6f41f6b1fcb5f914d2169a2b72b17fd7bdb516807.jpg) + +![](images/45eb745961823e8fa07cd51ed687baa84be2d31630114ede8c6c1e929da85a4e.jpg) + +![](images/dfd145f576a0d6d1b5c7e405dddb70f1ab42fa37d37a62295333e106e542e9e7.jpg) + +![](images/d0fb183808ef13a21336daad903ee6b516bc6029aee141513a5cd30c07d65b51.jpg) + +![](images/7b11d17c437d5a5cad2e37f62a9ecd21b5c8a3a1b0a8a883fd163a81ef4d7ab3.jpg) +Room + +![](images/ffb6562319b82468b6959edc7df44d6650172944cdf78ddf5cbf77a3e65958f2.jpg) + +![](images/2d7cf32d5f4cf00dacd7a56cd393ab6f808144fe36f0545c5b324338445b7ed2.jpg) + +![](images/5568311ecc85d6010d156e7e864715ef0223d2601d134faa105d2114d574a0f3.jpg) + +![](images/94681655781eec750f1b8d81bdfb8fac68138a791bb57182da87f48673057a50.jpg) + +![](images/478b6c0cab8976abb3a276eddb7020b85bda552f96df5ba79fd7bd8423fe1972.jpg) + +![](images/617ef88c90cbf9f2020b0c4beb07a4524705e57504cf0f64ce5bc1a1d5e8e8dc.jpg) + +![](images/183b1bf73e0973e4be06c81566ff03b4da5aed09132f7d5015dc2ddb674bef13.jpg) + +![](images/7f3c5c1eb7638a03205a1acb231979330f1ba61361bfbc1e1d25d3fb371b4412.jpg) + +![](images/a2b22b6d73436704d5deee341642fa1ca4ccb7c493f21a6c901307c72dd12573.jpg) +T-rex + +![](images/7f9e065f9262efcd927e7037650931f2956863075d46347bd728e14cd0c825a9.jpg) + +![](images/42965c4469f91c8f59238c4efabeed56ea28c1daf952f0d94288c6baddc2ef45.jpg) + +![](images/0b873517d75d2ce89411369771ab55a6839359573c26a90adae27ef7239c5870.jpg) + +![](images/67c18f41f3d6fae6b990135a8d7d1043ca2bf8c23ec8f3d339f1ab7669034a2e.jpg) + +![](images/c6f73b703e1cab6b49c455e57fd130a0b60a40fd28a158e4cb0fdd8928cfcf69.jpg) + +![](images/5f69b6e29eeb02129858d9740565b0821e21487d083240c1bbe78749f73a9eef.jpg) + +![](images/1843c745aa397ac2f6f88fbf07ea65ab4e1c26a858085dd77114db7004207abd.jpg) + +![](images/2ec8cb9c4af5d2f46c0b1f1e4334e4b7f32b4d253bb43900ace36e6a4bc6754e.jpg) + +![](images/cc2396cd977c17f75a808941bb6dc3dfe00f2b29dc9610b6c09a951b8b3c6a98.jpg) +Horns + +![](images/9a3f5b70bc70f869ab6ea48df53b8e0a2b91f148ab1d545d570cdcb08e8c040c.jpg) + +![](images/9f3bae48ada1d34754ef816bfce9786ec030922a60b26c4e4ec1df788f0e4575.jpg) +Ground Truth + +![](images/d094c85b6b05e874ab057a5e4a6340ed33d37205b11588f83500a589e0d42edf.jpg) + +![](images/c176b0f801ffc2936b86e3f92393eac903885c0ae3ca82e1b20bbeb25aa0706f.jpg) +Spiking NeRF + +![](images/ccb002675cb89f9a64726cffac2b080fc6f79f0e85e65acd52b2f5c4daffb1bf.jpg) + +![](images/7729a2de1423f1c41c7b126fc88136d6ec7b7bbf32d7cd4cb7ae591135271182.jpg) +Figure 3: The rendering performance in the realistic dataset. + +![](images/84387eb4e0db7a30c9046fb02bbec56c70e9799a9a32482c2cddded077428d73.jpg) + +![](images/0adae800bdc632ed3d3044418284908896c9d8483fbb6a005c6e31c0d3ac3a16.jpg) +Spik-NeRF + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes]. + +Justification: We clearly state the claims made and the contributions made in both the abstract and introduction. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. 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However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. 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sha256:e119e9a0c8ef07ce75961cce3c9da7dedad0273149c0a4760a5d99d030051477 +size 5492826 diff --git a/NeurIPS/2025/Spike-RetinexFormer_ Rethinking Low-light Image Enhancement with Spiking Neural Networks/full.md b/NeurIPS/2025/Spike-RetinexFormer_ Rethinking Low-light Image Enhancement with Spiking Neural Networks/full.md new file mode 100644 index 0000000000000000000000000000000000000000..b9abbfbdce08b410ff8a2a5c9db61d943d29be82 --- /dev/null +++ b/NeurIPS/2025/Spike-RetinexFormer_ Rethinking Low-light Image Enhancement with Spiking Neural Networks/full.md @@ -0,0 +1,610 @@ +# Spike-RetinexFormer: Rethinking Low-light Image Enhancement with Spiking Neural Networks + +Hongzhi Wang + +School of Software Technology + +Zhejiang University + +Ningbo, China + +whzzju@zju.edu.cn + +Xiubo Liang* + +School of Software Technology + +Zhejiang University + +Ningbo, China + +xiubo@zju.edu.cn + +Jinxing Han + +School of Software Technology + +Zhejiang University + +Ningbo, China + +hanjx@zju.edu.cn + +Weidong Geng + +College of Computer Science and Technology + +Zhejiang University + +Hangzhou, China + +gengwd@zju.edu.cn + +# Abstract + +Low-light image enhancement (LLIE) aims to improve the visibility and quality of images captured under poor illumination. However, existing deep enhancement methods often underemphasize computational efficiency, leading to high energy and memory costs. We propose Spike-RetinexFormer, a novel LLIE architecture that synergistically integrates Retinex theory, spiking neural networks (SNNs) and a Transformer-based design. Leveraging sparse spike-driven computation, the model reduces theoretical compute energy and memory traffic relative to ANN counterparts. Across standard benchmarks, the method matches or surpasses strong ANNs (25.50 dB on LOL-v1; 30.37 dB on SDSD-out) with comparable parameters and lower theoretical energy. Our work pioneers the synergistic integration of SNNs into Transformer architectures for LLIE, establishing a compelling pathway toward powerful, energy-efficient low-level vision on resource-constrained platforms. + +# 1 Introduction + +Capturing images in low-light conditions is challenging due to limited photons, sensor noise, and constrained dynamic range. These factors produce dark, noisy images that hinder downstream tasks like detection or recognition. LLIE techniques seek to brighten such images and restore details, enabling better visual quality and utility. Early approaches relied on heuristic image processing (gamma correction, histogram equalization) and the Retinex theory of human vision [1], which decomposes an image into reflectance and illumination. Traditional Retinex-based algorithms [2, 3] improved visibility by estimating a smooth illumination map and enhancing the reflectance, but often suffered from artifacts (haloing, color distortion) and were limited by manual parameter tuning. + +The advent of deep learning spurred significant advances in LLIE. Convolutional neural networks (CNNs) have been trained to directly map dark images to brighter ones, outperforming classical methods in handling noise and complex artifacts. Representative works include LLNet [4], one of the first deep autoencoders for natural low-light enhancement, and LightenNet [5] for weak illumination images. More specialized models integrated Retinex theory, such as RetinexNet [6] which learned to decompose an image into reflectance and illumination and enhance them with + +![](images/1df96ecdbf0b1684f31bc996f78019bacdb4967dd67df3d54310277ee7ef14ed.jpg) +Figure 1: Architecture of the Spike-RetinexFormer. It consists of two main stages: (1) the Spiking Illumination Estimator, which predicts illumination features and a light-up map to adjust initial brightness, (2) the Spiking Corruption Restorer, which utilizes a multi-scale U-Net structure with Spiking Illumination-Guided Attention. + +deep networks. Many subsequent methods built on this idea, proposing improved Retinex-based networks with attention or refinement modules [7, 8, 9]. Unsupervised and zero-reference techniques have also emerged: Zero-DCE [10, 11] optimizes a deep curve mapping without ground truth, while EnlightenGAN [12] uses generative adversarial learning to enhance lighting without paired data. Recent trends include normalizing flow models [13] and diffusion models [14] for LLIE, and Transformer-based architectures [15] to capture global illumination context. Despite improved enhancement quality, most deep LLIE methods are computationally heavy and energy-demanding[16, 17], which limits deployment on resource-constrained devices. + +Meanwhile, SNNs have gained interest as the third generation of neural networks, offering event-driven computation inspired by biological neurons [18]. Neurons communicate via sparse binary spikes, leading to significantly reduced power consumption on neuromorphic hardware [19]. SNNs have achieved competitive performance on classification [20] and object detection [21] tasks with far lower energy usage than standard ANN models. However, relatively few works have applied SNNs to low-level vision: LLIE is a continuous-tone regression problem (enhancing pixel intensities), whereas most prior SNN works focused on classification or recognition with discrete labels. Directly applying SNNs to image enhancement must handle fine-grained color and illumination adjustments, requiring high precision despite the coarse spike-based computations. [22] encoded pixel intensities into spike latencies and used a recurrent SNN (with ConvLSTM) for unsupervised LLIE. This demonstrates that SNNs can gradually capture image structure via spike timing. + +Transformers [23] have revolutionized many vision tasks due to their self-attention mechanism capturing long-range dependencies. Vision Transformers [24] and related models have shown excellent performance in recognition and even low-level image restoration [25, 26]. For LLIE, Transformers can globally model illumination variation and object context, yielding more balanced enhancement [15]. Merging the energy efficiency of SNNs with the representational strength of Transformers is a promising direction that remains underexplored. Recent studies have started integrating spiking neurons into Transformer models [27, 28, 29, 30, 31], mainly for classification or sequential data processing. These spiking Transformers achieve comparable accuracy to standard Transformers while greatly reducing Multiply-Accumulate operations via spike-based attention. This progress motivates our approach to design a spiking Transformer for LLIE. + +In this paper, we propose Spike-RetinexFormer, a SNN architecture for LLIE. Spike-RetinexFormer is a spike-driven variant of RetinexFormer that re-instantiates the one-stage Retinex parameterization with temporal spike coding, Leaky Integrate-and-Fire (LIF [32]) neurons, surrogate-gradient training, + +and an event-driven attention mechanism. To our knowledge, this is among the first feedforward spiking Transformer-style network tailored to LLIE. By coupling Retinex-based illumination modeling with event-driven computation, the approach aims at high-fidelity enhancement under constrained compute and energy budgets. Specifically, a spiking illumination estimator predicts a light-up map and illumination features consistent with the Retinex reparameterization, while a Retinex restoration module employs spiking illumination-guided attention (SIGA) to aggregate long-range context and suppress noise. Our network processes images over $T$ time steps, accumulating an enhanced output from spiking neuron responses. In summary, our contributions to the community include: + +- We introduce Spike-RetinexFormer, which instantiates a one-stage Retinex enhancement pipeline entirely using spiking primitives and is trained with surrogate gradients, demonstrating that continuous image restoration can be effectively addressed in the spike domain. +- We develop a spiking illumination-guided attention mechanism—implemented via spike-coincidence affinities, illumination-gated sparsification, and binary value routing—and integrate it into the Retinex framework to enable long-range interactions without forming dense $QK^{\top}$ matrices or softmax normalization. +- Across standard LLIE benchmarks, Spike-RetinexFormer attains competitive enhancement quality relative to representative ANN counterparts while using comparable parameters and fewer theoretical FLOPs. + +# 2 Related Works + +# 2.1 Low-Light Image Enhancement + +Early methodologies for low-light image enhancement drew upon traditional image processing techniques and models of human visual perception, prominently featuring the Retinex theory [1], which models an image as a product of reflectance and illumination components. Subsequent developments, such as Multi-Scale Retinex with Color Restoration (MSRCR) [2] and LIME [3], focused on estimating illumination maps to improve image visibility; however, these approaches necessitated manual parameter tuning and were susceptible to artifact generation. The emergence of deep learning marked a paradigm shift, introducing data-driven approaches that often yielded superior performance. CNN-based models, exemplified by LLNet [4] and LightenNet [5], utilized large-scale datasets (LOL [6]) to learn the image enhancement mapping directly from data. While not exclusively focused on LLIE, Ignatov et al. [33] explored broader applications of CNNs in image enhancement, contributing to the foundational understanding in the field. The principles of Retinex theory were subsequently incorporated into deep learning frameworks. Notable models include RetinexNet [6] and its variants [7, 8], which explicitly perform image decomposition into illumination and reflectance, with applications extending to specialized domains such as underwater imaging [9] and back-lit scene enhancement. + +To address the limitations associated with paired training data, unsupervised and semi-supervised methods were developed. For instance, Zero-DCE [10] and its subsequent refinement [11] utilized non-reference loss functions, while EnlightenGAN [12] and DeepExposure [34] employed generative models for unpaired learning scenarios. Yang et al. [35] proposed a semi-supervised framework designed to strike a balance between image fidelity and perceptual quality. Recent investigations have focused on leveraging sophisticated neural architectures. Models based on normalizing flows [13] and diffusion probabilistic models [14] have demonstrated capabilities for high-fidelity image enhancement, albeit often at a significant computational cost. Inspired by their success in other computer vision tasks such as super-resolution [25, 26], Transformer architectures and attention mechanisms have also been adapted for LLIE, as evidenced by works like [15]. In a related vein, Tang et al. [36] introduced a method focusing on disentangling various image components to achieve more flexible and controllable enhancement. + +# 2.2 Spiking Neural Networks + +SNNs are a class of biologically-inspired computational models that process information through discrete temporal events, termed spikes, contrasting with the continuous activation values characteristic of conventional Artificial Neural Networks (ANNs)[37]. Within SNNs, individual neurons integrate input currents over time, generating an output spike when their internal membrane potential surpasses + +a predefined threshold. This event-driven operational paradigm offers the potential for substantial energy efficiency, as computationally expensive multiply-accumulate operations prevalent in ANNs are often replaced by simpler arithmetic operations (additions), and power consumption is predominantly associated with active spiking events. Maass [18] conceptualized SNNs as the third generation of neural network models, and subsequent research has established their viability as an alternative to ANNs in specific application domains. For instance, [19] demonstrated that neuromorphic hardware platforms executing SNNs can achieve energy efficiencies orders of magnitude greater than those of Graphics Processing Units (GPUs) processing functionally equivalent ANN models. + +The practical training of SNNs was initially impeded by the non-differentiable nature of the spike generation mechanism. However, methodologies such as ANN-to-SNN conversion and surrogate gradient learning have facilitated the effective training of deep SNN architectures [20]. Consequently, SNNs have demonstrated competitive performance on established image classification benchmarks [38, 20, 39, 40, 41, 42] and have been successfully applied to more complex vision tasks, including object detection (Spiking-YOLO [21]). Notwithstanding these advancements, the application of SNNs to low-level vision tasks requiring continuous-valued outputs, such as image enhancement, remains relatively underexplored. A principal challenge in this context is the generation of high-resolution, analog-like outputs (an enhanced image) from discrete spike trains. This often necessitates sophisticated encoding schemes for pixel intensities, such as rate or temporal coding, thereby introducing additional representational and computational complexity. Addressing this, [22] investigated LLIE by encoding pixel intensities into spike latencies, which were then processed by a recurrent SNN. Their findings demonstrated the feasibility of image enhancement using this paradigm, achieving notable computational reductions compared to some ANN counterparts. These results provide preliminary evidence for the capability of SNNs in image enhancement tasks. + +# 3 Method + +# 3.1 Spiking Retinex-Based Enhancement Framework + +Following Retinex theory, a low-light RGB image $I\in [0,1]^{H\times W\times 3}$ is modeled as the Hadamard product of a reflectance image $R\in [0,1]^{H\times W\times 3}$ and an illumination map $L\in [0,1]^{H\times W}$ : + +$$ +I = R \odot L. \tag {1} +$$ + +Directly obtaining a lit image by element-wise division is numerically fragile when $L$ is small. Instead, we introduce a light-up map $\bar{L}$ that approximates $L^{-1}$ and enforces the constraint $\bar{L} \odot L \simeq 1$ . Multiplying by $\bar{L}$ yields a lit-up image + +![](images/a21d0f6d5a84e91c6ea2eab8218c31761742f7a7283aa0684d7aff80d88d34c6.jpg) +Figure 2: Spiking illumination estimator $\mathcal{E}$ + +$$ +I _ {\mathrm {l u}} = I \odot \bar {L} = R + C, \tag {2} +$$ + +where $C \in \mathbb{R}^{H \times W \times 3}$ collects the overall corruption introduced by sensor noise, quantization, color shifts, and the light-up process itself. We realize (2) in one stage with a spiking illumination estimator and a spiking corruption restorer. We adopt a one-stage Retinex-based framework with two spike-driven modules: + +$$ +\left(I _ {\mathrm {l u}}, F _ {\mathrm {l u}}\right) = E \left(I, L _ {p}\right), \quad I _ {\mathrm {e n}} = G \left(I _ {\mathrm {l u}}, F _ {\mathrm {l u}}\right), \tag {3} +$$ + +where $E$ denotes the spiking illumination estimator and $G$ the spiking corruption restorer. The illumination prior $L_{p}\in \mathbb{R}^{H\times W\times 1}$ is the channel-average mean of the input image: + +$$ +L _ {p} = \operatorname {m e a n} (I), \tag {4} +$$ + +and both $E$ and $G$ are implemented with LIF spiking neurons and operate over $T$ discrete time steps; gradients are estimated via surrogate functions. + +As illustrated in Fig. 2, $E$ is a shallow spike-driven CNN that maps $[I\parallel L_p]\in \mathbb{R}^{H\times W\times 4}$ to a light-up image and illumination features: + +C1: $1 \times 1$ fusion $\rightarrow$ C2: depth-wise $5 \times 5$ (light-up features) $\rightarrow$ C3: $1 \times 1$ projection. + +$G$ is a U-Net style encoder-decoder with skip connections, implemented entirely with spiking layers. At each decoder stage, we insert a SIGA block that consumes the current feature maps together with the scale-aligned $F_{\mathrm{lu}}$ (from (3)), injecting long-range, lighting-aware context without dense softmax attention. A lightweight spike-driven head predicts a residual $\Delta I$ so that + +$$ +I _ {\mathrm {e n}} = I _ {\mathrm {l u}} + \Delta I. \tag {5} +$$ + +The whole pipeline (3) is trained end-to-end with surrogate gradients. + +# 3.2 Spiking Neuron Model and Training + +We use LIF neurons with soft reset and a single, shared time budget of $T$ steps for both training and inference. For each layer $\ell$ and time step $t \in \{1, \dots, T\}$ , + +$$ +U ^ {(\ell)} [ t ] = \lambda^ {(\ell)} H ^ {(\ell)} [ t - 1 ] + I ^ {(\ell)} [ t ], \quad S ^ {(\ell)} [ t ] = \Theta \left(U ^ {(\ell)} [ t ] - V _ {\mathrm {t h}} ^ {(\ell)}\right), \quad H ^ {(\ell)} [ t ] = U ^ {(\ell)} [ t ] - V _ {\mathrm {t h}} ^ {(\ell)} S ^ {(\ell)} [ t ], \tag {6} +$$ + +with $H^{(\ell)}[0] = 0$ , leak $\lambda^{(\ell)} \in (0,1)$ , and a fixed threshold $V_{\mathrm{th}}^{(\ell)}$ . + +# Input encoding and $T$ -normalization. + +The illumination estimator $E$ (see (3)) is executed first and receives a time-shared, $T$ -normalized current + +$$ +I _ {E} ^ {(0)} [ t ] = \frac {1}{T} \Phi_ {\mathrm {i n}} (I, L _ {p}), \quad \Phi_ {\mathrm {i n}} (X, Y) = \kappa \phi (X) \| \kappa_ {p} \phi (Y), \quad \phi (z) = \log (1 + \zeta z), \tag {7} +$$ + +where $\parallel$ denotes channel concatenation. After $E$ predicts the light-up map $\bar{L}$ and features $F_{\mathrm{lu}}$ (cf. (3)), we form $I_{\mathrm{lu}} = I\odot \bar{L}$ (cf. (2)) and its prior $L_{p}^{\mathrm{lu}} = \mathrm{mean}(I_{\mathrm{lu}})$ . The decoder $G$ then uses + +$$ +I _ {G} ^ {(0)} [ t ] = \frac {1}{T} \Phi_ {\mathrm {i n}} \left(I _ {\mathrm {l u}}, L _ {p} ^ {\mathrm {l u}}\right), \tag {8} +$$ + +which keeps the total injected charge approximately invariant to $T$ . We use $\zeta = 6$ and $(\kappa, \kappa_p) = (1.0, 0.5)$ in all experiments. + +Illumination-guided FiLM. For $\ell \geq 1$ in the decoder $G$ , the synaptic current is a convolution on spikes modulated by per-channel FiLM parameters: + +$$ +I ^ {(\ell)} [ t ] = \alpha^ {(\ell)} \odot \left(W ^ {(\ell)} * S ^ {(\ell - 1)} [ t ]\right) + \beta^ {(\ell)}, \tag {9} +$$ + +where $*$ is a standard convolution and $\alpha^{(\ell)}, \beta^{(\ell)} \in \mathbb{R}^{C_{\ell}}$ are time-shared (constant over $t$ ) and broadcast spatially. They are predicted once from temporally aggregated illumination features (from (3)): + +$$ +\bar {F} _ {\mathrm {l u}} = \frac {1}{T} \sum_ {t = 1} ^ {T} F _ {\mathrm {l u}} [ t ], \quad \hat {f} = \operatorname {G A P} \left(\bar {F} _ {\mathrm {l u}}\right), \quad \left[ \tilde {\alpha} ^ {(\ell)} \| \tilde {\beta} ^ {(\ell)} \right] = W _ {g} ^ {(\ell)} \hat {f} + b _ {g} ^ {(\ell)}, \tag {10} +$$ + +$$ +\alpha^ {(\ell)} = \alpha_ {\min } + \left(\alpha_ {\max } - \alpha_ {\min }\right) \sigma \left(\tilde {\alpha} ^ {(\ell)}\right), \quad \beta^ {(\ell)} = s \tanh \left(\tilde {\beta} ^ {(\ell)}\right), \tag {11} +$$ + +with $(\alpha_{\mathrm{min}},\alpha_{\mathrm{max}}) = (0.5,2.0)$ , $s = 0.05$ , and $\sigma (\cdot)$ the logistic sigmoid. When $\overline{F}_{\mathrm{lu}}$ is lower-resolution, we bilinearly upsample inside the FiLM head; BatchNorm (if used) runs in eval mode with EMA statistics to remain stable under sparse spikes. + +Surrogate gradients and readout. We train end-to-end with a simple piecewise-linear surrogate derivative around the firing threshold. The network uses a single-step readout identical in training and inference: + +$$ +Y = \operatorname {c l i p} \left(I _ {\mathrm {l u}} + \operatorname {H e a d} \left(H ^ {(\text {o u t})} [ T ]\right), 0, 1\right), \tag {12} +$$ + +where Head is a $1 \times 1$ convolution to 3 channels; gradients through clip use a straight-through estimator on $[0,1]$ . + +Objective. The loss combines an $L_{1}$ image term, a firing-rate regularizer, and a temporal TV on the pre-readout state to suppress flicker without over-smoothing spatial details: + +$$ +\mathcal {L} = \| Y - J \| _ {1} + \lambda_ {\mathrm {s p k}} \cdot \operatorname {A v g R a t e} (S) + \lambda_ {\mathrm {t v}} \cdot \mathrm {T V} _ {t} \left(H ^ {(\text {o u t})} [ 1: T ]\right), \tag {13} +$$ + +with $(\lambda_{\mathrm{spk}},\lambda_{\mathrm{tv}}) = (10^{-4},5\times 10^{-4})$ + +# 3.3 Illumination-Guided Spiking Transformer + +We build on the U-Net-style backbone summarized in Sec. 3.1 and Fig. 1. In this subsection we focus on how illumination guidance is injected at the decoder scales: the scale-aligned illumination features $F_{\mathrm{lu}}$ condition both the attention (via SIGA) and FiLM modulation. + +At each decoder scale, we deploy a SIGA module as the core long-range dependency unit. SIGA operates on spike trains with multi-head processing: query-key spike coincidences open hard binary gates that route value spikes, while per-head FiLM parameters—predicted from scale-aligned illumination features—modulate the $Q / K / V$ projections and gating thresholds and are shared over time. All attention computations are realized by LIF neurons unrolled across $T$ steps; BatchNorm layers (when present) run in evaluation mode with EMA statistics to maintain stable activations under spike sparsity. The feed-forward branch is a two-layer spiking MLP with LIF activations, forming a standard attention+FFN block in spiking form and serving as the decoder's illumination-aware context aggregator. + +SIGA mechanism: Each SIGA takes spike-encoded query, key, and value streams $(Q[t], K[t], V[t])$ and a scale-aligned illumination feature $F_{\mathrm{lu}}$ . Multi-head processing is used: channels are split into $H$ heads, computed in parallel, and concatenated. For each head, $F_{\mathrm{lu}}$ drives FiLM-style parameters that modulate the $Q / K / V$ projections and set illumination-aware gating thresholds; these parameters are predicted once and shared across time steps. Spike coincidences between $Q$ and $K$ open binary gates that route $V$ spikes, yielding an event-driven hard-attention pattern implemented by LIF neurons unrolled over $T$ steps. + +![](images/e5974542430599433c3004424cb24d09bc398e1e060adc360c9baaf8ce883954.jpg) +Figure 3: Schematic of the SIGA module: illustrating spike-based $Q$ , $K$ , $V$ interactions, modulated by the illumination features (from $\bar{L} / F_{\mathrm{lu}}$ ), to generate a binary attention mask for value routing. + +To compute attention in SIGA, we forego explicit floating-point matrix multiplication and softmax. Instead, spike-driven interactions implement hard gating: when a query neuron at position $i$ fires at time $t$ , synapses to candidate keys are activated; if a key neuron at position $j$ simultaneously fires, the coincidence $Q_{i}[t] = 1 \land K_{j}[t] = 1$ opens an instantaneous hard gate $A_{ij}[t]$ whose threshold is modulated by illumination features. Routing uses the current $A_{ij}[t]$ (preserving causality), while optional co-firing accumulators over $t = 1 \dots T$ integrate statistics. Owing to spike sparsity (and optionally a local neighborhood), each query selects a small subset of keys, enforcing efficient hard attention without forming dense $QK^{\top}$ ; complexity scales with spike rate and neighborhood size rather than quadratic token pairs. + +Given the gates at time $t$ , value spikes are routed accordingly: a value spike $V_{j}[t]$ contributes to query position $i$ iff $A_{ij}[t]$ is open. Implementationally, each head employs LIF neurons receiving value + +![](images/e82566359bdb9ef680db0e3e6abe155d4d2c9ee5ac1da70d11f18900cf3415b2.jpg) +Figure 4: Low-light image enhancement results on the LOL-v1 dataset. Compared methods: URetinex-Net, EnlightenGAN, Diff-Retinex, RetinexFormer and Spike-RetinexFormer(Ours) + +inputs masked by $A_{ij}[t]$ ; absent a gate, $V_{j}[t]$ has no effect on $i$ . The SIGA head output at position $i$ is a spike train emitted by a LIF unit that fires when its integrated (binary-weighted) input crosses threshold; illumination features can shift this threshold via FiLM-style bias. + +After the attention stage, head outputs are concatenated and passed through a spiking feed-forward module (two layers of linear spikes with an intermediate LIF activation), analogous to a Transformer FFN. Residual connections wrap the SIGA+FFN block with proper temporal alignment—residuals are added in the membrane-potential domain at each time step—so that the training-inference readout remains identical to the rest of the network. Multiple SIGA blocks are placed across decoder stages; at each scale, illumination-guided spiking attention supplies long-range, lighting-aware context while preserving event-driven efficiency. + +# 4 Experiments + +# 4.1 Experimental Setup + +We evaluate Spike-RetinexFormer on a comprehensive suite of standard benchmarks for LLIE, largely following the protocol of [43]. These include: LOL-v1 [6], LOL-v2 (real and synthetic) [44], SID [45], SMID [46], SDSD (indoor and outdoor) [47], MIT-Adobe FiveK [48], and LIME[49] which commonly used for qualitative evaluation. For datasets with paired ground truth, we train a separate model per dataset using the official or widely adopted splits; for RAW datasets, we convert to sRGB via the standard ISP pipeline before computing losses and metrics to ensure comparability. + +All experiments share the same backbone and training hyperparameters as in Sec. 3. Unless otherwise stated, we use the AdamW optimizer (base learning rate $2 \times 10^{-4}$ ) with cosine annealing (optional warm-up), unrolling the spiking dynamics for $T$ time steps and applying global-norm clipping at 1.0. Each model is trained until the validation performance saturates. We report the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity (SSIM) [50] on sRGB results for quantitative comparison. For model complexity, we provide parameter counts (Millions) and FLOPs at $256 \times 256$ input resolution. + +# 4.2 Comparison with ANN models + +We compare Spike-RetinexFormer with recent ANN methods, including Retinex-based CNNs and Transformer-based models: RetinexNet [6], KinD [51], DeepUPE [52], RUAS [53], MIRNet [54], Uformer [55], Restormer [26], SNR-Net [56], and RetinexFormer [43]. Quantitative results on benchmarks with ground truth are summarized in Tab. 1 (LOL) and Tab. 2 (SID, SMID, SDSD). Overall, Spike-RetinexFormer delivers competitive performance across most datasets and often matches strong ANN baselines. Notably, on the widely used LOL-v1 dataset, our method attains + +Table 1: Quantitative comparison on the datasets of LOL-v1, LOL-v2 (real and synthetic). + +
MethodFLOPs (G)Params (M)LOL-v1 PSNR/SSIMLOL-v2-R PSNR/SSIMLOL-v2-S PSNR/SSIM
RetinexNet [6]587.50.8416.77/0.56015.47/0.56717.13/0.798
DeepUPE [52]21.11.0214.38/0.44613.27/0.45215.08/0.623
KinD [51]35.08.0220.86/0.79014.74/0.64113.29/0.578
RUAS [53]0.80.00318.23/0.72018.37/0.72316.55/0.652
MIRNet [54]785.031.6024.14/0.83020.02/0.82021.94/0.876
Restormer [26]144.326.1022.43/0.82319.94/0.82721.41/0.830
SNR-Net [56]26.34.0124.61/0.84221.48/0.84924.14/0.928
RetinexFormer [43]15.61.6125.16/0.84522.80/0.84025.67/0.930
Ours16.21.5025.50/0.84223.38/0.84826.47/0.938
+ +Table 2: Quantitative comparison on SID, SMID, and SDSD (indoor/outdoor) datasets. Our method achieves top performance across these diverse benchmarks. + +
MethodSID +PSNR/SSIMSMID +PSNR/SSIMSDSD-in +PSNR/SSIMSDSD-out +PSNR/SSIMFLOPs +(G)Params +(M)
KinD [51]18.02/0.58322.18/0.63421.95/0.67221.97/0.65435.08.02
MIRNet [54]20.84/0.60525.66/0.76224.38/0.86427.13/0.837785.031.60
Uformer [55]18.54/0.57727.20/0.79223.17/0.85923.85/0.74812.05.29
Restormer [26]22.27/0.64926.97/0.75825.67/0.82724.79/0.802144.326.10
SNR-Net [56]22.87/0.62528.49/0.80529.44/0.89428.66/0.86626.34.01
RetinexFormer [43]24.44/0.68029.15/0.81529.77/0.89629.84/0.87715.61.61
Ours24.68/0.68129.43/0.82030.45/0.90330.37/0.88516.21.50
+ +25.5 dB PSNR and 0.842 SSIM, which is on par with RetinexFormer (25.2 dB, 0.845) and clearly ahead of earlier Retinex-based CNNs such as RetinexNet (16.8 dB) and KinD (20.8 dB). On the more challenging LOL-v2 (synthetic), Spike-RetinexFormer reaches 26.5 dB PSNR, representing a 0.8 dB gain over RetinexFormer with comparable SSIM; trends on SID/SMID/SDSD are similar. We attribute this competitiveness to the synergy between Retinex decomposition and spiking neural dynamics: illumination-guided spiking attention helps balance contrast, while iterative spike integration aids noise suppression under extreme low light. In terms of perceptual quality, our results typically exhibit fewer color shifts and artifacts; as shown in Fig. 4, our method yields natural, well-exposed images with preserved details, whereas some baselines may over-smooth or leave residual noise. + +As seen in Tab. 1 and 2, our spiking approach shows competitive or improved performance across the LOL variants and the extremely dark, noisy datasets (SID, SMID, SDSD). On average, SpikeRetinexFormer improves PSNR by $\sim 0.5$ dB over strong prior methods, with gains up to $\sim 0.8$ dB depending on the benchmark. In particular, for the dark indoor scenes of SDSD, our method recovers additional details, reaching $30.45\mathrm{dB}$ . Meanwhile, the model is compact—1.5M parameters and 16.2 G FLOPs—noticeably lower than heavy Transformers such as Restormer (26M, 144G). We attribute this efficiency to the one-stage design and the sparsity of spiking computation (neurons do not fire at every time step), making the approach practical for deployment. On neuromorphic hardware, energy consumption is expected to decrease because operations are triggered by spikes rather than dense activations. For example, if a 32-bit MAC costs an order of magnitude more energy than an accumulate (AC) [19], and only $\sim 18\%$ of neurons fire per time step in our network (Tab. 5), a back-of-the-envelope estimate suggests on-the-order-of $5\times -10\times$ energy reduction versus comparable ANN models under these assumptions. While these savings are theoretical (we have not measured power on a neuromorphic chip), they highlight the potential advantage of event-driven processing for power efficiency. + +# 4.3 Ablation Studies + +We systematically ablate the one-stage Retinex formulation (ORF), the proposed SIGA, and the influence of the time step $T$ . All ablations are conducted on LOL-v1 using a single RTX 3090. + +Table 3: Backbone ablations on LOL-v1 (full-resolution). + +
VariantPSNRSSIMParams (M)FLOPs (G)
Baseline (Spike-U-Net)23.110.7891.209.5
+ ORF24.560.8221.4511.8
+ ORF + W-MSA25.070.8351.6213.2
+ ORF + G-MSA25.290.8371.7516.8
+ ORF + SIGA (ours)25.500.8421.6814.0
+ +Table 4: Ablation on neuron type and architecture (LOL-v1, $T = 8$ ). + +
ActivationArchitecturePSNRSSIMEnergy (mJ)
ReLURetinexFormer25.160.84571.6
LIFRetinexFormer20.310.7147.5
LIFSpike-RetinexFormer25.500.84216.7
IFSpike-RetinexFormer25.310.83118.9
PLIF [57]Spike-RetinexFormer25.490.83315.8
CLIF [58]Spike-RetinexFormer25.610.84816.3
+ +Tab. 3 compares five progressively enhanced variants. **Baseline** is a pure spiking U-Net (no illumination branch, no SIGA). Introducing the illumination estimator and light-up operation (+ ORF) yields a $+1.45$ dB PSNR gain $(23.11\rightarrow 24.56)$ , indicating that explicit exposure modeling is critical in the spiking regime. Augmenting the ORF backbone with either a local-window MSA (+ W-MSA) or a global MSA (+ G-MSA) brings additional improvements. Our SIGA attains the highest fidelity, outperforming W-MSA and G-MSA by $+0.43$ and $+0.21$ dB. Qualitatively, the advantage is most evident in extreme shadows and mixed-lighting regions, where illumination-aware gating improves detail recovery and noise suppression. + +We ablate neuron types on LOL-v1 ( $T = 8$ ) with: (i) ANN RetinexFormer; (ii) RetinexFormer with LIF (ReLU $\rightarrow$ LIF); (iii) Spike-RetinexFormer (LIF+ORF+SIGA); and (iv) the same spiking backbone with IF/PLIF/CLIF (Tab. 4). Swapping ReLU $\rightarrow$ LIF in the ANN cuts compute energy to $7.5\mathrm{mJ}$ ( $10.5\%$ of baseline) but reduces fidelity (PSNR $-4.85\mathrm{dB}$ ; SSIM $0.845 \rightarrow 0.714$ ). Adding ORF+SIGA in the full spiking model restores accuracy while keeping energy low: PSNR $25.50\mathrm{dB}$ ( $+0.34\mathrm{dB}$ vs ANN), SSIM $\approx$ ANN, and $16.7\mathrm{mJ}$ (down from $71.6\mathrm{mJ}$ , $-77\%$ ). Fixing the architecture, neuron choice yields small but consistent gaps: CLIF is best (25.61/0.848) at energy close to LIF; IF/PLIF are slightly lower, $15.8 - 18.9\mathrm{mJ}$ . + +Tab. 5 reports performance and efficiency under varying time-step configurations. Using fewer steps $(T = 4)$ provides limited temporal evidence, leading to a modest PSNR drop and a higher average spike firing rate (SFR). Increasing to $T = 8$ achieves a favorable accuracy-latency balance; pushing to $T = 12$ offers only + +Table 5: Impact of time steps $T$ . Avg. SFR is the percentage of active neurons per step; latency is wall-time on RTX 3090 (ms); $\Delta$ Energy is normalized to $T = 8$ . + +
TPSNRSSIMAvg. SFRLatencyΔEnergy
425.210.8360.23816.50.55×
825.500.8420.18731.01.00×
1225.590.8410.18546.21.32×
+ +marginal PSNR gains while increasing wall-time nearly linearly. We therefore adopt $T = 8$ as the default trade-off. + +# 5 Limitations and Future Work + +Hardware-validated efficiency and scalability. Although SIGA avoids dense softmax attention, computing spike co-firing statistics can still scale as $O(N^2)$ in the number of spatial tokens $N$ when firing rates increase or sparsity collapses in high-illumination regions. To address this, we will design an event-driven sparse attention kernel that constructs co-firing pairs from time-bucketed inverted indices, enumerating only observed spike events and thereby avoiding any dense map materialization. In addition, we will investigate top- $k$ gating, block/tile-level sparsity, and kernel-level fusion of FiLM-style illumination modulation to reduce compute, memory traffic, and latency, and will release kernels and power traces to facilitate reproducibility. + +Robustness and cross-domain generalization. We will pursue RAW-aware training via a differentiable imaging pipeline and camera-conditional normalization to account for sensor variability; cross-dataset adaptation with self-supervised consistency constraints across RAW-sRGB pairs; and uncertainty-aware exposure control that disentangles aleatoric and epistemic components to mitigate over- and under-enhancement. We also plan to extend the framework to video using temporal spike-consistency losses and frame-adaptive time steps, and to broaden evaluation with no-reference and perceptual metrics (NIQE, BRISQUE, LPIPS) alongside stress tests for flicker, haloing, and color fidelity to more rigorously assess deployment robustness. + +# 6 Conclusion + +We introduced Spike-RetinexFormer, a low-light image enhancer that unifies spiking neural networks with a Retinex-inspired Transformer architecture. 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In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 17683-17693, 2022. +[56] Xiaogang Xu, Ruixing Wang, Chi-Wing Fu, and Jiaya Jia. Snr-aware low-light image enhancement. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 17714-17724, 2022. +[57] Wei Fang, Zhaofei Yu, Yanqi Chen, Timothee Masquelier, Tiejun Huang, and Yonghong Tian. Incorporating learnable membrane time constant to enhance learning of spiking neural networks. In Proceedings of the IEEE/CVF international conference on computer vision, pages 2661-2671, 2021. +[58] Yulong Huang, Xiaopeng Lin, Hongwei Ren, Haotian Fu, Yue Zhou, Zunchang Liu, Biao Pan, and Bojun Cheng. Clif: Complementary leaky integrate-and-fire neuron for spiking neural networks. arXiv preprint arXiv:2402.04663, 2024. + +# NeurIPS Paper Checklist + +# 1. 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We relate a Hebbian spike-timing-dependent plasticity rule to noisy gradient descent with respect to a non-convex loss function on the probability simplex. Despite the constant injection of noise and the non-convexity of the underlying optimization problem, one can rigorously prove that the considered Hebbian learning dynamic identifies the presynaptic neuron with the highest activity and that the convergence is exponentially fast in the number of iterations. This is non-standard and surprising as typically noisy gradient descent with fixed noise level only converges to a stationary regime where the noise causes the dynamic to fluctuate around a minimiser. + +# 1 Introduction + +Hebbian learning is a fundamental concept in computational neuroscience, dating back to Hebb [16]. In this work, we provide a rigorous analysis of a Hebbian spike-timing-dependent plasticity (STDP) rule. Those are learning rules for the synaptic strength parameters that only depend on the spike times of the involved neurons. More precisely, we consider a neural network composed of $d$ presynaptic输入 neurons, which are connected to one postsynaptic/output neuron. The presynaptic neurons communicate with the postsynaptic neuron by sending spike sequences, the so-called spike-trains. Reweighted by synaptic strength parameters $w_{1},\ldots ,w_{d}\geq 0$ , they contribute to the postsynaptic membrane potential. Whenever the postsynaptic membrane potential exceeds a threshold, the postsynaptic neuron emits a spike, and the membrane potential is reset to zero. Experiments have shown the following stylised facts, which lie at the core of Hebbian learning based on spikes: (1) Locality: The change of the synaptic weight $w_{i}$ depends only on the spike-train of neuron $i$ and the postsynaptic spike-train. (2) Spike-timing: The change of the synaptic weight $w_{i}$ depends on the relative timing of presynaptic spikes of neuron $i$ and of + +the postsynaptic neuron. More precisely, a pre-post spike sequence tends to increase $w_{i}$ , whereas a post-pre sequence tends to decrease $w_{i}$ . We refer to Morrison et al. [27] for a more comprehensive list of experimental results on STDP rules. + +Hebbian learning rules are well-studied if instead of the precise timings of pre- and postsynaptic spikes, only the mean firing rates are taken into account. These rate-based models exhibit many desirable properties, including performing streaming PCA [22, 19] and receptive field development [13, Section 11.1.4]. Much less is known if the precise timing of pre- and postsynaptic spikes is considered, since the intrinsic randomness of the dynamics complicates the mathematical analysis. + +![](images/797b1bee0e5a5157f3d0b9fab4a5c833d8f2224b815560d5155f4c8998b2ce64.jpg) +Figure 1: Neural network with a single output neuron + +Our main contribution lies in connecting STDP to noisy gradient descent and providing a rigorous convergence analysis of the noisy learning scheme. To this end, we introduce a learning rule for the weights $w_{1}, \ldots, w_{d}$ , which captures the locality and spike-time dependence of Hebbian STDP. We rewrite the learning rule as a noisy gradient descent scheme with respect to a suitable loss function. The connection to noisy gradient descent and stochastic approximation [23, 33] paves the way for applying mathematical tools from stochastic process theory to analyse the STDP rule. Our analysis of STDP is inspired by the work on noisy gradient descent for non-convex loss functions of Mertikopoulos et al. [26]. By refining their arguments and carefully tracking the error terms, we show an exponentially fast alignment of the output neuron with the input neuron of the highest mean firing rate on an event of high probability. The specialisation of the output neuron to the input neuron of the highest intensity is related to the winner-take-all mechanism in decision making [12, 48, 30, 24, 39]. The competitive nature of Hebbian STDP has been observed by [38, 37, 15] and the specialisation to few input neurons is important for receptive field development [9]. By connecting Hebbian STDP to noisy gradient descent, we are able to provide a mathematical analysis beyond ensemble averages and to quantify the speed of convergence. + +Taking into account the intrinsic geometry of the probability simplex, we also relate our learning rule to noisy mirror descent, more precisely to noisy entropic gradient descent, which has been proposed for brain-like learning by Neumann et al. [29], Cornford et al. [10]. + +The key contributions are: + +1. STDP as noisy gradient descent. We deduce a new framework, in which Hebbian STDP is interpreted as noisy gradient descent. This connection allows us to employ powerful tools from the theory of stochastic processes for analysing Hebbian STDP. +2. Linear convergence. We prove the alignment of the output neuron with the input neuron of highest intensity at exponential rate on an event of high probability. +3. Connection to noisy mirror descent. We relate our learning rule to noisy mirror descent, more specifically to entropic gradient descent. This connection facilitates the integration of techniques from both areas, potentially leading to future synergistic effects. + +# Related literature + +Common approaches to understanding STDP restrict to the mean behaviour after taking the ensemble average, e.g. [13, 21, 15], or compute the full distribution using the master equation of the Markov process [13, Section 11.2.4]. Unfortunately, the latter is only feasible in specific scenarios. In [20], the authors consider a general noisy spike-time dependent dynamic which is transformed into a deterministic ODE by imposing a slow learning rate and using the self-averaging effect of the system. A stability analysis reveals structure formation and output stabilisation. One major difference to our work is the influence of the noise. In [20], the variance of the weights grows linearly and a careful comparison of time scales is required. In our work, despite a constant injection of noise into the system, the dynamic for the spike-triggering probabilities converges to a deterministic limit. Secondly, the use of recent ideas from the analysis of noisy SGD allows us to track the influence of the realised noise in every step. A considerable number of previous works derived STDP rules based on the minimisation of a loss function, typically corresponding to the minimization/maximization of some notion of energy or information, see [8, 6, 7, 43, 41, 31, 42, 35]. While this approach is appealing, the mathematical analysis of these learning rules is challenging due to the modifications required to achieve biological plausibility. In contrast, we start with a biologically plausible learning rule and utilise the arising loss function to derive mathematical convergence guarantees of the learning rule. The importance of the choice of a suitable metric for the derivation of the learning rule is laid out in [40]. We refer to [1, 47, 36, 14, 44] and the references therein for further results on STDP. + +# 1.1 Notation + +Linear algebra. For a positive integer $d$ , we write $[d] := \{1, \dots, d\}$ and $\mathbf{1} := (1, \dots, 1)^{\top} \in \mathbb{R}^{d}$ . For $i \in [d]$ we denote by $\mathbf{e}_i$ the $i^{\text{th}}$ standard basis vector of $\mathbb{R}^d$ . The Hadamard product between two vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^d$ is denoted by $\mathbf{a} \odot \mathbf{b} := (a_1 b_1, \dots, a_d b_d)^{\top} \in \mathbb{R}^d$ . We write $\mathbb{I} \in \mathbb{R}^{d \times d}$ for the identity matrix on $\mathbb{R}^d$ and $\| \mathbf{u} \|^2 = \sum_{i=1}^{d} u_i^2$ for the squared Euclidean norm of a vector $\mathbf{u} \in \mathbb{R}^d$ . + +Probability. $M(1, \mathbf{p})$ denotes the multinomial distribution with one trial ( $n = 1$ ) and probability vector $\mathbf{p} = (p_1, \dots, p_d)^\top$ , that is $\xi \sim M(1, \mathbf{p})$ if only only if $\mathbb{P}(\xi = i) = p_i$ for any $i \in [d]$ . We denote by + +$$ +\mathfrak {P} := \left\{\mathbf {p} \in \mathbb {R} ^ {d}: p _ {i} \geq 0 \forall i \in [ d ], \sum_ {i = 1} ^ {d} p _ {i} = 1 \right\} +$$ + +the probability simplex in $\mathbb{R}^d$ . We denote by $\mathbb{1}_{\mathcal{A}}$ the indicator function of a set $\mathcal{A}$ . + +# 1.2 Hebbian inspired learning rule + +Inspired by Hebbian learning, we consider an unsupervised learning dynamic with $d$ input (or presynaptic) neurons and one output (or postsynaptic) neuron. The $i^{\mathrm{th}}$ input neuron has a mean firing rate $\lambda_{i} > 0$ describing the expected number of spikes per time unit. The vector $\pmb{\lambda} = (\lambda_1,\dots ,\lambda_d)$ contains the $d$ mean firing rates. The strength of the connection between the $i^{\mathrm{th}}$ input neuron and the output neuron is modulated by the weight parameter $w_{i}\geq 0$ , and changes to encode the information of the input firing rates. + +We introduce a Hebbian STDP rule in Subsection 2.3 and show that, under some assumptions on the spike-trains, it is equivalent to the following dynamics. If $\mathbf{w}(0) = (w_{1}(0),\dots ,w_{d}(0))^{\top}$ are the $d$ weights at initialisation, the updating rule from $\mathbf{w}(k)$ to $\mathbf{w}(k + 1)$ is given by + +$$ +\mathbf {w} (k + 1) = \mathbf {w} (k) \odot \left(\mathbf {1} + \alpha (\mathbf {B} (k) + \mathbf {Z} (k))\right), \tag {1} +$$ + +where $\alpha > 0$ is the learning rate and $k = 0,1,\ldots$ denotes the postsynaptic spike time. The $d$ -dimensional vector $\mathbf{B}(k)$ is the standard basis vector pointing to the presynaptic neuron, which triggered the $(k + 1)^{\mathrm{st}}$ postsynaptic spike. It is given as $\mathbf{B}(k) = \sum_{i=1}^{d} \mathbb{1}_{\zeta_k = i} \mathbf{e}_i$ , the one-hot encoding of independent multinomial random variables $\zeta_k \sim M(1, \mathbf{p}(k))$ , with $k$ -dependent probability vector + +$$ +\mathbf {p} (k) = \frac {\boldsymbol {\lambda} \odot \mathbf {w} (k)}{\boldsymbol {\lambda} ^ {\top} \mathbf {w} (k)} \in \mathbb {R} ^ {d}, \quad k = 0, 1, \dots \tag {2} +$$ + +Since the probabilities $p_i(k)$ model the probability that the $(k + 1)^{\mathrm{st}}$ postsynaptic spike is triggered by neuron $i = 1, \ldots, d$ , we call them (postsynaptic) spike-triggering-probabilities. The i.i.d. $d$ -dimensional vectors $\mathbf{Z}(k)$ , $k = 0, 1, \ldots$ model the contribution of presynaptic spikes, which did not trigger the $(k + 1)^{\mathrm{st}}$ postsynaptic spike, to the weight change. They are modelled to have i.i.d. components $Z_1(k), \ldots, Z_d(k)$ , which are supported in $[-(Q - 1), (Q - 1)]$ , for some $Q > 1$ , and centred such that $\mathbb{E}[\mathbf{Z}(k)] = 0$ . + +In the remainder of the paper, we analyse the long-run behaviour of $\mathbf{p}(k)$ as $k\to \infty$ under the learning rule Eq. (3). We say that the output neuron aligns with the $j^{\mathrm{th}}$ input neuron if $p_j(k)\rightarrow 1$ as $k\rightarrow \infty$ . Since the input intensities $\lambda_1,\ldots ,\lambda_d > 0$ are fixed throughout the dynamic, this condition is equivalent to $w_{j}(k) / \sum_{i = 1}^{d}w_{i}(k)\rightarrow 1$ as $k\rightarrow \infty$ . Figure 1 visualises the learning rule Eq. (2). + +# 2 Representation as noisy gradient descent + +We continue by relating the learning rule Eq. (1) to noisy gradient descent. For notational simplicity, define $\mathbf{Y}(k)\coloneqq \mathbf{B}(k) + \mathbf{Z}(k)$ for $k = 0,1,\ldots$ . Combining the weight updates Eq. (1) with the formula for the probabilities $\mathbf{p}$ from Eq. (2), we find + +$$ +\mathbf {p} (k + 1) = \frac {\boldsymbol {\lambda} \odot (\mathbf {w} (k) \odot (\mathbf {1} + \alpha \mathbf {Y} (k)))}{\boldsymbol {\lambda} ^ {\top} (\mathbf {w} (k) \odot (\mathbf {1} + \alpha \mathbf {Y} (k)))} = \frac {\mathbf {p} (k) \odot (\mathbf {1} + \alpha \mathbf {Y} (k))}{\mathbf {p} (k) ^ {\top} (\mathbf {1} + \alpha \mathbf {Y} (k))}, \quad k = 0, 1, \dots . \tag {3} +$$ + +The normalisation in the denominator and the multiplicative nature of the update ensures that the dynamic of $\mathbf{p}(k)$ is restricted to the probability simplex. By a Taylor expansion around $\alpha = 0$ , we find + +$$ +\mathbf {p} (k + 1) = \mathbf {p} (k) \odot \left(\mathbf {1} + \alpha (\mathbf {Y} (k) - \mathbf {p} (k) ^ {\top} \mathbf {Y} (k) \mathbf {1})\right) + \mathcal {O} \left(\alpha^ {2}\right). \tag {4} +$$ + +Since + +$$ +\mathbb {E} \left[ \mathbf {Y} (k) - \mathbf {p} (k) ^ {\top} \mathbf {Y} (k) \mathbf {1} \mid \mathbf {p} (k) \right] = \mathbf {p} (k) - \| \mathbf {p} (k) \| ^ {2} \mathbf {1}, +$$ + +the random vectors + +$$ +\boldsymbol {\xi} (k) := \mathbf {p} (k) \odot (\mathbf {Y} (k) - \mathbf {p} (k) ^ {\top} \mathbf {Y} (k) \mathbf {1} - \mathbf {p} (k) + \| \mathbf {p} (k) \| ^ {2} \mathbf {1}), \quad k = 0, 1, \dots . +$$ + +![](images/cff369129ae3f2a59191c7b9c7c9ea651c79db677c715e69670e5efe37794202.jpg) +Figure 2: Contour plot of the loss function $L$ from Eq. (6) on the probability simplex $\mathfrak{P}$ for $d = 3$ with different overlays. Left: Three sample trajectories of Eq. (3) with different initial configurations $\mathbf{p}(0)$ . Middle: Stream plot of the gradient field given by Eq. (7). Right: 100 sample trajectories of Eq. (3) with $\mathbf{p}(0) = (0.3, 0.3, 0.4)^{\top}$ . All trajectories are simulated with 2000 iteration steps, learning rate $\alpha = 0.01$ and $\mathbf{Z}(k) \sim \mathrm{Unif}([-1, 1]^d)$ . + +![](images/9255be9d155f650d0be66814c159e6a51727b5487145d7fbbb048a6dfbde4585.jpg) + +![](images/509c3ea93de9cdbb6640a86535703c54c4c0f93e812af5de996800ef178d2841.jpg) + +are centred. The distribution of $\pmb{\xi}(k)$ depends on $\mathbf{w}(k)$ and $\mathbf{p}(k)$ . Up to $\mathcal{O}(\alpha^2)$ -terms, we can write the learning rule Eq. (3) as a noisy gradient descent scheme + +$$ +\begin{array}{l} \mathbf {p} (k + 1) = \mathbf {p} (k) \odot (\mathbf {1} + \alpha (\mathbf {p} (k) - \| \mathbf {p} (k) \| ^ {2} \mathbf {1})) + \alpha \boldsymbol {\xi} (k) \\ = \mathbf {p} (k) - \alpha \nabla L (\mathbf {p} (k)) + \alpha \boldsymbol {\xi} (k), \quad k = 0, 1, \dots \tag {5} \\ \end{array} +$$ + +for the loss function + +$$ +L (\mathbf {p}) := - \frac {1}{3} \sum_ {i = 1} ^ {d} p _ {i} ^ {3} + \frac {1}{4} \left(\sum_ {i = 1} ^ {d} p _ {i} ^ {2}\right) ^ {2} = - \frac {1}{3} \mathbf {p} ^ {\top} (\mathbf {p} \odot \mathbf {p}) + \frac {1}{4} \| \mathbf {p} \| ^ {4}, \quad \mathbf {p} \in \mathbb {R} ^ {d} \tag {6} +$$ + +with gradient + +$$ +\nabla L (\mathbf {p}) = - \mathbf {p} \odot \left(\mathbf {p} - \| \mathbf {p} \| ^ {2} \mathbf {1}\right) \in \mathbb {R} ^ {d}, \quad \mathbf {p} \in \mathbb {R} ^ {d}. \tag {7} +$$ + +Dropping $\mathcal{O}(\alpha^2)$ terms is only done for illustrative purposes. Our main result (Theorem 2.2) applies to the original learning rule Eq. (3). The subsequent lemma summarises the key properties of the loss function $L$ from Eq. (6). For $d = 3$ , Figure 2 visualises the loss function $L$ and the learning dynamics Eq. (3). + +Lemma 2.1. All critical points of the loss function Eq. (6) can be written as $\mathbf{p}^{*} = \frac{1}{|S|}\sum_{j\in S}\mathbf{e}_{j}$ for some $S\subseteq [d]$ . Every critical point with $|S|\geq 2$ is a saddle point. The local minima of the loss function $L$ from Eq. (6) are the standard basis vectors $\{\mathbf{e}_1,\dots ,\mathbf{e}_d\}$ . Furthermore, every local minimum of $L$ is also a global minimum. + +# 2.1 Linear convergence of the learning rule + +We state the convergence guarantee for the learning rule Eq. (3). Renaming the indices, we can assume that $p_1(0)$ is the largest initial probability. Provided that $p_1(0)$ is strictly larger than each other component of $\mathbf{p}(0)$ , the following theorem shows linear convergence of the first component to 1 on an event $\Theta$ in expectation. The probability of $\Theta$ can be chosen arbitrarily close to 1 by reducing the learning rate $\alpha$ . + +Theorem 2.2. Given $\varepsilon \in (0,1)$ , assume + +$$ +\Delta := p _ {1} (0) - \max _ {i = 2, \dots , d} p _ {i} (0) > 0 \quad a n d \quad 0 < \alpha \leq \frac {\Delta^ {2}}{1 6 Q ^ {2}} \Big ((1 - Q \alpha) ^ {3} \wedge \frac {1}{2 5 6 (1 - p _ {1} (0))} \Big (4 \frac {\Delta}{d} + \Delta^ {2} \Big) \varepsilon \Big). +$$ + +Then there exists an event $\Theta$ with probability $\geq 1 - \varepsilon /2$ such that + +$$ +\mathbb {E} \left[ \| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \mathbb {1} _ {\Theta} \right] \leq 2 \left(1 - p _ {1} (0)\right) \exp \left(- \frac {\alpha}{1 6} \left(4 \frac {\Delta}{d} + \Delta^ {2}\right) k\right), \quad f o r a l l k = 0, 1, \dots +$$ + +Consequently, given $\delta > 0$ , + +$$ +\mathbb {P} \left(\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \geq \delta\right) \leq \varepsilon \quad f o r a l l \quad k \geq \frac {1 6 d}{\alpha \Delta (4 + d \Delta)} \log \left(\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta}\right). +$$ + +Remark 2.3. If all weights are equal at the starting point of the learning algorithm, the assumption $p_1(0) - \max_{i=2,\dots,d} p_i(0) > 0$ , is equivalent to requiring $\lambda_1(0) - \max_{i=2,\dots,d} \lambda_i(0) > 0$ . In this case, the convergence of $\mathbf{p}(k)$ to $\mathbf{e}_1$ corresponds to the network performing a winner-take-all mechanism [12, 48, 30, 39, 24]. The competitive selection of input neurons in Hebbian STDP has been observed by [38, 37, 15], among others. Our results extend existing findings by going beyond ensemble averages and also provide a rate of convergence for the random dynamics. + +The convergence on an event of high probability is in line with other recent results on noisy/stochastic gradient descent for non-convex loss functions, see e.g. [26, 45] or [11, Theorem 2.5]. Contrary to these results, we can choose a constant learning rate and obtain linear convergence. To illustrate the reason for this, we give a brief overview of the proof of Theorem 2.2. The full proof can be found in Section A.1. + +1. We restrict the analysis to the event $\Theta$ on which + +$$ +p _ {1} (k) - \max _ {i = 2, \dots , d} p _ {i} (k) \geq c > 0, +$$ + +holds for all iterates $k$ . On this event, the derivative of the first component can be bounded from below by + +$$ +p _ {1} (k) \left(p _ {1} (k) - \| \mathbf {p} (k) \| ^ {2}\right) \gtrsim 1 - p _ {1} (k). \tag {8} +$$ + +2. As described in Eq. (3), we apply a Taylor approximation to the original dynamics. We bound the error term for the $i^{\mathrm{th}}$ component $p_i(k + 1)$ by the order $\alpha^2 p_i(k)(1 - p_i(k))$ . The approximation error is dominated by the gradient update on $\Theta$ , if the learning rate is small enough (see Eq. (8)). +3. Similarly as in Eq. (5), we restate the learning increments of the dynamics as the sum of the true gradient and a centred noise vector $\xi(k)$ . By Eq. (8), this decomposition yields linear convergence of $p_1(k) \to 1$ on $\Theta$ . +4. To find a lower bound for the probability of the chosen event $\Theta$ , we employ a similar strategy as Mertikopoulos et al. [26]. Through the representation Eq. (5), we can show that $\Theta$ occurs, as soon as $\mathbf{M}(k) \coloneqq \alpha \sum_{i=1}^{k} \xi(i)$ is uniformly bounded by some sufficiently small constant. As $(\mathbf{M}(k))_{k \in \mathbb{N}}$ is a martingale, the probability of the latter event can be controlled through Doob's submartingale inequality (see Eq. (27)). +5. To apply Doob's submartingale inequality, we bound the second moment of $\mathbf{M}(k)$ . Since the variance of the components of $\pmb{\xi}(k)$ is also dominated by $1 - p_{1}(k)$ , we achieve a bound of the order $\alpha^{2}\sum_{i=1}^{\infty}(1 - p_{1}(i))\mathbb{1}_{\Theta}$ . This series is summable as we have linear convergence to 0, which allows us to choose a constant learning rate $\alpha$ . + +# 2.2 Associated gradient flow + +In this subsection, we consider the associated gradient flow of probabilities $\mathbf{p}(t)$ as a vector-valued function, which solves the ODE + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} \mathbf {p} (t) = \mathbf {p} (t) \odot (\mathbf {p} (t) - \| \mathbf {p} (t) \| ^ {2} \mathbf {1}) = - \nabla L (\mathbf {p} (t)), \quad t \geq 0 \tag {9} +$$ + +and is initialized by the probability vector $\mathbf{p}(0)$ . By definition, $\frac{\mathrm{d}}{\mathrm{d}t}\sum_{i=1}^{d}p_i(t) = 0$ , such that $\sum_{i=1}^{d}p_i(t) = \sum_{i=1}^{d}p_i(0) = 1$ . Since the updating rule is multiplicative, the gradient flow produces for all $t \geq 0$ a probability vector. The gradient flow Eq. (9) also occurs as a specific replicator equation in evolutionary game theory, see Hofbauer and Sigmund [17, Chapter 7]. Elementary properties and an explicit solution for the gradient flow with $d = 2$ are derived in Section 2.2. Although the loss function $L(\mathbf{p})$ in Eq. (6) does not satisfy a global Polyak-Łojasiewicz condition, in particular it is not globally convex, we can deduce the following convergence for the ODE Eq. (9). + +Theorem 2.4. Assume + +$$ +p _ {1} (0) \geq \max _ {i = 2, \dots , d} p _ {i} (0) + \Delta , +$$ + +for some $\Delta > 0$ . Then + +$$ +\left\| \mathbf {e} _ {1} - \mathbf {p} (t) \right\| _ {1} \leq 2 \big (1 - p _ {1} (0) \big) \exp \Big (- \frac {\Delta}{d} (1 + (d - 1) \Delta) t \Big), +$$ + +that is linear convergence of $\mathbf{p}(t) \to \mathbf{e}_1$ as $t \to \infty$ . + +# 2.3 Biological plausibility of the proposed learning rule + +We study a biological neural network consisting of $d$ input (or presynaptic) neurons, which are connected to one output (or postsynaptic) neuron. For the subsequent argument, we assume that the spike times of the $d$ input neurons are given by the corresponding jump times of $d$ independent Poisson processes $(X_{t}^{(1)})_{t\geq 0},\ldots ,(X_{t}^{(d)})_{t\geq 0}$ with respective intensities $\lambda_1,\dots ,\lambda_d$ . All neurons are excitatory and each connection between an input neuron $j\in [d]$ and the output neuron has a time varying and nonnegative synaptic strength parameter, which we denote by $w_{j}(t)\geq 0$ . + +An idealized model is that a spike of the $j^{\mathrm{th}}$ input neuron at time $\tau$ causes an exponentially decaying contribution to the postsynaptic membrane potential of the form $t\mapsto w_{j}(\tau)Ce^{-c(t - \tau)}\mathbb{1}_{t\geq \tau}$ . We set the parameters $c,C > 0$ to one, as this can always be achieved by a time change $t\mapsto tc$ and a change of units of the voltage in the membrane potential. + +![](images/6520edae240369dd1401bbd6e05f9432e6600412b3a6e6d9debeb7d4eedc2a3d.jpg) +Figure 3: Considered biological neural network with spike trains and membrane potential $Y_{t}$ of the postsynaptic neuron. + +If $\mathcal{T}_j$ denotes the spike times of neuron $j\in [d]$ , the post + +synaptic membrane potential $(Y_{t})_{t\geq 0}$ is given by $Y_{0} = 0$ and $Y_{t} = \sum_{j = 1}^{d}\sum_{\tau \in \mathcal{T}_{j},\tau \leq t}w_{j}(\tau)e^{-(t - \tau)}$ for all $t\geq 0$ until $Y_{t}\geq S$ , where $S > 0$ is a given threshold value. Once the threshold $S$ is surpassed, the postsynaptic neuron emits a spike and its membrane potential is reset to its rest value, which we assume to be 0. Afterwards, the incoming spikes will contribute to rebuilding the postsynaptic membrane potential. If $\mathfrak{t}_0\coloneqq 0 < \mathfrak{t}_1 < \mathfrak{t}_2 < \ldots$ denote the postsynaptic spike times, the membrane potential at arbitrary time is therefore given by + +$$ +Y _ {t} = \sum_ {j = 1} ^ {d} \sum_ {\tau \in \mathcal {T} _ {j} \cap \left(\mathrm {t} _ {k}, t \right]} w _ {j} (\tau) e ^ {- (t - \tau)} \quad \text {f o r a l l} \mathrm {t} _ {k} < t \leq \mathrm {t} _ {k + 1}. \tag {10} +$$ + +We consider the following pair-based spike-timing-dependent plasticity (STDP) rule ([14, Section 19.2.2]): A spike of the $j^{\mathrm{th}}$ presynaptic neuron at time $\tau$ causes the weight parameter function $t\mapsto w_{j}(t)$ to decrease at $\tau$ by $\alpha e^{-(\tau -t_{-})}$ , where $t_{-}$ is the last postsynaptic spike time before $\tau$ and to increase at any postsynaptic spike time $t_k$ by $\alpha \sum_{\tau \in \mathcal{T}_j\cap (\mathfrak{t}_k,\mathfrak{t}_{k + 1}]}e^{-(\tau -\mathfrak{t}_k)}$ , with $\alpha >0$ the learning rate. As common in the literature, spike times that occurred before $t_k$ only have a minor influence and are neglected in the updating of the weights after $t_k$ . The term $\sum_{\tau \in \mathcal{T}_j\cap (\mathfrak{t}_k,\mathfrak{t}_{k + 1}]}e^{-(\tau -\mathfrak{t}_k)}$ is then the trace ([14, Equation (19.12)]) of the $j^{\mathrm{th}}$ presynaptic neuron at time $t_{k + 1}$ . + +For mathematical convenience, we will assume that all weight-updates in $(\mathfrak{t}_k,\mathfrak{t}_{k + 1}]$ are delayed to the postsynaptic spike times $\mathfrak{t}_k$ in the sense that the learning rule becomes + +$$ +w _ {j} \left(\mathrm {t} _ {k + 1}\right) = w _ {j} \left(\mathrm {t} _ {k}\right) \left(1 + \alpha \left(\sum_ {\tau \in \mathcal {T} _ {j} \cap \left(\mathrm {t} _ {k}, \mathrm {t} _ {k + 1} \right]} e ^ {- \left(\mathrm {t} _ {k + 1} - \tau\right)} - e ^ {- (\tau - \mathrm {t} _ {k})}\right)\right), \quad k = 0, 1, \dots. \tag {11} +$$ + +The postsynaptic spike times $\mathfrak{t}_k$ are the moments at which the postsynaptic membrane potential $Y_{t}$ reaches the threshold $S$ . They depend on the presynaptic spike times, however, the exact dependence is hard to characterise in the assumed model. For mathematical tractability, we will instead work with an adjusted rule to select the postsynaptic spike times $\mathfrak{t}_1, \mathfrak{t}_2, \ldots$ . Since $Y_{t}$ only increases at the presynaptic spike times, $\mathfrak{t}_{k + 1}$ has to happen at a presynaptic spike time. Denote by $\tau_{j1}, \tau_{j2}, \ldots$ the spike times of the $j$ -th presynaptic neuron after the previous postsynaptic spike time $\mathfrak{t}_k$ in increasing order. The distribution of $\mathfrak{t}_{k + 1} | \mathfrak{t}_k$ is completely determined by the probabilities + +$$ +\begin{array}{l} \mathbb {P} \left(\mathrm {t} _ {k + 1} = \tau_ {j \ell}\right) = \mathbb {P} \left(\mathrm {t} _ {k + 1} = \tau_ {j \ell} \mid \mathrm {t} _ {k + 1} \in (\tau_ {j m}) _ {m \geq 1}\right) \mathbb {P} \left(\mathrm {t} _ {k + 1} \in (\tau_ {j m}) _ {m \geq 1}\right), j = 1, \dots , d, \ell = 1, \dots \\ \text {B a s e d o n p r o b a b i l i s t i c a r g u m e n t s r e l a t e d t o t h e u b l y i n g P o i s s o n p r o b a b i l i s t i c a r g u m e s t h a t w e o u l t i n e n S e c t i o n A . 3}, \text {w e r e p r o b a l i s t i c a r g u m e s} P (\mathrm {t} _ {k + 1} \in (\tau_ {j m}) _ {m \geq 1}) \text {b y t h e p r o b a b i l i t i e s} \end{array} +$$ + +$$ +\frac {\lambda_ {j} w _ {j} \left(\mathbf {t} _ {k}\right)}{\sum_ {\ell = 1} ^ {d} \lambda_ {\ell} w _ {\ell} \left(\mathbf {t} _ {k}\right)}. \tag {12} +$$ + +Working with Eq. (12) instead of $\mathbb{P}(\mathfrak{t}_{k + 1}\in (\tau_{jm})_{m\geq 1})$ results in an approximation of the distribution of $\mathfrak{t}_{k + 1}$ . Lemma A.7 describes a setting, where Eq. (12) is exact. If all weights are much larger than the threshold $S$ , every presynaptic spike causes a postsynaptic spike. The proof of Lemma A.7 can be adapted to this case to show that the probability that the $j^{\mathrm{th}}$ neuron emits the first spike is $\lambda_j / \sum_{\ell = 1}^d\lambda_\ell$ . Since Hebbian learning is intrinsically unstable, we argue that the proposed approximation describes the dynamic at the beginning of the learning process. This view is corroborated by experimental results, see point (vi) of Morrison et al. [27, Section 2.1]. + +Compared to the original definition of $\mathfrak{t}_{k + 1}$ , the proposed sampling scheme has the advantage that the presynaptic spike times, which were not selected as postsynaptic spike time, add centred noise to the updates. More precisely, one can show that by the construction of $\mathfrak{t}_k$ and the properties of the underlying Poisson processes, the conditional distribution $\tau |\{\tau \in (\mathfrak{t}_k,\mathfrak{t}_{k + 1})\}$ is uniformly distributed on $(\mathfrak{t}_k,\mathfrak{t}_{k + 1})$ . By the symmetry relation $e^{-(b - u)} - e^{-(u - a)} = -(e^{-(b - v)} - e^{-(v - a)})\in [-1,1]$ which holds for all real numbers $a\leq u\leq b$ with $v = b + a - u\in [a,b]$ , this implies that conditionally on $\tau \in (\mathfrak{t}_k,\mathfrak{t}_{k + 1})$ , the random variable $e^{-(\mathfrak{t}_{k + 1} - \tau)} - e^{-(\tau -\mathfrak{t}_k)}$ is centred and supported on $[-1,1]$ . The update rule Eq. (11) then becomes + +$$ +w _ {j} \left(\mathbf {t} _ {k + 1}\right) = w _ {j} \left(\mathbf {t} _ {k}\right) \left(1 + \alpha \mathbb {1} _ {\{j = j ^ {*} (k + 1) \}} \left(1 - e ^ {\mathbf {t} _ {k} - \mathbf {t} _ {k + 1}}\right) + \alpha \sum_ {\tau \in \mathcal {T} _ {j} \cap \left(\mathbf {t} _ {k}, \mathbf {t} _ {k + 1}\right)} Z (\tau , j)\right), \tag {13} +$$ + +with centred random variables $Z(\tau, j)$ satisfying $|Z(\tau, j)| \leq 1$ . Assuming that the postsynaptic firing rate is slow compared to the learning dynamic, we discard the term $e^{\mathbf{t}_k - \mathbf{t}_{k + 1}} \ll 1$ . Since $j^{*}(k + 1)$ follows a multinomial distribution with parameters $\lambda_{j} w_{j}(\mathbf{t}_{k}) / (\sum_{\ell = 1}^{d} \lambda_{\ell} w_{\ell}(\mathbf{t}_{k}))$ , the term $\mathbb{1}_{\{j = j^{*}(k + 1)\}}$ corresponds to the $j^{\mathrm{th}}$ component of $\mathbf{B}(k)$ in Eq. (1). This motivates the learning rule Eq. (1). Additional details on the derivation are given in Subsection A.3 of the supplementary material. + +# 3 A mirror descent perspective + +In this section, we rewrite the gradient flow Eq. (9) as natural gradient descent on the probability simplex and relate the discrete-time learning rule Eq. (3) for the probabilities $\mathbf{p}$ to noisy mirror gradient descent. + +Recall from Eq. (5) that the learning rule Eq. (3) can be interpreted as noisy gradient descent with respect to the loss function $L$ from Eq. (6) in the Euclidean geometry. As we consider a flow on probability vectors, a different perspective is to use the natural geometry of the probability simplex. To this end, we consider the interior of the probability simplex $\mathcal{M} \coloneqq \operatorname{int}(\mathfrak{P})$ as a Riemannian manifold with tangent space $\mathcal{T}_{\mathbf{p}}\mathcal{M} = \{\mathbf{x} \in \mathbb{R} : \mathbf{1}^{\top}\mathbf{x} = 0\}$ for every $\mathbf{p} \in \mathcal{M}$ . A natural metric on $\mathcal{M}$ is given by the Fisher information metric / Shahshahani metric [4, 17], which is induced by the metric tensor $d_{\mathbf{p}} \colon \mathcal{T}_{\mathbf{p}}\mathcal{M} \times \mathcal{T}_{\mathbf{p}}\mathcal{M} \to \mathbb{R}$ , $(\mathbf{u},\mathbf{v}) \mapsto \mathbf{u}^\top \mathrm{diag}(\mathbf{p})^{-1}\mathbf{v}$ at $\mathbf{p} \in \mathcal{M}$ . Here, $\mathrm{diag}(\mathbf{p}) \in \mathbb{R}^{d\times d}$ is the diagonal matrix with diagonal entries given by $\mathbf{p}$ . We refer to Figure 1 of Mertikopoulos and Sandholm [25] for an illustration of unit balls in this metric. The (Riemannian) gradient of the loss function $\widetilde{L}(\mathbf{p}) = -\|\mathbf{p}\|^2/2$ with respect to $d_{\mathbf{p}}$ is given by + +$$ +\nabla_ {d _ {\mathbf {p}}} \widetilde {L} (\mathbf {p}) = \mathrm {d i a g} (\mathbf {p}) \nabla \widetilde {L} (\mathbf {p}) \in \mathcal {T} _ {\mathbf {p}} \mathcal {M}, +$$ + +where we denote by $\nabla \widetilde{L}$ the Euclidean gradient of $\widetilde{L}$ . The Riemannian gradient flow is called natural gradient flow in information geometry [3] and Shahshahani gradient flow in evolutionary game theory [18]. When transforming the Euclidean gradient flow for $L$ to a Riemannian gradient flow on the probability simplex, the part $+\| \mathbf{p}\|^2\mathbf{1}$ is orthogonal to $\mathcal{T}_{\mathbf{p}}\mathcal{M}$ . Consequently, it does not contribute a direction on the probability simplex. Consequently, the Riemannian gradient flow of $\widetilde{L}$ and the Euclidean gradient flow of $L$ coincide. + +The mirror descent algorithm [28] prescribes the discrete-time optimisation algorithm + +$$ +\mathbf {p} (k + 1) \in \underset {\mathbf {p} \in \mathcal {M}} {\operatorname {a r g m i n}} \left\{\mathbf {p} ^ {\top} \nabla f (\mathbf {p} (k)) + \frac {1}{\alpha} \Phi (\mathbf {p}, \mathbf {p} (k)) \right\}, \quad k = 0, 1, \dots , \tag {14} +$$ + +where $f\colon \mathcal{M}\to \mathbb{R}$ is the function to be minimised and $\Phi \colon \mathcal{M}\times \mathcal{M}\to \mathbb{R}_+$ is a suitable proximity function. Euclidean gradient descent is recovered by the choice $\Phi (\mathbf{p},\mathbf{p}(k)) = \| \mathbf{p} - \mathbf{p}(k)\| ^2$ . It + +is well-known that the natural gradient flow is the continuous-time analogue of the exponentiated gradient descent or entropic mirror descent, where $\Phi (\mathbf{p},\mathbf{p}(k)) = \mathrm{KL}(\mathbf{p}\| \mathbf{p}(k))$ is chosen as the Kullback-Leibler divergence between $\mathbf{p}$ and $\mathbf{p}(k)$ [2, 46, 32]. Consequently, the gradient flow Eq. (9) can also be viewed as continuous-time version of entropic mirror descent with respect to $f = \widetilde{L}$ . This connection transfers to the discrete-time and noisy updating rule Eq. (3). An alternative approach for connecting our proposed discrete-time learning rule Eq. (3) to entropic mirror descent is included in Subsection A.4 of the supplementary material. + +# 4 Multiple weight vectors + +The learning rule Eq. (1) aligns the output neuron with the input neuron of the highest intensity, but no information about the remaining input neurons is unveiled. As a proof-of-concept, we generalise the learning algorithm Eq. (1) to estimate the order of the intensities $\lambda_1,\dots ,\lambda_d$ . To this end, we consider $d$ different output neurons, which are connected to the $d$ input neurons via the weight vectors $\mathbf{w}_1,\ldots ,\mathbf{w}_d\in \mathbb{R}^d$ . The weights at time $k$ are combined into the matrix + +![](images/cba9839f2b837fb23387fc8355774377724fdbeaa6e0c216b5d6a96143ac1e10.jpg) +Figure 4: Neural network with $d$ input/output neurons. + +$$ +\mathbf {W} (k) = \left[ \begin{array}{c c c} \mathbf {w} _ {1} (k) & \dots & \mathbf {w} _ {d} (k) \end{array} \right] \in \mathbb {R} ^ {d \times d}, \quad k = 0, 1, \ldots +$$ + +and the corresponding probabilities $\mathbf{p}_1, \ldots, \mathbf{p}_d$ are combined into the matrix $\mathbf{P}(k) = [\mathbf{p}_1(k) \cdots \mathbf{p}_d(k)] \in \mathbb{R}^d$ . By reordering the neurons, we can achieve $\lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_d$ . If the intensities are strictly ordered, our goal is the alignment of the $j^{\text{th}}$ output neuron with the $j^{\text{th}}$ input neuron, which amounts to the convergence of $\mathbf{P}(k)$ to the identity matrix $\mathbb{I} \in \mathbb{R}^{d \times d}$ as time + +increases. If multiple intensities are equal, convergence is up to permutations within the group of equal intensities. We propose Algorithm 1, which constitutes an STDP rule as lines 3 - 4 can be implemented using the spike-trains and the learning rule Eq. (11). + +Algorithm 1: Aligning multiple output neurons +Input: $K\in \mathbb{N}$ : number of iterations, $\mathbf{W}(0)\in \mathbb{R}^{d\times d}$ : weight initialisation, $\alpha_{1},\ldots ,\alpha_{d}$ .. learning rates of the output neurons. +for $k = 0,1,\dots ,K - 1$ do +for $j = 1,\dots ,d$ do +Receive $\mathbf{B}_j(k)\sim M(1,\mathbf{p}_j(k))$ with $\mathbf{p}_j(k)\gets \lambda \odot \mathbf{w}_j(k) / \lambda^\top \mathbf{w}_j(k)$ and + $Z_{j}(k)\sim \mathrm{Unif}([ - 1,1]^{d})$ from spike trains; +Compute the base change $\Delta \mathbf{w}_j(k)\leftarrow \alpha_j\mathbf{w}_j(k)\odot (\mathbf{B}_j(k) + \mathbf{Z}_j(k));$ Update + $\mathbf{w}_j(k + 1)\gets \Delta \mathbf{w}_j(k) - \sum_{i = 1}^{j - 1}\frac{(\Delta\mathbf{w}_j(k))^{\top}\mathbf{w}_i(k)}{\|\mathbf{w}_i(k)\|^2}\mathbf{w}_i(k);$ +end + +Output: The weight evolution $\mathbf{W}(k) = [\mathbf{w}_1(k)\dots \mathbf{w}_d(k)], k = 0,\ldots ,K$ and probability evolution $\mathbf{P}(k) = [\mathbf{p}_1(k)\dots \mathbf{p}_d(k)], k = 1,\ldots ,K.$ + +Algorithm 1 is inspired by Sanger's rule [34] for learning $d$ principal components in streaming PCA. The first weight vector $\mathbf{w}_1(k)$ aligns with $\mathbf{e}_1$ by Theorem 2.2 since its dynamic equals the learning rule Eq. (1). By removing the components of the change $\Delta \mathbf{w}_j(k)$ in the direction of $\mathbf{w}_1(k),\ldots ,\mathbf{w}_{j - 1}(k)$ in line 5 of Algorithm 1, the weight vector $\mathbf{w}_j(k)$ is forced to converge to $\mathbf{e}_j$ , similarly to the Gram-Schmidt algorithm. + +Simulations of the corresponding probability matrix $\mathbf{P}(k)$ with varying learning rates and $\mathbf{Z}$ drawn i.i.d. from $\mathrm{Unif}([ - 1,1]^d)$ are included in Figure 5. We choose different learning rates for the $d$ vectors satisfying $\alpha_{1} > \dots >\alpha_{d} > 0$ . This ordering ensures fast convergence of the lower order weight vectors to the correct standard basis vector and counteracts the impact of initial misalignments of the higher order weight vectors. The simulation of Algorithm 1 shown in Figure 5 displays a + +![](images/c506ee7b4285a5f6f0230751276ac792f571c152fdc85c1b9e3be2e23a65f651.jpg) +Figure 5: Probability matrix $\mathbf{P}(k)$ arising from the weight dynamic $\mathbf{W}(k)$ of Algorithm 1 for dimensions $n = d = 3$ . The weights are initialised equally, and the intensities are given by $\lambda = (10,7.5,5)^{\top}$ . The resulting initial probabilities are $\mathbf{p}_1(0) = \mathbf{p}_2(0) = \mathbf{p}_3(0) = (4/9,1/3,2/9)^{\top}$ . Left: A single trajectory with learning rates $10^{-3}(1,0.75,0.5)^{\top}$ and $4\times 10^{4}$ iterations. The markers $\times$ and $\bullet$ correspond to the probabilities at $k = 4\times 10^{3}$ and $k = 10^{4}$ . Middle & right: The Frobenius error $\| \mathbf{P}(k) - \mathbb{I}\| ^2 /2$ of 100 trajectories with learning rates $10^{-3}(1,0.75,0.5)^{\top}$ and $10^{-4}(1,0.75,0.5)^{\top}$ , respectively. + +![](images/a1e3708131dcf806d3d795156165b2509b55333f7a6e68061d29e9a1ae960d3c.jpg) + +![](images/5e5a96b30aeb4aa777e13ab363a9483009815ba5cbf2cab969b3306785ec6201.jpg) + +decrease of the Frobenius error $\| \mathbf{P}(k) - \mathbb{I}\|^2 / 2$ over the iteration index $k$ , when averaged (blue line). Nevertheless, we observe that for a single trajectory, the error can plateau around 1 and 2. Given that the probability vectors tend to converge to standard basis vectors $\{\mathbf{e}_1, \dots, \mathbf{e}_d\}$ , a non-vanishing error is due to an incorrect ordering or duplicates. Consequently, the error $\| \mathbf{P}(k) - \mathbb{I}\|^2 / 2$ corresponds to the number of output neurons aligning with the incorrect input neuron, and plateaus at 1, 2 and 3 can arise. Theorem 2.2 shows that this phenomenon can be mitigated by slower learning rates, which is corroborated by decreasing the base learning rate from $10^{-3}$ to $10^{-4}$ in the simulation. A rigorous mathematical analysis of the Algorithm 1 is challenging due to joint updates in all read-out neurons. In Section A.5 we show that Theorem 2.2 is applicable if the learning is split into disjoint learning periods, and we derive theoretical convergence guarantees. + +# 5 Extensions, discussion and limitations + +Time-inhomogeneous intensities. We considered input spike trains generated from Poisson point processes with fixed intensity. It is natural to extend this to time-inhomogeneous intensities. Here we assume that the intensities of the input neurons are constant on the interval $(k, k + 1]$ and are stored in the vector $\boldsymbol{\lambda}(k)$ . The intensities $\boldsymbol{\lambda}(k)$ and weights $\mathbf{w}(k)$ determine the spike-triggering-probabilities $\mathbf{p}(k) = \boldsymbol{\lambda}(k) \odot \mathbf{w}(k) / \boldsymbol{\lambda}(k)^{\top} \mathbf{w}(k) = \mathbb{E}[\mathbf{Y}(k)]$ and the update formula Eq. (2) becomes + +$$ +\mathbf {p} (k + 1) = \frac {\boldsymbol {\lambda} (k + 1) \odot (\mathbf {w} (k) \odot (\mathbf {1} + \alpha \mathbf {Y} (k)))}{\boldsymbol {\lambda} (k + 1) ^ {\top} (\mathbf {w} (k) \odot (\mathbf {1} + \alpha \mathbf {Y} (k)))} = \frac {\widetilde {\mathbf {p}} (k) \odot (\mathbf {1} + \alpha \mathbf {Y} (k))}{\widetilde {\mathbf {p}} (k) ^ {\top} (\mathbf {1} + \alpha \mathbf {Y} (k))}, \quad k = 0, 1, \dots , \tag {15} +$$ + +with $\widetilde{\mathbf{p}}(k) \coloneqq \boldsymbol{\lambda}(k+1) \odot \mathbf{w}(k)/(\boldsymbol{\lambda}(k+1)^{\top} \mathbf{w}(k))$ . A first order Taylor expansion yields + +$$ +\mathbf {p} (k + 1) = \widetilde {\mathbf {p}} (k) \odot \left(\mathbf {1} + \alpha \left(\mathbf {Y} (k) - \widetilde {\mathbf {p}} (k) ^ {\top} \mathbf {Y} (k) \mathbf {1}\right)\right) + O \left(\alpha^ {2}\right). \tag {16} +$$ + +Since $\mathbb{E}[\mathbf{Y}(k)] = \mathbf{p}(k)$ , this means that + +$$ +\mathbf {p} (k + 1) = \widetilde {\mathbf {p}} (k) \odot \left(\mathbf {1} + \alpha \left(\mathbf {p} (k) - \widetilde {\mathbf {p}} (k) ^ {\top} \mathbf {p} (k) \mathbf {1}\right)\right) + \text {c e n t e r e d n o i s e} + O \left(\alpha^ {2}\right). \tag {17} +$$ + +Extending the gradient flow derivation to the time-inhomogeneous case, one can identify the ODE + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} \mathbf {p} (t) = \mathbf {p} (t) \odot \left(\frac {\mathrm {d}}{\mathrm {d} t} \log (\boldsymbol {\lambda} (t)) + \mathbf {p} (t) - \mathbf {p} (t) ^ {\top} \left(\frac {\mathrm {d}}{\mathrm {d} t} \log (\boldsymbol {\lambda} (t)) + \mathbf {p} (t)\right) \mathbf {1}\right), \tag {18} +$$ + +where the logarithm is taken componentwise, as corresponding deterministic dynamic in continuous time, see Section A.6 for a derivation. The ODE can be interpreted as a replicator equation with (time-varying) fitness $\frac{\mathrm{d}}{\mathrm{d}t}\log (\pmb {\lambda}(t)) + \mathbf{p}(t)$ , see e.g. Chapter 7 of [18]. An interesting scenario which lies beyond our mathematical analysis amounts to considering time-dependent mean firing rates that are piecewise constant, corresponding to the successive exposition to different input patterns [9]. + +Correlated inputs. Correlated inputs facilitate simultaneous spiking of different input spikes. The probability that input $i$ and $j$ spike at the same time is denoted by $\Gamma_{ij}$ , and naturally $\Gamma_{ii} := 1$ , for all $i \in [d]$ . We introduce the $d \times d$ random symmetric matrix $\mathbf{C}(k) = (C_{i,j}(k))_{i,j \in [d]}$ with independently sampled entries + +$$ +C _ {j, i} (k) := C _ {i, j} (k) \sim \operatorname {B e r} \left(\boldsymbol {\Gamma} _ {i, j}\right), \quad \text {f o r a l l} i \leq j \in [ d ]. +$$ + +$\mathbf{C}(k)$ describes the simultaneous spiking of the different inputs at the $k$ -th post-synaptic spike, i.e. if $C_{i,j}(k)$ is 1 then inputs $i$ and $j$ both spike at the $k$ -th post-synaptic spike if either $i$ or $j$ caused the post-synaptic spike. Compared to the original model, the random vector $\mathbf{Z}(k)$ remains the same, but the random vector $\mathbf{B}(k) = (B_1(k), \ldots, B_d(k))^{\top}$ that encapsulates which of the presynaptic neurons caused the postsynaptic spike is replaced by $\mathbf{S}(k) = \mathbf{C}(k)\mathbf{B}(k)$ . $\mathbf{S}(k)$ encodes which of the inputs spike at a post-synaptic spike, and in particular it holds + +$$ +\mathbb {P} \big (S _ {i} (k) = 1 | \mathbf {p} (k) \big) = \boldsymbol {\Gamma} _ {i,}. (k) \mathbf {p} (k), \quad \text {f o r a l l} i \in [ d ], +$$ + +where $\Gamma_{i..}(k)$ denotes the $i$ -th row of $\Gamma(k)$ . Since the only change in the dynamic of $\mathbf{p}(k)$ is replacing $\mathbf{B}(k)$ by $\mathbf{S}(k)$ in the definition of $\mathbf{Y}(k)$ , Eq. (4) still holds true, and we obtain + +$$ +\begin{array}{l} \mathbb {E} \big [ \mathbf {p} (k + 1) \big | \mathbf {p} (k) \big ] \approx \mathbb {E} \Big [ \mathbf {p} (k) \odot \left(\mathbf {1} + \alpha \left(\mathbf {Y} (k) - (\mathbf {p} (k)) ^ {\top} \mathbf {Y} (k) \mathbf {1}\right)\right) \Big | \mathbf {p} (k) \Big ] \\ = \mathbf {p} (k) \odot \left(\mathbf {1} + \alpha \left(\boldsymbol {\Gamma} \mathbf {p} (k) - (\mathbf {p} (k)) ^ {\top} \boldsymbol {\Gamma} \mathbf {p} (k) \mathbf {1}\right)\right), \\ \end{array} +$$ + +which induces the following gradient flow + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} \mathbf {p} (t) = \mathbf {p} (t) \odot (\boldsymbol {\Gamma} \mathbf {p} (t) - (\mathbf {p} (t)) ^ {\top} \boldsymbol {\Gamma} \mathbf {p} (t) \mathbf {1}). \tag {19} +$$ + +This is again a replicator equation with fitness $\Gamma \mathbf{p}$ , compare Section 7 of [18]. The associated Shahshahani-loss is given by $\mathbf{x} \mapsto -\frac{1}{2} \mathbf{x}^{\top} \boldsymbol{\Gamma} \mathbf{x}$ . Thus, in the correlated model, the probabilities $\mathbf{p}$ follow a flow restricted to the probability simplex aimed at maximising the quadratic form associated to the matrix $\boldsymbol{\Gamma}$ . This property is similar to principal component analysis (PCA), as the goal there is to recover the eigenvector corresponding to the largest eigenvalue of the underlying covariance matrix. Thus, the behaviour of the correlated model can be interpreted as a form of PCA restricted to the probability simplex. Theorem A.13 in the supplementary material generalizes Theorem 2.2 to weakly correlated input neurons and is accompanied by simulations in Figure 6. + +Small biological neural network. In this paper, we mathematically analyse the convergence behaviour of a small biological neural network with one layer composed of excitatory presynaptic/in input neurons and multiple postsynaptic/output neurons. It is natural to generalise the setting to account for inhibitory neurons and more than one layer. + +Weight explosion. The learning rule Eq. (1) for the weights $\mathbf{w}(k)$ causes them to increase without bound as the iteration index $k$ increases. When the weights exceed the spike threshold, the model becomes biologically implausible and the derivation of the probabilities Eq. (12) is no longer valid. This unstable nature is well-known to be intrinsic to Hebbian learning algorithms and is commonly countered by soft or hard bounds, or by including mean-reverting terms to the dynamic Gerstner et al. [14, pages 497-498]. We follow a different route, namely viewing Hebbian learning as a temporal phase of limited length, which is followed by a stabilising homeostatic learning phase. This view is corroborated by experimental results, compare Point (vi) in Morrison et al. [27, Section 2.1]. + +Beyond Pair-based STDP rules. While pair-based learning rules such as Eq. (1) only account for the relative timing of one pre- and one postsynaptic spike time, also the voltage at the location of the synapse should be taken into account [36]. A natural generalization of our framework would be to extend the results to the model proposed in [9]. + +# Acknowledgments and Disclosure of Funding + +All authors acknowledge support from ERC grant A2B (grant agreement no. 101124751). S. G. has been partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - CRC/TRR 388 "Rough Analysis, Stochastic Dynamics and Related Fields" - Project ID 516748464, Project B07. 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Almost sure convergence of stochastic gradient methods under gradient domination. Transactions on Machine Learning Research, 2025. +[46] A. Wibisono, A. C. Wilson, and M. I. Jordan. A variational perspective on accelerated methods in optimization. Proceedings of the National Academy of Sciences, 113(47), 2016. +[47] Yang Dan and Mu-ming Poo. Spike timing-dependent plasticity of neural circuits. Neuron, 44 (1):23-30, 2004. +[48] A. L. Yuille and N. M. Grzywacz. A Winner-Take-All mechanism based on presynaptic inhibition feedback. Neural Computation, 1(3):334–347, 1989. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Claim 1 (connection between Hebbian STDP and noisy gradient descent) is justified in Section 2. Claim 2 (alignment of postsynaptic neuron with input neuron of highest intensity) is proven in Theorem 2.2. Claim 3 (connection to mirror descent) is addressed in Section 3. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: Our results were obtained without the use of LLM. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Technical Appendix + +We first study the loss landscape + +$$ +\mathbf {p} \mapsto L (\mathbf {p}) = - \frac {1}{3} \sum_ {i = 1} ^ {d} p _ {i} ^ {3} + \frac {1}{4} \left(\sum_ {i = 1} ^ {d} p _ {i} ^ {2}\right) ^ {2}. +$$ + +Lemma 2.1 identifies the stationary points if we view the landscape as a function on $\mathbb{R}^d$ . + +Proof of Lemma 2.1. The formula Eq. (7) for the gradient $\nabla L(\mathbf{p})$ shows that the set of critical points is given by + +$$ +\begin{array}{l} \operatorname {C r i t} := \left\{\mathbf {p} \in \mathfrak {P}: \nabla L (\mathbf {p}) = 0 \right\} \\ = \left\{\mathbf {p} \in \mathfrak {P}: p _ {i} \in \{0, \| \mathbf {p} \| ^ {2} \} \forall i \in [ d ] \right\} \\ = \left\{\mathbf {p} \in \mathfrak {P}: \exists n \in \{1, \dots , d \}, S \subset [ d ] \text {w i t h} \# S = n \text {s u c h t h a t} \mathbf {p} = \frac {1}{n} \sum_ {j \in S} \mathbf {e} _ {j} \right\}. \\ \end{array} +$$ + +To identify local the extrema, we compute the Hessian matrix + +$$ +J (\mathbf {p}) = 2 \mathbf {p} \mathbf {p} ^ {\top} + \| \mathbf {p} \| ^ {2} \mathbb {I} - 2 \operatorname {d i a g} (\mathbf {p}) \in \mathbb {R} ^ {d \times d}, \quad \mathbf {p} \in \mathbb {R} ^ {d}, +$$ + +where $\mathrm{diag}(\mathbf{p})\in \mathbb{R}^{d\times d}$ is the diagonal matrix with diagonal entries given by $\mathbf{p}$ . Substituting a critical point $\mathbf{p}^*$ with $n\in [d]$ non-zero entries yields (up to permutations of rows and columns) + +$$ +J (\mathbf {p} ^ {*}) = \frac {1}{n} \left( \begin{array}{c c} J _ {n} & 0 \\ 0 & \mathbf {I} _ {d - n} \end{array} \right), \quad J _ {n} = - \mathbf {I} _ {n} + \frac {2}{n} \mathbf {1} _ {n \times n}, \quad n \in [ d ], +$$ + +where $\mathbf{1}_{n\times n}$ is the $n\times n$ matrix consisting of ones. The corresponding eigenvalues are + +$$ +\left\{ \begin{array}{l l} \frac {1}{n} > 0 & \text {w i t h m u l t i p l i c i t y 1 a n d e i g e n s p a c e} \mathbf {E} _ {n} := \operatorname {s p a n} \left(\sum_ {i = 1} ^ {d} \mathbf {e} _ {i}\right), \\ - \frac {1}{n} < 0 & \text {w i t h m u l t i p l i c i t y} n - 1 \text {a n d e i g e n s p a c e} \mathbf {E} _ {n} ^ {\perp}, \\ \frac {1}{n} > 0 & \text {w i t h m u l t i p l i c i t y} d - n \text {a n d e i g e n s p a c e} \operatorname {s p a n} \left(\mathbf {e} _ {n + 1}, \dots , \mathbf {e} _ {d}\right), \end{array} \right. \tag {20} +$$ + +where $\mathbf{E}_n^\perp$ is the orthogonal complement of $\mathbf{E}_n$ in $\operatorname{span}(\mathbf{e}_1, \ldots, \mathbf{e}_n)$ . Consequently, only those critical points $\mathbf{p}^* \in \mathrm{Crit}$ are local minima, which have $n = 1$ , i.e. $\mathbf{p}^* \in \{\mathbf{e}_1, \ldots, \mathbf{e}_d\}$ . Since all local minima attain the same loss $-1/12$ and $L(\mathbf{p}) \to \infty$ as $\|\mathbf{p}\| \to \infty$ , every local minimum is also a global minimum. + +Remark A.1. The eigenvalues of the Hessian of the loss function computed in Eq. (20) also imply that if $n \geq 2$ , then $\mathbf{p}^* \in \mathrm{Crit}$ is a saddle point in $\mathbb{R}^d$ . Interestingly, when restricting to directions within the probability simplex, the case $n = d$ is not a saddle point, but a maximum, since the direction $\sum_{i=1}^{d} \mathbf{e}_i$ is orthogonal to $\mathfrak{P}$ . + +# A.1 Proofs for Subsection 2.1 + +In the following we will always assume that + +$$ +p _ {1} (0) \geq \max _ {i = 2, \dots , d} p _ {i} (0) + \Delta , \tag {21} +$$ + +for some $\Delta \in (0,1)$ . This is a deterministic constraint. The randomness occurs because of the noise in the updates. We assume that all random variables are defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and denote by $\mathcal{F}_n$ , $n = 0,1,\ldots$ , the natural filtration of $(\mathbf{B}(n), \mathbf{Z}(n))_{n \in \mathbb{N}}$ . By a slight abuse of notation we also introduce $\mathcal{F}_{-1} = \{\emptyset, \Omega\}$ . In particular it then holds that $\mathbf{p}_n$ is $\mathcal{F}_{n-1}$ -measurable for $n = 0,1,\ldots$ . The starting point for the proof of the linear convergence of STDP is given by the following Lemma, which explicitly bounds the error term in the Taylor approximation contained in Eq. (3). Recall that $|Y_i(k)| \leq Q$ , for all $i \in [d], k = 0,1,\ldots$ . + +Lemma A.2. For $i\in [d]$ and $k = 0,1,\ldots$ define + +$$ +\xi_ {i} (k) := p _ {i} (k) \left(\mathbb {E} \left[ Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \mid \mathcal {F} _ {k - 1} \right] - \left(Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right)\right), \tag {22} +$$ + +and assume $\alpha < 1 / Q$ . Then for any $i \in [d]$ and $k = 0, 1, \ldots$ , there exists a random variable $\theta_i(k)$ satisfying + +$$ +\left| \theta_ {i} (k) \right| \leq \alpha^ {2} \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} p _ {i} (k) \big (1 - p _ {i} (k) \big), \quad \text {a l m o s t s u r e l y}, +$$ + +such that + +$$ +p _ {i} (k + 1) = p _ {i} (k) + \alpha p _ {i} (k) \left(p _ {i} (k) - \| \mathbf {p} (k) \| ^ {2}\right) - \alpha \xi_ {i} (k) - \theta_ {i} (k). +$$ + +Proof. By definition + +$$ +p _ {i} (k + 1) = p _ {i} (k) \frac {1 + \alpha Y _ {i} (k)}{1 + \alpha \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)}, \quad k = 0, 1, \dots , \quad i \in [ d ]. +$$ + +Now, for $a,b\in [-Q,Q]$ the first two derivatives of the function + +$$ +f (x) \colon \left(0, \frac {1}{Q}\right) \to \mathbb {R}, \quad x \mapsto \frac {1 + a x}{1 + b x}, +$$ + +are given by + +$$ +f ^ {\prime} (x) = \frac {a (1 + b x) - b (1 + a x)}{(1 + b x) ^ {2}} = \frac {a - b}{(1 + b x) ^ {2}} +$$ + +$$ +f ^ {\prime \prime} (x) = - 2 b \frac {a - b}{(1 + b x) ^ {3}}. +$$ + +Thus, a Taylor expansion around $x = 0$ gives that there exists some $\gamma \in (0,x)$ , such that + +$$ +f (x) = 1 + (a - b) x - b \frac {a - b}{(1 + b \gamma) ^ {3}} x ^ {2}. +$$ + +Hence we obtain that for some $\gamma \in (0,\alpha)$ + +$$ +\begin{array}{l} p _ {i} (k + 1) = p _ {i} (k) + \alpha p _ {i} (k) \left(Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right) \\ - \alpha^ {2} p _ {i} (k) \frac {\sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \left(Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right)}{\left(1 + \gamma \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right) ^ {3}}. \\ \end{array} +$$ + +Using that $|Y_{i}(k)| \leq Q$ almost surely for all $i \in [d], k = 0,1,\ldots$ , the absolute value of the error term can be bounded as follows + +$$ +\begin{array}{l} \left| \alpha^ {2} p _ {i} (k) \frac {\sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \left(Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right)}{\left(1 + \gamma \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right) ^ {3}} \right| \\ \leq \alpha^ {2} p _ {i} (k) \frac {Q}{(1 - Q \alpha) ^ {3}} \left| Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \right| \\ \leq \alpha^ {2} p _ {i} (k) \frac {Q}{(1 - Q \alpha) ^ {3}} \left(\left(1 - p _ {i} (k)\right) \left| Y _ {i} (k) \right| + \sum_ {j = 1, j \neq i} ^ {d} p _ {j} (k) \left| Y _ {j} (k) \right|\right) \\ \leq \alpha^ {2} \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} p _ {i} (k) (1 - p _ {i} (k)). \\ \end{array} +$$ + +Since $p_i(k)$ is $\mathcal{F}_{k-1}$ -measurable for any $i \in [d]$ and $\mathbb{E}[Y_i(k) | \mathcal{F}_{k-1}] = p_i(k)$ , we also obtain + +$$ +\xi_ {i} (k) = p _ {i} (k) \Big (p _ {i} (k) - \| \mathbf {p} (k) \| ^ {2} - Y _ {i} (k) + \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \Big), +$$ + +which concludes the proof. + +For $\Delta$ given in Eq. (21), we define a sequence of benign events + +$$ +\Omega (k) := \left\{p _ {1} (u) \geq \max _ {i = 2, \dots , d} p _ {i} (u) + \frac {\Delta}{2}, \forall u \in [ k ] \right\}, \quad k = 1, 2, \dots , +$$ + +and due to the assumption Eq. (21) we set $\Omega(0) = \Omega$ . On the above events, the gradient is bounded away from zero and + +$$ +p _ {1} (k) = \frac {1}{d} \sum_ {j = 1} ^ {d} \left(p _ {1} (k) - p _ {j} (k)\right) + \frac {1}{d} \geq \frac {(d - 1) \Delta}{2 d} + \frac {1}{d}. \tag {23} +$$ + +Using these properties, we can prove a recursive upper bound for $1 - p_{1}(k)$ . + +Proposition A.3. If + +$$ +0 < \alpha \leq \frac {(1 - Q \alpha) ^ {3}}{8 Q ^ {2}} \Delta , +$$ + +then, on the event $\Omega (k)$ + +$$ +1 - p _ {1} (k + 1) \leq \left(1 - \alpha \frac {\Delta}{4 d} \left(1 + \frac {\Delta}{2} (d - 1)\right)\right) \left(1 - p _ {1} (k)\right) + \alpha \xi_ {1} (k). +$$ + +Proof. By definition, we have on the event $\Omega (k)$ + +$$ +\begin{array}{l} p _ {1} (k) - \| \mathbf {p} (k) \| ^ {2} = \sum_ {j = 1} ^ {d} p _ {j} (k) \left(p _ {1} (k) - p _ {j} (k)\right) \\ \geq \frac {\Delta}{2} (1 - p _ {1} (k)). \tag {24} \\ \end{array} +$$ + +The constraint imposed on the learning rate implies that $\alpha < 1 / Q$ and Lemma A.2 becomes applicable. Now, combining the previous inequality with the assumption on $\alpha$ and applying Lemma A.2 with $i = 1$ , as well as Eq. (23), gives, on the event $\Omega(k)$ + +$$ +\begin{array}{l} 1 - p _ {1} (k + 1) \\ \leq 1 - p _ {1} (k) - \alpha \frac {\Delta}{2} p _ {1} (k) \left(1 - p _ {1} (k)\right) + \alpha \xi_ {1} (k) + \theta_ {1} (k) \\ \leq 1 - p _ {1} (k) - \alpha \left(\frac {\Delta}{2} - \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} \alpha\right) p _ {1} (k) \left(1 - p _ {1} (k)\right) + \alpha \xi_ {1} (k) \\ \leq 1 - p _ {1} (k) - \frac {\Delta}{4} \alpha p _ {1} (k) \left(1 - p _ {1} (k)\right) + \alpha \xi_ {1} (k) \\ \leq \left(1 - \alpha \frac {\Delta}{4 d} \left(1 + \frac {\Delta}{2} (d - 1)\right)\right) \left(1 - p _ {1} (k)\right) + \alpha \xi_ {1} (k). \\ \end{array} +$$ + +This concludes the proof. + +![](images/27f442222caceb3a3bf5cf05f915f4ebe28bd9c06cd4b46225864ab53686840a.jpg) + +Having understood the dynamics of $\mathbf{p}$ on the favourable event $\Omega(k)$ , we aim for a lower bound for its probability. A key step is the following Lemma, which states that $\Omega(k)$ is fulfilled as soon as + +$$ +M _ {j} (k) := \sum_ {\ell = 0} ^ {k} \alpha \xi_ {j} (\ell) \mathbb {1} _ {\Omega (\ell)}, \quad k = 0, 1, \dots \tag {25} +$$ + +with $\xi_j(\ell)$ defined in Eq. (22), exhibit a uniform concentration behaviour. + +Lemma A.4. Define the sets + +$$ +E _ {j} (k) := \left\{\max _ {u \in [ k ]} | M _ {j} (u) | \leq \frac {\Delta}{4} \right\}, \quad E (k) := \bigcap_ {j = 1} ^ {d} E _ {j} (k), \quad k = 0, 1, \dots . +$$ + +Then if + +$$ +0 < \alpha \leq \Delta^ {2} \frac {(1 - Q \alpha) ^ {3}}{1 6 Q ^ {2}}, +$$ + +the following set inclusion holds for any $k = 0,1,\ldots$ + +$$ +E (k) \subseteq \Omega (k + 1). +$$ + +Proof. Let $u \in \{2, \dots, d\}$ be arbitrary. It follows by Lemma A.2, that on $\Omega(k)$ the bound + +$$ +\begin{array}{l} p _ {u} (k + 1) = p _ {u} (k) + \alpha p _ {u} (k) \left(p _ {u} (k) - \| \mathbf {p} (k) \| ^ {2}\right) - \alpha \xi_ {u} (k) - \theta_ {u} (k) \\ \leq p _ {u} (k) + \alpha p _ {u} (k) \left(p _ {1} (k) - \| \mathbf {p} (k) \| ^ {2}\right) - \alpha \xi_ {u} (k) - \theta_ {u} (k) \\ \end{array} +$$ + +holds. Consequently, on $\Omega (k)$ , we have + +$$ +\begin{array}{l} p _ {1} (k + 1) - p _ {u} (k + 1) \\ = p _ {1} (k) - p _ {u} (k) + \alpha \left(p _ {1} (k) - p _ {u} (k)\right) \left(p _ {1} (k) - \| \mathbf {p} (k) \| ^ {2}\right) + \alpha \left(\xi_ {j} (k) - \xi_ {1} (k)\right) - \theta_ {1} (k) + \theta_ {u} (k). \\ \end{array} +$$ + +We have $p_u(k) \leq 1 - p_1(k)$ and thus, on $\Omega(k)$ , + +$$ +\begin{array}{l} - \theta_ {1} (k) + \theta_ {u} (k) \geq - \alpha^ {2} \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} \left(p _ {1} (k) \left(1 - p _ {1} (k)\right) + p _ {u} (k) \left(1 - p _ {u} (k)\right)\right) \\ \geq - \alpha^ {2} \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} \left(1 - p _ {1} (k) + p _ {u} (k)\right) \\ \geq - \alpha^ {2} \frac {4 Q ^ {2}}{(1 - Q \alpha) ^ {3}} (1 - p _ {1} (k)) \\ \geq - \alpha \frac {\Delta^ {2}}{4} (1 - p _ {1} (k)), \\ \end{array} +$$ + +invoking the constraint on the learning rate in the last step. From the assumptions on $\alpha$ we deduce that on the event $\Omega (k)$ + +$$ +\begin{array}{l} p _ {1} (k + 1) - p _ {u} (k + 1) - \alpha (\xi_ {j} (k) - \xi_ {1} (k)) \\ \geq p _ {1} (k) - p _ {u} (k) + \alpha \left(p _ {1} (k) - p _ {u} (k)\right) \left(p _ {1} (k) - \| \mathbf {p} (k) \| ^ {2}\right) - \alpha \frac {\Delta^ {2}}{4} \left(1 - p _ {1} (k)\right) \\ \geq p _ {1} (k) - p _ {u} (k) + \alpha \frac {\Delta}{2} \left(p _ {1} (k) - p _ {u} (k)\right) \left(1 - p _ {1} (k)\right) - \alpha \frac {\Delta^ {2}}{4} \left(1 - p _ {1} (k)\right) \\ \geq p _ {1} (k) - p _ {u} (k) + \alpha \frac {\Delta^ {2}}{4} \left(1 - p _ {1} (k)\right) - \alpha \frac {\Delta^ {2}}{4} \left(1 - p _ {1} (k)\right) \\ \geq p _ {1} (k) - p _ {u} (k), \\ \end{array} +$$ + +where we applied Eq. (24) in the third to last inequality. Because of $\Omega(k) \subseteq \Omega(k - 1)$ it then follows, + +$$ +\begin{array}{l} (p _ {1} (k + 1) - p _ {u} (k + 1)) \mathbb {1} _ {\Omega (k)} \geq (p _ {1} (k) - p _ {u} (k)) \mathbb {1} _ {\Omega (k)} + \alpha (\xi_ {j} (k) - \xi_ {1} (k)) \mathbb {1} _ {\Omega (k)} \\ = \mathbb {1} _ {\Omega (k)} \Big ((p _ {1} (k) - p _ {u} (k)) \mathbb {1} _ {\Omega (k - 1)} + \alpha (\xi_ {j} (k) - \xi_ {1} (k)) \mathbb {1} _ {\Omega (k)} \Big). \\ \end{array} +$$ + +This gives + +$$ +\begin{array}{l} (p _ {1} (k + 1) - p _ {u} (k + 1)) \mathbb {1} _ {\Omega (k)} \\ \geq \mathbb {1} _ {\Omega (k)} \left(p _ {1} (0) - p _ {u} (0) + \sum_ {\ell = 0} ^ {k} \alpha \left(\xi_ {\ell} ^ {j} - \xi_ {\ell} ^ {1}\right) \mathbb {1} _ {\Omega (\ell)}\right) \tag {26} \\ \geq \Delta \mathbb {1} _ {\Omega (k)} - | M _ {j} (k) | - | M _ {1} (k) |. \\ \end{array} +$$ + +We want to prove by induction that $E(k) \subseteq \Omega(k + 1)$ for all $k = 0, 1, \ldots$ . For $k = 0$ , this directly follows from Eq. (26), since $\Omega(0) = \Omega$ due to assumption Eq. (21). Assume the assertion holds for some $k = 0, 1, \ldots$ . Hence, it holds $E(k + 1) \subseteq E(k) \subseteq \Omega(k + 1)$ , such that for any $u \in \{2, \ldots, d\}$ it holds on $E(k + 1)$ by Eq. (26) + +$$ +\begin{array}{l} (p _ {1} (k + 2) - p _ {u} (k + 2)) \geq \Delta - | M _ {j} (k + 1) | - | M _ {1} (k + 1) | \\ \geq \frac {\Delta}{2}, \\ \end{array} +$$ + +which proves the assertion. + +![](images/0e4166dfcfb27aef82ac1d99f3e0c690dfba9969693ae6e73f62766ccdde1eb8.jpg) + +Having assembled the previous results, we are able to prove the linear convergence of STDP stated in Theorem 2.2. As Proposition A.3 already suggests the desired behaviour of $\mathbf{p}$ on $\Omega (k)$ , the main + +part of the proof is to show that $\Omega(k)$ is satisfied with large probability. For that we deploy Doob's submartingale inequality, which states that for a martingale $(X_{n})_{n\in \mathbb{N}}$ , any $p\geq 1$ , and any $u > 0$ + +$$ +\mathbb {P} \left(\max _ {i \in [ n ]} | X _ {i} | \geq u\right) \leq \frac {\mathbb {E} \left[ | X _ {n} | ^ {p} \right]}{u ^ {p}}. \tag {27} +$$ + +This will be applied to derive lower bounds for the event $E(k)$ defined in Lemma A.4, which are also lower bounds for the probability of $\Omega (k + 1)$ by the same Lemma. For the reader's convenience we restate Theorem 2.2 before giving its proof. + +Theorem 2.2. Given $\varepsilon \in (0,1)$ , assume + +$$ +\Delta := p _ {1} (0) - \max _ {i = 2, \dots , d} p _ {i} (0) > 0 \quad a n d \quad 0 < \alpha \leq \frac {\Delta^ {2}}{1 6 Q ^ {2}} \Big ((1 - Q \alpha) ^ {3} \wedge \frac {1}{2 5 6 (1 - p _ {1} (0))} \Big (4 \frac {\Delta}{d} + \Delta^ {2} \Big) \varepsilon \Big). +$$ + +Then there exists an event $\Theta$ with probability $\geq 1 - \varepsilon /2$ such that + +$$ +\mathbb {E} \left[ \| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \mathbb {1} _ {\Theta} \right] \leq 2 (1 - p _ {1} (0)) \exp \left(- \frac {\alpha}{1 6} \left(4 \frac {\Delta}{d} + \Delta^ {2}\right) k\right), \quad f o r a l l k = 0, 1, \dots +$$ + +Consequently, given $\delta > 0$ , it holds + +$$ +\mathbb {P} \Big (\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \geq \delta \Big) \leq \varepsilon \quad f o r a l l \quad k \geq \frac {1 6 d}{\alpha \Delta (4 + d \Delta)} \log \Big (\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta} \Big). +$$ + +Proof of Theorem 2.2. The recursive definition ensures that $\mathbf{p}(k)$ is $\mathcal{F}_{k - 1}$ -measurable. Thus, also $\Omega (k)\in \mathcal{F}_{k - 1}$ for any $k = 0,1,\ldots$ . One can check that $(M_{i}(k))_{k = 0,1,\dots}$ , defined in Eq. (25), forms a martingale for each $i\in [d]$ . This allows us to apply Doob's submartingale inequality. To apply it with $p = 2$ , we deduce the following bound on the second moment, + +$$ +\begin{array}{l} \mathbb {E} [ M _ {1} (k) ^ {2} ] \\ = \alpha^ {2} \mathbb {E} \left[ \left(\sum_ {\ell = 0} ^ {k} \xi_ {1} (\ell) \mathbb {1} _ {\Omega (\ell)}\right) ^ {2} \right] \\ = \alpha^ {2} \left(\sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left(\xi_ {1} (\ell) \mathbb {1} _ {\Omega (\ell)}\right) ^ {2} \right] + \sum_ {i, j = 0, i \neq j} ^ {k} \mathbb {E} \left[ \xi_ {1} (i) \mathbb {1} _ {\Omega (i)} \xi_ {1} (j) \mathbb {1} _ {\Omega (j)} \right]\right) \\ = \alpha^ {2} \left(\sum_ {\ell = 0} ^ {k} \mathbb {E} [ (\xi_ {1} (\ell) \mathbb {1} _ {\Omega (\ell)}) ^ {2} ] + \sum_ {i, j = 0, i \neq j} ^ {k} \mathbb {E} \left[ \xi_ {1} (i \wedge j) \mathbb {1} _ {\Omega (i \wedge j)} \mathbb {E} [ \xi_ {1} (i \vee j) \mathbb {1} _ {\Omega (i \vee j)} | \mathcal {F} _ {i \wedge j} ] \right]\right) \\ = \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} [ (\xi_ {1} (\ell) \mathbb {1} _ {\Omega (\ell)}) ^ {2} ] \\ = \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ (p _ {1} (\ell)) ^ {2} \left(\mathbb {E} \left[ Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell) \mid \mathcal {F} _ {\ell - 1} \right] - \left(Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \mathbb {E} \left[ \left(\mathbb {E} \left[ Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell) \mid \mathcal {F} _ {\ell - 1} \right] - \left(Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right)\right) ^ {2} \mid \mathcal {F} _ {\ell - 1} \right] \mathbb {1} _ {\Omega (\ell)} \right] \\ = \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \Big [ \mathbb {E} \Big [ \Big (Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell) \Big) ^ {2} \Big | \mathcal {F} _ {\ell - 1} \Big ] \mathbb {1} _ {\Omega (\ell)} - \mathbb {E} \Big [ Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell) \Big | \mathcal {F} _ {\ell - 1} \Big ] ^ {2} \mathbb {1} _ {\Omega (\ell)} \Big ] \\ \leq \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left(Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ = \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left((1 - p _ {1} (\ell)) Y _ {1} (\ell) - \sum_ {j = 2} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \end{array} +$$ + +$$ +\begin{array}{l} \leq 2 \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left((1 - p _ {1} (\ell)) Y _ {1} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} + \left(\sum_ {j = 2} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq 2 Q ^ {2} \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left((1 - p _ {1} (\ell)) ^ {2} + \left(\sum_ {j = 2} ^ {d} p _ {j} (\ell)\right) ^ {2}\right) \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq 4 Q ^ {2} \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left(1 - p _ {1} (\ell)\right) \mathbb {1} _ {\Omega (\ell)} \right], \\ \end{array} +$$ + +where we used that $\mathbb{E}[\xi_1(k_2) \mid \mathcal{F}_{k_1}] = 0$ , for any $k_2 > k_1 = 0, 1, \ldots$ , together with $|Y_1(k)| \leq Q$ , the inequality $(a + b)^2 \leq 2(a^2 + b^2)$ for $a, b \in \mathbb{R}$ and $1 - p_\ell^1 \leq 1$ . Arguing similarly we obtain for $u \in \{2, \ldots, d\}$ , + +$$ +\begin{array}{l} \mathbb {E} \left[ \left(M _ {u} (k) ^ {2} \right] = \alpha^ {2} \mathbb {E} \left[ \sum_ {\ell = 0} ^ {k} \left(\xi_ {u} (\ell) \mathbb {1} _ {\Omega (\ell)}\right) ^ {2} \right] \right. \\ \leq \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left(p _ {u} (\ell)\right) ^ {2} \left(Y _ {u} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq 4 Q ^ {2} \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ p _ {u} (\ell) \mathbb {1} _ {\Omega (\ell)} \right]. \\ \end{array} +$$ + +Hence, applying a union bound, Doob's submartingale inequality Eq. (27) with $p = 2$ gives for any $k = 0,1,\ldots$ + +$$ +\begin{array}{l} \mathbb {P} (E (k)) = 1 - \mathbb {P} \left(\bigcup_ {j = 1} ^ {d} \max _ {u \in [ k ]} | M _ {j} (u) | \geq \Delta / 4\right) \\ \geq 1 - \sum_ {j = 1} ^ {d} \mathbb {P} \left(\max _ {u \in [ k ]} | M _ {j} (u) | \geq \Delta / 4\right) \\ \geq 1 - 64 Q ^ {2} \frac {\alpha^ {2}}{\Delta^ {2}} \Big (\sum_ {\ell = 0} ^ {k} \mathbb {E} \Big [ (1 - p _ {1} (\ell)) \mathbb {1} _ {\Omega (\ell)} \Big ] + \sum_ {j = 2} ^ {d} \sum_ {\ell = 0} ^ {k} \mathbb {E} \Big [ p _ {j} (\ell) \mathbb {1} _ {\Omega (\ell)} \Big ]. \Big) \\ = 1 - 1 2 8 Q ^ {2} \frac {\alpha^ {2}}{\Delta^ {2}} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ (1 - p _ {1} (\ell)) \mathbb {1} _ {\Omega (\ell)} \right]. \\ \end{array} +$$ + +Proposition A.3 gives for any $k = 0,1,\ldots$ the bound + +$$ +\begin{array}{l} \mathbb {E} \left[ \left(1 - p _ {1} (k + 1)\right) \mathbb {1} _ {\Omega (k + 1)} \right] \leq \mathbb {E} \left[ \left(1 - p _ {1} (k + 1)\right) \mathbb {1} _ {\Omega (k)} \right] \\ \leq \mathbb {E} \left[ \left(1 - \alpha \frac {\Delta}{4 d} \left(1 + \frac {\Delta}{2} (d - 1)\right)\right) \left(1 - p _ {1} (k)\right) \mathbb {1} _ {\Omega (k)} + \alpha \xi_ {1} (k) \mathbb {1} _ {\Omega (k)} \right] \\ = \left(1 - \alpha \frac {\Delta}{4 d} \left(1 + \frac {\Delta}{2} (d - 1)\right)\right) \mathbb {E} \left[ \left(1 - p _ {1} (k)\right) \mathbb {1} _ {\Omega (k)} \right], \\ \end{array} +$$ + +which implies + +$$ +\mathbb {E} \left[ \left(1 - p _ {1} (k)\right) \mathbb {1} _ {\Omega (k)} \right] \leq \left(1 - p _ {1} (0)\right) \left(1 - \alpha \frac {\Delta}{4 d} \left(1 + \frac {\Delta}{2} (d - 1)\right)\right) ^ {k}. \tag {28} +$$ + +We set + +$$ +\Theta := \bigcap_ {k = 0} ^ {\infty} \Omega (k) = \left\{p _ {1} (u) \geq \max _ {i = 2, \dots , d} p _ {i} (u) + \frac {\Delta}{2}, \forall u \in \mathbb {N} \right\}. +$$ + +The continuity of probability measures and Lemma A.4 then imply + +$$ +\begin{array}{l} \mathbb {P} (\Theta) = \lim _ {k \rightarrow \infty} \mathbb {P} (\Omega (k)) \\ \geq \lim _ {k \rightarrow \infty} \mathbb {P} (E (k)) \\ \end{array} +$$ + +$$ +\begin{array}{l} \geq 1 - 1 2 8 Q ^ {2} \frac {\alpha^ {2}}{\Delta^ {2}} \sum_ {\ell = 0} ^ {\infty} \mathbb {E} \left[ (1 - p _ {1} (\ell)) \mathbb {1} _ {\Omega (\ell)} \right] \\ \geq 1 - 128 Q ^ {2} \left(1 - p _ {1} (0)\right) \frac {\alpha^ {2}}{\Delta^ {2}} \sum_ {\ell = 0} ^ {\infty} \left(1 - \alpha \frac {\Delta}{4 d} \left(1 + \frac {\Delta}{2} (d - 1)\right)\right) ^ {\ell} \\ = 1 - 1 0 2 4 Q ^ {2} \left(1 - p _ {1} (0)\right) \frac {d \alpha}{\Delta^ {3} \left(2 + \Delta (d - 1)\right)} \\ \geq 1 - 2 0 4 8 Q ^ {2} (1 - p _ {1} (0)) \frac {\alpha}{\Delta^ {3} \left(\frac {4}{d} + \Delta\right)} \\ \geq 1 - \frac {\varepsilon}{2}, \\ \end{array} +$$ + +where we used that we can assume $d \geq 2$ without loss of generality. Additionally, Eq. (28) and the elementary inequality $1 - x \leq \exp(-x)$ , which is valid for any real number $x$ , give + +$$ +\begin{array}{l} \mathbb {E} \left[ \left(1 - p _ {1} (k)\right) \mathbb {1} _ {\Theta} \right] \leq \mathbb {E} \left[ \left(1 - p _ {1} (k)\right) \mathbb {1} _ {\Omega (k)} \right] \\ \leq \left(1 - p _ {1} (0)\right) \left(1 - \alpha \frac {\Delta}{4 d} \left(1 + \frac {\Delta}{2} (d - 1)\right)\right) ^ {k} \\ \leq \left(1 - p _ {1} (0)\right) \exp \Big (- \alpha \frac {\Delta}{4 d} \Big (1 + \frac {\Delta}{2} (d - 1) \Big) k \Big). \\ \end{array} +$$ + +When $d = 1$ , the right hand side of this inequality is 0. For $d \geq 2$ , we can also use the bound $d - 1 \geq d / 2$ . Together with + +$$ +\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} = 1 - p _ {1} (k) + \sum_ {i = 2} ^ {d} p _ {i} (k) = 2 \big (1 - p _ {1} (k) \big), +$$ + +this concludes the proof of the first statement. For the proof of the second statement, we apply Markov's inequality to obtain + +$$ +\begin{array}{l} \mathbb {P} \left(\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \geq \delta\right) \\ \leq \mathbb {P} (\boldsymbol {\Theta} ^ {\mathbf {C}}) + \mathbb {P} \left(\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \mathbb {1} _ {\boldsymbol {\Theta}} \geq \delta\right) \\ \leq \frac {\varepsilon}{2} + 2 (1 - p _ {1} (0)) \exp \left(- \frac {\alpha}{1 6} \left(4 \frac {\Delta}{d} + \Delta^ {2}\right) k\right) \delta^ {- 1}. \\ \end{array} +$$ + +Hence, if + +$$ +\begin{array}{l} k \geq \left(\frac {\alpha}{1 6} \left(4 \frac {\Delta}{d} + \Delta^ {2}\right)\right) ^ {- 1} \log \left(\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta}\right) \\ = \frac {1 6 d}{\alpha \Delta (4 + d \Delta)} \log \left(\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta}\right), \\ \end{array} +$$ + +then, + +$$ +\mathbb {P} \Big (\left\| \mathbf {p} (k) - \mathbf {e} _ {1} \right\| _ {1} \geq \delta \Big) \leq \varepsilon . +$$ + +Lemma A.5. Given an initialization of the weights $\mathbf{w}(0) = (w_{1}(0),\dots ,w_{d}(0))^{\top}$ consider two $d$ dimensional intensity vectors $\lambda = (\lambda_1,\ldots ,\lambda_d)^\top$ $\widetilde{\lambda} = (\widetilde{\lambda}_1,\dots ,\widetilde{\lambda}_d)^\top$ with positive entries. Assume that $\lambda_{1}w_{1}(0) > \max_{i = 2,\dots ,d}\lambda_{i}w_{i}(0)$ and $\widetilde{\lambda}_d w_d(0) > \max_{i = 1,\dots ,d - 1}\widetilde{\lambda}_i w_i(0)$ . Assume we run the learning rule Eq. (3) with intensity $\pmb{\lambda}$ until time $K^{*}$ and then, for $k > K^{*}$ , change the intensity to $\widetilde{\lambda}$ and run the learning dynamic until time $k\to \infty$ . If $K^{*}$ is small, in particular, if $K^{*} = 0$ , the above convergence result can be applied to show that the dynamic converges to the corner $\mathbf{e}_d$ . However, for any $\varepsilon \in (0,1)$ and all sufficiently large $K^{*}$ (depending on $\varepsilon$ ), the dynamics will converge to $\mathbf{e}_1$ with probability $1 - 2\varepsilon$ . + +This result shows that the dynamic can be primed at the beginning to end up in one regime. Despite the noise and the infinite amount of data, the dynamic is unable to escape this domain of attraction. From the proof, one can derive quantitative bounds for $K^{*}$ . + +Proof. Given $\varepsilon \in (0,1)$ , choose $\delta \in (0,1)$ such that + +$$ +\max _ {i \neq 1} \frac {\widetilde {\lambda} _ {i} \lambda_ {1} \delta}{\lambda_ {i} \widetilde {\lambda} _ {1} (1 - \delta)} < 1. \tag {29} +$$ + +Let $\Delta = p_{1}(0) - \max_{i\neq 1}p_{i}(0)$ . Given $\varepsilon, \delta, \Delta$ choose + +$$ +K ^ {*} \geq \frac {1 6 d}{\alpha \Delta (4 + d \Delta)} \log \left(\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta}\right). +$$ + +By Theorem 2.2, this guarantees that + +$$ +\mathbb {P} \left(\| \mathbf {p} (K ^ {*}) - \mathbf {e} _ {1} \| _ {1} \geq \delta\right) \leq \varepsilon . +$$ + +On the event $\| \mathbf{p}(K^{*}) - \mathbf{e}_{1}\|_{1} < \delta$ , we have + +$$ +1 - \frac {\lambda_ {1} w _ {1} (K ^ {*})}{\boldsymbol {\lambda} ^ {\top} \mathbf {w} (K ^ {*})} < \delta , \quad \text {a n d} \quad \max _ {i \neq 1} \frac {\lambda_ {i} w _ {i} (K ^ {*})}{\boldsymbol {\lambda} ^ {\top} \mathbf {w} (K ^ {*})} < \delta , +$$ + +which can be combined into + +$$ +\max _ {i \neq 1} \lambda_ {i} w _ {i} (K ^ {*}) < \delta \boldsymbol {\lambda} ^ {\top} \mathbf {w} (K ^ {*}) < \frac {\delta}{1 - \delta} \lambda_ {1} w _ {1} (K ^ {*}). +$$ + +Using the inequality Eq. (29), we obtain + +$$ +\max _ {i \neq 1} \widetilde {\lambda} _ {i} w _ {i} (K ^ {*}) < \frac {\widetilde {\lambda} _ {1} (1 - \delta)}{\lambda_ {1} \delta} \frac {\delta}{1 - \delta} \lambda_ {1} w _ {1} (K ^ {*}) = \widetilde {\lambda} _ {1} w _ {1} (K ^ {*}). +$$ + +This means that restarting the learning rule Eq. (3) at time $K^{*}$ with intensities $\widetilde{\lambda}_1, \ldots, \widetilde{\lambda}_d$ and weights $w_{1}(K^{*}), \ldots, w_{d}(K^{*})$ , shows that $p_1(K^*) > \max_{i \neq 2} p_i(K^*)$ . Applying Theorem 2.2 again shows convergence to $\mathbf{e}_1$ with probability $1 - 2\varepsilon$ . + +![](images/ad40f3c47486e4d4d2f19bb5107969c27f1f5e755235132f28d6300ef1c43193.jpg) + +# A.2 Proofs for Subsection 2.2 + +This section contains additional material on the gradient flow Eq. (9), as well as the proofs for Lemma A.6 and Theorem 2.4. + +For $d = 2$ , the gradient flow admits an explicit solution. In this case, $\mathbf{p}(t) = (p_1(t), p_2(t))^{\top}$ . If $\mathbf{p}(0) = (1/2, 1/2)^{\top}$ , then this is a stationary solution and $\mathbf{p}(t) = \mathbf{p}(0) = (1/2, 1/2)^{\top}$ for all $t \geq 0$ . If $p_1(0) > 1/2$ , then, + +$$ +p _ {1} (t) = \frac {1}{2} + \frac {1}{2 \sqrt {C e ^ {- t} + 1}}, \quad \text {w i t h} C := \frac {1}{\left(2 p _ {1} (0) - 1\right) ^ {2}} - 1. \tag {30} +$$ + +If $p_1(0) < 1/2$ , then $p_2(t) = 1 - p_1(t) > 1/2$ follows the dynamic in Eq. (30). This formula immediately implies that $p_1(t)$ converges exponentially fast to 1. + +Proof of Formula Eq. (30). Throughout the proof we set $p(t) \coloneqq p_1(t)$ and do not use the previous notation $p_1(t), p_2(t)$ for the first and second probability. For $d = 2$ , the gradient flow ODE Eq. (9) becomes + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} p (t) = p (t) ^ {2} - p (t) \left(p (t) ^ {2} + (1 - p (t)) ^ {2}\right) = 3 p (t) ^ {2} - 2 p (t) ^ {3} - p (t). \tag {31} +$$ + +Rewriting this in the variable $u(t) = 1 - 2p(t)$ gives the dynamic, + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} u (t) = - 2 \frac {\mathrm {d}}{\mathrm {d} t} p (t) = - 6 p (t) ^ {2} + 4 p (t) ^ {3} + 2 p (t) = \frac {1}{2} \big (1 - u (t) ^ {2} \big) u (t). +$$ + +This is solved by $u(t) = -1 / \sqrt{Ce^{-t} + 1}$ since + +$$ +- \frac {\mathrm {d}}{\mathrm {d} t} \frac {1}{\sqrt {C e ^ {- t} + 1}} = - \left(- \frac {1}{2}\right) \cdot \frac {- C e ^ {- t}}{(C e ^ {- t} + 1) ^ {3 / 2}} = \frac {1}{2} (1 - u (t) ^ {2}) u (t). +$$ + +Thus, $p(t) = \frac{1}{2} (1 - u(t))$ solves Eq. (31). Finally, $C$ is determined by the initial condition $p(0) = \frac{1}{2} (1 - 1 / \sqrt{C + 1})$ . + +The following lemma summarises different properties of the gradient flow Eq. (9). In its statement, differentiable on $[0,1]$ means differentiable on $(0,1)$ and continuous on $[0,1]$ . + +Lemma A.6. The gradient flow Eq. (9) exhibits the following properties. + +(a) If $\phi \colon [0,1] \to \mathbb{R}$ is a convex and differentiable function, then $t \mapsto \sum_{i=1}^{d} \phi(p_i(t))$ is monotone increasing. +(b) Let $i, j \in [d]$ with $i \neq j$ . If $p_i(0) > p_j(0)$ , respectively $p_i(0) = p_j(0)$ , then $p_i(t) > p_j(t)$ , respectively $p_i(t) = p_j(t)$ , for all $t \geq 0$ . Moreover, if + +$$ +\Delta := p _ {1} (0) - \max _ {i = 2, \dots , d} p _ {i} (0) > 0, +$$ + +then + +$$ +p _ {1} (t) \geq \max _ {i = 2, \dots , d} p _ {i} (t) + \Delta \text {f o r a l l} t \geq 0. +$$ + +Lemma A.6 implies that the $q$ -norm $t \mapsto |\mathbf{p}(t)|_q$ is monotonically increasing whenever $1 \leq q < \infty$ . The result also implies that if instead $\phi$ is concave and differentiable, then $t \mapsto \sum_{i=1}^{d} \phi(p_i(t))$ is monotonically decreasing. + +Proof of Lemma A.6. + +(a) Since $\phi$ is convex, $\phi'$ is monotonically increasing. Thus, for a probability vector $\mathbf{q} = (q_1, \ldots, q_d)$ , we have + +$$ +\begin{array}{l} \sum_ {i = 1} ^ {d} q _ {i} \phi^ {\prime} (q _ {i}) \left(q _ {i} - \| \mathbf {q} \| _ {2} ^ {2}\right) = \sum_ {i = 1} ^ {d} q _ {i} \phi^ {\prime} (q _ {i}) \left(q _ {i} \sum_ {j = 1} ^ {d} q _ {j} - \sum_ {j = 1} ^ {d} q _ {j} ^ {2}\right) \\ = \sum_ {i, j = 1} ^ {d} q _ {i} q _ {j} \phi^ {\prime} (q _ {i}) \left(q _ {i} - q _ {j}\right) \\ = \sum_ {1 \leq i < j \leq d} q _ {i} q _ {j} \left(\phi^ {\prime} \left(q _ {i}\right) - \phi^ {\prime} \left(q _ {j}\right)\right) \left(q _ {i} - q _ {j}\right) \\ \geq 0. \\ \end{array} +$$ + +(If $\phi$ is strictly convex, then strict equality holds if and only if $\mathbf{q}$ is one of the stationary points described above.) Using this and the gradient flow formula + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} \sum_ {i = 1} ^ {d} \phi (p _ {i} (t)) = \sum_ {i = 1} ^ {d} \phi^ {\prime} (p _ {i} (t)) \frac {\mathrm {d}}{\mathrm {d} t} p _ {i} (t) = \sum_ {i = 1} ^ {d} p _ {i} (t) \phi^ {\prime} (p _ {i} (t)) \left(p _ {i} (t) - \| \mathbf {p} (t) \| _ {2} ^ {2}\right) \geq 0, +$$ + +proving the result. + +(b) By definition it holds for $i,j\in [d]$ + +$$ +\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \left(p _ {i} (t) - p _ {j} (t)\right) = p _ {i} (t) \left(p _ {i} (t) - \| \mathbf {p} (t) \| ^ {2}\right) - p _ {j} (t) \left(p _ {j} (t) - \| \mathbf {p} (t) \| ^ {2}\right) \\ = \left(p _ {i} (t) - p _ {j} (t)\right) \left(p _ {i} (t) + p _ {j} (t) - \| \mathbf {p} (t) \| ^ {2}\right). \tag {32} \\ \end{array} +$$ + +If $p_i(0) > p_j(0)$ , we can apply Gronwall's inequality together with Eq. (32) to obtain for any $t \geq 0$ , + +$$ +p _ {j} (t) - p _ {i} (t) \leq \left(p _ {j} (0) - p _ {i} (0)\right) \exp \left(\int_ {0} ^ {t} p _ {i} (s) + p _ {j} (s) - \| \mathbf {p} (s) \| ^ {2} d s\right) < 0. +$$ + +Similarly, if $p_i(0) = p_j(0)$ , we obtain + +$$ +p _ {j} (t) - p _ {i} (t) \leq \left(p _ {j} (0) - p _ {i} (0)\right) \exp \left(\int_ {0} ^ {t} p _ {i} (s) + p _ {j} (s) - \| \mathbf {p} (s) \| ^ {2} d s\right) = 0, +$$ + +and + +$$ +p _ {i} (t) - p _ {j} (t) \leq \left(p _ {i} (0) - p _ {j} (0)\right) \exp \left(\int_ {0} ^ {t} p _ {i} (s) + p _ {j} (s) - \| \mathbf {p} (s) \| ^ {2} d s\right) = 0, +$$ + +concluding the proof. + +To prove the second statement, we have + +$$ +p _ {1} (t) = p _ {1} (0) + \int_ {0} ^ {t} p _ {1} (s) \left(p _ {1} (s) - \| \mathbf {p} (s) \| ^ {2}\right) \mathrm {d} s \geq p _ {1} (0) - t, +$$ + +and similarly, for any $i\in \{2,\dots ,d\}$ + +$$ +p _ {i} (t) \leq p _ {i} (0) + t \leq p _ {1} (0) + t - \Delta . +$$ + +Hence, whenever $t \in [0, \Delta/2]$ , we find + +$$ +p _ {1} (t) \geq \max _ {i = 2, \dots , d} p _ {i} (t), +$$ + +which also implies + +$$ +p _ {1} (t) \geq p _ {1} (t) ^ {2} + \max _ {i = 2, \dots , d} p _ {i} (t) \big (1 - p _ {1} (t) \big) \geq \| \mathbf {p} (t) \| ^ {2}, +$$ + +for all $t\in [0,\Delta /2]$ . Therefore for any $t\in [0,\Delta /2],i\in [d]$ it holds + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} \big (p _ {1} (t) - p _ {i} (t) \big) = \big (p _ {1} (t) - p _ {i} (t) \big) \big (p _ {1} (t) + p _ {i} (t) - \| \mathbf {p} (t) \| ^ {2} \big) \geq 0, +$$ + +which implies for any $i\in \{2,\ldots ,d\}$ and $t\in [0,\Delta /2]$ + +$$ +p _ {1} (t) - p _ {i} (t) \geq p _ {1} (0) - p _ {i} (0) \geq \Delta , +$$ + +Applying this argument iteratively concludes the proof. + +![](images/13ce9ee283016c8ce8847bae204b83d31e85336114fa0240608957e74a9736c5.jpg) + +For the reader's convenience we restate Theorem 2.4 before giving its proof. + +Theorem 2.4. Assume + +$$ +p _ {1} (0) \geq \max _ {i = 2, \dots , d} p _ {i} (0) + \Delta , +$$ + +for some $\Delta > 0$ . Then + +$$ +\left\| \mathbf {e} _ {1} - \mathbf {p} (t) \right\| _ {1} \leq 2 (1 - p _ {1} (0)) \exp \Big (- \frac {\Delta}{d} (1 + (d - 1) \Delta) t \Big), +$$ + +that is linear convergence of $\mathbf{p}(t) \to \mathbf{e}_1$ as $t \to \infty$ . + +Proof of Theorem 2.4. Arguing as in Eq. (24) and Eq. (23), Lemma A.6 Eq. (b) implies + +$$ +\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} (1 - p _ {1} (t)) = - p _ {1} (t) \left(p _ {1} (t) - \| \mathbf {p} (t) \| ^ {2}\right) \\ \leq - \mu (1 - p _ {1} (t)), \\ \end{array} +$$ + +where + +$$ +\mu := \frac {\Delta}{d} ((d - 1) \Delta + 1). +$$ + +Gronwall's inequality entails + +$$ +1 - p _ {1} (t) \leq (1 - p _ {1} (0)) \exp (- \mu t), +$$ + +which gives + +$$ +\| \mathbf {p} (t) - \mathbf {e} _ {1} \| _ {1} = 1 - p _ {1} (t) + \sum_ {i = 2} ^ {d} p _ {i} (t) = 2 (1 - p _ {1} (t)) \leq 2 (1 - p _ {1} (0)) \exp (- \mu t). +$$ + +# A.3 Proofs for Subsection 2.3 + +In this subsection, we heuristically derive the expression for the probabilities + +$$ +\frac {\lambda_ {j} w _ {j} \left(\mathrm {t} _ {k}\right)}{\sum_ {\ell = 1} ^ {d} \lambda_ {\ell} w _ {\ell} \left(\mathrm {t} _ {k}\right)}, \quad j = 1, \dots , d, \tag {33} +$$ + +in Eq. (12). To this end, we assume that the weights are small compared to the threshold $S$ , that the weights are only updated at the postsynaptic spike times, and that $\sum_{\ell = 1}^{d}\lambda_{\ell}w_{\ell}(\mathfrak{t}_{k})\gg S$ . For convenience, we write $w_{\ell}$ for $w_{\ell}(\mathfrak{t}_k)$ and all $\ell \in [d]$ . The constraint $\sum_{\ell = 1}^{d}\lambda_{\ell}w_{\ell}\gg S$ guarantees that after the postsynaptic spike time $\mathfrak{t}_k$ , the membrane potential $Y_{t}$ will again reach $S$ and thus emit another spike at time $\mathfrak{t}_{k + 1}$ . + +Taking the expectation of the membrane potential $Y_{t} = \sum_{j=1}^{d} \sum_{\tau \in \mathcal{T}_{j} \cap (\mathfrak{t}_{k}, t]} w_{j} e^{-(t - \tau)}$ with respect to all except the $j^{\text{th}}$ spike-train, gives + +$$ +\begin{array}{l} Z _ {t} := \sum_ {\tau \in \mathcal {T} _ {j} \cap \left(\mathrm {t} _ {k}, t \right]} w _ {j} e ^ {- (t - \tau)} + \sum_ {\ell \neq j} w _ {\ell} \lambda_ {\ell} \int_ {\mathrm {t} _ {k}} ^ {t} e ^ {- (t - s)} \mathrm {d} s \\ = \sum_ {\tau \in \mathcal {T} _ {j} \cap (\mathfrak {t} _ {k}, t ]} w _ {j} e ^ {- (t - \tau)} + \sum_ {\ell \neq j} w _ {\ell} \lambda_ {\ell} \big (1 - e ^ {- (t - \mathfrak {t} _ {k})} \big), \\ \end{array} +$$ + +for all $\mathbf{t}_k\leq t < \mathbf{t}_{k + 1}$ + +Introduce $\mathfrak{t}^* \coloneqq \inf \{t \geq \mathfrak{t}_k : Z_t \geq S - w_j\}$ and write $\mathfrak{t}^+$ for the first time after $\mathfrak{t}^*$ where + +$$ +t \mapsto Z _ {\mathfrak {t} ^ {*}} + \underbrace {e ^ {- (\mathfrak {t} ^ {*} - \mathfrak {t} _ {k})} \sum_ {\ell \neq j} ^ {d} w _ {\ell} \lambda_ {\ell} \big (1 - e ^ {- (t - \mathfrak {t} ^ {*})} \big)} _ {=: V _ {t}} +$$ + +reaches the threshold $S$ . If there are sufficiently many neurons, the probability that the $j^{\text{th}}$ presynaptic neuron spikes at time $\mathfrak{t}^*$ is small and will be neglected. We have $Z_{\mathfrak{t}^*} = S - w_j$ such that $V_{\mathfrak{t}^+} = w_j$ . Approximating $1 - e^{-(\mathfrak{t}^+ - \mathfrak{t}^*)} \approx \mathfrak{t}^+ - \mathfrak{t}^*$ gives + +$$ +\mathsf {t} ^ {+} - \mathsf {t} ^ {*} \approx e ^ {\mathsf {t} ^ {*} - \mathsf {t} _ {k}} \frac {w _ {j}}{\sum_ {\ell \neq j} w _ {\ell} \lambda_ {\ell}}. +$$ + +The $j^{\mathrm{th}}$ presynaptic neuron causes the next postsynaptic spike if and only if it spikes in the interval $(\mathfrak{t}^{*},\mathfrak{t}^{+})$ . The spike times of the $j^{\mathrm{th}}$ presynaptic neuron are generated from a Poisson process with intensity $\lambda_{j}$ . Thus, if $U\sim \mathrm{Poisson}(\lambda_j(\mathfrak{t}^+ -\mathfrak{t}^*))$ , the probability that the $j^{\mathrm{th}}$ presynaptic neuron spikes in $(\mathfrak{t}^{*},\mathfrak{t}^{+})$ is given by + +$$ +\mathbb {P} \big (U \neq 0 \big) = 1 - \mathbb {P} \big (U = 0 \big) = 1 - \exp \big (- \lambda_ {j} (\mathfrak {t} ^ {+} - \mathfrak {t} ^ {*}) \big) \approx \lambda_ {j} (\mathfrak {t} ^ {+} - \mathfrak {t} ^ {*}) \approx e ^ {\mathfrak {t} ^ {*} - \mathfrak {t} _ {k}} \frac {w _ {j} \lambda_ {j}}{\sum_ {\ell \neq j} w _ {\ell} \lambda_ {\ell}}. +$$ + +We can moreover approximate the denominator on the right hand side by the full sum $\sum_{\ell=1}^{d} w_{\ell} \lambda_{\ell}$ . Since the probabilities add up to one, we must have $e^{t^* - t_k} \approx 1$ . This shows that the probability of the $j^{\text{th}}$ presynaptic neuron triggering the first postsynaptic spike after $t_k$ is approximately given by Eq. (33). + +Lemma A.7. Consider the setting outlined in Subsection 2.3. If at some time point $\mathfrak{t} > 0$ , all weights are the same, then the probability that the $j^{th}$ neuron triggers the next postsynaptic spike after $\mathfrak{t}$ is given by + +$$ +\frac {\lambda_ {j}}{\sum_ {\ell = 1} ^ {d} \lambda_ {\ell}}. +$$ + +Proof. Since all weights are the same, we can denote their value by $w$ . The $j^{\text{th}}$ neuron causes a postsynaptic spike if and only if it is the first one to spike after the postsynaptic membrane potential $Y_{t}$ has reached a level $\geq S - w$ . As $\mathfrak{t}^{*} = \inf \{t \geq \mathfrak{t} : Y_{t} \geq S - w\}$ is a jump time and a stopping + +time, we can restart the process at $\mathfrak{t}^*$ . As the increments of Poisson processes are independent, and the time between the jumps is exponentially distributed with parameters $\lambda_{j}$ , the probability that the $j^{\mathrm{th}}$ neuron causes the next presynaptic spike is given by + +$$ +\mathbb {P} \left(X _ {j} = \min \left(X _ {1}, \dots , X _ {d}\right)\right), +$$ + +where $(X_{i})_{i\in [d]}$ are independent random variables satisfying $X_{i}\sim \mathrm{Exp}(\lambda_{i})$ . If $U\sim \mathrm{Exp}(\lambda)$ and $V\sim \mathrm{Exp}(\lambda^{\prime})$ are independent, then, $U\wedge V\sim \mathrm{Exp}(\lambda +\lambda^{\prime})$ and $\mathbb{P}(U\leq V) = \lambda /(\lambda +\lambda ')$ . Thus, $\min_{i\neq j}X_i\sim \mathrm{Exp}(\sum_{i\neq j}\lambda_i)$ , and + +$$ +\mathbb {P} \big (X _ {j} = \min (X _ {1}, \dots , X _ {d}) \big) = \mathbb {P} \Big (X _ {j} \leq \min _ {i \neq j} X _ {i} \Big) = \frac {\lambda_ {j}}{\sum_ {\ell = 1} ^ {d} \lambda_ {\ell}}. +$$ + +![](images/402f0be56a878188c691d135d279c052f355c18afe1b679c79fb36de49db3d90.jpg) + +The above lemma and the previous discussion give a motivation for the form of the probabilities in settings, where the weights are small compared to the threshold $S$ or equal. We now give another heuristic, motivating our modelling choice. Let $Y$ be the postsynaptic membrane potential. Then, in expectation $Y$ grows linearly with slope $\lambda^{\top}\mathbf{w}$ . Furthermore, input $i$ causes a postsynaptic spike if, and only if, it spikes at a time at which $Y \geq S - w_{i}$ , where $S > 0$ is the threshold level. As $Y$ 's growth is approximately linear, the amount of time in which $Y \geq S - w_{i}$ , holds is approximately equal to $w_{i} / \lambda^{\top}\mathbf{w}$ . Now the probability that input $i$ jumps in an interval of length $w_{i} / \lambda^{\top}\mathbf{w}$ is given by $1 - \exp(-\lambda_i w_i / \lambda^{\top}\mathbf{w}) \approx \lambda_i w_i / \lambda^{\top}\mathbf{w}$ , which is exactly our modelling choice in the independent setting. + +# A.4 On the connection to entropic mirror descent + +An alternative approach to connecting our proposed learning rule Eq. (3) for the probabilities $\mathbf{p}$ and the entropic mirror descent in discrete-time is as follows. The entropic mirror descent step Eq. (14) with Kullback-Leibler divergence and potential $f$ can be solved explicitly and yields + +$$ +p _ {i} (k + 1) = \frac {p _ {i} (k) \exp (- \alpha (\nabla f (\mathbf {p} (k))) _ {i})}{\sum_ {j = 1} ^ {d} p _ {j} (k) \exp (- \alpha (\nabla f (\mathbf {p} (k))) _ {j})}, \quad i \in [ d ], k = 0, 1, \dots , \tag {34} +$$ + +see Section 5 of Beck and Teboulle [5] for details. With $f(\mathbf{p}) = \widetilde{L} (\mathbf{p}) = \| \mathbf{p}\| ^2 /2$ and the first order approximation $\exp (x)\approx 1 + x$ for small $x$ we deduce + +$$ +\begin{array}{l} p _ {i} (k + 1) = \frac {p _ {i} (k) \exp (- \alpha (\nabla \widetilde {L} (\mathbf {p} (k))) _ {i})}{\sum_ {j = 1} ^ {d} p _ {j} (k) \exp (- \alpha (\nabla \widetilde {L} (\mathbf {p} (k))) _ {j})} = \frac {p _ {i} (k) \exp (\alpha p _ {i} (k))}{\sum_ {j = 1} ^ {d} p _ {j} (k) \exp (\alpha p _ {j} (k))} \\ = \frac {p _ {i} (k) \exp \left(\alpha \left(p _ {i} (k) - \| \mathbf {p} (k) \| ^ {2}\right)\right)}{\sum_ {j = 1} ^ {d} p _ {j} (k) \exp \left(\alpha \left(p _ {j} (k) - \| \mathbf {p} (k) \| ^ {2}\right)\right)} \\ \approx \frac {p _ {i} (k) (1 + \alpha (\nabla L (\mathbf {p} (k))) _ {i})}{\sum_ {j = 1} ^ {d} p _ {j} (k) (1 + \alpha (\nabla L (\mathbf {p} (k))) _ {j})} \\ \end{array} +$$ + +for any $i = 1, \ldots, d$ and $k = 0, 1, \ldots$ . As our proposed learning rule Eq. (3) is a noisy version of the last line, it is naturally connected to noisy entropic gradient descent. + +# A.5 Theoretical results for the alignment of multiple read-out neurons + +For a weight vector $\mathbf{w} \in \mathbb{R}^d$ with nonnegative entries let $i^* := \operatorname{argmax}_{i=1,\dots,d} \mathbf{w}^\top \mathbf{e}_i = \operatorname{argmax}_{i=1,\dots,d} w_i$ and define the cosine-projection + +$$ +\mathcal {P} _ {\mathbf {w}} := \| \mathbf {w} \| \mathbf {e} _ {i ^ {*}}. +$$ + +Assume that $\lambda_1 > \dots > \lambda_d$ and consider first learning the first weight vector $\mathbf{w}_1$ using the learning rule in Eq. (1) while the remaining weight vectors $\mathbf{w}_2, \ldots, \mathbf{w}_d$ are fixed. Theorem 2.2 implies that after $K$ iterations, we have $\mathbf{w}_1 \approx \mathbf{e}_1$ . By projecting onto $\mathbf{w}_1^* := \mathcal{P}\mathbf{w}_1(K)$ and setting $\mathbf{w}_2(0) \gets \mathbf{w}_2(0) - \mathbf{w}_1^*\mathbf{w}_2(0)^\top \mathbf{w}_1^* / \| \mathbf{w}_1^*\|$ , we ensure that $[\mathbf{p}_2(0)]_2 = \max_{i=1,\ldots,d} [\mathbf{p}_2(0)]_i$ and Theorem 2.2 applies and yields $\mathbf{p}_2(K) \approx \mathbf{e}_2$ . Proceeding successively, we arrive at Algorithm 2. + +Theorem A.8. Consider Algorithm 2. Assume that $\lambda_1 > \dots > \lambda_d$ and the minimal gap $\Delta = \min_{i=1,\ldots,d-1}([\mathbf{p}_i(0)]_i - \max_{j>i}[\mathbf{p}_i(0)]_j) > 0$ is positive. Then we have + +$$ +\mathbb {P} (\mathbf {P} ^ {*} \neq \mathbb {I}) \to 0 +$$ + +as $K\to \infty$ . More precisely, let $\delta < \kappa /(1 + \kappa)$ for $\kappa = \min_{i = 1,\ldots ,d}\lambda_i / \max_{i = 1,\ldots ,d}\lambda_i$ and $\varepsilon >0$ Then + +$$ +\mathbb {P} \left(\mathbf {P} ^ {*} = \mathbb {I}\right) \geq (1 - \varepsilon) ^ {d} \quad f o r K \geq \frac {1 6 d}{\alpha \Delta (4 + d \Delta)} \log \left(\frac {4}{\varepsilon \delta}\right). +$$ + +# Algorithm 2: Sequential alignment of multiple output neurons + +Input: $K\in \mathbb{N}$ : number of iterations for each learning period, $\mathbf{W}(0)\in \mathbb{R}^{d\times d}$ : weight initialisation. +1 for $j = 1,\ldots ,d$ do +2 if $j\geq 2$ then +3 $\begin{array}{r}\mathbf{w}_j(0)\leftarrow \mathbf{w}_j(0) - \sum_{i = 1}^{j - 1}\frac{\mathbf{w}_j(0)^{\top}\mathbf{w}_i^*}{\|\mathbf{w}_i^*\|^{2}}\mathbf{w}_i^*; \end{array}$ +4 end +5 for $k = 0,1,\dots ,d$ do +6 Receive $\mathbf{B}_j(k)\sim M(1,\mathbf{p}_j(k))$ with $\mathbf{p}_j(k)\gets \lambda \odot \mathbf{w}_j(k) / \lambda^\top \mathbf{w}_j(k)$ and +7 $\mathbf{Z}_j(k)\sim \mathrm{Unif}([-1,1]^d)$ from spike trains; +Update +8 Set $\mathbf{w}_j^* := \mathcal{P}\mathbf{w}_j(K)$ to obtain $\mathbf{p}_j^* = \boldsymbol {\lambda}^\top \mathbf{w}_j^*$ +9 end +Output: The weight evolution $\mathbf{W}(k) = [\mathbf{w}_1(k)\dots \mathbf{w}_d(k)],k = 0,\dots ,K$ probability evolution $\mathbf{P}(k) = [\mathbf{p}_1(k)\dots \mathbf{p}_d(k)],k = 1,\dots ,K$ and projections $\mathbf{P}^{*} = [\mathbf{p}_{1}^{*}\dots \mathbf{p}_{d}^{*}]$ + +Before proving Theorem A.8, we start with an auxiliary result on the order of the weights when the probability vector is close to a standard unit vector. + +Lemma A.9. Assume that $0 < \lambda_{\mathrm{min}} = \min_{i=1,\dots,d}\lambda_i \leq \lambda_{\mathrm{max}} = \max_{i=1,\dots,d}\lambda_i < \infty$ and let $\kappa = \lambda_{\mathrm{min}} / \lambda_{\mathrm{max}}$ . Consider a weight vector $\mathbf{w} \in \mathbb{R}^d$ with corresponding probability vector $\mathbf{p} = \mathbf{w} \odot \boldsymbol{\lambda} / \mathbf{w}^\top \boldsymbol{\lambda}$ and let $0 < \delta < 1$ . Then the condition $1 - p_1 < \delta$ implies that $\max_{i=2,\dots,d}w_i \leq w_1\kappa^{-1}\delta/(1-\delta)$ . + +Proof. Since $1 - p_{1} \leq \delta$ we know that $\sum_{i=2}^{d} \lambda_{i} w_{i} \leq \frac{\delta}{1 - \delta} \lambda_{1} w_{1}$ . By bounding the $\lambda_{i}$ from above and below we find + +$$ +\lambda_ {\min } \sum_ {i = 2} ^ {d} w _ {i} \leq \frac {\delta}{1 - \delta} \lambda_ {\max } w _ {1}, +$$ + +such that + +$$ +\max _ {i = 2, \dots , d} w _ {i} \leq \frac {\delta}{1 - \delta} \kappa^ {- 1} w _ {1}. +$$ + +□ + +Proof of Theorem A.8. By Theorem 2.2 we know that $\mathbb{P}(\| \mathbf{p}_1(K) - \mathbf{e}_1\|_1 \leq \delta) \geq 1 - \varepsilon$ for our choice of $K$ . Note that the projection picks out the direction $\operatorname{argmax}_{i=1,\dots,d} \mathbf{w}_1(K)^\top \mathbf{e}_i = \operatorname{argmax}_{i=1,\dots,d} [\mathbf{w}_1(K)]_i$ . Consequently, Lemma A.9 and $\delta < \kappa/(1 - \kappa)$ imply that $\mathbf{w}_1(K)$ projects in the direction of $\mathbf{e}_1$ with probability at least $1 - \varepsilon$ . We find that $\mathbb{P}(\mathbf{p}_1^* \neq \mathbf{e}_1) \leq 1 - \varepsilon$ . By the adjustment of the weight vector $\mathbf{w}_2(0)$ we remove its first component, such that $[\mathbf{p}_2(0)]_1 = 0$ and $[\mathbf{p}_2(0)]_2 = \max_{j=1,\dots,d} [\mathbf{p}_2(0)]_i$ . By the assumption on $[\mathbf{p}_2(0)]_2 - \max_{j=3,\dots,d} [\mathbf{p}_2(0)]_j \geq \Delta > 0$ we find that $\mathbb{P}(\| \mathbf{p}_2(K) - \mathbf{e}_2\|_1 \leq \delta) \geq 1 - \varepsilon$ . By iteration and independence of the training windows we conclude the proof. + +# A.6 Additional material for the extension to time-inhomogeneous intensities + +Derivation of Eq. (18): Eq. (17) yields the deterministic update scheme + +$$ +\mathbf {p} (k + 1) = \widetilde {\mathbf {p}} (k) \odot \left(\mathbf {1} + \alpha \left(\mathbf {p} (k) - \widetilde {\mathbf {p}} (k) ^ {\top} \mathbf {p} (k) \mathbf {1}\right)\right) + \mathcal {O} \left(\alpha^ {2}\right). \tag {35} +$$ + +Let $T > 0$ be the time horizon and assume that the time-inhomogeneous intensities $\lambda_{\alpha}$ change on the same scale as the weights and are given by $\lambda_{\alpha}(k) = \lambda(k\alpha)$ , $k = 0,1,\ldots,\lfloor T/\alpha \rfloor$ , where $\lambda \in C_b^2([0,T])$ is a universal, twice differentiable function with bounded derivatives and $\lambda(t) \geq \lambda_{\min} > 0$ , $t \in [0,T]$ , componentwise. For vectors $\mathbf{w}, \lambda, \lambda'$ that are of the same length, and $\Delta := \lambda' - \lambda$ , we have + +$$ +\frac {\boldsymbol {\lambda} ^ {\prime} \odot \mathbf {w}}{\boldsymbol {\lambda} ^ {\prime \top} \mathbf {w}} = \frac {\Delta \odot \mathbf {w} + \boldsymbol {\lambda} \odot \mathbf {w}}{\Delta^ {\top} \mathbf {w} + \boldsymbol {\lambda} ^ {\top} \mathbf {w}} = \frac {\Delta \odot \mathbf {w}}{\Delta^ {\top} \mathbf {w} + \boldsymbol {\lambda} ^ {\top} \mathbf {w}} + \frac {\boldsymbol {\lambda} \odot \mathbf {w}}{\boldsymbol {\lambda} ^ {\top} \mathbf {w}} \Big (1 - \frac {\Delta^ {\top} \mathbf {w}}{\Delta^ {\top} \mathbf {w} + \boldsymbol {\lambda} ^ {\top} \mathbf {w}} \Big). +$$ + +Applying this with $\lambda' = \lambda_{\alpha}(k + 1), \lambda = \lambda_{\alpha}(k), \mathbf{w} = \mathbf{w}(k)$ , yields + +$$ +\begin{array}{l} \widetilde {\mathbf {p}} (k) = \frac {\boldsymbol {\lambda} _ {\alpha} (k + 1) \odot \mathbf {w} (k)}{\boldsymbol {\lambda} _ {\alpha} (k + 1) ^ {\top} \mathbf {w} (k)} \\ = \mathbf {p} (k) + \frac {\Delta \boldsymbol {\lambda} _ {\alpha} (k) \odot \mathbf {w} (k)}{\Delta \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k) + \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k)} - \mathbf {p} (k) \frac {\Delta \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k)}{\Delta \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k) + \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k)}. \\ \end{array} +$$ + +Our assumptions on $\lambda_{\alpha}$ imply that $\widetilde{\mathbf{p}}(k) - \mathbf{p}(k) \lesssim \|\Delta \lambda_{\alpha}(k)\| \lesssim \alpha$ for all $k = 0,1,\ldots,\lfloor T / \alpha \rfloor$ . This gives $\widetilde{\mathbf{p}}(k) \odot \mathbf{p}(k) - \widetilde{\mathbf{p}}(k)\widetilde{\mathbf{p}}(k)^{\top}\mathbf{p}(k) = \mathbf{p}(k) \odot \mathbf{p}(k) - \mathbf{p}(k)\mathbf{p}(k)^{\top}\mathbf{p}(k) + \mathcal{O}(\alpha)$ . In combination with Eq. (35) we find + +$$ +\begin{array}{l} \frac {\mathbf {p} (k + 1) - \mathbf {p} (k)}{\alpha} = \frac {\Delta \boldsymbol {\lambda} _ {\alpha} (k)}{\alpha} \odot \frac {\mathbf {w} (k)}{\Delta \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k) + \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k)} \\ - \frac {\mathbf {p} (k)}{\Delta \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k) + \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k)} \frac {\Delta \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top}}{\alpha} \mathbf {w} (k) \\ + \widetilde {\mathbf {p}} (k) \odot \mathbf {p} (k) - \widetilde {\mathbf {p}} (k) \widetilde {\mathbf {p}} (k) ^ {\top} \mathbf {p} (k) \\ = \frac {\Delta \boldsymbol {\lambda} _ {\alpha} (k)}{\alpha} \odot \frac {\mathbf {w} (k)}{\mathcal {O} (\alpha) + \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k)} \\ - \frac {\mathbf {p} (k)}{\mathcal {O} (\alpha) + \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top} \mathbf {w} (k)} \frac {\Delta \boldsymbol {\lambda} _ {\alpha} (k) ^ {\top}}{\alpha} \mathbf {w} (k) \tag {36} \\ + \mathbf {p} (k) \odot \mathbf {p} (k) - \mathbf {p} (k) \mathbf {p} (k) ^ {\top} \mathbf {p} (k) + \mathcal {O} (\alpha). \\ \end{array} +$$ + +Sending $\alpha \to 0$ and using that $\Delta \pmb{\lambda}_{\alpha}(k) / \alpha -\frac{\mathrm{d}}{\mathrm{d}t}\pmb {\lambda}(k\alpha)\to 0$ as $\alpha \rightarrow 0$ we recognize Eq. (36) as an Euler-type scheme for the ODE + +$$ +\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \mathbf {p} (t) = \frac {\frac {\mathrm {d}}{\mathrm {d} t} [ \boldsymbol {\lambda} (t) ] \odot \mathbf {w} (t)}{\boldsymbol {\lambda} ^ {\top} (t) \mathbf {w} (t)} - \mathbf {p} (t) \frac {\frac {\mathrm {d}}{\mathrm {d} t} [ \boldsymbol {\lambda} (t) ] ^ {\top} \mathbf {w} (t)}{\boldsymbol {\lambda} ^ {\top} (t) \mathbf {w} (t)} + \mathbf {p} (t) \odot (\mathbf {p} (t) - \| \mathbf {p} (t) \| ^ {2} \mathbf {1}) \\ = \mathbf {p} (t) \odot \left(\frac {\mathrm {d}}{\mathrm {d} t} \log (\boldsymbol {\lambda} (t)) - \mathbf {p} ^ {\top} (t) \frac {\mathrm {d}}{\mathrm {d} t} \log (\boldsymbol {\lambda} (t)) \mathbf {1}\right) + \mathbf {p} (t) \odot (\mathbf {p} (t) - \| \mathbf {p} (t) \| ^ {2} \mathbf {1}) \\ = \mathbf {p} (t) \odot \left(\frac {\mathrm {d}}{\mathrm {d} t} \log (\boldsymbol {\lambda} (t)) + \mathbf {p} (t) - \mathbf {p} (t) ^ {\top} \left(\frac {\mathrm {d}}{\mathrm {d} t} \log (\boldsymbol {\lambda} (t)) + \mathbf {p} (t)\right) \mathbf {1}\right), \\ \end{array} +$$ + +where the logarithm is taken componentwise. + +# A.7 Analysis of the correlated model + +In the following we always assume that the off-diagonal entries of $\Gamma$ are strictly smaller than 1, that is, $\Gamma_{i,j} < 1$ for all $i \neq j = 1, \ldots, d$ . This assumption is reasonable since a perfect correlation between input $i$ and input $j$ corresponds to one single input neuron with Poisson process intensity $\lambda_i + \lambda_j$ . Under this assumption it is easy to see that the quadratic form associated to $\Gamma$ is maximised on the probability simplex by the basis vectors $\mathbf{e}_1, \ldots, \mathbf{e}_d$ . Thus, the natural question to investigate is the + +![](images/154dad19a935d02f12265320b1cedb94d000bfa1cc5c6e9075ed58ef7bce0054.jpg) + +![](images/21be77c98ff57d2060e418d24082f2eadd4d953db1560fd3cc136acccec22153.jpg) + +![](images/1023e977281aff332fd5be5d9c4b85e7810870f5dedf34d4531b426d0384e0cb.jpg) + +![](images/a35cc52e17b7d410a29ed483bd6292cf49dcd0b7b56d2a63df4af2f851d1a62e.jpg) + +![](images/b07ffaae25253ba308518d7fa6486593b426ab51d0939b1bb641997fa5f9caa0.jpg) + +![](images/5ccde676f01494f919e09018a4d97fda17aef322933d4074154a075a1c7f718c.jpg) + +![](images/985ca2588c64b892daeef9235835cbc040b7e778bd99de809d86e4085b09fc97.jpg) +Figure 6: Correlated inputs with + +![](images/e07aa45ee0e86a7132c8094376a7db6f3cc75eadc83e0df46ecdd82b5e898319.jpg) + +![](images/fa2af69ea49227628d87b3e8d4c4d54049f72b06fa4cdb2d157c0c3e8e8f1c3d.jpg) + +$$ +\boldsymbol {\Gamma} = \left( \begin{array}{c c c} 1 & 1 / 2 & 0 \\ 1 / 2 & 1 & 1 / 2 \\ 0 & 1 / 2 & 1 \end{array} \right) (\text {t o p}), \boldsymbol {\Gamma} = \left( \begin{array}{c c c} 1 & 3 / 4 & 0 \\ 3 / 4 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) (\text {m i d d l e}), \left( \begin{array}{c c c} 1 & 1 / 1 0 & 1 / 1 0 \\ 1 / 1 0 & 1 & 0 \\ 1 / 1 0 & 0 & 1 \end{array} \right) (\text {b o t t o m}). +$$ + +Contour plot of the Shahshahani loss function $L(\mathbf{p}) = -\frac{1}{2}\mathbf{p}^{\top}\boldsymbol{\Gamma}\mathbf{p}$ on the probability simplex $\mathfrak{P}$ for $d = 3$ with different overlays. Left: Three sample trajectories of Eq. (3) with different initial configurations $\mathbf{p}(0)$ . Middle: Stream plot of the gradient field given by Eq. (7). Right: 100 sample trajectories of Eq. (3) with $\mathbf{p}(0) = (0.3, 0.3, 0.4)^{\top}$ . All trajectories are simulated with 2000 iteration steps, learning rate $\alpha = 0.01$ and $\mathbf{Z}(k) \sim \mathrm{Unif}([-1, 1]^d)$ . + +same as in the original model: To which basis vector does the model converge? In the following section we investigate this question and show results for the weakly dependent case. + +The analysis of the correlated version of the model follows the same steps as in the independent case. For this we assume that all random variables are defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and denote by $\mathcal{F}_n, n = 1, 2, \ldots$ the natural filtration of $(\mathbf{B}(n), \mathbf{Z}(n), \mathbf{C}(n))_{n \in \mathbb{N}}$ , and set $\mathcal{F}_{-1} = \{\emptyset, \Omega\}$ . The proof follows exactly the same steps as the proof of Theorem 2.2. We start with the following result, which bounds the error of the Taylor approximation of the random dynamics. + +Lemma A.10. For $i\in [d]$ and $k = 0,1,\ldots$ define + +$$ +\xi_ {i} (k) := p _ {i} (k) \left(\mathbb {E} \left[ Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \mid \mathcal {F} _ {k - 1} \right] - \left(Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right)\right), \tag {37} +$$ + +and assume $\alpha < 1 / Q$ . Then for any $i \in [d]$ and $k = 0, 1, \ldots$ , there exists a random variable $\theta_i(k)$ satisfying + +$$ +\left| \theta_ {i} (k) \right| \leq \alpha^ {2} \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} p _ {i} (k) \big (1 - p _ {i} (k) \big), \quad \text {a l m o s t s u r e l y}, +$$ + +such that + +$$ +p _ {i} (k + 1) = p _ {i} (k) + \alpha p _ {i} (k) \left(\mathbf {\Gamma} _ {i,}. \mathbf {p} (k) - (\mathbf {p} (k)) ^ {\top} \mathbf {\Gamma} \mathbf {p} (k)\right) - \alpha \xi_ {i} (k) - \theta_ {i} (k). +$$ + +Proof. As the dynamic of $\mathbf{p}$ still follows Eq. (4) as in the uncorrelated model, we can argue as in the proof of Lemma A.2 to obtain that for some $\gamma \in (0,1)$ + +$$ +\begin{array}{l} p _ {i} (k + 1) = p _ {i} (k) + \alpha p _ {i} (k) \left(Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right) \\ - \alpha^ {2} p _ {i} (k) \frac {\sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \Big (Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \Big)}{\big (1 + \gamma \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \big) ^ {3}}. \\ \end{array} +$$ + +Then, since $|Y_{i}(k)| \leq Q$ we obtain the same bound on the error term as in the proof of Lemma A.2 + +$$ +\begin{array}{l} \left| \alpha^ {2} p _ {i} (k) \frac {\sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k) \left(Y _ {i} (k) - \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right)}{\left(1 + \gamma \sum_ {j = 1} ^ {d} p _ {j} (k) Y _ {j} (k)\right) ^ {3}} \right| \\ \leq \alpha^ {2} \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} p _ {i} (k) (1 - p _ {i} (k)). \\ \end{array} +$$ + +Since the $p_i(k)$ are $\mathcal{F}_{k-1}$ -measurable and $\mathbf{C}(k)$ is independent of $\mathcal{F}_{k-1}$ , we finally obtain $\mathbb{E}[Y_i(k)|\mathcal{F}_{k-1}] = \Gamma_{i,\mathbf{p}}(k)$ . + +In the following we always assume for $\Delta_p, \Delta_{\Gamma} > 0$ + +$$ +p _ {1} (0) \geq p _ {i} (0) + \Delta_ {p}, \quad \forall i = 2, \dots , d, +$$ + +$$ +(\mathbf {\Gamma} \mathbf {p} (0)) _ {1} \geq (\mathbf {\Gamma} \mathbf {p} (0)) _ {i} + \Delta_ {\mathbf {r}}, \quad \forall i = 2, \dots , d, \tag {38} +$$ + +$$ +c _ {\star} := \frac {\Delta_ {p} \Delta_ {\Gamma}}{4} - \nu \left(1 + \frac {\Delta_ {p} \Delta_ {\Gamma}}{4}\right) > 0, \quad \text {w h e r e}, \quad \nu := \max _ {i \neq j \in [ d ]} \Gamma_ {i, j}. +$$ + +These assumptions are for example fulfilled in the following case + +$$ +\mathbf {p} (0) = \left( \begin{array}{c} 0. 8 \\ 0. 1 \\ 0. 1 \end{array} \right), \quad \boldsymbol {\Gamma} = \left( \begin{array}{c c c} 1 & 0. 1 & 0. 1 \\ 0. 1 & 1 & 0 \\ 0. 1 & 0 & 1 \end{array} \right), +$$ + +since we can choose $\Delta_p = 0.7$ , $\Delta_{\Gamma} = 0.64$ , $\nu = 0.1$ , and hence $c_{\star} = 8 * 10^{-4}$ . In the independent case, which corresponds to $\nu = 0$ , the assumptions given in Eq. (38) reduce to the original assumption given in Eq. (21). For $k = 1, 2, \ldots$ , we define a sequence of benign events + +$$ +\Omega (k) := \Big \{p _ {1} (u) \geq \max _ {i = 2, \dots , d} p _ {i} (u) + \frac {\Delta_ {p}}{2}, \big (\mathbf {\Gamma} \mathbf {p} (u) \big) _ {1} \geq \max _ {i = 2, \dots , d} \big (\mathbf {\Gamma} \mathbf {p} (u) \big) _ {i} + \frac {\Delta_ {\mathbf {r}}}{2}, \forall u \in [ k ] \Big \}. +$$ + +Due to assumption Eq. (38), $\Omega(0) = \Omega$ . Additionally, since the assumptions in Eq. (38) imply the gradient to be bounded away from 0, we can prove the following recursive upper bound for $1 - p_1(k)$ . + +Proposition A.11. If + +$$ +0 < \alpha \leq \frac {\left(1 - Q \alpha\right) ^ {3}}{8 Q ^ {2}} \Delta_ {\Gamma}, +$$ + +then, on the event $\Omega (k)$ + +$$ +1 - p _ {1} (k + 1) \leq \left(1 - \alpha \frac {\Delta \mathbf {r}}{4 d} \left(1 + \frac {\Delta_ {p}}{2} (d - 1)\right)\right) \left(1 - p _ {1} (k)\right) + \alpha \xi_ {i} (k). +$$ + +Proof. By definition, we have on the event $\Omega (k)$ + +$$ +\begin{array}{l} \left(\boldsymbol {\Gamma} \mathbf {p} (k)\right) _ {1} - \mathbf {p} (k) ^ {\top} \boldsymbol {\Gamma} \mathbf {p} (k) = \sum_ {j = 1} ^ {d} p _ {j} (k) \bigl (\left(\boldsymbol {\Gamma} \mathbf {p} (k)\right) _ {1} - \left(\boldsymbol {\Gamma} \mathbf {p} (k)\right) _ {j} \bigr) \\ \geq \frac {\Delta_ {\Gamma}}{2} (1 - p _ {1} (k)). \tag {39} \\ \end{array} +$$ + +The assumption on $\alpha$ implies $\alpha < 1 / Q$ and thus Lemma A.10 becomes applicable. Now applying Lemma A.10 with $i = 1$ , gives on the event $\Omega (k)$ + +$$ +\begin{array}{l} 1 - p _ {1} (k + 1) \\ \leq 1 - p _ {1} (k) - \alpha \frac {\Delta \mathbf {r}}{2} p _ {1} (k) (1 - p _ {1} (k)) + \alpha \xi_ {i} (k) + \theta_ {i} (k) \\ \leq 1 - p _ {1} (k) - \alpha \left(\frac {\Delta_ {\Gamma}}{2} - \alpha \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}}\right) p _ {1} (k) (1 - p _ {1} (k)) + \alpha \xi_ {i} (k) \\ \leq 1 - p _ {1} (k) - \alpha \frac {\Delta \mathbf {r}}{4} p _ {1} (k) (1 - p _ {1} (k)) + \alpha \xi_ {i} (k) \\ \leq \left(1 - \alpha \frac {\Delta \Gamma}{4 d} \Big (1 + \frac {\Delta p}{2} (d - 1) \Big)\right) (1 - p _ {1} (k)) + \alpha \xi_ {i} (k) \\ \end{array} +$$ + +This concludes the proof. + +![](images/1fa55b4503e2c61ee298404bb74509915d65ab83cb0eeb5eec4187838e8f8385.jpg) + +As in the uncorrelated setting our goal is now to derive a lower bound for the probability of the favourable event $\Omega(k)$ . For this we again follow the same strategy and rely on uniform concentration inequalities for martingales. In order apply those, we require the following lemma, which states that $\Omega(k)$ is fulfilled as soon as + +$$ +M _ {j} (k) := \sum_ {\ell = 0} ^ {k} \alpha \xi_ {j} (\ell) \mathbb {1} _ {\Omega (\ell)}, \quad k = 0, 1, \dots \tag {40} +$$ + +with $\xi_{j}(\ell)$ defined in Eq. (22), exhibits a uniform concentration behaviour. In the following we denote by $\| \boldsymbol{\Gamma}\|_{\infty}$ the row-sum norm of $\boldsymbol{\Gamma}$ , i.e. $\| \boldsymbol{\Gamma}\|_{\infty} = \max_{i\in [d]}\sum_{j = 1}^{d}\boldsymbol{\Gamma}_{i,j}$ . + +Lemma A.12. Define the sets + +$$ +(E) _ {j} (k) := \left\{\max _ {u \in [ k ]} | M _ {j} (u) | \leq \frac {1}{4} \left(\Delta_ {p} \wedge \frac {\Delta_ {\Gamma}}{\| \boldsymbol {\Gamma} \| _ {\infty}}\right) \right\}, \quad E (k) := \bigcap_ {j = 1} ^ {d} (E) _ {j} (k), \quad k = 0, 1, \dots . +$$ + +Then if + +$$ +0 < \alpha \leq \frac {(1 - Q \alpha) ^ {3}}{4 Q ^ {2}} c _ {\star}, +$$ + +the following set inclusion holds for any $k = 0,1,\ldots$ + +$$ +E (k) \subseteq \Omega (k + 1). +$$ + +Proof. Let $u \in \{2, \ldots, d\}$ be arbitrary. By Lemma A.10, on the event $\Omega(k)$ , + +$$ +\begin{array}{l} p _ {1} (k + 1) - p _ {u} (k + 1) \\ \geq p _ {1} (k) - p _ {u} (k) + \alpha \left(p _ {1} (k) - p _ {u} (k)\right) \left(\left(\boldsymbol {\Gamma} \mathbf {p} (k)\right) _ {1} - \mathbf {p} (k) ^ {\top} \boldsymbol {\Gamma} \mathbf {p} (k)\right) \\ + \alpha (\xi_ {j} (k) - \xi_ {1} (k)) - \theta_ {1} (k) + \theta_ {u} (k). \\ \end{array} +$$ + +Since $p_u(k) \leq 1 - p_1(k)$ we also obtain on $\Omega(k)$ , + +$$ +\begin{array}{l} - \theta_ {1} (k) + \theta_ {u} (k) \geq - \alpha^ {2} \frac {2 Q ^ {2}}{(1 - Q \alpha) ^ {3}} \left(p _ {1} (k) \left(1 - p _ {1} (k)\right) + p _ {u} (k) \left(1 - p _ {u} (k)\right)\right) \\ \geq - \alpha^ {2} \frac {4 Q ^ {2}}{(1 - Q \alpha) ^ {3}} (1 - p _ {1} (k)). \\ \end{array} +$$ + +The assumptions on $\alpha$ then imply that on the event $\Omega (k)$ + +$$ +\begin{array}{l} p _ {1} (k + 1) - p _ {u} (k + 1) - \alpha (\xi_ {j} (k) - \xi_ {1} (k)) \\ \geq p _ {1} (k) - p _ {u} (k) + \alpha \big (p _ {1} (k) - p _ {u} (k) \big) \big ((\boldsymbol {\Gamma} \mathbf {p} (k)) _ {1} - \mathbf {p} (k) ^ {\top} \boldsymbol {\Gamma} \mathbf {p} (k) \big) - \alpha^ {2} \frac {4 Q ^ {2}}{(1 - Q \alpha) ^ {3}} \big (1 - p _ {1} (k) \big) \\ \geq p _ {1} (k) - p _ {u} (k). \\ \end{array} +$$ + +Because of $\Omega (k)\subseteq \Omega (k - 1)$ it then follows, + +$$ +\begin{array}{l} (p _ {1} (k + 1) - p _ {u} (k + 1)) \mathbb {1} _ {\Omega (k)} \geq (p _ {1} (k) - p _ {u} (k)) \mathbb {1} _ {\Omega (k)} + \alpha (\xi_ {j} (k) - \xi_ {1} (k)) \mathbb {1} _ {\Omega (k)} \\ = \mathbb {1} _ {\Omega (k)} \Big ((p _ {1} (k) - p _ {u} (k)) \mathbb {1} _ {\Omega (k - 1)} + \alpha (\xi_ {j} (k) - \xi_ {1} (k)) \mathbb {1} _ {\Omega (k)} \Big). \\ \end{array} +$$ + +This gives + +$$ +\begin{array}{l} (p _ {1} (k + 1) - p _ {u} (k + 1)) \mathbb {1} _ {\Omega (k)} \\ \geq \mathbb {1} _ {\Omega (k)} \left(p _ {1} (0) - p _ {u} (0) + \sum_ {\ell = 0} ^ {k} \alpha \left(\xi_ {j} (\ell) - \xi_ {1} (\ell)\right) \mathbb {1} _ {\Omega (\ell)}\right) \tag {41} \\ \geq \Delta_ {p} \mathbb {1} _ {\Omega (k)} - | M _ {j} (k) | - | M _ {1} (k) |. \\ \end{array} +$$ + +Now again let $u \in \{2, \ldots, d\}$ be given. Then it holds, on $\Omega(k)$ + +$$ +\begin{array}{l} (\Gamma \mathbf {p} (k + 1)) _ {1} - (\Gamma \mathbf {p} (k + 1)) _ {u} \\ = (\mathbf {\Gamma} \mathbf {p} (k)) _ {1} - (\mathbf {\Gamma} \mathbf {p} (k)) _ {u} + \alpha \sum_ {j = 1} ^ {d} \mathbf {\Gamma} _ {1, j} p _ {j} (k) ((\mathbf {\Gamma} \mathbf {p} (k)) _ {j} - \mathbf {p} (k) \mathbf {\Gamma} \mathbf {p}) \\ - \alpha \sum_ {j = 1} ^ {d} \boldsymbol {\Gamma} _ {u, j} p _ {j} (k) ((\boldsymbol {\Gamma} \mathbf {p} (k)) _ {j} - \mathbf {p} (k) \boldsymbol {\Gamma} \mathbf {p}) - \sum_ {j = 1} ^ {d} (\boldsymbol {\Gamma} _ {1, j} - \boldsymbol {\Gamma} _ {u, j}) (\alpha \xi_ {j} (k) + \theta_ {j} (k)) \\ \geq (\boldsymbol {\Gamma} \mathbf {p} (k)) _ {1} - (\boldsymbol {\Gamma} \mathbf {p} (k)) _ {u} + \alpha (1 - \boldsymbol {\Gamma} _ {1, u}) (p _ {1} (k) - p _ {u} (k)) ((\boldsymbol {\Gamma} \mathbf {p} (k)) _ {1} - \mathbf {p} (k) \boldsymbol {\Gamma} \mathbf {p}) \\ + \alpha \sum_ {j = 2, j \neq u} ^ {d} \left(\boldsymbol {\Gamma} _ {1, j} - \boldsymbol {\Gamma} _ {u, j}\right) p _ {j} (k) \left(\left(\boldsymbol {\Gamma} \mathbf {p} (k)\right) _ {j} - \mathbf {p} (k) \boldsymbol {\Gamma} \mathbf {p}\right) \\ - \sum_ {j = 1} ^ {d} \left(\boldsymbol {\Gamma} _ {1, j} - \boldsymbol {\Gamma} _ {u, j}\right) \left(\alpha \xi_ {j} (k) + \theta_ {j} (k)\right) \\ \geq (\mathbf {\Gamma} \mathbf {p} (k)) _ {1} - (\mathbf {\Gamma} \mathbf {p} (k)) _ {u} + \alpha (1 - \mathbf {\Gamma} _ {1, u}) \frac {\Delta_ {p} \Delta_ {\mathbf {r}}}{4} (1 - p _ {1} (k)) - \alpha \nu (1 - p _ {1} (k)) \\ - \sum_ {j = 1} ^ {d} \left(\boldsymbol {\Gamma} _ {1, j} - \boldsymbol {\Gamma} _ {u, j}\right) \left(\alpha \xi_ {j} (k) + \theta_ {j} (k)\right) \\ \geq (\mathbf {\Gamma} \mathbf {p} (k)) _ {1} - (\mathbf {\Gamma} \mathbf {p} (k)) _ {u} + \alpha \left((1 - \nu) \frac {\Delta_ {p} \Delta_ {\mathbf {r}}}{4} - \nu - \alpha \frac {4 Q ^ {2}}{(1 - Q \alpha) ^ {3}}\right) (1 - p _ {1} (k)) \\ - \alpha \sum_ {j = 1} ^ {d} \left(\boldsymbol {\Gamma} _ {1, j} - \boldsymbol {\Gamma} _ {u, j}\right) \xi_ {j} (k) \\ \geq (\mathbf {\Gamma} \mathbf {p} (k)) _ {1} - (\mathbf {\Gamma} \mathbf {p} (k)) _ {u} - \alpha \sum_ {j = 1} ^ {d} (\mathbf {\Gamma} _ {1, j} - \mathbf {\Gamma} _ {u, j}) \xi_ {j} (k). \\ \end{array} +$$ + +Hence, arguing as in the derivation of Eq. (41) gives + +$$ +\begin{array}{l} \left(\left(\boldsymbol {\Gamma} \mathbf {p} (k + 1)\right) _ {1} - \left(\boldsymbol {\Gamma} \mathbf {p} (k + 1)\right) _ {u}\right) \mathbb {1} _ {\Omega (k)} \\ \geq \mathbb {1} _ {\Omega (k)} \left(\left(\boldsymbol {\Gamma} \mathbf {p} (0)\right) _ {1} - \left(\boldsymbol {\Gamma} \mathbf {p} (0)\right) _ {u} - \alpha \sum_ {\ell = 0} ^ {k} \sum_ {j = 1} ^ {d} \left(\boldsymbol {\Gamma} _ {1, j} - \boldsymbol {\Gamma} _ {u, j}\right) \xi_ {j} (\ell) \mathbb {1} _ {\Omega (\ell)}\right) \tag {42} \\ = \mathbb {1} _ {\Omega (k)} \Big ((\mathbf {T p} (0)) _ {1} - (\mathbf {T p} (0)) _ {u} - \sum_ {j = 1} ^ {d} (\mathbf {T} _ {1, j} - \mathbf {T} _ {u, j}) M _ {j} (k) \Big) \\ \geq \mathbb {1} _ {\Omega (k)} \Big ((\boldsymbol {\Gamma} \mathbf {p} (0)) _ {1} - (\boldsymbol {\Gamma} \mathbf {p} (0)) _ {u} - 2 \| \boldsymbol {\Gamma} \| _ {\infty} \max _ {j \in [ d ]} | M _ {j} (k) | \Big). \\ \end{array} +$$ + +With the above results we can now begin proving that $E(k) \subseteq \Omega(k + 1)$ for all $k = 0, 1, \ldots$ . We do this by induction. For $k = 0$ , this directly follows from Eq. (41) and Eq. (42), since + +$\Omega(0) = \Omega$ by assumption. Now, assume the assertion holds for some $k = 0,1,\ldots$ . This implies $E(k + 1) \subseteq E(k) \subseteq \Omega(k + 1)$ , such that for any $u \in \{2,\dots,d\}$ it holds on $E(k + 1)$ by Eq. (41) + +$$ +\begin{array}{l} (p _ {1} (k + 2) - p _ {u} (k + 2)) \geq \Delta_ {p} - | M _ {j} (k + 1) | - | M _ {1} (k + 1) | \\ \geq \frac {\Delta_ {p}}{2}, \\ \end{array} +$$ + +and additionally by Eq. (42) it holds on $E(k + 1)$ + +$$ +\begin{array}{l} (\mathbf {\Gamma} \mathbf {p} (k + 2)) _ {1} - (\mathbf {\Gamma} \mathbf {p} (k + 2)) _ {u} \geq (\mathbf {\Gamma} \mathbf {p} (0)) _ {1} - (\mathbf {\Gamma} \mathbf {p} (0)) _ {u} - 2 \| \boldsymbol {\Gamma} \| _ {\infty} \max _ {j \in [ d ]} | M _ {j} (k + 1) | \\ \geq \frac {\Delta_ {\Gamma}}{2}, \\ \end{array} +$$ + +which proves the assertion. + +![](images/04bf9b25ac2e4c55cddf26a940d0169fb514be2bd1006bd1d8fd7a88de416fa8.jpg) + +With the above results we are now able to prove and state the main theorem for the correlated case. + +Theorem A.13. Given $\varepsilon \in (0,1)$ , assume + +$$ +\begin{array}{l} \Delta_ {p} := p _ {1} (0) - \max _ {i = 2, \dots , d} p _ {i} (0) > 0, \quad \Delta_ {\Gamma} := (\mathbf {\Gamma} \mathbf {p} (0)) _ {1} - \max _ {i = 2, \dots , d} (\mathbf {\Gamma} \mathbf {p} (0)) _ {1} > 0, \\ c _ {\star} := \frac {\Delta_ {p} \Delta \mathbf {r}}{4} - \nu \left(1 + \frac {\Delta_ {p} \Delta \mathbf {r}}{4}\right) > 0, \quad \text {w h e r e}, \quad \nu := \max _ {i \neq j \in [ d ]} \mathbf {\Gamma} _ {i, j}, \\ \end{array} +$$ + +and + +$$ +0 < \alpha \leq \frac {1}{4 Q ^ {2}} \Big ((1 - Q \alpha) ^ {3} c _ {\star} \wedge \frac {1}{1 0 2 4 (1 - p _ {1} (0))} \Big (\Delta_ {p} \wedge \frac {\Delta_ {\mathbf {r}}}{\| \mathbf {r} \| _ {\infty}} \Big) ^ {2} \Delta_ {\mathbf {r}} \Big (\frac {4}{d} + \Delta_ {p} \Big) \Big). +$$ + +Then there exists an event $\Theta$ with probability $\geq 1 - \varepsilon /2$ such that + +$$ +\mathbb {E} \left[ \| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \mathbb {1} _ {\Theta} \right] \leq 2 (1 - p _ {1} (0)) \exp \left(- \alpha \frac {\Delta \mathbf {r}}{1 6} \left(\frac {4}{d} + \Delta_ {p}\right) k\right), \quad f o r a l l k = 0, 1, \dots . +$$ + +Consequently, given $\delta > 0$ , it holds + +$$ +\mathbb {P} \Big (\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \geq \delta \Big) \leq \varepsilon f o r a l l k \geq \frac {1 6 d}{\alpha \Delta_ {\Gamma} (4 + d \Delta_ {p})} \log \Big (\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta} \Big). +$$ + +Before giving the proof of the above theorem, we want to remark that Theorem A.13 exactly recovers the result of Theorem 2.2 in the independent case. Indeed, in the independent case $\Gamma$ is equal to the identity matrix and thus $\Delta_p = \Delta_\Gamma$ , $\nu = 0$ and $c_\star = \Delta_p^2 /4$ hold true, which gives the result of Theorem 2.2. + +Proof of Theorem A.13. As in the uncorrelated case, the recursive definition ensures that $\mathbf{p}(k)$ is $\mathcal{F}_{k - 1}$ -measurable. Thus, also $\Omega (k)\in \mathcal{F}_{k - 1}$ for any $k = 0,1,\ldots$ . Then $(M_{i}(k))_{k = 0,1,\ldots}$ , defined in Eq. (40), is a martingale for each $i\in [d]$ . This allows us to apply Doob's submartingale inequality. For this, we deduce the following bound on the second moment, + +$$ +\begin{array}{l} \mathbb {E} [ M _ {1} (k) ^ {2} ] \\ = \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left(\xi_ {1} (\ell) \mathbb {1} _ {\Omega (\ell)}\right) ^ {2} \right] \\ = \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ (p _ {1} (\ell)) ^ {2} \left(\mathbb {E} \left[ Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell) \mid \mathcal {F} _ {\ell - 1} \right] - \left(Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \mathbb {E} \left[ \left(\mathbb {E} \left[ Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell) \mid \mathcal {F} _ {\ell - 1} \right] - \left(Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right)\right) ^ {2} \mid \mathcal {F} _ {\ell - 1} \right] \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left(Y _ {1} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \end{array} +$$ + +$$ +\begin{array}{l} = \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left((1 - p _ {1} (\ell)) Y _ {1} (\ell) - \sum_ {j = 2} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq 2 Q ^ {2} \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left((1 - p _ {1} (\ell)) ^ {2} + \left(\sum_ {j = 2} ^ {d} p _ {j} (\ell)\right) ^ {2}\right) \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq 4 Q ^ {2} \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ (1 - p _ {1} (\ell)) \mathbb {1} _ {\Omega (\ell)} \right], \\ \end{array} +$$ + +where we argued similarly as for the independent case. Furthermore, we obtain for $u \in \{2, \ldots, d\}$ , + +$$ +\begin{array}{l} \mathbb {E} [ (M _ {u} (k)) ^ {2} ] = \alpha^ {2} \mathbb {E} \left[ \sum_ {\ell = 0} ^ {k} (\xi_ {u} (\ell) \mathbb {1} _ {\Omega (\ell)}) ^ {2} \right] \\ \leq \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ \left(p _ {u} (k)\right) ^ {2} \left(Y _ {u} (\ell) - \sum_ {j = 1} ^ {d} p _ {j} (\ell) Y _ {j} (\ell)\right) ^ {2} \mathbb {1} _ {\Omega (\ell)} \right] \\ \leq 4 Q ^ {2} \alpha^ {2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ p _ {u} (\ell) \mathbb {1} _ {\Omega (\ell)} \right]. \\ \end{array} +$$ + +Hence, applying a union bound, Doob's submartingale inequality Eq. (27) with $p = 2$ gives for any $k = 0,1,\ldots$ + +$$ +\begin{array}{l} \mathbb {P} (E (k)) = 1 - \mathbb {P} \left(\bigcup_ {j = 1} ^ {d} \max _ {u \in [ k ]} | M _ {j} (u) | \geq \frac {1}{4} \left(\Delta_ {p} \wedge \frac {\Delta_ {\Gamma}}{\| \Gamma \| _ {\infty}}\right)\right) \\ \geq 1 - \sum_ {j = 1} ^ {d} \mathbb {P} \left(\max _ {u \in [ k ]} | M _ {j} (u) | \geq \frac {1}{4} \left(\Delta_ {p} \wedge \frac {\Delta_ {\Gamma}}{\| \boldsymbol {\Gamma} \| _ {\infty}}\right)\right) \\ \geq 1 - 64 Q ^ {2} \alpha^ {2} \left(\Delta_ {p} \wedge \frac {\Delta_ {\mathbf {r}}}{\| \mathbf {r} \| _ {\infty}}\right) ^ {- 2} \left(\sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ (1 - p _ {1} (\ell)) \mathbb {1} _ {\Omega (\ell)} \right] + \sum_ {j = 2} ^ {d} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ p _ {j} (\ell) \mathbb {1} _ {\Omega (\ell)} \right]\right) \\ = 1 - 1 2 8 Q ^ {2} \alpha^ {2} \left(\Delta_ {p} \wedge \frac {\Delta_ {\mathbf {r}}}{\| \mathbf {r} \| _ {\infty}}\right) ^ {- 2} \sum_ {\ell = 0} ^ {k} \mathbb {E} \left[ (1 - p _ {1} (\ell)) \mathbb {1} _ {\Omega (\ell)} \right]. \\ \end{array} +$$ + +Proposition A.11 gives for any $k = 0,1,\ldots$ the bound + +$$ +\begin{array}{l} \mathbb {E} \left[ (1 - p _ {1} (k + 1)) \mathbb {1} _ {\Omega (k + 1)} \right] \\ \leq \mathbb {E} \left[ \left(1 - p _ {1} (k + 1)\right) \mathbb {1} _ {\Omega (k)} \right] \\ \leq \mathbb {E} \left[ \left(1 - \alpha \frac {\Delta \mathbf {r}}{4 d} \left(1 + \frac {\Delta_ {p}}{2} (d - 1)\right)\right) \left(1 - p _ {1} (k)\right) \mathbb {1} _ {\Omega (k)} + \alpha \xi_ {1} (k) \mathbb {1} _ {\Omega (k)} \right] \\ = \left(1 - \alpha \frac {\Delta_ {\Gamma}}{4 d} \left(1 + \frac {\Delta_ {p}}{2} (d - 1)\right)\right) \mathbb {E} \left[ \left(1 - p _ {1} (k)\right) \mathbb {1} _ {\Omega (k)} \right], \\ \end{array} +$$ + +which implies + +$$ +\mathbb {E} \left[ (1 - p _ {1} (k)) \mathbb {1} _ {\Omega (k)} \right] \leq (1 - p _ {1} (0)) \left(1 - \alpha \frac {\Delta_ {\mathbf {r}}}{4 d} \left(1 + \frac {\Delta_ {p}}{2} (d - 1)\right)\right) ^ {k}. \tag {43} +$$ + +We set + +$$ +\Theta := \bigcap_ {k = 0} ^ {\infty} \Omega (k). +$$ + +The continuity of probability measures and Lemma A.12 then imply + +$$ +\mathbb {P} (\Theta) = \lim _ {k \rightarrow \infty} \mathbb {P} (\Omega (k)) +$$ + +$$ +\begin{array}{l} \geq \lim _ {k \rightarrow \infty} \mathbb {P} (E (k)) \\ \geq 1 - 1 2 8 Q ^ {2} \alpha^ {2} \left(\Delta_ {p} \wedge \frac {\Delta_ {\mathbf {r}}}{\| \boldsymbol {\Gamma} \| _ {\infty}}\right) ^ {- 2} \sum_ {\ell = 0} ^ {\infty} \mathbb {E} \left[ (1 - p _ {1} (\ell)) \mathbb {1} _ {\Omega (\ell)} \right] \\ \geq 1 - 1 2 8 Q ^ {2} \alpha^ {2} (1 - p _ {1} (0)) \left(\Delta_ {p} \wedge \frac {\Delta_ {\mathbf {r}}}{\| \mathbf {r} \| _ {\infty}}\right) ^ {- 2} \sum_ {\ell = 0} ^ {\infty} \left(1 - \alpha \frac {\Delta_ {\mathbf {r}}}{4 d} \left(1 + \frac {\Delta_ {p}}{2} (d - 1)\right)\right) ^ {\ell} \\ = 1 - 1 0 2 4 Q ^ {2} \alpha (1 - p _ {1} (0)) \left(\Delta_ {p} \wedge \frac {\Delta_ {\mathbf {r}}}{\| \mathbf {r} \| _ {\infty}}\right) ^ {- 2} \frac {d}{\Delta_ {\mathbf {r}} \left(2 + \Delta_ {p} (d - 1)\right)} \\ = 1 - 2 0 4 8 Q ^ {2} \alpha \left(1 - p _ {1} (0)\right) \left(\Delta_ {p} \wedge \frac {\Delta_ {\mathbf {r}}}{\| \mathbf {r} \| _ {\infty}}\right) ^ {- 2} \frac {1}{\Delta_ {\mathbf {r}} \left(4 / d + \Delta_ {p}\right)} \\ \geq 1 - \varepsilon / 2, \\ \end{array} +$$ + +where we used that we can assume $d \geq 2$ without loss of generality. Additionally, Eq. (43) and the elementary inequality $1 - x \leq \exp(-x)$ , which is valid for any real number $x$ , give + +$$ +\begin{array}{l} \mathbb {E} \left[ (1 - p _ {1} (k)) \mathbb {1} _ {\Theta} \right] \leq \mathbb {E} \left[ (1 - p _ {1} (k)) \mathbb {1} _ {\Omega (k)} \right] \\ \leq \left(1 - p _ {1} (0)\right) \left(1 - \alpha \frac {\Delta \Gamma}{4 d} \left(1 + \frac {\Delta_ {p}}{2} (d - 1)\right)\right) ^ {k} \\ \leq \left(1 - p _ {1} (0)\right) \exp \Big (- \alpha \frac {\Delta_ {\Gamma}}{4 d} \Big (1 + \frac {\Delta_ {p}}{2} (d - 1) \Big) k \Big). \\ \end{array} +$$ + +When $d = 1$ , the right hand side of this inequality is 0. For $d \geq 2$ , we can also use the bound $d - 1 \geq d / 2$ . Together with + +$$ +\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} = 1 - p _ {1} (k) + \sum_ {i = 2} ^ {d} p _ {i} (k) = 2 \big (1 - p _ {1} (k) \big), +$$ + +this concludes the proof of the first statement. For the proof of the second statement, we apply Markov's inequality to obtain + +$$ +\begin{array}{l} \mathbb {P} \left(\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \geq \delta\right) \\ \leq \mathbb {P} ((\Theta) ^ {\mathbf {C}}) + \mathbb {P} \left(\| \mathbf {p} (k) - \mathbf {e} _ {1} \| _ {1} \mathbb {1} _ {\Theta} \geq \delta\right) \\ \leq \frac {\varepsilon}{2} + 2 (1 - p _ {1} (0)) \exp \left(- \alpha \frac {\Delta_ {\mathbf {r}}}{1 6} \left(\frac {4}{d} + \Delta_ {p}\right) k\right) \delta^ {- 1}. \\ \end{array} +$$ + +Hence, if + +$$ +\begin{array}{l} k \geq \left(\frac {\alpha \Delta_ {\Gamma}}{1 6} \left(\frac {4}{d} + \Delta_ {p}\right)\right) ^ {- 1} \log \left(\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta}\right) \\ = \frac {1 6 d}{\alpha \Delta_ {\mathbf {r}} (4 + d \Delta_ {p})} \log \left(\frac {4 (1 - p _ {1} (0))}{\varepsilon \delta}\right), \\ \end{array} +$$ + +then, + +$$ +\mathbb {P} \left(\left\| \mathbf {p} (k) - \mathbf {e} _ {1} \right\| _ {1} \geq 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Scene Rendering with 4D Gaussian Splatting via a Spike Camera Array/full.md new file mode 100644 index 0000000000000000000000000000000000000000..758a0f5222ec5eb765e6b8419d4fc9de0d31f1fd --- /dev/null +++ b/NeurIPS/2025/Spike4DGS_ Towards High-Speed Dynamic Scene Rendering with 4D Gaussian Splatting via a Spike Camera Array/full.md @@ -0,0 +1,659 @@ +# Spike4DGS: Towards High-Speed Dynamic Scene Reconstruction with 4D Gaussian Splatting via a Spike Camera Array + +Qinghong Ye $^{1,3,*}$ , Yiqian Chang $^{2,3,*}$ , Jianing Li $^{4}$ , Haoran Xu $^{3,5}$ , Xuan Wang $^{2}$ , Wei Zhang $^{3,\dagger}$ , Yonghong Tian $^{1,3,4,\dagger}$ , Peixi Peng $^{1,3,\dagger}$ + +1Shenzhen Graduate School of Peking University + +2 Shenzhen Campus of Harbin Institute of Technology + +3Peng Cheng Laboratory + +$^{4}$ School of Computer Science, Peking University + +5 Shenzhen Campus of Sun Yat-sen University + +# Abstract + +Spike camera with high temporal resolution offers a new perspective on high-speed dynamic scene rendering. Most existing rendering methods rely on Neural Radiance Fields (NeRF) or 3D Gaussian Splatting (3DGS) for static scenes using a monocular spike camera. However, these methods struggle with dynamic motion, while a single camera suffers from limited spatial coverage, making it challenging to reconstruct fine details in high-speed scenes. To address these problems, we propose Spike4DGS, the first high-speed dynamic scene rendering framework with 4D Gaussian Splatting using spike camera arrays. Technically, we first build a multi-view spike camera array to validate our solution, then establish both synthetic and real-world multi-view spike-based reconstruction datasets. Then, we design a multi-view spike-based dense initialization module that obtains dense point clouds and camera poses from continuous spike streams. Finally, we propose a spike-pixel synergy constraint supervision to optimize Spike4DGS, incorporating both rendered image quality loss and dynamic spatiotemporal spike loss. The results show that our Spike4DGS outperforms state-of-the-art methods in terms of novel view rendering quality on both synthetic and real-world datasets. More details are available at the project page. + +# 1 Introduction + +Novel view synthesis [16] is a cornerstone of many cutting-edge applications, enabling the creation of precise novel views from ideal image sequences. However, conventional cameras struggle in high-speed motion scenarios, where rapid movement causes motion blur and significantly degrades reconstruction quality [39]. While some approaches [30, 44] attempt to improve reconstruction from motion-blurred images, they remain fundamentally limited by the sampling rates of RGB cameras. As a result, the use of new vision sensors for high-quality rendering in high-speed scenes has garnered increasing attention. + +Spike cameras [9, 20, 48] with high temporal resolution offer a new perspective on high-speed scene rendering. Unlike conventional cameras, they asynchronously encode absolute light intensity into continuous spike streams at rates of up to $20\mathrm{kHz}$ . This unique property makes them particularly + +effective for preserving high-speed scene textures with greater detail. Existing studies [64, 65, 24, 67, 66, 62, 60, 58, 57, 46, 59, 47] have demonstrated their capability for fine-grained 2D reconstruction. The advantages of spike cameras also indicate their immense potential for advancing 3D scene reconstruction and novel view synthesis. + +Extensive works [15, 61] have explored neuromorphic cameras for rendering in high-speed scenarios. Some studies [19, 38, 21, 28, 3, 31, 52, 36] utilize Radiance Fields (NeRF) and 3D Gaussian Splatting (3DGS) for scene representation and novel view synthesis using event cameras. However, event cameras capture only light changes rather than absolute brightness, making them challenging for fine-grained 3D reconstruction. Other efforts [69, 27, 18, 51, 54] have explored 3DGS and NeRF with alternative spike cameras to overcome these limitations and enhance rendering quality. For example, pioneering works with SpikeNeRF [69] and Spike-NeRF [18] have demonstrated the feasibility of using spike streams to reconstruct 3D scenes. Nevertheless, NeRF-based methods suffer from time-consuming training and inference processes, and their implicit representations limit scene editing capabilities. In contrast, Gaussian Splatting offers a compelling alternative, providing high accuracy and fast inference speeds. For instance, SpikeGS [51, 54], SpikeNVS [7], and USP-Gaussian [5] achieve impressive results in photorealistic 3D reconstruction using spike cameras. However, these 3DGS-based methods may face difficulties in handling high-speed dynamic scenes. Additionally, relying solely on monocular cameras may present 3D reconstruction challenges in areas with weak textures or when using static cameras. In fact, camera arrays could enhance texture information through multi-view perspectives, enabling dynamic scene rendering even without camera motion. + +To address the aforementioned challenges, we propose Spike4DGS, the first high-speed dynamic scene rendering framework utilizing 4D Gaussian Splatting with spike camera arrays. We aim at overcoming the following challenges: (i) Camera setup and dataset - How could we establish a multi-view spike camera array and build high-quality spike-based reconstruction datasets? (ii) Effective model - How could we design an efficient dynamic scene rendering model that directly processes multi-view spike streams with 4D Gaussian splatting(4DGS)? + +To be specific, we first build a multi-view spike camera array and then establish both synthetic and real-world spike-based reconstruction datasets. Then, we design a novel multi-view spike-based dense initialization module to generate dense point clouds and camera poses from continuous spike streams. Finally, we propose a pixel-spike synergy supervision strategy to optimize Spike4DGS, which incorporates both reconstructed image quality loss and dynamic spatiotemporal spike loss. Experimental results show that our Spike4DGS outperforms state-of-the-art methods in terms of rendering quality on both synthetic and real-world datasets. We further verify that spike cameras achieve higher rendering quality than event cameras and RGB cameras in high-speed scenes. Meanwhile, rendering quality improves as the number of spike cameras increases. + +The main contributions of this work are summarized as: + +- We introduce Spike4DGS, the first framework that combines spike camera arrays with 4DGS, enabling novel view synthesis in high-speed dynamic scenarios. +- We present a Spike-Pixel Synergy Supervision strategy to optimize the parameters of our Spike4DGS for enhanced rendering quality. +- We build a spike camera array, along with a highly realistic synthetic and real-world datasets that contain multi-view spike streams. We believe two standardized datasets open up opportunities for research in this novel problem. + +# 2 Related Work + +# 2.1 NVS for Dynamic Scenes + +Novel View Synthesis (NVS) tasks aim to generate unknown views of an object or scene from a set of images of known views. Representative papers include Neural Radiance Fields (NeRF) [32] and 3D Gaussian Splatting (3DGS) [23]. Recently, a large number of static 3DGS-based techniques [4, 22, 53] have been proposed due to their high quality and real-time rendering without using neural networks like NeRF. However, the assumption of static scenes prevents application to real-world scenarios with moving objects. Therefore, several works extend the 3DGS to dynamic scenes[45, 50, 29, 26, 2]. These methods are usually divided into two main lines. For instance, De3DGS [50] and 4DGS [45] + +model spatial-temporal deformation with an implicit deformation field as the first line. On the other hand, the second line is based on the idea that scenes' motion could be encoded into the 3D Gaussian representation straightly, such as STG [26] and D3DGS [29], which represents changes in 3D Gaussian over time through a temporal opacity and a polynomial function for each Gaussian. The above approaches could perform well on synthetic datasets and simple real-world datasets. However, when there are some high-speed objects in the scene, traditional dynamic reconstruction methods using RGB image data may suffer motion blur since most standard RGB cameras have limited frame rates. This motivates us to use the neuromorphic cameras to avoid this problem. + +# 2.2 Neuromorphic Cameras on 3DGS + +There are two types of bio-inspired sensors: event cameras [15] and spike cameras [9]. Event cameras are based on the temporal contrast sampling method and generate events asynchronously when pixel brightness changes exceed a threshold. Early event-based reconstruction approaches [28, 3, 31, 38, 21, 42] have been proposed to derive neural radiance fields directly from event streams. E2nerf [36] and evagaussian [52] achieved sharp reconstruction from blurry images. E-4DGS [13] achieved high-fidelity dynamic reconstruction from the multi-view event cameras. GS2E [25] has introduced an effective event stream generator by gaussian splatting. Another type of neuromorphic camera is called the spike camera. Spike cameras record the absolute light intensity at a fairly high frame rate and provide a more explicit input format for detailed reconstruction. Some nef-based methods [69, 27, 18] have verified the feasibility of reconstruction with spike, but they suffer suboptimal training and rendering speeds due to the complex spike simulation network. Some 3DGS-based spike reconstruction methods have emerged to optimize this defect. Yu's SpikeGS[51] reconstructed view synthesis results from a continuous spike stream captured by a moving spike camera. In a harder setting, Zhang's SpikeGS[54] reconstructed scenes via a single spike stream with monocular high-speed camera motion. However, there is no established spike-based method for addressing the challenge of rendering high-speed dynamic scenes using multi-view spike streams. On this basis, our work aims to overcome the limitations and construct a spike-based 3D Gaussian Splatting model for high-speed dynamic scenes via a spike camera array. + +# 3 Methodology + +# 3.1 Preliminaries + +Spike Camera. Spike camera is a bio-inspired sensor which records and converts the absolute light intensity at a fairly high frame rate (up to $20\mathrm{kHz}$ ) into accumulated voltage through photoreceptors [64, 8]. If the accumulated voltage $V$ reaches the scheduling threshold $\Theta$ , a spike will be triggered and $V$ is reset to zero, mathematically formulated as follows: + +$$ +V (t) = \int_ {t _ {s}} ^ {t} \sigma \cdot L (t) d t \bmod \Theta , \tag {1} +$$ + +where $L(t)$ represents the instant light intensity at time $t$ , $t_s$ is the moment when the previous spike was emitted, and $\sigma$ is the constant photoelectric conversion coefficient. + +DUSt3R Initialization. DUSt3R [41] is a dense initialization method used for 3D reconstruction. Compared with COLMAP Initialization [40], DUSt3R provides more accurate point clouds under low-quality image input with less time, which is more suitable for high-speed scenes. Specifically, given a pair of images $(I_1, I_2)$ , DUSt3R utilizes a ViT for the encoder and decoder [11] and a DPT head [37] for estimating point clouds $\mathcal{P}\mathcal{C}$ : + +$$ +\mathcal {P C} = \mathrm {D P T} (\mathrm {V i T} (I _ {1}, I _ {2})). \tag {2} +$$ + +Although dUST3r and its improved versions [49, 55] could generate dense point clouds based on multi-view images, however, the quality of point clouds depends on the quality of input images. + +4D Gaussian Splatting. 4D Gaussian Splatting (4DGS) [45] is used for rendering dynamic scenes. It proposes a network that learns the Gaussian deformation field to predict the deformation of each 3D Gaussian. For input 3D Gaussian $\mathcal{G}$ and time $t$ , a spatial-temporal structure encoder $\mathcal{H}$ and a multi-head Gaussian deformation decoder $\mathcal{D}$ are used for calculating the deformations $\Delta \mathcal{G}$ : + +$$ +\Delta \mathcal {G} = \mathcal {D} (\mathcal {H} (\mathcal {G}, t)). \tag {3} +$$ + +![](images/86c73405eab82aaea2c6557e7fbad1e54def624601b1e7e0a9c13a32dcbd4920.jpg) +Figure 1: The Framework of our Spike4DGS, we establish the connection between the real-world spike streams and the dynamic scene rendering images. The input multi-view spike streams are sent to a Multi-view Spike-based Dense Initialization to estimate point cloud and camera poses. Based on the initialization, a 4DGS with Spike-Pixel Synergy Supervision consisting of a Pixel Loss and a Spike Variation Loss is utilized for rendering. Our Spike4DGS could reconstruct high-speed dynamic scenes with delicate motion and texture details. + +# 3.2 Problem Formulation + +We aim to reconstruct high-speed dynamic scenes with delicate motion and texture details. To reach this goal, we build a multi-view spike camera array to capture the high-speed motion and propose a novel method called Spike4DGS for rendering based on continuous spike streams. The problem could be denoted as: + +$$ +\text {S p i k e 4 D G S}: \left\{\left(S _ {1}, S _ {2}, \dots , S _ {N - 1}\right), V _ {N}, t \right\}\rightarrow \hat {I} _ {t} \tag {4} +$$ + +where $S_{1}, S_{2}, \ldots, S_{N-1}$ are the spike streams captured from N-1 views of spike cameras for training, $\hat{I}_{t}$ is the rendering image of novel view $V_{N}$ (the view of $N$ -th spike camera) at time $t$ . + +To solve this problem, Spike4DGS first builds an end-to-end Multi-view Spike-based Dense Initialization (MSDI) method to estimate the dense point cloud and camera poses from input spike streams, as detailed in Sec. 3.3. Then, the initial point cloud and camera poses are sent to a 4D Gaussian Splatting to generate novel view rendering results. To get more delicate motion and texture details, Spike-Pixel Synergy Supervision (SPSS) is proposed for the constraint of 4D Gaussian Splatting in Sec. 3.4. The total framework of Spike4DGS is shown in Fig. 1. + +# 3.3 Multi-view Spike-based Dense Initialization + +Previous rendering tasks based on spike cameras, such as SpikeNeRF [69], employ a two-step initialization method. They first use spike-to-image methods such as TFI [64], TFP [64] and spk2img [6] to get images, and then utilize COLMAP [40] or DUSt3R [41] on these images for the scene initialization. However, this two-step method is complex and requires high-quality spike-to-image results. When dealing with high-speed dynamic scenes, the lack of texture details in converted images may influence the final point cloud estimation. In contrast, we propose an end-to-end framework, called Multi-view Spike-based Dense Initialization (MSDI), which consists of a spike feature extractor, a point cloud and pose estimator, and an image generator. Given a series of spike streams from $N$ spike cameras, MSDI aims to estimate 3D point clouds, camera poses, and their corresponding images: + +$$ +\mathbf {M S D I}: (S _ {1}, \dots , S _ {N}) \rightarrow \{\mathcal {P C}, ([ R | T ] _ {1}, \dots , [ R | T ] _ {N}), (I _ {1}, \dots , I _ {N}) \}. \tag {5} +$$ + +where $\mathcal{PC}$ is the estimated point cloud, $[R|T]_1,\dots,[R|T]_N$ are the camera rotation and translation parameters of $N$ spike cameras, $I_{1},\ldots,I_{N}$ are the images generated from input spike streams. As + +an end-to-end network, MSDI is fine-tuned together with pre-trained weights on multi-view spike streams captured from the Carla [10] simulator. + +Spike Feature Extractor. Firstly, we build a feature extractor for the multi-view spike streams. Assuming $\Gamma_{t}$ is a time interval around frame time $t$ . For an input spike stream $S_{i}(\Gamma)$ which lasts for a time interval $\Gamma_{t}$ and is captured from the $i$ -th camera view, MSDI rotates $S_{\Gamma_t}$ four times to obtain a complete receptive field in four directions and concentrate them together: + +$$ +R \left(S _ {i} \left(\Gamma_ {t}\right)\right) = \operatorname {C A T} \left\{\operatorname {R o t} \left(S _ {i} \left(\Gamma_ {t}\right), \theta\right) \right\} \mid \theta \in \left\{0 ^ {\circ}, 9 0 ^ {\circ}, 1 8 0 ^ {\circ}, 2 7 0 ^ {\circ} \right\}, \tag {6} +$$ + +where CAT and Rot mean concatenation and rotation operations respectively. Then, we utilize a shift-based $3 \times 3$ convolution layer from [54] to extract features: + +$$ +\hat {f} _ {i} \left(\Gamma_ {t}\right) = \mathcal {M} \left(R \left(S _ {i} \left(\Gamma_ {t}\right)\right)\right), \tag {7} +$$ + +where $\mathcal{M}$ is the shift-based convolution layer, $\hat{f}_{\Gamma_t}$ is the extracted spike features for input spike streams $S_{i}(\Gamma_{t})$ . Replacing the input $S_{i}(\Gamma_{t})$ to $N$ views of spike streams, we can get a series of extracted features $(\hat{f}_1(\Gamma_t),\dots,\hat{f}_N(\Gamma_t))$ . + +Point Cloud and Pose Estimator. To estimate the point cloud and camera poses, the problem could be formulated as: + +$$ +\operatorname {E s t i m a t o r}: \left(\hat {f} _ {1} \left(\Gamma_ {t}\right), \dots , \hat {f} _ {N} \left(\Gamma_ {t}\right)\right)\rightarrow \mathcal {P C}, \left([ R | T ] _ {1}, \dots , [ R | T ] _ {N}\right). \tag {8} +$$ + +To solve the problem, MSDI builds an estimator consisting of $N$ ViT encoders [11] (equals to the number of views) and a fusion transformer. For $N$ ViT encoders, each encoder $\mathrm{ViT}_i$ handles a camera view with shared weights: + +$$ +\hat {F} _ {i} \left(\Gamma_ {t}\right) = \operatorname {V i T} _ {i} \left(\hat {f} _ {i} \left(\Gamma_ {t}\right)\right), i \in (1, \dots , N). \tag {9} +$$ + +For the fusion transformer FusionTF, it is a 24-layer transformer which is the same as [49]: + +$$ +G _ {1} (\Gamma_ {t}), G _ {2} (\Gamma_ {t}), \dots , G _ {N} (\Gamma_ {t}) = \text {F u s i o n T F} \left(\hat {F} _ {1} (\Gamma_ {t}), \hat {F} _ {2} (\Gamma_ {t}), \dots , \hat {F} _ {N} (\Gamma_ {t})\right), \tag {10} +$$ + +where $G_{i}(\Gamma_{t})$ is the temporal feature of $i$ -th camera view. This operation generates temporal features with global contextual understanding from all views. + +Then, a DPT [37] decoding head is utilized for decoding the temporal features into a point cloud $\mathcal{PC}$ and its corresponding confidence map $\Sigma_{\mathcal{PC}}$ : + +$$ +\left(\mathcal {P C}, \Sigma_ {\mathcal {P C}}\right) = \mathrm {D P T} \left(G _ {1} \left(\Gamma_ {t}\right), G _ {2} \left(\Gamma_ {t}\right), \dots , G _ {N} \left(\Gamma_ {t}\right)\right). \tag {11} +$$ + +Finally, we utilize the global point cloud $\mathcal{PC}$ to estimate camera rotations and translations $([R|T]_{1},\dots,[R|T]_{N})$ via RANSAC-PnP [14, 1]. + +Image Generator. In addition, MSDI also generates discrete images from continuous spike streams. For the time interval $\Gamma_{t}$ around frame time $t$ , the image at time $t$ in $i$ -th camera view could be generated from: + +$$ +\bar {I} _ {i} (t) = \operatorname {B C O N V} \left(\hat {f} _ {i} \left(\Gamma_ {t}\right)\right), i \in (1, \dots , N), \tag {12} +$$ + +where $\bar{I}_i(t)$ is the generated image, BCONV is a network consisting of three $1\times 1$ convolutions followed from BSN [6]. + +Compared with the previous two-step initialization methods like TFI [64] + COLMAP [40], our end-to-end MSDI method could avoid the errors of the final estimations which occur from the low-quality images generated in the intermediate steps. + +# 3.4 4DGS with Spike-Pixel Synergy Supervision + +After MSDI, we could get the initial point cloud and camera poses. Then the initial 3D Gaussian $\mathcal{G}$ could be obtained from them. Thus, we could generate the deformed Gaussians from 4DGS [45] and render an image $\hat{I}(t)$ at $i$ -th view $p_i$ : + +$$ +\hat {I} _ {i} (t) = \operatorname {R e n d e r} \left(4 \mathrm {D G S} (\mathcal {G}, t), p _ {i}\right). \tag {13} +$$ + +According to 4DGS, the rendered image loss could be formulated as follows to offer our Spike4DGS pixel supervision: + +$$ +\mathcal {L} _ {t} ^ {\text {p i x e l}} = \left| \left| \hat {I} _ {i} (t) - \bar {I} _ {i} (t) \right| \right| _ {1}, \tag {14} +$$ + +where $\hat{I}_i(t)$ is the rendered image at the time $t$ , $\bar{I}_i(t)$ is the generated images from the above MSDI. However, this pixel loss concentrates only on image similarity but ignores the texture and motion details, which are contained in spike streams. To take advantage of the spike characteristics, we propose a spike variation loss which first translates the rendered images into spike streams and then compares the variation between generated and real spike streams. + +Translate from Rendered Image to Spike Stream. Let us denote the intensity values of the pixel $(x,y)$ in rendered images at time $t$ as $\hat{I}_i(x,y,t)$ . After getting the real light of the scene, we convert the scene light intensity into spike streams using an Integrate-and-Fire (IF) [17, 12] mechanism. Following intensity translation method [68], we could establish the following relationship: + +$$ +\hat {S} _ {i} (x, y, t) = \operatorname {I F} \left(\hat {I} _ {i} (x, y, t) \cdot n (x, y)\right), \tag {15} +$$ + +where $n(x,y)$ is the deviation matrix corresponding to the response nonuniformity noise which could be obtained by capturing a uniform light scene and recording the intensity. IF(·) is a IF neuron, and $\hat{S}_i(x,y,t)$ is the predicted spike values of the pixel $(x,y)$ at time $t$ . + +Spike Variation Supervision. Different from static scenes, objects in high-speed moving scenes are constantly changing, thus the segment of spikes $S_{i}(\Gamma_{t})$ in a time interval $\Gamma_{t}$ around $t$ is a continuously changing sequence. For high-speed moving objects, naive L1 or L2 loss treats each frame separately, without considering how these objects move over time. In contrast, we propose spike variation supervision, in order to concentrate on the dynamic changes of objects. Since high-speed motions are continuous, spike streams in a short time interval around time $t$ could be generated from a single spike at time $t$ . Thus, we simply design a one-layer MLP network $\phi_{s}$ to map a single spike $\hat{S}_{i}(t)$ to a spike sequence $\hat{S}_{i}(\Gamma_{t})$ in a time interval $\Gamma_{t}$ around $t$ : + +$$ +\hat {S} _ {i} \left(\Gamma_ {t}\right) = \phi_ {s} \left(\hat {S} _ {i} (t)\right), \tag {16} +$$ + +where the shape of $\hat{S}_i(t)$ is $(H,W,1)$ and the shape of $\hat{S}_i(\Gamma_t)$ is $(H,W,\Gamma_t)$ . $H$ and $W$ are the height and width of the spikes. Then, we calculate the average value and the standard deviation of this spike sequence $\hat{S}_i(\Gamma_t)$ and compare them with ground truth. Therefore, the Spike Variation Loss could be presented as: + +$$ +\mathcal {L} _ {t} ^ {\text {s p i k e V}} = \left| \left| \operatorname {a v g} \left(\hat {\mathrm {S}} _ {\mathrm {i}} \left(\Gamma_ {\mathrm {t}}\right)\right) - \operatorname {a v g} \left(\mathrm {S} _ {\mathrm {i}} ^ {\mathrm {g t}} \left(\Gamma_ {\mathrm {t}}\right)\right) \right| \right| _ {1} + \left| \left| \operatorname {s t d} \left(\hat {\mathrm {S}} _ {\mathrm {i}} \left(\Gamma_ {\mathrm {t}}\right)\right) - \operatorname {s t d} \left(\mathrm {S} _ {\mathrm {i}} ^ {\mathrm {g t}} \left(\Gamma_ {\mathrm {t}}\right)\right) \right| \right| _ {1}. \tag {17} +$$ + +Combining this regularization with the original Pixel Loss, we get the final synergy training loss for our Spike4DGS: + +$$ +\mathcal {L} _ {t} ^ {\text {t o t a l}} = \mathcal {L} _ {t} ^ {\text {s p i k e V}} + \mathcal {L} _ {t} ^ {\text {p i x e l}}. \tag {18} +$$ + +# 4 Experiment + +# 4.1 Datasets + +To verify the validity of our proposed Spike4DGS, we create two datasets. The first dataset is a real-world object dataset collected by the aforementioned spike camera array. The second dataset is a high-speed synthetic outdoor dataset generated by the CARLA simulator [10]. All our experiments are conducted on these two self-made datasets. We achieve the best results on both two datasets, which demonstrates the superiority of our Spike4DGS. + +Real-world Object Dataset. As shown in Fig. 2, the spike array consists of 9 spike cameras that are capable of capturing spike streams with a spatial resolution of $250 \times 400$ and a temporal resolution of $20\mathrm{kHz}$ . During the data collection process, synchronized recordings were made from all 9 cameras, ensuring that motion was consistently represented across different + +![](images/f79d813326373ac65832cb227d80e07c3fb9acdbdd005cac55dc0b162274f04f.jpg) +Figure 2: Our spike camera array. + +![](images/4fba6dc45b95c9e63542924376698c7a443448038f034a38d0a7eafd7df8e412.jpg) + +Table 1: Quantitative evaluation on synthetic outdoor dataset. Unit: PSNR-dB ↑, SSIM ↑, LPIPS ↓. + +
MethodJaywalkBicycleMotorCarVan
PSNRSSIMLPIPSPSNRSSIMLPIPSPSNRSSIMLPIPSPSNRSSIMLPIPSPSNRSSIMLPIPS
TFI[64]+D3DGS[29]20.6575.70.38421.7377.70.38519.9876.20.37720.0976.20.38719.2475.30.402
TFP [67]+D3DGS[29]20.8876.50.35721.7878.90.34619.8576.00.37220.0776.60.36119.2076.40.389
Spk2img[57]+D3DGS[29]20.9476.60.30421.6877.50.37920.1276.50.38020.1576.30.38918.9375.90.392
TFI[64]+STG[26]24.4884.20.22424.5284.10.22124.4784.30.22723.5084.00.22326.5588.70.201
TFP [67]+STG[26]24.4584.00.22024.5084.20.22524.5384.10.22224.6084.30.22626.4889.00.205
Spk2img[57]+STG[26]24.7284.60.22124.7584.50.22024.7084.70.22323.7384.80.21925.9488.40.202
TFI[64]+4DGS[45]26.8887.70.21327.0289.70.20225.5290.50.19825.3886.60.21925.2186.10.214
TFP [67]+4DGS[45]27.9691.20.19227.1590.40.19626.5788.30.20625.0186.50.21325.2787.00.220
Spk2img[57]+4DGS[45]27.0490.90.20826.1788.60.21926.3588.90.21824.7587.80.22924.8787.90.227
Dy-SpikeGS[51]23.040.8520.24322.170.8460.24923.350.8570.23822.750.8490.24422.870.8510.242
Dy-SpikeNeRF[69]17.040.7840.37216.170.7730.38416.350.7760.37914.750.7590.39515.870.7680.388
Our Spike4DGS28.2993.10.18927.6991.30.18527.9291.60.19327.7491.20.19927.1390.10.197
+ +![](images/b03d2208d042a705f9415dd911cd3cb87449fd7073d961e74a5fefceaae5976e.jpg) +Figure 3: Quantitative comparison with other methods on the dataset on our synthetic outdoor dataset. We mainly compare our method with some SOTA approaches. In contrast, our method delivers both superior outlines and clear details. + +views. When capturing spike streams, we fix the position of our spike camera array and place the high-speed dynamic objects in front of the cameras. Then we record 9 spike streams of approximately 0.5 seconds for each real-world scene with high-speed objects. Firstly, we choose high-speed dynamic objects such as "the collapse of bricks" (Bricks) and "the spin of a turntable"(Turntable) and put them before our spike camera array. Secondly, we minimize noise by providing the spike camera with ideal light intensity and obtaining multiple ideal spike streams. These spike streams could be converted to images by our MSDI module in the training process, and we use them as image supervision. + +Synthetic Outdoor Dataset. To quantitatively analyze our superiority, we create a synthetic outdoor dataset using the Carla [10] simulator, which includes scenarios like Jaywalk, Bicycle, Van, Car, and Motor. These scenarios feature objects of varying sizes and speeds, with objects appearing from the left side of the image and moving to the right. An array of 9 camera sets with an overhead view was set up to capture different views of the objects. Each view setting consists of a spike camera at 20k FPS for training and an RGB camera at 1k FPS for evaluation. + +# 4.2 Experimental Setup + +Competitors. Due to the relative lack of methods for dynamic novel view synthesis based on spike cameras, we completed the comparison using some two-stage rendering approaches. First, we choose some direct spike-to-image approaches: TFI[64], TFP[64], and spk2img[57], and then combine them with previous multiview dynamic NVS methods[45, 26, 29]. For a more comprehensive experiment, we also manually integrated a deformation network from 4DGS into SpikeGS and SpikeNeRF (denoted as Dy-SpikeGS and Dy-SpikeNeRF) as comparison. Those comparisons are initialized by + +Table 2: Quantitative evaluation on real-world object dataset. Unit: Brisque $\downarrow$ ,NIQE $\downarrow$ ,MetaIQA $\uparrow$ . + +
MethodBricksBrisqueChipsTurntableBird
BrisqueNIQEMetaIQABrisqueNIQEMetaIQABrisqueNIQEMetaIQABrisqueNIQEMetaIQA
TFI[64]+D3DGS[29]57.3516.710.11461.3216.430.12456.5315.340.11757.8714.870.112
TFP [67]+D3DGS[29]56.8715.370.12760.1515.010.13155.5714.930.12356.3714.530.126
Spk2img[57]+D3DGS[29]58.7815.980.13359.2715.040.13556.9415.870.12957.1414.650.137
TFI[64]+STG [26]45.3313.880.14343.7713.930.14936.4510.530.15033.3212.640.143
TFP [67]+STG [26]44.8313.210.14844.4611.080.15037.9410.870.16134.099.980.155
Spk2img[57]+STG [26]45.6113.630.14744.6712.080.15238.0110.920.15734.029.270.149
TFI[64]+4DGS [45]39.5610.230.16545.1510.010.16237.579.930.17033.379.530.163
TFP [67]+4DGS [45]40.5810.420.16444.4611.080.15037.9410.870.16134.099.980.155
Spk2img[57]+4DGS [45]40.5310.550.16744.6712.080.15238.0110.920.15734.029.270.149
Dy-SpikeGS[51]48.7314.230.13649.1513.840.13947.9213.570.13346.8813.220.137
Dy-SpikeNeRF[69]61.8417.250.11662.1716.930.11360.5816.710.11859.7416.350.115
Our Spike4DGS34.299.950.17633.528.030.18326.869.860.17923.868.360.180
+ +![](images/8562daf54967dafb71f8315a086bf469ff74292018875fb9b11599d47c3c49f5.jpg) +Figure 4: Qualitative comparison on real-world object datasets with SOTA rendering methods. We present the RGB photos of real scenes in the last column due to the lack of ground truth. + +DUSt3R [41] to obtain point clouds. In our framework, the training data are multiple spike streams for both synthetic and real datasets. + +Implementation Details. Firstly, we use the MSDI in Sec. 3.3 to estimate initialize point clouds, camera poses, and generate images. We fine-tune this end-to-end framework together with pre-trained weights using multi-view spike streams captured from dynamic scenes in the Carla simulator. The fine-tuning adopts L1 loss supervised by Carla images and confidence-aware pointmap regression loss in DUSt3R [41]. We utilize Adam optimizer with a learning rate of 0.0001 to effectively minimize both losses and ensure stable convergence during training. Secondly, for 4DGS with Spike-Pixel Synergy supervision in Sec. 3.4, the learning rate is the same as 4DGS [45]. At each optimization iteration, we randomly sample a batch of views from the same time $t$ . The total experiments are conducted on a single NVIDIA GTX 4090 with PyTorch and the optimization for a single scene typically takes about 20 minutes to converge and 40 FPS when rendering. For the metrics of the synthetic outdoor dataset, we employ three widely-used image quality assessment metrics, PSNR [45], SSIM [43] and LPIPS [56]. For the real dataset, which lacks corresponding ground truth images, we employ NIQE [35], BRISQUE [33, 34] and MetaQA [63] as no-reference image quality evaluation metrics, the same as state-of-the-art works [69, 51]. + +# 4.3 Performance Comparison + +# 4.3.1 Synthetic Outdoor Data Experiments + +Quantitative Performance. We present a detailed comparison of our method against some two-stage rendering approaches, such as TFI [64] + STG [26], TFI[64] + D3DGS[29], TFI[64] + 4DGS [45], on our synthetic outdoor dataset. As demonstrated in Table 1, our method outperforms the SOTA two-stage rendering methods across all five distinct scenes. Note that our Spike4DGS improves more on the higher-speed scenes like "Car", which proves the effect in high-speed scenarios. + +Qualitative Performance. We also present the qualitative results in Fig. 3, where: (I) TFI [64]+D3DGS [29] almost failed to reconstruct the outlines and details of each scene. (II) TFI [64]+4DGS [45] could only reconstruct the outlines with texture details missing. (iii) In contrast, our method demonstrates superior performance, generating clearer and more detailed novel views, especially in challenging high-speed scenes like "Van". + +# 4.3.2 Real-world Object Data Experiments + +Quantitative Performance. As shown in Table 2, our method achieves superior performance compared to those SOTA rendering approaches. Our method achieves an average improvement in BRISQUE [33, 34], NIQE [35] and MetaIQA [63] by $15.4\%$ , $4.6\%$ and $6.7\%$ respectively, indicating higher-quality novel view rendering. + +Qualitative Performance. In qualitative experiments, we focus on the generated texture details. Fig. 4 presents that our method could generate high-quality texture details. For example, in the third line of Fig. 4, our rendering could generate clear alphabets "A, B, C..." on the Turntable dataset while others could not. + +# 4.3.3 Analysis + +Performance Analysis. The quantitative and qualitative results above show that our Spike4DGS could significantly surpass the SOTA approaches. Moreover, on the synthetic outdoor dataset, our method proves the ability to render the outdoor high-speed scenes. While on the real-world object dataset, our method could also reconstruct indoor high-speed objects. This highlights the robustness and comprehensive improvements of our Spike4DGS on different high-speed dynamic scenes. + +# 4.4 Ablation Study + +Contribution of Each Component. We conduct an ablation study to assess the contribution of each component in our Spike4DGS. As shown in Table 3, we combine 4DGS with some initialization frameworks. The results demonstrate that our MSDI initialization achieves the best performance, and the SPSS module consistently improves reconstruction quality under different initialization frameworks. Results in the last row demonstrate that the full model (4DGS+MSDI+SPSS) achieves the best performance, validating the effectiveness of our design for spike-based 3D reconstruction. + +Ablation on Supervision. To evaluate the effect of our supervision strategy, we compare our SPSS in Sec. 3.4 with pixel loss in vanilla 4DGS [45], spike loss in SpikeNerf [69], and the combination of the above. The quantitative results of novel view synthesis are listed in Table 4. Our SPSS, consisting of pixel loss and spike variation loss, obtains the highest PSNR performance. + +Ablation on View Numbers. In this part, we investigate the relationship between view numbers and performance. As shown in Fig. 5, we present both quantitative and qualitative comparisons. For the quantitative experiments, we use Brisque [33, 34] and NIQE [35] as the evaluation metric. Our Spike4DGS achieves the best performance + +Table 3: The contribution of each component. + +
MethodsPSNR↑SSIM↑LPIPS↓
TFI [64]+COLMAP [40]21.0884.90.253
TFI [64]+DUSt3R [41]25.6689.30.205
MSDI26.6690.30.198
TFI [64]+COLMAP [40]+SPSS22.7685.20.233
TFI [64]+DUSt3R [41]+SPSS26.3490.20.201
MSDI+SPSS(Full model)27.7491.20.190
+ +Table 4: Ablation on supervision strategy. + +
Supervision StrategyPSNR↑SSIM↑LPIPS↓
PixelLoss [45]21.2285.00.251
SpikeLoss [69]13.7976.30.408
PixelLoss [45]+SpikeLoss [69]26.6989.50.203
SPSS (Ours)27.7491.20.190
+ +![](images/8cad6fa70d3c4ca39cf14a4688704b2e021fb029e3192f34933fa7073c116d26.jpg) + +![](images/d84248504628988f70bf6e19f86e008260b25ea70cd0ceab69cb62098f2184de.jpg) + +![](images/86ade2e59229082975649111249dd4ecd973a95c30145d756ef0ab21ed07b805.jpg) + +![](images/dd865c3cf698ad14d85ff20fd9210d00a57de85fb94e19753cbf8d441a116ccf.jpg) + +![](images/0fe26edeb778afdf87ed0c05d4ec42c13a85b2f1dde4a19d3e4000aa4dd5aa53.jpg) + +![](images/a31e0e61a7836a72f67a68879e53805f51fac27df8249d93fcbea347fb650fa9.jpg) + +![](images/666fcac4c6dde0910a53b544e80e44dcd3cd7ee1bd08f7f2c53fa95582d5c87d.jpg) +5 views + +![](images/336d6e53703e05dc38208c8f92114c608ae0b309161ef37c0697f095764e6d97.jpg) +7 views + +![](images/13483df0a015dbf13bbfa3b7063bfb43d4d71fb9e619442bf6ab6c20451079b7.jpg) +9 views +Figure 5: Quantitative and qualitative ablation of view numbers on real-world object datasets. + +Table 5: Comparison of different methods under varying numbers of views. (Brisque/NIQE) + +
Methods3 Views4 Views5 Views6 Views7 Views8 Views
TFI+D3DGS [29]83.4/23.8773.6/21.6765.2/20.8162.9/18.6059.8/16.2056.53/15.34
TFI+STG [26]71.0/22.8061.2/17.5057.0/16.7052.1/14.6042.0/13.1036.45/10.53
TFI+4DGS [45]70.3/22.7060.8/18.4056.4/16.5051.8/14.4043.6/13.9037.57/9.93
Ours63.43/18.5453.62/15.6745.23/13.8132.91/11.6729.84/10.2026.86/9.86
+ +in both visual quality and quantitative score. The quantitative comparisons across different methods are summarized in Table 5. It proves that more view numbers will lead to higher rendering quality. This means that our method could be extended to more views in the future. + +The effect of MSDI's Generator. To validate the effectiveness of our MSDI module for spike2image initialization, we conducted quantitative comparisons with representative spike-to-image methods, including TFI, TFP, and spk2img. On the synthetic dataset where ground truth images are available, we evaluated the spike-to-image quality using average PSNR and SSIM. The results are summarized in Tab 6. Note the reconstruction images in test have the same view with the training images, hence the improvements are more clearly than NVS. + +Table 6: Quantitative comparison on average quality. + +
MetricTFI [3]TFP [3]TVS [4]Spike2img [5]MSDI's images
Avg PSNR↑24.5125.3722.4330.7634.90
Avg SSIM↑0.8500.8520.8210.9040.961
+ +The effect of Whole MSDI. This part provides a more detailed evaluation of the module MSDI. Specifically, we feed the MSDI outputs (i.e., reconstructed images, point clouds, and poses) into the standard 4DGS procedure, respectively, to validate the contribution of MSDI. In addition, the traditional spike-to-image methods (e.g., TFI) and $\mathrm{MSDI + SPSS}$ are also listed for comparison. The experiments are conducted on the synthetic dataset, and the average performances are reported in the following Tab. 7. It indicates MSDI's contribution positively. + +Table 7: Focus Comparison of MSDI without SPSS. + +
MethodAvg PSNR↑Avg SSIM↑Avg LPIPS↓
TFI + COLMAP + 4DGS21.080.8490.253
TFI + DUST3R + 4DGS25.660.8930.205
MSDI IMAGE + COLMAP + 4DGS22.430.8540.233
MSDI IMAGE + DUST3R + 4DGS25.760.8980.203
MSDI + 4DGS26.660.9030.198
MSDI + SPSS (Ours)27.740.9120.190
+ +# 5 Conclusions + +This paper introduces Spike4DGS, a novel framework that seamlessly integrates multiple spike streams captured by a spike array into 4DGS training, effectively addressing the challenges of reconstructing high-speed dynamic scenes. Spike4DGS designs a novel Multi-view Spike-based Dense Initialization module to obtain dense point clouds from continuous spike streams and a SpikePixel Synergy Supervision strategy to optimize the parameters for enhanced rendering quality. We contribute two novel datasets and conduct comprehensive evaluations. The results on the datasets demonstrate that our Spike4DGS surpasses previous SOTA dynamic reconstruction approaches in high-speed dynamic scenes, with almost no sacrifice in training cost and rendering FPS. + +Limitation. While our spike camera array is functional, the approach has limitations which we will solve in our future works. The images rendered by our Spike4DGS are grayscale due to the lack of RGB information. Additionally, the spike array is not portable, which limits its use in mobile or field-based applications. + +Acknowledgements The study was funded by the National Natural Science Foundation of China under contracts Shenzhen Science and Technology Program(KQTD20240729102051063), No. 62422602, No. 62425101, No. 62332002, No. 62372010, No. 62027804, No. 62088102, No. 62206281, and the major key project of the Peng Cheng Laboratory (PCL2021A13 and PCL2024AS204). Computing support was provided by Pengcheng Cloudbrain. + +# References + +[1] Alex M Andrew. Multiple view geometry in computer vision. Kybernetes, 30(9/10):1333-1341, 2001. +[2] Jeongmin Bae, Seoha Kim, Youngsik Yun, Hahyun Lee, Gun Bang, and Youngjung Uh. 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Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We claim the contributions of this work at the end of the Introduction, and elaborate on how we achieve them in the Methodology. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: See supplementary materials. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. 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Attention_ Efficient Remote Sensing Image Super-Resolution with Attention Spiking Neural Networks/full.md @@ -0,0 +1,670 @@ +# Spiking Meets Attention: Efficient Remote Sensing Image Super-Resolution with Attention Spiking Neural Networks + +Yi Xiao $^{1}$ Qiangqiang Yuan $^{2*}$ Kui Jiang $^{3}$ Wenke Huang $^{2}$ Qiang Zhang $^{4}$ Tingting Zheng $^{3}$ Chia-Wen Lin $^{5}$ Liangpei Zhang $^{2}$ + +$^{1}$ School of Computer and Artificial Intelligence, Zhengzhou University + $^{2}$ Wuhan University $^{3}$ Harbin Institution of Technology $^{4}$ Dalian Maritime University + $^{5}$ National Tsinghua University +yixiao@zzu.edu.cn + +# Abstract + +Spiking neural networks (SNNs) are emerging as a promising alternative to traditional artificial neural networks (ANNs), offering biological plausibility and energy efficiency. Despite these merits, SNNs are frequently hampered by limited capacity and insufficient representation power, yet remain underexplored in remote sensing image (RSI) super-resolution (SR) tasks. In this paper, we first observe that spiking signals exhibit drastic intensity variations across diverse textures, highlighting an active learning state of the neurons. This observation motivates us to apply SNNs for efficient SR of RSIs. Inspired by the success of attention mechanisms in representing salient information, we devise the spiking attention block (SAB), a concise yet effective component that optimizes membrane potentials through inferred attention weights, which, in turn, regulates spiking activity for superior feature representation. Our key contributions include: 1) we bridge the independent modulation between temporal and channel dimensions, facilitating joint feature correlation learning, and 2) we access the global self-similar patterns in large-scale remote sensing scenarios to infer spatial attention weights, incorporating effective priors for realistic and faithful reconstruction. Building upon SAB, we proposed SpikeSR, which achieves state-of-the-art performance across various remote sensing benchmarks such as AID, DOTA, and DIOR, while maintaining high computational efficiency. Code of SpikeSR will be available at https://github.com/XY-boy/SpikeSR. + +High-resolution remote sensing images (RSIs) contain fine-grained object structures and textures, which are critical for accurate interpretation in downstream tasks [42, 11, 6]. However, limited by the intrinsic resolution of airborne sensors, RSI can merely capture partial spatial details, resulting in suboptimal scene representation and visual quality. Image super-resolution (SR) aims to alleviate this problem by reconstructing high-resolution (HR) images from low-resolution (LR) observations [53, 60]. Despite this, SR remains a challenging ill-posed issue, as a degraded input may correspond to multiple plausible outputs. + +Early efforts rely on hand-crafted priors to tame the ill-posedness, e.g., nonlocal mean [67, 7] and gradient profile [48], but they are often trapped in limited performance and scalability. Recent advances in artificial neural networks (ANNs), e.g., CNNs and Transformers, have witnessed remarkable progress in SR with large-capacity models [49, 50, 3, 4, 71]. However, they often come with a + +![](images/6e3882250ceebe9e9f0c5937dfd3961071011901c4c7b1f8255d62000564fbcd.jpg) +(a) Pixel Intensity (top) and Neuron Voltage (down) Maps + +![](images/9e44355450041780d0a61ce30020eea3b2d7690bb44cddb40605c831948689cb.jpg) +(b) Performance and Efficiency Comparison +Figure 1: (a) The visualization of pixel intensity and neuron voltage in images under various degradation factors reveals important insights. The pixel intensity map illustrates that the high-frequency components of the image tend to be smooth, indicating a reduction in sharp details during progressive downsampling. Neuron intensity maps, derived from a LIF model [37, 14], show that high-frequency details persist with drastic fluctuations, suggesting that the neurons remain in an active state. (b) FLOPs and PSNR performance comparison. The circle sizes represent the number of parameters. Our SpikeSR outperforms SOTA efficient SR methods with high efficiency. PSNR results are averaged on the AID, DOTA, and DIOR datasets. + +trade-off of increased computational overhead and growing storage costs, making them less efficient in practical scenarios, particularly when reconstructing large-scale RSIs. + +More recently, brain-inspired spiking neural networks (SNNs), as the third generation of neural networks, have emerged as a promising alternative for energy-efficient intelligence [25, 59, 40]. Different from ANNs that encode features as continuous values, SNNs can emulate biological communication with discrete spiking signals and propagate them by neurons, thus enjoying lower power consumption. As depicted in Fig. 1(a), our experiments reveal a novel finding that spiking neurons maintain an active learning state across LR RSIs, even in severely damaged textures. Specifically, we observed that degraded RSIs exhibit smoothed pixel intensities and obscured sharp details, posing a significant challenge to characterize high-frequency representations. In contrast, spiking signals retain drastic responses and pronounced spike rates, highlighting that neurons remain in an active learning state. This naturally arises a question: Can SNNs leverage their inherent properties to handle image degradation for efficient yet high-quality RSI SR? + +In fact, to effectively grasp complex and diverse spatial details in RSIs, the network must possess adequate capacity and representation power. Unfortunately, there are two critical challenges when adapting SNNs for SR tasks. Firstly, spiking activity in SNNs inevitably causes pixel-wise information loss, which hampers the representation capacity of SNNs, especially when the network deepens. This stems from the discrete nature of binary spiking signals, leading to undesirable spiking degradation problems [61, 15]. Secondly, SNNs remain constrained by suboptimal membrane potential dynamics, restricting effective exploration of global context during spiking communications. This necessitates a customized strategy to optimize membrane potentials, but is barely explored before. + +To address these limitations, we propose SpikeSR, an SNN-based framework inspired by human visual attention mechanisms, which can actively represent image degradation and modulate synaptic weights to focus on salient regions, which, in turn, regulate the spiking activity for improved capacity and representation power. Specifically, SpikeSR employs a concise yet effective spiking attention block (SAB) to optimize feature emphasis through spiking response dynamics, which integrates three key innovations: 1) the combination of CNN and SNN layers to mitigate information loss induced by discrete spiking activity; 2) introducing hybrid dimension attention (HDA) to recalibrate spiking response across both temporal and channel dimensions, facilitating a joint feature correlations learning; 3) accessing global self-similarity patterns in RSIs to infer spatial attention weights, incorporating effective priors for realistic and faithful reconstruction. Compared to state-of-the-art (SOTA) ANN-based efficient SR models, our SpikeSR demonstrates lower model complexity and superior performance, as shown in Fig. 1(b). + +Our contributions are summarized as follows: + +- We pioneer an attention spiking neural network for efficient SR of RSIs, providing a new perspective on developing efficient models in large-scale Earth observation scenarios. +- We devise a concise yet effective SAB, which mitigates the information loss and regulates membrane potentials of spiking activity for improved representation of SNNs. +- Extensive experimental results on various remote sensing datasets demonstrate that our SpikeSR achieves competitive SR performance against SOTA ANN-based methods. + +# 1 Related Work + +Deep Networks for SR. Inspired by the pioneering SRCNN [10], CNN-based SR methods have achieved remarkable progress, dominating the field for years. They mainly elaborated on the network design to tame the ill-posedness, with notable advances in residual connections [21, 32] and attention mechanisms [69, 39]. However, these methods often suffer from high computational complexity, e.g., exhaustive non-local modeling [28, 38], making them less efficient in large-scale RSIs. + +Recently, transformer-based SR models have demonstrated impressive performance, benefiting from their ability to model long-range dependencies. IPT [2] first introduces Transformers in SR field, but requires massive parameters and laborious pre-training processes. SwinIR [31] effectively reduces the model size by partitioning the image into smaller windows when applying multi-head attention mechanisms, while maintaining favorable performance. Against transformer, Mamba-based SR methods achieve comparable global model capacity with linear complexity [18, 57, 17]. Despite these advancements, advanced SR models are often trapped by rising computational overhead and growing storage costs, posing significant concerns in real-world applications, particularly in remote sensing scenarios. + +Efficient SR Models. To reduce computational budget, CARN [1] utilizes grouped convolutions and a cascading mechanism to improve the residual architecture. IMDN [20] progressively distills useful information during feature extraction and applies network pruning to further decrease complexity. FMEN [12] optimizes residual modules to accelerate inference. In Transformer-based SR methods, SPIN [66] enhances long-range modeling by combining self-attention with pixel clustering, facilitating interactions between superpixels. HiT-SR [68] expands the self-attention receptive field by applying different window sizes of hierarchical layers. Despite these successes, there is still room to further boost SR performance. Moreover, the potential of energy-efficient SNNs for SR tasks remains largely unexplored. + +Spiking Neural Networks. Recent advances in neuromorphic computing have shown the great potential of SNNs in computational efficiency and power as CNNs [63, 64]. Currently, SNNs have been successfully applied to various tasks, such as image classification [25, 44], object detection and tracking [22, 59], optical flow estimation [26, 40], etc. + +A common solution to build SNNs is converting pre-trained ANN models [8, 62]. Li et al. [30] proposed a layer-wise calibration to minimize activation mismatch during conversion. Ding et al. [9] replaced ReLU with the rate norm layer, enabling direct conversion from a trained ANN to an SNN. Stockl et al. [46] used time-varying multi-bit spikes to better approximate activation functions. However, conversion-based methods face accuracy gaps and high latency due to extensive time-step simulations, resulting in increased latency and energy consumption. + +An alternative involves using agent gradient functions for continuous relaxation of non-smooth spike activities, enabling direct training via backpropagation through time. Lee et al. [27] treated membrane potential as a signal to overcome discontinuities, enabling direct training from spikes. Wang et al. [54] introduced an iterative LIF model and proposed spatiotemporal backpropagation based on approximate peak activity derivatives. Later, Zheng et al. [70] proposed temporal delay batch normalization, which significantly enhanced the depth of SNNs. To bridge the performance gap between ANNs and SNNs, some methods borrowed insights from CNNs, applying residual learning [15, 19] and attention mechanisms [65, 41] to SNNs. Nonetheless, there has been limited exploration of pixel-level regression tasks, such as SR. + +![](images/9b39d325480a6bee9050dee94fc6d43d2c719fe4af83da742bd181c5fb729edc.jpg) +Figure 2: Overall network architecture of SpikeSR. The LR input is replicated along the temporal dimension and then processed through a convolution to extract shallow features. The core module of SpikeSR is SAG, which employs SABs to capture deep spiking representations. Each SAB contains three main components: (1) SCB, (2) HDA, and (3) DSA. The fusion block (FB) aggregates the spatial-temporal sequences, and pixelshuffle is used to reconstruct the SR output. + +# 2 Method + +The architecture of the proposed SpikeSR is illustrated in Fig. 2, which mainly consists of SAGs. Before SAGs, we utilize a $3 \times 3$ convolution to extract high-dimensional features from the LR input. These features are then processed through $m$ stacked SAGs to explore deeper representations. Each SAG includes $n$ SABs, a SCB, and a residual connection. In the SCB, leaky integrate-and-fire (LIF) neurons [37, 14] are used to convert the inputs into binary spike sequences (i.e., 0 or 1). As shown in Fig. 2, the output of the LIF neuron is 1 when the membrane potential exceeds the threshold, and 0 otherwise. To optimize the membrane potential, we introduce HDA, which refines the spiking activity using an efficient temporal-channel joint attention [72]. Furthermore, the proposed DSA is employed to introduce global context for accurate SR. After the terminal SAG, an FB is utilized to convert the spike sequence features into continuous values. Finally, SpikeSR generates super-resolved output from the fused features by applying pixel-shuffle [43] and a $3 \times 3$ convolutional layer. + +# 2.1 Spiking Attention Block + +As evidenced in Fig. 1, regions degraded by different factors exhibit noticeable fluctuations when encoded by LIF, highlighting pronounced firing spike rates of neurons. This provides robust and latent informative spiking cues from LR images. Unlike ANNs that encode images into continuous decimal values, SNNs use discrete binary spike values for neuronal communication, and thus demonstrate undesirable information loss [24, 65], resulting in limited capacity to represent degraded LR images. To address this, the design philosophy of the SAB is focused on leveraging CNNs and attention mechanisms to regulate membrane potentials, facilitating high-quality feature representation for SR, which in turn affects the spiking activity. + +As shown in Fig. 2, in particular, the output of the $n$ -th SAB at the $t$ -th time step is denoted as $\mathbf{X}^{t,n}$ and can be obtained by the following: + +$$ +\mathbf {X} ^ {t, n} = \mathbf {X} ^ {t, n - 1} + \operatorname {D S A} \left(\operatorname {H D A} \left(\bar {\mathbf {X}} _ {1} ^ {t, n} + \bar {\mathbf {X}} _ {2} ^ {t, n}\right)\right), \tag {1} +$$ + +where $\bar{\mathbf{X}}_1^{t,n}$ and $\bar{\mathbf{X}}_2^{t,n}$ are two feature representations obtained from parallel branches, defined by: + +$$ +\begin{array}{l} \bar {\mathbf {X}} _ {1} ^ {t, n} = \operatorname {S C B} (\operatorname {S C B} (\mathbf {X} ^ {t, n - 1})), \\ \bar {\bar {\mathbf {X}}} _ {n} ^ {t, n} = \operatorname {D N} (\mathbf {C}, (\mathbf {X} ^ {t, n - 1})) \end{array} \tag {2} +$$ + +$$ +\bar {\mathbf {X}} _ {2} ^ {\hat {t}, n} = \operatorname {t d B N} \left(\operatorname {C o n v} \left(\mathbf {X} ^ {t, n - 1}\right)\right), +$$ + +where Conv represents a $3 \times 3$ convolution layer, and tdBN means the threshold-dependent batch normalization. + +Different from previous works that focus solely on separate temporal and channel modulation [61, 65, 45], SAB adheres to temporal-channel joint attention [72] to realize joint adjustment of the spike response in HDA, effectively achieving interdependencies between the temporal and channel scopes. More details of HDA can be found in the Appendix. + +![](images/557e49b8a3e88fbe90fba8291354ca30bb5f5f9f4737fcb228ce83541f3d520a.jpg) +Figure 3: The illustration of our DSA. Note that we set the diagonal elements of the similarity matrix to zero before selecting the indices of the highest scores. The deformable convolution operates at the patch level, alleviating the mismatch between the most similar patches. + +# 2.2 Deformable Similarity Attention + +Non-local self-similarity has been recognized as an effective prior for SR tasks [47]. However, existing non-local attention mechanisms are computationally expensive due to exhaustive non-local operations, which impedes their efficiency in large-scale RSIs. In contrast, the proposed DSA efficiently grasps complex self-similar patterns in RSIs at the patch level to infer intricate spatial weights. Then, we utilize the cross-attention (CA) paradigm to enhance long-range communication, facilitating the fusion of useful context. + +The details of DSA are shown in Fig. 3. Considering that the object scale exhibits explicit diversity in RSIs, the input feature $\mathrm{F}$ is downsampled using bilinear interpolation, forming a multi-scale feature pyramid. For clarity, we demonstrate this process by dividing the initial features into 9 patches. Following the design in [33], the final DSA exploits a cascaded patch division strategy. Specifically, each patch is first average-pooled to capture its spatial characteristics, then reshaped and subjected to self-similarity computation, yielding a similarity matrix. The final self-similarity scores are fused via matrix multiplication to enhance the multi-scale representation. The best-matching patch $\bar{\mathbf{P}}_i$ with $\mathbf{P}_i$ can be obtained by: + +$$ +\bar {\mathbf {P}} _ {i} = \underset {\mathbf {P} _ {j}} {\operatorname {a r g m a x}} E \left(\mathbf {P} _ {i}\right) ^ {\mathrm {T}} E \left(\mathbf {P} _ {j}\right), j \neq i, \tag {3} +$$ + +where $\bar{\mathbf{P}}_i$ is the patch in $\bar{\mathbf{F}}$ , and $E$ means the operation of average pooling and feature reshaping. We adopt the Gumbel-Softmax [52] to achieve the non-differentiable argmax function. + +Although matched patches contain highly relevant similarity, they are inevitably subject to mismatches and geometric transformations. Hence, we use deformable convolution (DConv) to reduce the generation of hallucinated textures. The deformable feature $\mathbf{F}^{\mathrm{D}}$ at location $p_0$ is computed as follows: + +$$ +\mathbf {F} ^ {\mathrm {D}} \left(p _ {0}\right) = \sum_ {p _ {m} \in \mathcal {R}} \omega \left(p _ {m}\right) \cdot \mathbf {F} \left(p _ {0} + p _ {m} + \Delta p _ {m}\right), \tag {4} +$$ + +where $\omega(p_m)$ is the convolution weight at relative location $p_m$ , $\Delta p_m$ is a 2D vector that represents the learnable offsets, $\mathcal{R}$ is a regular grid that determines the receptive field of the convolution kernel. For a $3 \times 3$ kernel, $\mathcal{R} = \{(-1, -1), (-1, 0), \dots, (1, 1)\}$ . To fuse the self-similar features $\mathbf{F}^{\mathrm{D}}$ with $\overline{\mathbf{F}}$ , we embed $\mathbf{F}^{\mathrm{D}}$ to $\mathbf{Q}$ , and $\overline{\mathbf{F}}$ to $\mathbf{K}$ , $\mathbf{V}$ using fully connected layers, then perform aggregation by: + +$$ +\bar {\mathbf {V}} = \operatorname {s o f t m a x} \left(\mathbf {Q} \mathbf {K} ^ {\mathrm {T}} / \sqrt {d}\right) \mathbf {V}, \tag {5} +$$ + +Finally, the fused features are summarized with the original features $\mathbf{F}$ and fed into the multilayer perceptron (MLP) to obtain the final output: + +$$ +\tilde {\mathbf {F}} = \operatorname {M L P} (\mathbf {F} + \bar {\mathbf {V}}). \tag {6} +$$ + +Table 1: Quantitative comparison of SpikeSR with SOTA methods on three remote sensing datasets. FLOPs are measured corresponding to an LR image of $160 \times 160$ pixels. Note that we set $T = 1$ to evaluate the model complexity of SpikeSR for fair comparison. + +
Methods#Param.FLOPsAID [56]DOTA [55]DIOR [29]Average
PSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIM
Bicubic--28.860.738231.160.794728.570.743229.530.7587
SRCNN [10]20K0.512G29.700.774132.100.826429.490.776830.430.7924
VDSR [21]667K17.08G30.440.800433.220.856930.360.803631.340.8203
EDSR [32]1518K50.77G30.650.808633.640.864830.630.811631.640.8283
CARN [1]1112K40.39G30.660.806833.660.863330.640.810231.650.8268
IMDN [20]715K18.18G30.710.807633.700.864130.730.811531.710.8277
RFDN-L [34]681K16.49G30.690.807433.730.864230.720.811431.710.8277
LatticeNet [36]777K19.39G30.730.808933.750.865330.750.812631.740.8289
HNCT [13]364K8.48G30.790.810433.830.866430.800.813631.810.8301
FMEN [12]1046K26.72G30.650.806333.660.863130.660.810431.660.8266
RLFN [23]544K13.25G30.700.807433.690.863630.700.811031.700.8273
ESRT [35]752K26.06G30.770.810233.750.866830.810.814231.780.8304
SwinLR-light [31]897K23.56G30.830.811433.940.867730.850.814931.870.8313
Omni-SR [51]2803K70.98G30.890.814233.940.869530.890.817031.910.8336
NGswin [5]995K12.73G30.790.810733.870.866730.790.814031.820.8305
SPIN [66]555K18.91G30.780.809833.850.867330.820.813931.820.8303
HiT-SR [68]792K21.04G30.870.813833.930.868930.890.816731.900.8331
SpikeSR-S (Ours)472K15.21G30.860.812633.890.868730.890.816231.880.8325
SpikeSR-M (Ours)763K24.00G30.880.813333.920.868930.900.816331.900.8328
SpikeSR (Ours)1042K33.05G30.910.814233.980.870030.950.817531.950.8339
+ +![](images/10bd87134f8f69645c840e58db7ddd95029f00deead6d662e6c68441dba25791.jpg) +playground_270 + +![](images/66442e5abcce4b41c4bcd98724e54f7cd3d441ea0cb0d3c61a49d574cdb7314c.jpg) + +![](images/782070f9d3ee740c913956ee0ff83c9d55834cf0afc7cb22a59594a85a1d61f8.jpg) + +![](images/cc0a42232462d1980885f6042c139e24ff868dd3b7f46a5d366710e5974040f6.jpg) + +![](images/4e820087fa5a53ca82ee958dfb5698994cf2ac11b2405939b922bcead997297f.jpg) + +![](images/e143452a4503390be241623c1abf4a24f1a55b9ef13e51bd6d65d8d334eb06b3.jpg) + +![](images/e3bc55de1c1442abfdca94486c438e8f5e47999d56d852ece4f11ed08aac0e4d.jpg) +Bicubic +ESRT + +![](images/5d9d3a5af8d53872a51b96db0a5bfaba59b598aa744a811db6bd13629aea9c26.jpg) +EDSR +SwinIR-light +Figure 4: Qualitative comparison of SOTA efficient models for $\times 4$ SR task on AID test set. + +![](images/e041c2562e3982dbe816d1ab93a0ba7a09b818948f1fbb729a3ed1731a28cbb2.jpg) +CARN +SPIN + +![](images/c74136f549b45f9a35df06ba9cd965df8ef746c11d584d3682b82a38fed7fcc6.jpg) +IMDN +SpikeSR + +![](images/739ffa2ca90ff0fe3cdb5dffe76e8d6f0b3b603e75e54ba885394657d59e8303.jpg) +LatticeNet +Ground Truth + +# 2.3 Fusion Block + +To transform discrete spiking sequences into continuous pixel values, a common approach is to apply mean sampling along the time dimension. However, this naive process may lead to the loss of crucial spatial details, potentially affecting the SR quality. Therefore, we introduce a fusion block that adaptively aggregates spiking sequences and mitigates information loss. Given an input spike input $\mathbf{Y}$ , the computation process of FB can be formulated as: + +$$ +\mathbf {Y} _ {1} = \sigma (\operatorname {T A} (\mathbf {Y})) \otimes \mathbf {Y}, +$$ + +$$ +\mathbf {Y} _ {2} = \sigma (\operatorname {S A} (\mathbf {Y})) \otimes (1 - \mathbf {Y} _ {1}), \tag {7} +$$ + +where TA and SA denote temporal and spatial attention [65], $\sigma$ means a sigmoid function and $\otimes$ denotes feature multiplication. The final output of FB is obtained by summing $\mathbf{Y}_1$ and $\mathbf{Y}_2$ . + +# 3 Experiments + +Datasets. We use the AID dataset [56] as the training set, a large-scale remote sensing benchmark for scene classification, consisting of 30 different scene categories. The AID dataset includes 10,000 HR images, where we randomly select 3,000 for training and 900 for validation. The LR samples are generated by bicubic downsampling. Following TTST [58], we also evaluate our method on the DOTA [55] and DIOR [29] datasets, which contain 900 and 1,000 images, respectively. + +Implementation Details. During model training, the learning rate is fixed to $10^{-4}$ , and the training procedure stops after 1000 epochs with a batch size of 4. Adam optimizer is used with $\beta_{1} = 0.9$ + +Table 2: Quantitative comparison of SpikeSR with SOTA methods on 30 scene types of AID datasets. + +
Scene typesEDSR [32]CARN [1]IMDN [20]ESRT [35]SwinIR-L [31]SPIN [66]HiT-SR [68]SpikeSR
PSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIM
Airport29.930.828229.960.826430.000.827030.090.829230.140.830730.110.829530.210.832530.270.8336
Bare Land36.940.883736.920.882936.930.883436.900.884036.990.884136.960.884337.000.884636.980.8846
Baseball Field33.050.876533.060.875333.170.876333.210.877333.270.878233.140.875933.290.879133.320.8794
Beach34.180.872734.270.873734.290.873934.350.875534.390.875634.360.875734.380.876234.410.8764
Bridge32.930.880032.860.877432.930.878433.050.880333.150.881033.050.880333.140.882033.270.8827
Center28.770.792128.710.788128.790.789228.880.792329.000.795428.880.791929.030.797429.090.7985
Church26.300.746926.410.744926.460.746726.520.748926.590.751226.560.750726.640.754926.700.7560
Commercial29.010.794029.110.794429.170.795829.220.797529.280.799629.270.798929.330.802129.370.8033
D-Residential24.380.683924.560.685624.600.686424.670.689824.710.692424.630.687224.800.699824.800.6978
Desert40.200.926840.220.925940.170.926440.060.927640.310.927540.240.927940.270.928240.250.9283
Farmland35.000.868334.890.865634.940.866734.990.867935.020.868134.980.867835.090.869635.090.8700
Forest29.850.731529.880.730429.900.731229.990.736529.980.735029.970.735030.050.739530.050.7380
Industrial28.880.793128.840.789428.880.790428.980.794229.030.795928.980.793229.110.799829.160.8007
Meadow34.630.780434.530.776934.550.778434.630.781434.660.780734.640.781334.580.780734.680.7820
M-Residential28.340.736528.390.734728.420.734928.470.737028.560.740128.390.733428.640.744328.630.7436
Mountain30.630.788530.700.789230.700.789530.740.790930.780.792130.750.791130.790.793030.790.7930
Park30.540.813030.630.813630.650.814130.720.816930.760.817730.730.816230.810.819830.820.8203
Parking27.250.831727.080.824527.230.827027.470.835227.420.835427.500.836327.700.843527.720.8424
Playground35.370.894335.270.89235.420.892935.470.894235.490.894635.450.894635.590.896835.700.8976
Pond32.110.854232.100.853232.110.853432.170.854632.220.855332.150.854532.230.856132.250.8563
Port28.500.859628.610.859328.670.859728.750.862028.810.863128.790.862428.850.865128.940.8658
Railway Station28.720.773828.680.769928.770.771828.840.774428.920.777728.890.775928.970.780229.020.7816
Resort28.520.779928.590.779128.620.779528.680.781928.740.783728.660.780128.780.786428.820.7869
River31.550.789131.550.788231.570.788531.600.790031.640.790531.610.790431.660.791831.680.7922
School29.360.804429.410.803329.450.804129.510.806729.590.809129.500.804829.670.812329.680.8124
S-Residential27.710.672827.790.672327.800.672527.850.675227.880.675827.800.672827.910.678227.920.6775
Square30.840.820030.830.818130.870.818330.970.821831.060.823630.980.82131.110.825631.150.8266
Stadium29.630.838729.510.83429.620.835829.740.838829.820.841329.760.839429.800.842029.930.8439
Storage Tanks27.440.766427.500.764927.520.764827.580.767127.630.769227.580.76727.660.772027.680.7718
Viaduct28.990.775728.920.771128.960.772229.060.775929.140.778429.050.775329.160.781129.250.7831
Average30.650.808630.660.806830.710.807630.770.810230.830.811430.780.809830.870.813830.910.8142
+ +![](images/b98c8e68ee90d1c00b864ff596c30fd06ad31af65508532fd0fda13913ee0d3b.jpg) +DOTA: img_591 + +![](images/185593a446a8fee4ed2146b65eda63b4044500b57faa18166f0045b3b36a8664.jpg) +Bicubic + +![](images/169d01e9a259eee7c72b5e0c03a0c7440f4862440327176134d388b81538bf66.jpg) +SR + +![](images/f5f17f9b0973e0b49c8632f837b71187cca6417cc3546967d5d58ba8ddf5e06e.jpg) +ESRT + +![](images/c4989d0bb61da49d341b4dd70e23a2f9d5c69a42625f3a8862240027986777b0.jpg) +DOTA:img_675 +Figure 5: Qualitative comparison of SOTA efficient SR models for $\times 4$ SR task on DOTA dataset. + +DOTA: img_675 + +![](images/f06743c1f711bac2b4e0f4bf59c58ffbefe770a11717e293af3a63645fa71464.jpg) +Bicubic + +![](images/a69e5e8e858cbb3bf6ac1eb22ddae941cab0d53258ed533710b1d2a1d0a134e2.jpg) + +![](images/520b0ca2aee13bcf1383b0b69ce7f957b435fed13fa4ccba91c452eb920009f3.jpg) +CARN ESRT +RT + +![](images/770e3cc994721f72b5f02c433bcee36fc49134cef90b2f37b5e782b1a5702c19.jpg) +SwinIR + +![](images/49fd241ff22e3cc0e6953584e9184a870d59947921af3711d70c7b74c09b0a42.jpg) +SPIN + +![](images/7b54be896a22432955d7f9dfbd530c0861e5fc65489fb91f09d8e6bed76e3908.jpg) +Ours + +![](images/40b2029412e9bb1158866c2ac5bda29336e238cdb5f6a877bcd4c4e3ce9dad36.jpg) + +![](images/e47697b69a05e8e720679b17be92b88b38f3077083e619c94f98dd4ee4d400e1.jpg) +DOTA: img_596 + +![](images/645833d24eea300254481b03f945bcd35726548356f516540fd62a870562a71e.jpg) +Bicubic + +![](images/ef1e32ddac3765eb06638ad3dd97e3daacfc6d994082a70630d4980f112089d1.jpg) +EDSR + +![](images/eb48f81f246ff4c8375b1358aed8fde216062ab12a6abb212461f3d9d85b45e1.jpg) +CARN + +![](images/a3e829d313b1c6f50ffa99503d8c7e8ede519069042f69362dbbd5e89cc3819c.jpg) +ESRT + +![](images/f0380f3a99249dd44a2865b82f25b63fa22f41664b6b3521433f224b18b9d6d7.jpg) +SPIN + +![](images/0c30e52616c40ef0f2702d6c3e9846f8404dddd494648cc2958722666395485e.jpg) +Ours + +![](images/a91ab4526944fc891b4e3cbb3d2e28cbc4fa943ed52015973d5f1462212b6af8.jpg) +GT + +and $\beta_{2} = 0.999$ . Data augmentation includes random rotations of $90^{\circ}$ , $180^{\circ}$ , $270^{\circ}$ , and horizontal flips on $64 \times 64$ patches. The channel number, embedding dimension of cross-attention, and MLP rate of small, medium, and final SpikeSR are set to $\{40, 24, 72\}$ , $\{56, 24, 72\}$ , and $\{64, 32, 100\}$ , respectively. The number of SAGs is 4, with 2 SABs in each SAG. Time step is set to $\mathrm{T} = 4$ . + +Metrics. The widely used peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are used to evaluate SR performance. The results are measured on the Y channel after converting RGB to YCbCr space. For a fair comparison, all SR methods are retained on an RTX 4090 GPU from scratch using the AID dataset, adhering to their official implementation settings. + +# 3.1 Comparison with Efficient Models + +We compare our SpikeSR with state-of-the-art (SOTA) efficient SR methods, including CNN-based models of CARN [1], LatticeNet [36], RLFN [23], etc, and transformer-based approaches of SwinIR + +![](images/02d4942c7a51da8c295aa2d96f95fba12b32df5b548887ec00a5d136e9de4778.jpg) +Bicubic + +![](images/42ef66471acec21d5a144bf2554a211d7d2ed8924c30a00a58edeceb51064ed6.jpg) +EDSR + +![](images/8b00225b2826497fba56863c846278bb4131520ba6de9aae0bc3710ad40fe23f.jpg) +CARN + +![](images/99835af990634fa4767bfaf2c08a6b8e1aedc211cdb2e793960e90752e3e23be.jpg) +LatticeNet + +![](images/385ef0bb36ce454d8a583019313567bd3701682e6a5ee506b7ecf9799eebef9c.jpg) +IMDN + +![](images/d5741a6a6ea216d5b16a424ecd2169df70303148144bc92e9bad8c7147b74e9a.jpg) +FMEN + +![](images/e81a47b4b807036f172e69636ff68fc3d23852ed4a9ee6067b3f2d09fb6c0605.jpg) +ESRT +Figure 6: Qualitative comparison of SOTA efficient SR models for $\times 4$ SR task on DIOR dataset. + +![](images/f44d81c707d27c67c3468ab611bd7b98207506a6778a213fefdf5dd86b234f98.jpg) +SwinIR + +![](images/ce454734603f08c74c6b27c4b12624e646898ce0068365866a25fd025e158da4.jpg) +Omni-SR + +![](images/2bcee4a7c9609e5986b2921be521a15eee0df83a6553aa032be5a122c5391bf5.jpg) +SPIN + +![](images/567b63e2aa78fe7a4b4b98d0be787a3b0d83255dbe8b32d231d65aaa9d2f18a1.jpg) +SpikeSR + +![](images/4de18b9a2ee7d38ed42d2486977ce87244a7052ca886224c2b079f7a115732cf.jpg) +GT + +[31], SPIN [66], HiT-SR [68], etc. We also report results for the small and medium versions of our SpikeSR, denoted as SpikeSR-S and Spike-M, respectively. + +Quantitative comparisons. The quantitative results of various methods are reported in Table 1. We can observe that our SpikeSR achieves the best performance across three benchmarks, outperforming SOTA CNN- and transformer-based SR models. For example, on AID, DOTA, and DIOR datasets, SpikeSR improves PSNR by 0.08 dB, 0.04 dB, and 0.1 dB, respectively, compared to the impressive SwinIR. Moreover, the small version of SpikeSR requires fewer parameters (472K vs. 897K) and FLOPs (15.21G vs. 23.56G) than SwinIR, yet achieves superior average performance. + +Qualitative comparisons. Fig. 4, Fig. 5, and Fig. 6 present visual comparisons on AID, DOTA, and DIOR datasets, respectively. As shown in Fig. 4, SpikeSR effectively restores severely damaged textures, e.g., the runway line in the playground. By contrast, other SR models fail to recover such weak high-frequency details. In Fig. 5, the reconstruction of "img_591" highlights that SpikeSR produces results closest to the GT, while other methods like SPIN recovers unrealistic results. Moreover, Fig. 6 further demonstrates that SpikeSR consistently delivers superior visual quality, restoring more textural information compared to large-capacity ANN-based model like Omni-SR. + +# 4 Ablation Study + +We conduct ablation studies to assess the impact of key components in SpikeSR. The experimental results in Table 3 are measured on the AID-tiny dataset [58]. In particular, the Baseline model is constructed by removing the HDA and DSA and replacing them with standard temporal attention (TA), channel attention (CA), and spatial attention (SA) mechanisms. For a fair comparison, we increase the number of $m$ , $n$ , and channel dimensions to 8, 4, and 256, respectively, which ensures a similar number of parameters with our SpikeSR. Similarly, those settings of Variant-A are modified to 10, 5, and 128, respectively. + +Effectiveness of HDA and DSA. Table 3 indicates how the SR performance is influenced by the HDA and DSA. Comparing the PSNR values of the Baseline and Variant-A reveals that HDA contributes a $0.12\mathrm{dB}$ improvement. This suggests that enhancing the correlation between the temporal and channel dimensions delivers better recalibration of the membrane potentials. More intuitively, we visualize the feature maps to highlight the impact of HDA, as shown in Fig. 7(b). The results illustrate that HDA effectively refines the feature representation by emphasizing salient details and suppressing irrelevant information. + +By introducing the proposed DSA to the Baseline model, the PSNR results can be improved by a large margin of $0.21\mathrm{dB}$ . When employing HDA and DSA simultaneously, the resulting SpikeSR achieves an additional $0.08\mathrm{dB}$ improvement compared to Variant-B. To better demonstrate its effectiveness in capturing global self-similarity priors, we provide the LAM visualization and diffusion index in Fig. 7(a). As observed, DSA generates more pronounced LAMs and significantly increases the DI, indicating that the model activates more valuable pixels for accurate SR. + +![](images/b77a397682abc41f9219916dfd9626367675efc41224d23e6cbd99330c14d43c.jpg) +Figure 7: (a) Analysis of local attribution maps (LAMs) [16] and diffusion index (DI). The proposed DAS helps SpikeSR exploit more useful information against CARN and HiT-SR. (b) Feature visualizations. The feature obtained by HDA is sharper and preserves more details, indicating high-quality feature representations. + +![](images/0ad2ad306a4e4ddf583ad21c4a6504c6b7d3429f94e9f8a2e420ba982379def5.jpg) + +Table 3: Ablation on different variants of our SpikeSR. + +
MethodsTACASAHDADSA#Param.PSNR (dB)
Baseline✓✓✓1120K27.80
Variant-A✓✓1062K27.92
Variant-B✓✓✓1009K28.11
SpikeSR✓✓1042K28.19
+ +Table 4: Ablation on feature pyramid and deformable convolution. + +
Methodsw/o Pyramidw/o DConvDSA (Full)
#Param.1042K918K1042K
FLOPs31.41G28.86G33.00G
PSNR (dB)28.1428.0728.19
+ +Table 5: Performance and complexity analysis of SpikeSR with different numbers of blocks. + +
Blocks23456
#Param.558K800K1042K1284K1526K
FLOPs17.47G25.26G33.00G40.84G48.60G
PSNR (dB)28.1128.1728.1928.1628.20
+ +Feature Pyramid and DConv. Due to the scale diversity of objects in RSIs, we constructed a feature pyramid to grasp the self-similarity in multiple levels. As listed in Table 4, the use of feature pyramid improves the performance by $0.05\mathrm{dB}$ without introducing additional parameters. Furthermore, to demonstrate the effectiveness of deformable convolution, we remove this component, which leads to a severe performance drop by $0.12\mathrm{dB}$ . This illustrates that the self-similar patches contain massive irrelevant and misaligned contents, and direct fusion may introduce interference, thus resulting in suboptimal performance. + +Network Depth. We evaluate the impact of the network depth by changing the number of SAGs of our SpikeSR from 2 to 6 blocks. As reported in Table 5, SpikeSR achieves the highest SR performance when $m = 3$ . While increasing the number of $m$ may further improve the reconstruction, it also brings larger model size. Therefore, we set $m = 4$ in our final model, considering the trade-off between performance and computational complexity. + +# 5 Conclusion and Limitation + +In this paper, we investigate the application of SNNs for efficient SR of remote sensing images. Motivated by the observation that LIF neurons exhibit a higher spike rate in degraded images, we integrate SNNs with convolutions for improved feature representation. Besides, a hybrid dimension attention is employed to modulate the spike response, further refining salient information. To incorporate valuable prior knowledge for more accurate SR, we propose a deformable similarity attention module, capturing global self-similarity across multiple feature levels. Extensive experiments on various remote sensing datasets demonstrate the efficacy and effectiveness of the proposed SpikeSR model. We believe our exploration can facilitate the practical application of energy-efficient models in remote sensing area. + +SpikeSR still has some limitations. First, its representational capacity remains improbable. Second, the reliance on attention mechanisms introduces additional computational overhead, limiting efficiency. We leave these issues for future exploration. + +Acknowledgment. This work is supported in part by the National Natural Science Foundation of China (423B2104, 623B2080), and in part by the Natural Science Foundation of Heilongjiang Province of China for Excellent Youth Project (YQ2024F006), and in part by the Open Research Fund from Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ) (GML-KF-24-09). + +# References + +[1] Namhyuk Ahn, Byungkon Kang, and Kyung-Ah Sohn. Fast, accurate, and lightweight super-resolution with cascading residual network. 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Neural Networks promise brain-inspired and energy-efficient computation by transmitting information through binary (0/1) spikes. Yet, their performance still lags behind that of artificial neural networks, often assumed to result from information loss caused by sparse and binary activations. In this work, we challenge this long-standing assumption and reveal a previously overlooked frequency bias: spiking neurons inherently suppress high-frequency components and preferentially propagate low-frequency information. This frequency-domain imbalance, we argue, is the root cause of degraded feature representation in SNNs. Empirically, on Spiking Transformers, adopting Avg-Pooling (low-pass) for token mixing lowers performance to $76.73\%$ on CIFar-100, whereas replacing it with Max-Pool (high-pass) pushes the top-1 accuracy to $79.12\%$ . Accordingly, we introduce Max-Former that restores high-frequency signals through two frequency-enhancing operators: (1) extra Max-Pool in patch embedding, and (2) Depth-Wise Convolution in place of self-attention. Notably, Max-Former attains $82.39\%$ top-1 accuracy on ImageNet using only 63.99M parameters, surpassing Spikformer $(74.81\%, 66.34\mathrm{M})$ by $+7.58\%$ . Extending our insight beyond transformers, our Max-ResNet-18 achieves state-of-the-art performance on convolution-based benchmarks: $97.17\%$ on CIFAR-10 and $83.06\%$ on CIFAR-100. We hope this simple yet effective solution inspires future research to explore the distinctive nature of spiking neural networks. Code is available: https://github.com/bic-L/MaxFormer. + +![](images/45bc220f92fde8dc9d4947dd1785928b88648aacced1c30dc60218bf5eafa079.jpg) +(a) + +![](images/b5468be31f770f734fdbfb8a4fea27909bf219b49f7d9eddd7f8b313b03aba5e.jpg) +(b) +Figure 1: Spiking Transformer architectures: (a) Avg-Pool vs. (b) Max-Pool for token mixing, with (c) detailed implementation of the Spiking MLP (S-MLP) block. In mainstream (non-spiking) Vision Transformer research, Avg-Pool that captures global low-frequency patterns is a more common token mixing strategy than Max-Pool (high-pass) [1, 2]. Surprisingly, in Spiking Transformers, replacing Avg-Pool with Max-Pool yields a $+2.39\%$ improvement on Cifar-100. + +![](images/0abfe7b150dcba618567cc312531378be267ef8ddb817759a7fd9a523e64483b.jpg) +(c) + +![](images/d8313398bfc3e8641498851f121783dd0203d3fa8ed8b44a12ac0da1994b45ca.jpg) +Spectrnnn + +![](images/7167699b95f8c073b43efc77ccaece696c59b90d32ed6670f57f68d8b81505dd.jpg) +Figure 2: Comparison between ReLU and spiking neuron (S-Neuron): (a) Input images; (b) Fourier spectrum analysis of output features processed as input $\rightarrow$ activation $\rightarrow$ weighting, with high-frequency regions marked (red dashed boxes: regions $>0.55\times$ max amplitude) and (c) the corresponding relative log amplitude; (d) GradCAM comparison with identical architectural setting following [9], with the converted Spiking Transformer using 256 timesteps. Spiking neurons cause the rapid dissipation of high-frequency components, which consequently leads to the degradation of feature representations. + +![](images/36803eadd290be7385f965042a6a06c7aa3196619bda86fb357cc720c40dac59.jpg) + +![](images/81888e8451a9c918eaa941a62e4c44897fab9131557fed35f7a0bdb1b775312f.jpg) + +![](images/8b5d95e00665686ff87fe371af599828878c1c331efcd34445c8faef9c1bd26d.jpg) + +![](images/fae73447bedf1085cd804e545ab07b4b94f3ec533556d1a509a912d7841d8e4f.jpg) + +# 1 Introduction + +Spiking neural networks (SNNs) are emerging as an energy-efficient alternative to conventional artificial neural networks (ANNs) [3, 4]. Their efficiency arises from spiking neurons that utilize spatiotemporal dynamics to mimic biological computation in the human brain [5]. In ANNs, all neurons within the same layer must await the complete processing of real-valued, dense tensors before any information can flow to the subsequent layer. SNNs, however, transmit information asynchronously, with spiking neurons consuming energy only when receiving or emitting spikes ("1"), otherwise remaining inactive [6, 7]. This binary activation pattern enables SNNs to replace the energy-intensive multiply-and-accumulate (MAC) operations that are essential in ANNs with much simpler spike-based accumulation. Leveraging the energy-efficiency benefits, modern SNN variants, such as Spiking Transformers that integrate the powerful Transformer architecture with spike-based computing, have gained growing attention [8, 9]. + +Despite their energy efficiency, the discrete nature of spike-based computation presents both opportunities and challenges. A major obstacle for SNNs remains their performance gap relative to ANNs. This gap is often attributed to the so-called "representation error" [10-12], which argues binary spike trains inherently limit the precision of feature representations compared to continuous activations. However, this seems inconsistent with the established consensus in the standard deep learning literature, where low-bit and even binary networks can still achieve comparable accuracy [13, 14]. Further, it should be noted that SNNs operate on temporal sequences: while spiking neurons strictly transmit binary signals at each individual time step, a train of spikes spanning $n$ simulation timesteps can encode activation values with at least $\log (n)$ -bit precision [15, 16]. The conflicting observation reveals an unexplored dimension in understanding SNN performance limitations. + +It is natural to think from the frequency domain. Spiking neurons produce discrete, pulse-like activations, fundamentally distinguishing their frequency response from continuous activation functions commonly used in standard networks (e.g., ReLU [17]). Prior works have suggested that spiking neurons may enrich signals with local details (high frequencies) [18, 19]. However, in Figure 2 (b-c), we observed a surprising phenomenon: when examining the end-to-end information flow of input $\rightarrow$ activation $\rightarrow$ weighting, rather than focusing solely on the property of activation functions, spiking neurons tend to propagate low-frequency information more prominently than ReLU. + +Feature degradation observed in SNNs may instead originate from the rapid dissipation of high-frequency components, which prevents the network from effectively capturing local, fine-grained information (Figure 2 (d)). To support this finding, we perform a simple experiment in which non-parametric pooling operators, i.e., Max-Pool and Avg-Pool, serve as token mixers in Spiking Transformers, shown in Figure 1. From the frequency domain perspective, Max-Pool excels at capturing local high-frequency details (e.g., local edges/textures), whereas Avg-Pool favors global low-frequency patterns. Intriguing, Spiking Transformers exhibit opposite preferences to ANNs in token mixing: while standard Transformers typically employ Avg-Pool for token mixing [1, 2], replacing it with Max-Pool in Spiking Transformers yields a $+2.39\%$ improvement on CIFAR-100, making it surpass the well-tuned Spikformer [8] baseline by $0.97\%$ . + +Overall, this work provides further theoretical and empirical evidence supporting the view that high-frequency information is essential for SNNs: + +- We provide the first theoretical proof that spiking neurons inherently act as low-pass filters at the network level, revealing their tendency to suppress high-frequency features. +- We propose Max-Former, which restores high-frequency information in Spiking Transformers via two lightweight modules: extra Max-Pool in patch embedding and Depth-Wise Convolution (DWC) in place of early-stage self-attention. +- Restoring high-frequency information significantly improves performance while saving energy cost. On ImageNet, Max-Former achieves $82.39\%$ top-1 accuracy $(+7.58\%)$ over Spikformer with $30\%$ energy consumption and lower parameter count (63.99M vs. 66.34M). +- Extending the insight beyond transformers, Max-ResNet-18 achieves state-of-the-art performance on convolution-based benchmarks: $97.17\%$ on CIFAR-10 and $83.06\%$ on CIFAR-100. + +We believe this straightforward yet powerful solution will motivate future research to explore the unique properties of SNNs, beyond the established practice in standard deep learning. + +# 2 Preliminary and Related Works + +# 2.1 Spiking Neuron Models + +SNNs implement spike-driven processing through biologically-inspired neuron models for non-linear activations. The Leaky Integrate-and-Fire (LIF) model is a widely adopted abstraction of this behavior, offering an effective balance between biological plausibility and computational efficiency [20]. The discretized LIF model under each simulation timestep $n$ can be formulated as: + +$$ +U [ n ] = f (V [ n - 1 ], I [ n ]), \tag {1} +$$ + +$$ +S [ n ] = H \left(U [ n ] - V _ {\mathrm {t h}}\right), \tag {2} +$$ + +$$ +V [ n ] = \left\{ \begin{array}{l l} U [ n ] - V _ {\text {t h}}, & S [ n ] = 1, \\ U [ n ], & S [ n ] = 0, \end{array} \right. \tag {3} +$$ + +where $\beta$ is the decay factor, $V_{\mathrm{th}}$ is the firing threshold, and $H(\cdot)$ refers to the Heaviside step function that determining spike generation: $S[n] = H\big(U[n] - V_{\mathrm{th}}\big)\big) = 1$ when $U[n]\geq V_{\mathrm{th}}$ , otherwise remains inactive ( $S[n] = 0$ ). The charging process of the LIF neuron is determined by $f(\cdot)$ : + +$$ +f (V [ n - 1 ], I [ n ]) = \beta V [ n - 1 ] + (1 - \beta) I [ n ] \tag {4} +$$ + +At each timestep $n$ , the current membrane potential $U[n]$ is updated by integrating the time-domain signal $I[n]$ corresponding to input data or intermediate operations like Conv and MLP. If $U[n]$ exceeds the threshold $V_{\mathrm{th}}$ , the neuron fires a spike $(S[n] = 1)$ . $V[n]$ records membrane potential over time given the decay factor $\beta$ and output spike activity. If the neuron does not fire, then $V[n] = U[n]$ . Notably, the LIF model simplifies to integrate-and-fire (IF) neurons when eliminating the membrane potential decay process between timesteps. Its charging process can be formulated as: + +$$ +f (V [ n - 1 ], I [ n ]) = V [ n - 1 ] + I [ n ] \tag {5} +$$ + +# 2.2 Spiking Neural Networks + +Drawing inspiration from biological neurons, SNNs extend conventional ANNs by incorporating temporal dynamics and discrete spike-based communication [5]. Leveraging this spike-driven mechanism, neuromorphic chips implement computation through event-driven spike routing and accumulation, which substitutes energy-intensive matrix-vector multiplications [7, 6]. This facilitates high parallelism, scalability, and exceptional power efficiency, with power consumption typically in the range of tens to hundreds of milliwatts [21]. + +Recently, the development of modern SNNs, e.g., Spiking Transformer, has demonstrated both attractive performance and reduced energy consumption [8, 9, 22]. Spikformer [8] pioneered the spike-based self-attention mechanism called Spiking Self Attention (SSA) that utilizes sparse spikeform Query, Key, and Value vectors to eliminate the need for energy-intensive softmax operations. Following its success, many works endeavor to enhance Spiking Transformers by adapting advanced ANN Transformer architectures [23, 24] or devising complicated spike coding mechanisms to reduce representation error (e.g., multi-threshold [25]/ multi-spike neurons [26, 27]). + +Here, we instead address a fundamental question: What truly limits SNNs' performance compared to ANNs? Our investigation reveals that the answer lies in frequency properties — specifically, that spiking neurons function as low-pass filters, impeding the propagation of high-frequency detail + +within the network. In addition to theoretical proof, we probe the importance of high-frequency information through our Max-Former, which features two frequency-enhancing operators: Max-Pool in patch embedding and DWC in place of early-stage self-attention. We further validate this principle in convolutional architectures with our proposed Max-ResNet. + +# 3 Methods + +In this section, we first present a theoretical analysis of the frequency properties of spiking neurons. We show that, although the raw output spike trains of spiking neurons appear spectrally all-pass due to their impulse-shaped spike waveform, the resulting high-frequency components are merely superficial and cannot be propagated through the network. In fact, spiking neurons act as low-pass filters at the network level. This is a fundamental problem that has been overlooked in previous works. Building on this insight, we probe the importance of high-frequency information in SNNs through Max-Former, which strategically employs high-pass operators (Max-Pool and DWC) to restore high-frequency details and avoid feature degradation. + +![](images/4b4373b064e37c8145360218616a8dd35af0c659fd95bffb6ec7b88060f5396a.jpg) +Figure 3: Time-frequency analysis of ReLU and spiking neurons. (a) Time-domain signals of input $x(t) = \frac{1}{3} (\sin (2\pi \cdot 100t) + \sin (2\pi \cdot 200t) + \sin (2\pi \cdot 300t))$ (blue), ReLU-processed: $r(t)$ (red), spiking output of a LIF neuron with the $\beta = 0.25$ : $s(t)$ (green). (b) Fourier analysis of $x(t)$ , $r(t)$ , and $s(t)$ . (c) Fourier analysis of linear transformed (CONV/MLP) activations, where ReLU expands the frequency bandwidth of the input signal, while the spiking neuron shows high-frequency attenuation. + +# 3.1 Spiking Neurons are Low-pass Filters + +We begin with an intuitive time-frequency analysis using an input $x(t) = \frac{1}{3} (\sin (2\pi \cdot 100t) + \sin (2\pi \cdot 200t) + \sin (2\pi \cdot 300t))$ as shown in Figure 3. The results reveal three key observations: (1) In the time domain, the ReLU output $(r(t))$ perfectly follows $x(t) > 0$ while spiking neurons selectively respond to $100\mathrm{Hz}$ (Figure 3 (a)); (2) However, the spiking outputs' spectral response $|\mathrm{S}(\mathrm{f})|$ still appears nearly all-pass, which contradicts the low-frequency behavior observed in the time domain. These spurious high-frequency components actually arise from the impulse-shaped spike waveform itself rather than from genuine (Figure 3 (a-b)); (3) The waveform-induced high-frequency components cannot be propagated across layers, resulting in a network-level low-pass behavior. When considering the whole process from input $\rightarrow$ activation $\rightarrow$ linear transform, ReLU expands the frequency bandwidth of $x(t)$ [28], whereas the spiking neuron exhibits strong high-frequency attenuation (Figure 3 (c)). + +We first examine the charging process of spiking neurons to theoretically analyze their frequency-selective properties. Given Equ. (3) and Equ. (4), this can be formulated as: + +$$ +V [ n ] = \beta V [ n - 1 ] + (1 - \beta) I [ n ] \tag {6} +$$ + +Applying the Z-transform with $\mathcal{Z}\{V[n - 1]\} = z^{-1}V(z)$ yields: + +$$ +V [ z ] = \beta z ^ {- 1} V [ z ] + (1 - \beta) I [ z ] \tag {7} +$$ + +which can be rearranged to formulate the transfer function from input current to membrane potential: + +$$ +H (z) = \frac {V (z)}{I (z)} = \frac {1 - \beta}{1 - \beta z ^ {- 1}}, \quad 0 \leq \beta < 1 \tag {8} +$$ + +Equ. (8) is exactly the form of a first-order infinite-impulse-response (IIR) low-pass filter with a single pole at $z = \beta$ . Accordingly, as $\beta$ , i.e., the pole, approaches 1, LIF neurons exhibit stronger + +low-frequency selectivity. Notably, the decay factor $\beta$ correlates to the membrane time constant $\tau$ as $\beta = 1 - \frac{1}{\tau}$ , with $\tau$ ranges from 1 to $+\infty$ , a smaller $\tau$ yields a smaller $\beta$ . From an intuitive standpoint, a shorter time constant allows the membrane potential to respond within narrower temporal windows, rendering the neuron more sensitive to higher-frequency inputs. + +From the individual neuron to the network-level information transmission, the average membrane potential is positively and consistently correlated with the possibility of spike firing during operation. We approximate the inherently nonlinear spike-generation process as linear around the firing threshold $V_{th}$ . Denoting the firing rate by $fr(V)$ , we define the local gain $k$ and approximate the Z-domain spike train $S(z)$ by: + +$$ +k = \left. \frac {\partial f r}{\partial V} \right| _ {V = V _ {t h}} \tag {10} +$$ + +When output spike trains are weighted by a causal synaptic kernel $w[n]$ , the Z-transformed output current $(y[n] = w[n] * s[n])$ can be denoted as $Y(z) = W(z)S(z)$ . The overall input-to-output transfer function is obtained by combining this with (8) and (10): + +$$ +H ^ {\prime} (z) = \frac {Y (z)}{I (z)} = S (z) W (z) H (z) = k W (z) \frac {1 - \beta}{1 - \beta z ^ {- 1}}. \tag {11} +$$ + +The first-order low-pass IIR characteristics of $H(z)$ make the system $Y(z)$ inherently favor low-frequency signal components, regardless of whether the synaptic kernel $W(z)$ or the spike coding process $S(z)$ changes the gain or phase response. The low-pass term $\left(\frac{1 - \beta}{1 - \beta z^{-1}}\right)^L$ further amplifies the system's frequency selectivity when the process $H'(z)$ is cascaded $L$ times (layers). The complete formula is as follows: + +$$ +H _ {L} ^ {\prime} (z) = \frac {Y _ {L} (z)}{I (z)} = \prod_ {i = 1} ^ {L} \left[ S _ {i} (z) W _ {i} (z) H (z) \right] = \left(\prod_ {i = 1} ^ {L} k _ {i} W _ {i} (z)\right) \left(\frac {1 - \beta}{1 - \beta z ^ {- 1}}\right) ^ {L} \tag {12} +$$ + +In the special case of the non-leaky IF neurons, which obey the charging process in (5), $H(z)$ is formulated as: + +$$ +H (z) = \frac {1}{1 - z ^ {- 1}}, \tag {13} +$$ + +This corresponds to an ideal discrete-time low-pass filter with a pole at $z = 1$ , which can yield a consistent conclusion with our previous analysis. + +# 3.2 Max-Former + +It remains unclear whether high-frequency information is truly important for SNNs and whether restoring it can improve performance. Therefore, we systematically investigate the low-pass filtering characteristics of spiking neurons through Max-Former. To decouple frequency effects from model complexity, we: (1) replace self-attention with high-frequency-preserving DWC in the early stages, and (2) add Max-Pool in patch embedding to compensate spiking neurons' low-pass preference. Notably, compared to the quadratic computational complexity of self-attention, DWC and Max-Pool only require linear complexity relative to the sequence length and are more parameter-efficient. We consistently adopt the LIF neuron model throughout this work. + +# 3.2.1 Overall Architecture + +Figure 4(a) illustrates the overall framework of Max-Former. The architecture consists of 3 stages with $\frac{H}{4} \times \frac{W}{4}$ , $\frac{H}{8} \times \frac{W}{8}$ , and $\frac{H}{16} \times \frac{W}{16}$ tokens respectively, where $H$ and $W$ denote the height and width of the input image. Critically, MaxFormer processes information through discrete spikes over time. This spike-driven computing paradigm supports two types of input: + +(1) Event Streams: Asynchronous events $e = [x,y,t,p]$ containing spatial coordinates $(x,y)$ , times-stamp $t$ , and polarity $p$ are converted to event frames through temporal binning. Given original resolution $dt_{o}$ and target $dt = \alpha dt_{o}$ , events are aggregated over $\alpha$ consecutive bins: + +$$ +\alpha (t + 1) - \overleftarrow {1} +$$ + +$$ +I _ {t} = \sum_ {k = \alpha t} S _ {k} \in \mathbb {R} ^ {2 \times h \times w} \tag {14} +$$ + +where $S_{k}$ denotes raw event data. The whole process denoises raw events and converts them to frame sequences at target temporal resolutions. + +(2) Static Images: Conventional images are converted to spike sequences by: 1) Repeating static frames $T$ times, 2) Encoding pixel intensities to spikes using spiking neurons. The resulting input is formulated as: $I = \text{Spiking\_Embed}(\{I_t\}_{t=1}^T)$ , which contains identical information across timesteps. + +![](images/fe56a67eb17c057439b48b2c8362e5cc793411206ad3c7c11c21a9f96ca496d5.jpg) + +![](images/2ab978976e05d6f4dc4545040480b8f1b4d77421eaa8ea9b859b1606940ecf1a.jpg) +Figure 4: (a) Overview of Max-Former: we restore high-frequency signals by using lightweight DWCs instead of self-attention in the early stages. Following the hierarchical design of [29], Max-Former adopts a 3-stage architecture. $D_{i}$ : feature dimensions of stage- $i$ . (b) In Max-Former's patch embedding stage, we propose three configurations (Embed-origin, Embed-Max, and Embed-Max+) to enhance high-frequency components. + +# 3.2.2 Patch Embedding + +To transform the input into a tokenized representation, given input $\{S\} \in \mathbb{R}^{T\times C\times H\times W}$ , the process of patch embedding is formulated as: + +$$ +\mathbf {Y} = \left(\mathcal {G} _ {1} (\{S \}) + G _ {2} (\{S \}), \quad \mathbf {Y} \in \mathbb {R} ^ {T \times C ^ {\prime} \times H ^ {\prime} \times W ^ {\prime}} \right. \tag {15} +$$ + +where $C' = 2C$ , and $H' = \lfloor H / P \rfloor$ , $W' = \lfloor W / P \rfloor$ with the patch size $P = 4$ . To address spiking neurons' inherent frequency preference, we present three patch embedding configurations as shown in Figure 4(b): + +$$ +\operatorname {E m b e d - O r i g}: \left(\mathcal {G} _ {1}, \mathcal {G} _ {2}\right) = (\operatorname {E m b e d}, \operatorname {E m b e d}) \tag {16} +$$ + +$$ +\operatorname {E m b e d} - \operatorname {M a x}: \left(\mathcal {G} _ {1}, \mathcal {G} _ {2}\right) = (\operatorname {M a x} - \operatorname {E m b e d}, \operatorname {E m b e d}) \tag {17} +$$ + +$$ +\operatorname {E m b e d} - \operatorname {M a x} +: \left(\mathcal {G} _ {1}, \mathcal {G} _ {2}\right) = (\operatorname {M a x} - \operatorname {E m b e d}, \operatorname {M a x} - \operatorname {E m b e d}) \tag {18} +$$ + +where: Embed $\equiv$ {LIF - CONV - BN}, and Max-Embed $\equiv$ {LIF - CONV - BN - MaxPool}. + +# 3.2.3 Token Mixing + +In Transformers, lower layers typically require more high-frequency details, while higher layers benefit from more global information [30, 31]. Like biological vision, high-frequency details enable early stages to learn low-level features while progressively building local-to-global representations. Accordingly, we replace early-stage self-attention with DWC to preserve high frequencies essential for local feature learning. Given input embedding $\mathbf{Y} \in \mathbb{R}^{T \times C \times H \times W}$ , the spiking DWC is defined as: + +$$ +\mathbf {Z} _ {c} (\mathbf {Y}) [ i ] = \operatorname {L I F} \left(\sum_ {j \in \Omega (i)} w _ {c, j} \cdot \mathbf {Y} _ {c} [ j ]\right) \tag {19} +$$ + +where $\Omega(i)$ denotes the local neighborhood of position $i$ , $w_{c,j}$ represents learnable convolution weights for channel $c$ , and $X_{c}, Z_{c} \in \mathbb{R}^{T \times H \times W}$ is the input and output slice for channel $c$ . For the final stage, we implement token mixing via Spiking Self-Attention (SSA) [8]. The SSA computation follows: + +$$ +\mathbf {Z} = \operatorname {L I F} (\mathrm {B N} (\mathbf {Y W})), \quad \mathbf {Z} \in \{Q, K, V \} \tag {20} +$$ + +$$ +\operatorname {S S A} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) = \operatorname {L I F} \left(\mathbf {Q} \mathbf {K} ^ {\mathrm {T}} \mathbf {V} \cdot s\right) \tag {21} +$$ + +where $\mathbf{Q},\mathbf{K},\mathbf{V}\in \mathbb{R}^{T\times N\times H\times W}$ are spike-form tensors generated by learnable linear layers, $s$ is a scaling factor. SSA eliminates floating-point multiplications, ensuring spike-driven compatibility. + +# 3.3 Membrane Shortcut + +Residual learning and shortcuts enable very deep networks to be trained effectively by providing identity shortcuts that preserve information flow and mitigate vanishing gradients [32]. For SNNs, a crucial consideration is maintaining compatibility with the spike-driven computing paradigm throughout all operations. As Fig. 5 shows, existing SNN shortcut implementations fall into three categories: (1) Vanilla Shortcut [33], (2) Pre-Spike Shortcut [34], and (3) Membrane Shortcut [35]. + +![](images/9ddba90875394622288ae536b95f34f6b2fa10a840128ec1fb99a7d3dc75cf72.jpg) +Figure 5: Shortcut connection in SNNs. (Left) Vanilla Shortcut that combines spike and membrane potential. (Middle) Pre-Spike Shortcut that adds spike signals before neuron charging. (Right) Membrane Shortcut that directly connects membrane potentials, ensuring identical potential mapping while strictly preserving the spike-driven computing paradigm throughout the network. + +The Vanilla Shortcut scheme [33] directly connects spikes (binary) to membrane potentials (continuous), leading to a distribution mismatch that inherently violates the identity mapping principle. The Pre-Spike Shortcut [34] adds spike signals before neuron charging, resulting in summation values that range from 0 to 2. This disrupts the binary spike representation and spike-driven flow in SNNs. + +In this work, we adopt the Membrane Shortcut [35] for its dual advantages: it preserves identity mapping by directly connecting membrane potentials while maintaining binary spike outputs that remain inherently compatible with spike-driven computation. Unlike Vanilla or Pre-Spike Shortcuts, this approach ensures both mathematical consistency with residual learning principles and seamless compatibility with spike-driven operations. We provide a detailed analysis of its impact on model performance and energy costs in Appendix D. + +# 4 Experiment + +As shown in Figure 6, Max-Former restores high-frequency information by grafting the merits of frequency-enhancing operators: Max-Pool and DWC. To empirically probe the importance of high-frequency information in Spiking Transformers, we evaluate Max-Former through comprehensive experiments on static datasets (CIFAR-10 [36], CIFAR-100 [37] and ImageNet [38]) and neuromorphic datasets (CIFAR10-DVS [39], DVS128 Gesture [40]), with architecture configurations detailed in Table 1. In addition, we design Max-ResNet to further investigate the effect of high-frequency restoration in convolutional architectures (model implementation is detailed in Appendix A.3). Experimental settings and energy analysis methods are detailed in Appendix A and B. + +![](images/0e2dd2f157fab2623da1901cb35afc8ae4e4d6ea9cda11835717150d8a32d142.jpg) +Spiking Neuron + +![](images/ab557292450bf616a88029336e758baaecab09d6d7a55d3f892b6428ad782d91.jpg) +Max-Pooling + +![](images/64d66c4c27b9cc0e32aa1acb610383108fc057d206445c5862e5e0bf285ac3ba.jpg) +Depth-Wise Conv +Figure 6: Fourier spectrum of Spiking Neurons, Spiking Max-Pool, Spiking Depth-Wise Convolution and Spiking Self-attention. + +![](images/9aa93b143f6c1a538980f4d173aaecc704f764f8a63ceb787ee5c3c72cfa1c7d.jpg) +Self-Attention + +Table 1: Max-Former architecture configurations for different classification tasks. Notation DWC-N denotes depth-wise convolution with kernel size $N\times N$ . For block settings: CIFAR-10/100: 3 stages (1/1/2 blocks); ImageNet: 3 stages (1/3/7 blocks); Neuromorphic: 2 stages (1/1 blocks). + +
DatasetStage 1Stage 2Stage 3
Patch EmbedToken MixPatch EmbedToken MixPatch EmbedToken Mix
Cifar10 [36]/100 [37]Embed-OrigIdentityEmbed-MaxDWC-3Embed-MaxSSA
ImageNet [38]Embed-OrigDWD-7Embed-MaxDWC-5Embed-MaxSSA
Neuromorphic [39, 40]Embed-Max+DWC-3Embed-MaxSSA——
+ +# 4.1 Results on CIFAR and Neuromorphic Datasets + +As shown in Table 2, Max-Former delivers performance improvements across both static datasets (CIFAR10/CIFAR100) and neuromorphic datasets (DVS128/CIFAR10-DVS). Notably, for CIFAR10/100 classification, its first stage only uses identity mapping for token mixing (Table 1, yet still attains attractive results. Max-Former achieves $97.04\%$ accuracy on CIFAR10 with only 6.57M parameters at $\mathrm{T} = 4$ , surpassing Spikformer $(95.51\%)$ , 9.32M), S-Transformer $(95.60\%)$ , 10.28M), and QKFormer $(96.18\%)$ , 6.74M). Similarly, on CIFAR100, Max-Former attains $82.65\%$ accuracy, significantly outperforming Spikformer $(78.21\%)$ , S-Transformer $(78.40\%)$ , and QKFormer $(81.57\%)$ + +Max-Former and QKFormer share a similar hierarchical architecture, though QKFormer originally employs pre-spike shortcuts [24]. For a fair comparison, we additionally implemented QKFormer with the Membrane Shortcut (denoted as MS-QKFormer in the table) using identical training configurations. Max-Former still outperforms MS-QKFormer by $0.2\%$ on CIFAR10 (97.04% vs. 96.84%) and by $1.08\%$ on CIFAR100 (82.65% vs. 81.57%), while requiring slightly fewer parameters (6.57M vs. 6.74M). For neuromorphic datasets, Max-Former maintains this performance advantage. On DVS128, it achieves $98.6\%$ accuracy, matching MS-QKFormer with the membrane shortcut. On CIFAR10-DVS, Max-Former reaches $84.2\%$ accuracy, exceeding MS-QKFormer (82.3%) by $1.9\%$ and surpassing other spike-driven models like S-Transformer (80.0%) and SWformer (83.9%). + +Table 2: Performance Comparison on CIFAR10 [36], CIFAR100 [37], DVS128 [40] and CIFAR10-DVS [39]. Param.: Parameter (M); Acc.:Top-1 Accuracy $(\%)$ $T$ : Simulation Timestep. Models trained from scratch using identical configurations. + +
MethodCIFAR10CIFAR100DVS128CIFAR10-DVSMembrane Shortcut
Param.TAcc.Param.TAcc.Param.TAcc.Param.TAcc.
ResNet-19 (ANN) [24]12.63194.9712.63175.35———————
Max-Former (ANN)6.57196.826.60182.41———————
Spikformer [8]9.32495.519.32478.212.571698.32.571680.9X
S-Transformer [41]10.28495.6010.28478.402.571699.32.571680.0✓
SWformer [19]7.51496.107.51479.30---2.051683.9✓
QKFformer [24]6.74496.186.74481.151.501698.61.501684.0X
MS-QKFormer*6.74496.846.74481.571.501698.61.501682.3✓
Max-Former*6.57497.046.60482.651.451698.61.451684.2✓
+ +Table 3: ImageNet performance comparison. Notation: A2S: ANN-to-SNN conversion; Model-L-D: models with L blocks and D channels. Input resolution: ${224} \times {224}$ . *Identical training configurations. + +
MethodsTypeArchitectureParam (M)Power (mJ)Time StepTop-1 Acc (%)Membrane Shortcut
ViT [42]ANNViT-L/16304.380.96179.70—
DeiT [43]ANNDeiT-B86.680.50181.80—
PVT [44]ANNPVT-Large61.445.08181.70—
MST [9]A2SSwin Transformer-T28.50—51278.51X
Spikformer [8]SNNSpikformer-8-38416.817.73470.24X
Spikformer-8-76866.3421.48474.81X
S-Transformer [41]SNNS-Transformer-8-38416.813.90472.28✓
S-Transformer-8-76866.346.10476.30✓
Meta-Spikformer [23]SNN—31.37.80175.4✓
—31.332.80477.2✓
SWformer [19]SNNSWformer-8-51227.65.08475.43✓
QKFormer [24]SNNHST-10-38416.4715.13478.80X
HST-10-76864.968.52181.69X
MS-QKFormer*SNNHST-10-38416.475.52476.48✓
HST-10-76864.966.79177.78✓
Max-Former*SNNMax-10-38416.234.89477.82✓
Max-10-51228.652.50175.47✓
Max-10-51228.657.49479.86✓
Max-10-76863.995.27178.60✓
Max-10-76863.9914.87482.39✓
+ +# 4.2 Results on ImageNet Classification + +Table 3 shows Max-Former's performance on ImageNet classification, demonstrating its effectiveness for complex visual tasks. Max-Former-10-768 (T=4) achieves $82.39\%$ accuracy $(+7.58\%$ over Spikformer) with $30\%$ lower energy (14.87mJ vs 21.48mJ), despite using only lightweight for early-stage token mixing. It also outperforms the ANN-to-SNN MST model $(78.51\%)$ that requires 512 timesteps. Training/inference speed and memory usage are analyzed in Appendix C. + +Our analysis focuses on models using the Membrane Shortcut, which eliminates the energy-inefficient ternary spike transmissions ( $\{0,1,2\}$ in Pre-Spike Shortcut) while maintaining full compatibility with standard neuromorphic hardware without additional hardware overhead (see Appendix D). For fair comparison, we implemented a membrane potential variant of QKFormer (denoted MS-QKFormer), where MS-QKFormer shows $64\%$ lower energy ( $5.52\mathrm{mJ}$ vs $15.13\mathrm{mJ}$ ) in HST-10-384. + +Max-Former-10-384 (16.23M, T=4) achieves 77.82% accuracy, outperforming MS-QKFormer-10-384 (16.47M, 76.48%), S-Transformer-8-768 (66.34M, 76.3%), and Meta-Spikformer (31.3M, 77.2%). For energy efficiency under identical settings, Max-Former-10-384 consumes 4.89mJ, significantly lower than MS-QKFormer-10-384(5.52mJ), S-Transformer(6.10mJ), and Meta-Spikformer (32.8mJ). Compared to conventional ANN models, Max-Former demonstrates concrete advantages in energy efficiency while maintaining competitive accuracy. Specifically, when compared to PVT-Large (a representative hierarchical ANN), Max-Former-10-768 (T=4) achieves comparable accuracy (82.34% vs. 81.70%) with 67% lower energy consumption (14.87mJ vs. 45.08mJ). These results confirm the importance of high-frequency information in Spiking Transformers: replacing energy-intensive self-attention with lightweight DWC in early stages actually produces better performance. + +Table 4: Ablation of Patch-Embedding/ Token-Mixing Strategies on CIFAR100 and CIFAR10-DVS. + +
CIFAR100CIFAR10-DVS
Patch EmbedToken MixAcc (%)Patch EmbedToken MixAcc (%)
Orig/Max/MaxIdentity/DWC-3/SSA82.65Max+/MaxDWC-3/SSA84.2
Orig/Orig/OrigIdentity/DWC-3/SSA81.63Orig/OrigDWC-3/SSA79.2
Orig/Max/OrigIdentity/DWC-3/SSA81.88Orig/MaxDWC-3/SSA81.5
Orig/Max/MaxIdentity/Identity/SSA81.28Max+/MaxDWC-1/SSA81.2
Orig/Max/MaxIdentity/DWC-5/SSA82.02Max+/MaxDWC-5/SSA82.7
Orig/Max/MaxDWC-7/DWC-5/SSA82.42Max+/MaxDWC-7/SSA82.1
Orig/Max/MaxSSA/SSA/SSA82.23Max+/MaxSSA/SSA83.9
Orig/Orig/OrigSSA/SSA/SSA81.43Orig/OrigSSA/SSA79.8
+ +# 4.3 Ablation Study + +We provide direct evidence for the critical role of high-frequency information in Spiking Transformers through in-depth ablation studies. + +(1) Ablation on Patch Embedding Strategies: Proper patch embedding strategies help to unlock the performance limit of Spiking Transformers. On CIFAR100, changing Max-Former's patch embedding strategy from the proposed Embed-Orig/Embed-Max/Embed-Max to the default Embed-Orig/Embed-Orig/Embed-Orig configuration reduces performance from $82.65\%$ to $81.63\%$ . Neuromorphic datasets exhibit stronger dependence on high-frequency components. On CIFAR10-DVS, replacing the first stage's Embed-Max+ with Embed-Orig causes a significant performance drop from $84.2\%$ to $81.5\%$ . Incorporating high-frequency information through patch embedding proves effective. When purely using SSA token mixing, optimizing the patch embedding strategy can improve accuracy by $+0.8\%$ on Cifar 100 and $+4.1\%$ on CIFAR10-DVS, highlighting its critical role in spiking architectures. + +(2) Ablation on Token Mixing Strategies: Max-Former further restores high-frequency information through the token mixing strategy. Despite SSA taking higher energy/parameter costs, Max-Former achieves better performance by simply replacing early-stage self-attention with lightweight DWC. On CIFar100, this substitution leads to $+0.42\%$ performance gain (All SSA: $82.23\%$ vs. Max-Former: $82.65\%$ ). On CIFAR10-DVS, Max-Former with hybrid token mixing (DWC-3 + SSA) achieves $84.2\%$ accuracy, outperforming the full-SSA variant $(83.9\%)$ by $0.3\%$ . The kernel size selection for high-frequency preservation is also of great significance. Larger kernels (DWC-5/7) degrade performance $(-0.63\%$ CIFAR100, $-2.1\%$ CIFAR10-DVS) due to excessive feature smoothing. Insufficient filtering + +with DWC-1 also underperforms (81.2% vs 84.2% on CIFAR10-DVS). This highlights the necessity of balancing high-frequency and low-frequency components in Spiking Transformers. + +Overall, Max-Former's performance gains stem from effective high-frequency information propagation, independent of parameter count, which is evidenced by: (1) Max-pooling patch embedding (Embed-Max/Embed-Max+) consistently outperforms original versions despite similar parameter budgets. (2) Larger DWC kernels (DWC-5/DWC-7) increasing parameters but degrading accuracy $(-0.63\%$ on CIFAR100, $-2.1\%$ on CIFAR10-DVS vs Max-Former with DWC-3). See Appendix D for visualizations and $\mathbf{E}$ for limitations discussion. + +# 4.4 Generality across Convolutional Architectures + +We further extend the effectiveness of Max-Former to convolutional architectures by proposing Max-ResNet. The key modification in our Max-ResNet lies in the inclusion of only two additional max-pooling operations compared to MS-ResNet [45]. The detailed implementation of Max-ResNet is provided in Appendix A.3. Training settings are listed in Appendix A. + +High-frequency information is essential for SNNs. As shown in Table 5, Max-ResNet achieves a remarkable performance improvement over the baseline MS-ResNet, despite having identical model sizes. Specifically, with the block configuration [2, 2, 2, 2], Max-ResNet improves the CIFAR-10 accuracy by $+2.41\%$ (from $94.4\%$ to $96.81\%$ ) and the CIFAR-100 accuracy by $+6.48\%$ . Similarly, under the [3, 3, 2] configuration, the accuracy increases by $+2.25\%$ and $+6.65\%$ on CIFAR-10 and CIFAR-100, respectively. + +In short, Max-ResNet-18 sets state-of-the-art benchmarks across convolutional baselines, with only a moderate model size and an extremely straightforward high-frequency restoration strategy. Therefore, preserving high-frequency information is fundamental to effective feature representation in SNNs, regardless of architecture. + +Table 5: Comparison of different ResNet architectures on CIFAR-10 and CIFAR-100. + +
ArchitectureTraining MethodBlock Config.Params (M)Time StepCIFAR-10 Acc. (%)CIFAR-100 Acc. (%)
KDSNN-ResNet-18 [46]Knowledge Distillation[2, 2, 2, 2]11.22495.7278.46
MS-ResNet-18 [45]Direct Training[2, 2, 2, 2]11.22494.4075.06
[3, 3, 2]12.50494.9276.41
MS-ResNet-34 [45][2, 2, 2, 2]21.33494.6975.34
Max-ResNet-18Direct Training[2, 2, 2, 2]11.22496.81 (+2.41)81.54 (+6.48)
[3, 3, 2]12.50497.17 (+2.25)83.06 (+6.65)
+ +# 5 Conclusion + +This work challenges the prevailing assumption that binary activation constraints are the primary cause of SNNs' performance gap. Through theoretical analysis and empirical validation, we demonstrate for the first time that spiking neurons inherently function as low-pass filters at the network level, resulting in the rapid attenuation of high-frequency components that critically degrade feature representation. We demonstrate that high-frequency information is crucial for effective spiking computation: Max-Former (63.99M parameters) achieves $82.39\%$ top-1 accuracy on ImageNet, outperforming Spikformer (74.81%, 66.34M parameters) by $+7.58\%$ , while reducing energy consumption by $30\%$ at a comparable model size; Max-ResNet-18 further achieves state-of-the-art performance among convolutional baselines $-97.17\%$ on CIFAR-10 and $83.06\%$ on CIFAR-100. Notably, all these improvements are achieved through extremely simple modifications, even slightly reducing the overall model size. We believe this simple yet effective solution will motivate future research to explore the unique properties of SNNs, beyond the established practices in ANN studies. + +# 6 Acknowledgements + +This work is supported by the Major Science and Technology Innovation 2030 "Brain Science and Brain-like Research" key project(No.2021ZD0201405), the Guangzhou-HKUST(GZ) Joint Funding Program (Grant No. 2023A03J0682), the National Natural Science Foundation of China (Grant No. 62405255), GuangDong Basic and Applied Basic Research Foundation (No. 2023A1515110679), and partially supported by the collaborative project with Brain Mind Innovation, Inc + +# References + +[1] Weihao Yu, Mi Luo, Pan Zhou, Chenyang Si, Yichen Zhou, Xinchao Wang, Jiashi Feng, and Shuicheng Yan. Metaformer is actually what you need for vision. 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In International Conference on Learning Representations, 2020. + +# A Analysis of High-Frequency Information in Spiking Transformers + +To validate our approach, we conduct experiments across three static and two neuromorphic datasets. In this section, we first provide the detailed experimental setup used to obtain the results presented in our paper. We then perform additional analysis on the importance of high-frequency information in Spiking Transformers. For complete parameter configurations, please refer to our public code repository at https://github.com/bic-L/MaxFormer. + +Table 6: Hyperparameters for image classification on different datasets. + +
Hyper parametersImageNetCIFAR-10CIFAR-100Neuromorphic
Model Size10–384 / 10–512 / 10–7684–3844–3842–256
Epochs200400400106
Resolution224 × 22432 × 3232 × 32128 × 128
Batch Size512 (8 × 64)1286416
OptimizerAdamWAdamWAdamWAdamW
Learning rate1.2 × 10-3(T=1) +1.35 × 10-3(T=4)1.50 × 10-31.50 × 10-36.00 × 10-3
Learning rate decayCosineCosineCosineCosine
Warmup epochs5202010
Weight decay0.050.060.060.06
Rand Augment9 / 0.59-n1 / 0.49-n1 / 0.4—
Mixup0.25 / 0.4/ 0.80.50.750.5
CutMix10.50.5—
Mixup prob0.5110.5
Erasing prob0.00.250.25—
Label smoothing0.10.10.10.1
+ +# A.1 Experimental Details + +Datasets: We evaluate Max-Former through comprehensive experiments on static datasets (CIFAR10 [36], CIFAR-100 [37] and ImageNet [38]) and neuromorphic datasets (CIFAR10-DVS [39], DVS128 Gesture [40]). The training and inference pipeline are implemented in SpikingJelly [47]. + +Static Datasets: For static image classification, we evaluate on three standard benchmarks. ImageNet-1k [38] is one of the most widely used datasets in computer vision. It contains 1.28 million training, 50,000 validation, and 100,000 test images covering the common 1K classes. Both CIFAR-10 [36] and CIFAR-100 [37] include 50,000 training images and 10,000 testing images with $32 \times 32$ resolution. The main difference between them is that CIFAR-10 has 10 categories for classification, while CIFAR-100 has 100 categories. + +Neuromorphic Datasets: For event-based vision tasks, we evaluate on two standard benchmarks. CIFAR10-DVS [39] is an event-based version of the CIFAR-10 dataset, created by capturing moving image samples using the Dynamic Vision Sensor (DVS). It includes 10,000 event-based images $(128\times 128$ pixels) spread across 10 classes, with 9,000 samples for training and 1,000 for testing. The DVS128 Gesture dataset [40] contains 1,342 event-based recordings of 11 different hand gesture types performed by 29 people under 3 different lighting conditions. Each gesture recording lasts about 6 seconds on average. + +Hyper Parameters: Our training scheme mainly follows [24] and [41]. Specifically, MixUp [48], CutMix [49] and RandAugment [50] are used for data augmentation. The models are trained using AdamW optimizer [51] with the weight decay of 0.05 for ImageNet-1K classification tasks and the weight decay of 0.06 for all other datasets. Label Smoothing [52] is set as 0.1. Detailed training + +hyperparameters are shown in Table 6. For our ImageNet experiments, we used 8 NVIDIA A30 GPUs to train most models. However, for the MaxFormer-10-512 (T=4) and MaxFormer-10-768 (T=4) models, we used 8 NVIDIA H20 GPUs instead. For the smaller datasets (CIFAR10, CIFAR100, DVS128 Gesture, and CIFAR10-DVS), we used a single A30 GPU for training. + +# A.2 Impact of High-Frequency Information on Model Performance + +Table 7: Patch embedding and token mixing schemes on CIFAR-100. Acc.:Top-1 Accuracy $(\%)$ . DWC-N: spiking depth-wise convolution with kernel size $N\times N$ SSA: Spiking Self-Attention. + +
Patch EmbedToken MixAcc. (%)
(1)Embed-Origin/Embed-Origin/Embed-OriginAvg-Pool/Avg-Pool/Avg-Pool76.73
Max-Pool/Max-Pool/Max-Pool79.12
(2)Embed-Origin/Embed-Origin/Embed-MaxIdentity/Identity/Identity80.11
Embed-Origin/Embed-Max/Embed-MaxIdentity/Identity/Identity80.46
(3)Embed-Origin/Embed-Max/Embed-MaxAvg-Pool/Avg-Pool/Avg-Pool77.61
Max-Pool/Max-Pool/Max-Pool79.78
Identity/Max-Pool/Identity79.99
Identity/Identity/Max-Pool80.12
(4)Embed-Origin/Embed-Max/Embed-MaxSSA/SSA/SSA82.13
DWC-3/DWC-5/SSA82.36
DWC-5/DWC-3/SSA82.46
DWC-5/DWC-7/SSA82.45
DWC-7/DWC-5/SSA82.42
DWC-3/DWC-3/SSA82.59
Identity/DWC-3/SSA82.65
(5)Embed-Origin/Embed-Max/Embed-MaxSSA+DWC-5/SSA+DWC-5/SSA+DWC-582.09
Embed-Origin/Embed-Origin/Embed-OriginSSA+DWC-3/SSA+DWC-3/SSA+DWC-382.56
Embed-Origin/Embed-Origin/Embed-OriginSSA+DWC-5/SSA+DWC-5/SSA+DWC-582.73
+ +In Table 7, we provide additional analysis on the impact of high-frequency information. We conducted these experiments on CIFAR-100 [37], using the experimental settings detailed in Table 6. We discuss the ablation below according to the following aspects: + +# (a) Extra Max-Pool in Patch Embedding/Token Mixing: + +High-frequency information plays a critical role in the performance of Spiking Transformer due to the inherent low-pass filter characteristics of spiking neurons. Experimental results shown in Table 7 (1) reveal that strategically preserving these frequencies through max-pooling operations significantly enhances model accuracy, with a $2.39\%$ improvement when replacing average pooling with max-pooling across all stages $(76.73\%)$ to $79.12\%$ . In Table 7 (2), the performance of Spiking Transformer increases progressively when extending Embed-Max from the last patch embedding blocks $(80.11\%)$ to include the middle block $(80.46\%)$ . + +However, excess high-frequency information will instead impair model performance. For instance, in Table 7 (3), switching from using avg-pool for all token mixing to max-pool improves top-1 accuracy from $77.61\%$ to $79.78\%$ . Yet we found even better results with a more targeted setting: when employing max-pooling exclusively in the middle stage increases accuracy to $79.99\%$ , while restricting it to only the last stage further pushes performance to $80.12\%$ . + +This happens because spiking neurons act like low-pass filters that naturally reduce high-frequency components as information moves deeper through the network. Therefore, strategically adding back high-frequency components at specific points in the network is crucial for pushing the performance limit of Spiking Transformers. + +# (b) Spiking Transformers Benefit from High-Frequency Information: + +In biological vision, high-frequency details help early processing stages learn elementary features, which are then gradually built from local to global representations. Similarly, in standard non-spiking Transformers, the lower layers typically need more high-frequency details, while higher layers work better with global information. Spiking Transformers follow the same design philosophy, but with + +an important difference: they need additional frequency-enhancing operations (e.g., max-pooling and depth-wise convolution) for restoring high-frequency information that would otherwise be lost. + +In Table 7 (4), we show that, a proper token mixing strategy can effectively restore high-frequency information in Spiking Transformer, resulting in significant performance gains. By replacing Spiking Self-Attention (SSA) with depth-wise convolution (DWC) for token mixing, we improved performance from $82.13\%$ to $82.65\%$ . Importantly, this improvement does not come from adding more parameters/computational burden. For example, the Identity/DWC-3/SSA combination works $0.29\%$ better than DWC-3/DWC-5/SSA, even though the former one has a lower computational cost. Our further experiments in Table 7 (4-5) confirm that these findings hold true in the full-SSA network: restoring high-frequency components can significantly optimize the performance from $82.13\%$ to $82.73\%$ $(+0.6\%)$ on Cifar100. The proper high-frequency enhancement strategy is essential to unlocking the full potential of Spiking Transformers. + +# A.3 Max-ResNet Implementation + +As shown in Figure 7, Max-ResNet introduces only a minor architectural change to MS-ResNet [45]: all layers are replaced with Max-ResNet layers, while the first layer remains unchanged. Code implementation is available at https://github.com/bic-L/MaxFormer. + +![](images/9c10e8781440f5eda486b42658a63d38cd6138f90dc78f1e3d1a06f082f4990b.jpg) +Figure 7: Overview of Max-ResNet. A single Max-Pool operation is added per block and layer. + +# B Energy Analysis + +To estimate the theoretical energy consumption of Spiking Transformers, we follow the methodology used in previous studies [8, 53, 41, 19]. It is worth noting that the batch normalization (BN) layers and linear scaling transformations that follow convolution layers can be combined directly into the convolution layers themselves with added bias terms during deployment. Thus, in common practice [8, 53, 41, 19], the energy consumption of BN is typically excluded when calculating theoretical energy usage. For fair comparison, our work adopts the same strategy. In Spiking Transformers, energy consumption is directly proportional to synaptic operations (SOPs), which can be calculated as: + +$$ +\operatorname {S O P s} (l) = f r \times \mathrm {T} \times \operatorname {F L O P s} (l) \tag {22} +$$ + +where $l$ represents a specific block or layer in the Spiking Transformer architecture, $fr$ refers to the firing rate of the input spike train for that particular block or layer, and $\mathrm{T}$ is the simulation time step of spiking neurons. Assuming the multiply-accumulate (MAC) and accumulate (AC) operations are implemented on the 45nm neuromorphic chip described in [7], where each MAC operation uses $E_{\mathrm{MAC}} = 4.6\mathrm{pJ}$ of energy and each AC operation uses $E_{\mathrm{AC}} = 0.9\mathrm{pJ}$ , we can estimate the total energy consumption of a Spiking Transformer by adding up the energy used by all MAC and AC operations across all layers: + +$$ +E _ {\mathrm {S N N}} = E _ {\mathrm {M A C}} \times \mathrm {F L O P} _ {\mathrm {C O N V}} ^ {1} + E _ {\mathrm {A C}} \times \left(\sum_ {n = 2} ^ {N} \mathrm {S O P} _ {\mathrm {S N N} \text {C o n v}} ^ {n} + \sum_ {j = 1} ^ {M} \mathrm {S O P} _ {\mathrm {S N N} \mathrm {F C}} ^ {j}\right) \tag {23} +$$ + +$\mathrm{FLOP}_{\mathrm{CONV}}^{1}$ represents the floating-point operations in the first layer, which converts non-spike inputs into spike form for static image classification tasks. Since this layer performs floating-point computations, we estimate its energy consumption using $E_{\mathrm{MAC}}$ . For all subsequent layers, which process spike data, we estimate energy consumption using $E_{\mathrm{AC}}$ . For mainstream non-spiking Transformers, the energy consumption is estimated through: + +$$ +E _ {\text {A N N}} = E _ {\text {M A C}} \times \text {F L O P s} \tag {24} +$$ + +# C Comparison on Train/Inference Time and Memory Consumption + +Table 8: Comparison of training/inference time and memory usage between QKFormer-10-768 and Max-Former-10-768 models. All measurements were conducted with the simulation timestep of 1 and the batch size of 32. MS-QKFormer indicates the QKFormer variant with the membrane shortcut. + +
Train Time (s)Train Memory (MB)Infer. Time (s)Infer. Memory (MB)
QKFormer (64.96M, T=1, B=32)0.214182270.0535000
MS-QKFormer* (64.96M, T=1, B=32)0.208174960.0484822
Max-Former* (63.99M, T=1, B=32)0.179154310.0444354
+ +Max-Former delivers faster training and inference speeds while consuming less memory. We compared the performance metrics of QKFormer [24], its membrane shortcut variant (MS-QKFormer), and Max-Former on ImageNet using $224 \times 224$ input resolution. All tests were conducted on a CentOS 7.9 server equipped with the Intel Xeon Gold 6348 CPU (2.60GHz) and the Nvidia A30 GPU. As Table 8 shows, when compared to MS-QKFormer with the same hierarchical architecture and shortcut configuration, Max-Former reduces training time by $14\%$ and both inference time and memory usage by $10\%$ . Additionally, our results indicate that the pre-spike shortcut strategy used in the original QKFormer increases both processing time and memory demands. + +# D Residual Connections in Spiking Neural Networks + +# (a) Performance and Energy Tradeoffs: + +The unique asynchronous nature of spike-based computation makes implementing residual connections challenging. As a result, the research community of spiking neural networks (SNNs) has not yet reached a consensus on the standardized residual learning approach, either in terms of algorithm or hardware implementation. While the main focus of our work is not related to residual learning, we want to offer a detailed comparison between the pre-spike shortcut [54] and the membrane shortcut [53], the two most representative residual learning methods that emerged in recent years. + +As explained in Section 3.3, the pre-spike shortcut [54] implements residual connections between spiking outputs, while the membrane shortcut [53] connects membrane potentials directly. In algorithm designs, membrane shortcuts have been reported to facilitate better performance in prior works [24, 41], especially on small datasets. However, our findings indicate this advantage is not universal across all scenarios. As shown in Table 9, the patch embedding stages of Spiking Transformers account for the majority of energy consumption. Consequently, the multiple patch embedding stages in hierarchical architectures like QKFormer [24], while enabling efficient feature learning with fewer parameters, also come at higher energy usage. This makes the choice of shortcut scheme particularly impactful on overall energy efficiency. When processing ImageNet images at $224 \times 224$ resolution, the pre-spike QKFormer consumes three times more energy than its membrane shortcut variant, reflecting the substantially higher SOPs required. Nevertheless, this increased computational overhead does translate to notable performance gains ( $+2.32\%$ accuracy when comparing QKFormer to MS-QKFormer). + +From the hardware perspective, implementing either shortcut type on neuromorphic chips is technically feasible but presents significant challenges [55, 56]. Implementing the pre-spike shortcut specifically requires the chips to support multi-spike operations, as the ternary spike transmissions (values of 0, 1, or 2) can occur. This results in either higher energy consumption or increased hardware complexity [57]. Yao et al. [41] proposed implementing membrane shortcuts through the addressing function that passes the membrane potential to corresponding neurons in subsequent layers for merging. While membrane shortcuts do strictly adhere to the spike-driven computing paradigm and could possibly be supported by standard neuromorphic hardware, transmitting membrane potentials can create substantial communication overhead, making practical implementation non-trivial. Many discussions on hardware deployment still advocate avoiding shortcuts as the current preferred approach [58, 59]. However, given the critical role of residual learning in modern deep learning, avoiding shortcuts altogether is not a sustainable long-term strategy for advancing neuromorphic computing. + +In our work, we primarily compare with MS-QKFormer using the membrane shortcut to ensure fair comparisons. We hope the SNN community can establish a consensus on standardized shortcut implementations in the near future, considering the significant impact of shortcut schemes on both energy efficiency and model performance. + +Table 9: Energy consumption comparison (in mJ) across processing stages for QKFormer, QKFormer with membrane shortcut (MS-QKFormer), and our proposed Max-Former model. * Identical training configurations. + +
ModelStage 1Stage 2Stage 3ClassifierTotal Energy
Patch EmbedToken MixMLPPatch EmbedToken MixMLPPatch EmbedToken MixMLP
QKFormer (16.47M, T=4, 78.80%)1.260.160.412.680.350.772.972.713.810.00615.13
MS-QKFormer* (16.47M, T=4, 76.48%)1.190.0520.150.960.110.250.880.881.060.0075.52
Max-Former* (16.23M, T=4, 77.82%)0.410.020.170.890.010.360.910.961.160.0014.89
+ +# E Visualization + +We demonstrate the GradCAM visualizations [60] of four Spiking Transformers with the membrane shortcut of similar size. Compared to Spike-Driven Transformer [41] and SWFormer [19], the hierarchical architecture used in both QKFormer [24] and our Max-Former allows for more precise focusing on target objects. Compared to MS-QKFormer, Max-Former shows more concentrated activation patterns. For instance, in the polar bear image, Max-Former completely skips the background and precisely focuses on the bear's key features (the head, rather than the outline or fur). + +![](images/e7f65cb142281896ede12d47e4f7206591b025cc5c5de33c7c7df654b5742b7a.jpg) +Figure 8: GradCAM visualizations [60] comparing four Spiking Transformers of similar size: Spike-Driven Transformer-8-512 [41] (29.68M), SWFormer-8-512 [19] (27.6M), QKFormer-10-512 with membrane shortcut (MS-QKFormer) (29.08M), and our Max-Former-10-512 (28.65M). + +# F Limitation + +In this work, we provide theoretical evidence that high-frequency information is essential for Spiking Transformers. To empirically probe the importance of high-frequency information, we design Max-Former. However, Max-Former should be considered just a good starting baseline for future architectural designs. Similar to [31], Max-Former requires manually balancing frequency components, which demands considerable expertise when adapting to different tasks. Incorporating direct frequency learning approaches like Fourier-based [61] or Wavelet-based[19] methods would offer more straightforward solutions. The main challenge, however, lies in developing efficient spike-based frequency representations. Overall, we believe our work will inspire more future research to advance neuromorphic computing through exploring the unique properties of spiking neurons, rather than expending excessive effort to adapt established practices from standard non-spiking neural networks. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction reflect the paper's contributions and scope clearly. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We discuss the limitations in Appendix E. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: All the theorems, formulas, and proofs are clearly stated in the main paper. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: We provide experiment details in the Appendix A, B. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [Yes] + +Justification: Code is available: https://github.com/bic-L/MaxFormer. + +# 6. Experimental setting/details + +Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? + +Answer: [Yes] + +Justification: All training and test details have been revealed in the Appendix A. + +# 7. Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [No] + +Justification: Error bars are not reported because it would be too computationally expensive. + +# 8. Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +Answer: [Yes] + +Justification: See Appendix A. + +# 9. Code of ethics + +Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? + +Answer: [Yes] + +Justification: This research is in every respect with the NeurIPS Code of Ethics. + +# 10. Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +Answer: [NA] + +Justification: This research belongs to foundational research and is not tied to particular applications. + +# 11. Safeguards + +Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? + +Answer: [NA] + +Justification: The paper poses no such risks. + +# 12. Licenses for existing assets + +Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? + +Answer: [Yes] + +Justification: The paper properly credits creators and mentions the license and terms of use for existing assets. + +# 13. New assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [NA] + +Justification: The paper does not release new assets. + +# 14. Crowdsourcing and research with human subjects + +Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? + +Answer: [NA] + +Justification: The paper does not involve crowdsourcing or research with human subjects. + +# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +Justification: The paper does not involve crowdsourcing or research with human subjects. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The paper does not involve LLMs as any important, original, or non-standard components. \ No newline at end of file diff --git a/NeurIPS/2025/Spiking Neural Networks Need High-Frequency Information/images.zip b/NeurIPS/2025/Spiking Neural Networks Need High-Frequency Information/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..f7e7e9c90fff3e6a6ea2e4b1d9179cd1ab9ac8a8 --- /dev/null +++ b/NeurIPS/2025/Spiking Neural Networks Need High-Frequency Information/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:31e99296c1557553d6100b97a35f7fab719c21ef0ed2d7b177a59a0ae1febb25 +size 1134543 diff --git a/NeurIPS/2025/Spiking Neural Networks Need High-Frequency Information/layout.json b/NeurIPS/2025/Spiking Neural Networks Need High-Frequency Information/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..2ebb6a3443732d8eb9375dad8b14db45d9da387e --- /dev/null +++ b/NeurIPS/2025/Spiking Neural Networks Need High-Frequency Information/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:28372d068db0f2b5ed2dbbc9e4139c52698cd2396c645e787aa64e684f51da83 +size 707856 diff --git a/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_content_list.json b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..30b2dc9ff2e9f3fc2afc8a42d47b66ca43286a88 --- /dev/null +++ b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:da35e44e428bf6ce714ac9eef12925a380d6b4be30fe7e7647bc39291542ab5c +size 142000 diff --git a/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_model.json b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_model.json new file mode 100644 index 0000000000000000000000000000000000000000..6f30fccee7db78157d42c08ffe8a5ba5d3de3eaf --- /dev/null +++ b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7451ff6606b2097b4d7088897d92623b7bfecfd3b5e78bfcd082d0c7314f0a1b +size 182944 diff --git a/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_origin.pdf b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb5383a52153b13541f9375b8bb182ab66bccc18 --- /dev/null +++ b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/8c44f49e-8272-4960-b386-246d261f2361_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0670d79f768801503b1267ddd1d1503c585493e55e9132566dafa847858be556 +size 1327696 diff --git a/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/full.md b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/full.md new file mode 100644 index 0000000000000000000000000000000000000000..9ae9e6957d4781a7da04c2caf82435e7ea06ff5d --- /dev/null +++ b/NeurIPS/2025/SpikingVTG_ A Spiking Detection Transformer for Video Temporal Grounding/full.md @@ -0,0 +1,646 @@ +# SpikingVTG: A Spiking Detection Transformer for Video Temporal Grounding + +Malyaban Bal $^{1,2}$ , Brian Matejek $^{2}$ , Susmit Jha $^{2}$ , Adam D. Cobb $^{2}$ + +1The Pennsylvania State University +2Computer Science Laboratory, SRI International + +# Abstract + +Video Temporal Grounding (VTG) aims to retrieve precise temporal segments in a video conditioned on natural language queries. Unlike conventional neural frameworks that rely heavily on computationally expensive dense matrix multiplications, Spiking Neural Networks (SNNs)—previously underexplored in this domain—offer a unique opportunity to tackle VTG tasks through bio-plausible spike-based communication and an event-driven accumulation-based computational paradigm. We introduce SpikingVTG, a multi-modal spiking detection transformer, designed to harness the computational simplicity and sparsity of SNNs for VTG tasks. Leveraging the temporal dynamics of SNNs, our model introduces a Saliency Feedback Gating (SFG) mechanism that assigns dynamic saliency scores to video clips and applies multiplicative gating to highlight relevant clips while suppressing less informative ones. SFG enhances performance and reduces computational overhead by minimizing neural activity. We analyze the layer-wise convergence dynamics of SFG-enabled model and apply implicit differentiation at equilibrium to enable efficient, BPTT-free training. To improve generalization and maximize performance, we enable knowledge transfer by optimizing a Cos-L2 representation matching loss that aligns the layer-wise representation and attention maps of a non-spiking teacher with those of our student SpikingVTG. Additionally, we present Normalization-Free (NF)-SpikingVTG, which eliminates non-local operations like softmax and layer normalization, and an extremely quantized 1-bit (NF)-SpikingVTG variant for potential deployment on edge devices. Our models achieve competitive results on QVHighlights, Charades-STA, TACoS, and YouTube Highlights, establishing a strong baseline for multi-modal spiking VTG solutions. + +# 1 Introduction + +The rapid expansion of various social medias and portable smart technologies has triggered an unprecedented surge in video content. This vast influx of data has intensified the need for efficient methods to retrieve and analyze video information. Consequently, the field of Video Temporal Grounding (VTG) [1] has emerged as an important area of research. The main objective of VTG is to identify the precise segment of a video that corresponds to a given natural language query, enabling accurate and context-driven video content retrieval. In this paper, we focus on two tasks: moment retrieval [2, 3], which aims to identify video intervals relevant to a given query, and highlight detection [4], which retrieves the best candidate segment of the video in response to the query. Our work involves analyzing multimodal data—combining video content with natural language queries—to develop an effective solution to the problem. With the rise of transformer based architectures the field of VTG has seen significant advancements [5, 6]. However, these models demand substantial + +power and energy [7] to operate. Furthermore, VTG is inherently resource-intensive, requiring the analysis of long video sequences, leading to significant computational overhead. Additionally, applications such as edge-based event detection in video—e.g., identifying accidents from traffic cameras—require deploying VTG models on resource-constrained devices near the data source, in order to reduce data transfer to the cloud. These devices often operate under limited energy availability. Inspired by neural dynamics in the brain, this paper leverages bio-plausible neuronal models and learning dynamics to develop an efficient and brain-inspired solution for VTG tasks. Our model enables flexible inference-time tradeoff between energy consumption and accuracy, making it well-suited for edge-based deployment in VTG tasks. + +SpikingVTG Model: We introduce SpikingVTG, a spiking detection transformer designed for efficient and accurate VTG. Built on the sparse, event-driven communication and accumulation-based computation of Spiking Neural Networks (SNNs) [8], SpikingVTG leverages the intrinsic dynamics of SNNs to offer a lightweight yet competitive alternative to conventional transformer-based approaches—eliminating the need for costly dense, real-valued matrix multiplications. The architecture comprises three key components: (i) a spiking transformer core, (ii) a Saliency Feedback Gating (SFG) mechanism, and (iii) a spiking decoder for output generation. The spiking transformer captures temporal and cross-modal dependencies, while the decoder produces task-specific outputs. The SFG mechanism addresses a critical challenge unique to VTG: given an input video composed of many clips, how can we identify those most relevant to the query? Tailored specifically for multi-modal VTG tasks, the SFG module leverages the temporal dynamics of SNNs to attend more towards the most salient segments while suppressing irrelevant clips. + +Saliency Feedback Gating Mechanism: Operating over discrete time steps, SpikingVTG uses the intermediate spiking activity of the transformer core to dynamically estimate the relevance of each video segment. We compute a feedback-based saliency score for each segment based on the average spiking rate (ASR) of the output of the transformer, conditioned on the query. These scores serve as soft attention masks within a multiplicative gating mechanism, suppressing less informative segments to reduce computational overhead while enhancing focus on the most relevant candidate clips. From a neuroscience perspective, feedback connections are known to play a critical role in object recognition in the visual cortex [9]. Furthermore, our feedback mechanism maintains layer-wise convergence of ASR to equilibrium, enabling us to adopt an efficient training mechanism leveraging the equilibrium dynamics [10]. This approach circumvents the need for computationally intensive BPTT [11], and instead updates model parameters using a single backward pass, significantly improving memory efficiency. + +Cos-L2 Representation Matching (CLRM): While our base SpikingVTG model achieves performance comparable to non-pretrained, non-spiking VTG models such as M-DETR and UniVTG [1, 6], achieving state-of-the-art results on VTG tasks typically requires transformer-based, non-spiking models like UniVTG to undergo extensive pre-training. This pre-training significantly boosts their generalization and task-specific capabilities. To enable our SpikingVTG to benefit from similar pre-training advantages—without incurring the high computational cost of training on large-scale datasets like Ego4D, Video-CC [12]—we propose a direction and scale aware CLRM loss for efficient knowledge transfer. Optimizing CLRM loss aligns the hidden states and attention score maps of a pre-trained non-spiking multi-modal transformer with the hidden state converged ASR and mean attention scores of our SpikingVTG model. By minimizing this alignment loss on the downstream task, we allow the student model to imbibe the generalization capability learnt by the pre-trained teacher without having to perform the extensive pre-training from scratch. + +Optimizations and Application to VTG Tasks: Traditional transformer-based VTG solutions [6, 13] rely heavily on non-local normalization operations, such as softmax and layer normalization, which pose significant challenges for efficient implementation on neuromorphic hardware [14]. To address this limitation, we develop and evaluate a Normalization-Free (NF)-SpikingVTG model, which eliminates all layer normalization operations and substitutes softmax spiking attention with a ReLU and scaling-based spiking attention mechanism. Although alternative Softmax-free attention approaches have been explored in the literature [15, 16], they have primarily been applied uni-modal vision tasks. We empirically demonstrate that while these optimizations enhance computational efficiency, they result in minimal performance degradation. + +To further reduce computational complexity and memory footprint [17, 18], we introduce 1-bit (NF)-SpikingVTG model, which rely primarily on integer accumulation operations and eliminate + +![](images/445d2cd84f1a25b626e6731cd26429585388100a86391fe83a40295649dd5a1b.jpg) +Figure 1: High-level overview of the proposed SpikingVTG model. The model employs a spiking transformer core that utilizes Saliency Feedback Gating through temporal feedback connections. The model also incorporates a spiking decoder module that takes the output of the transformer core to predict parameters for the VTG task. + +all non-local operations. This design makes it well-suited for deployment on resource-constrained edge devices. To our knowledge, this work is the first to evaluate an operational spiking detection transformer across VTG tasks, including moment retrieval and highlight detection, on datasets such as QVHighlights, Charades-STA, TACoS and Youtube Highlights. To further highlight the benefits of using SpikingVTG, we present an energy-accuracy tradeoff analysis. This demonstrates that our model is well-suited for deployment on edge devices, where its performance can be dynamically adjusted based on the available energy budget. + +# 2 Related Works + +VTG Advancements: Moment-DETR [1], a transformer encoder-decoder model introduced alongside the QVHighlights dataset, laid a strong foundation for subsequent VTG architectures. UMT [5] introduced an unified framework for solving both highlight detection and moment retrieval tasks. Due to the limited availability of trainable video data, UniVTG [6] proposed an innovative solution by unifying various VTG tasks and labels under a single formulation. This enables the development of an LLM-like pretraining framework, achieving state-of-the-art performance on VTG tasks. Although no fully spiking-based model has been explored for VTG tasks, SpikeMba [19]—primarily a non-spiking model—integrates SNN components to generate proposal sets from video data. However, since its core framework is derived from Mamba [20] and relies on floating-point matrix multiplications, SpikeMba cannot be considered a baseline for spiking models, which predominantly use accumulation-based operations. While recent VTG models have significantly improved task-specific performance, adopting a spiking framework enables us to leverage energy-accuracy tradeoff—making suitable solutions on edge devices with limited or dynamic energy supply. + +Spiking neural networks (SNNs): SNNs have been implemented in neuromorphic systems like IBM TrueNorth [21] and Intel Loihi 2 [22], demonstrating approximately $75 \times$ greater energy efficiency compared to traditional networks running on low-power GPUs [23]. SNNs, with their energy-efficient computational framework, offer a promising solution to the resource-intensive demands of multimodal VTG tasks. While SNNs for a long time were confined in simpler vision-based tasks [24] with simple architectures, recent developments have scaled them to transformer-based models for tasks ranging from vision to language modelling [25, 26, 27], however majority of them are uni-modal and rely on non-local operations not implementable on a neuromorphic chip. + +# 3 Video Temporal Grounding (VTG) + +For a given video $V$ and language query $Q$ , we start by segmenting $V$ into a sequence of $L_{v}$ fixed-length clips, denoted as $\{v_{1},\ldots ,v_{L_{v}}\}$ . Each clip $v_{i}$ has a length $l$ and is centered at timestamp $t_i$ . The textual query $Q$ consists of $L_{q}$ tokens, denoted as $Q = \{q_{1},\dots,q_{L_{q}}\}$ . Following previous studies on VTG [6], we define three parameters for each clip $v_{i} = (f_{i},d_{i},s_{i})$ , where $f_{i} = 1$ if the + +clip is in foreground, i.e. relevant else $f_{i} = 0$ . $d_{i} = [d_{s_{i}}, d_{e_{i}}] \in \mathbb{R}^{2}$ represent the temporal distance that converts the clip timestamp $t_{i}$ to its interval boundaries. Here, $d_{i}$ is valid when $f_{i} = 1$ . The term $d_{s_{i}}$ denotes the distance between the start of the interval and $t_{i}$ , while $d_{e_{i}}$ denotes the distance between the end of the interval and $t_{i}$ . $s_{i} \in [0,1]$ is a continuous score that quantifies the relevance between the visual content of clip $v_{i}$ and the query $Q$ . Our proposed SpikingVTG predicts these three parameters for each video clip. In this paper, we focus on the following VTG tasks: + +Moment Retrieval: We rank the predicted clip boundaries $\{\tilde{b}_i\}_{i=1}^{L_v}$ , where $b_i = [t_i - d_{s_i}, t_i + d_{e_i}]$ , based on their associated probabilities given by $\{\tilde{f}_i\}_{i=1}^{L_v}$ . Since the predicted $L_v$ boundaries are dense, we employ a 1-dimensional Non-Maximum Suppression (NMS) [28] with a threshold of 0.7 to eliminate highly overlapping boundary boxes, resulting in a final prediction. + +Highlight Detection For each clip, we rank all clips based on their combined scores $\{\tilde{f}_i + \tilde{s}_i\}_{i = 1}^{L_v}$ . This combined value represents how well the chip $i$ match with the underlying query. We then return the top clips (e.g., Top-1) as predictions. + +# 4 SpikingVTG: Architecture Overview + +The core computational unit of the proposed SpikingVTG model is a leaky integrate-and-fire (LIF) neuron [29]. Neurons communicate with each other using sparse, spike-based activations instead of real-valued signals, thus we can replace floating-point matrix multiplications with accumulative operations, resulting in improved computational efficiency. + +# 4.1 Spiking Neural Networks + +The discrete time dynamics of an LIF-based spiking neuron can be given as follows, + +$$ +u _ {i} [ t + \delta ] = \gamma u _ {i} [ t ] + W _ {(i - 1)} (s _ {(i - 1)} [ t ]) + b _ {i}, +$$ + +$$ +u _ {i} [ t + 1 ] = u _ {i} [ t + \delta ] - V _ {t h _ {i}} s _ {i} [ t + 1 ], \tag {1} +$$ + +$$ +a _ {i} [ t ] = \frac {\sum_ {\tau = 1} ^ {t} \gamma^ {t - \tau} s _ {i} [ \tau ]}{\sum_ {\tau = 1} ^ {t} \gamma^ {t - \tau}}. +$$ + +where, at time $t$ , $u_{i}[t]$ is the membrane potential of the $i^{th}$ neuronal layer; $b_{i}$ indicates a bias term and $\gamma$ is the leaky term. $W_{(i-1)}$ represents the layer-specific operation; $t + \delta$ is an intermediate time step to determine if the neuron fired; $V_{th_i}$ is the threshold of layer $i$ . We use a ternary spiking model [30] in our work for spike $(s[t+1])$ generation. This improves performance while avoiding the introduction of additional floating-point multiplicative and accumulative (FP-MAC) operations. The average spiking rate (ASR $a_{i}[t]$ ) of neurons within each layer $i$ at time $t$ can be defined as a weighted-average function. + +# 4.2 Spiking Transformer Core + +The high-level overview of each encoder block of our spiking transformer architecture is demonstrated in Fig. 1. The model consists of $N$ encoder layers, each consists of a spiking multi-headed attention block, followed by an intermediate layer and an output layer. Communication within and between encoder layers occurs via spikes. Furthermore, all matrix multiplications involved in linear layers and attention layer comprises of more efficient fp-accumulative (FP-ACC) operations instead of FP-MAC operations in conventional neural architectures. Detailed descriptions of each layer are provided in the Appendix A. Following architectural optimizations (Section 4.6), we replace softmax-based attention with a ReLU and scaling-based spiking attention mechanism, remove all layer normalization operations, and explore extreme quantization of linear weights. + +# 4.3 Saliency Feedback Gating (SFG) + +SpikingVTG operates over a specific number of convergence time steps $(T_{c})$ , with the convergence dynamics detailed in Section 4.5. This temporal processing allows us to leverage intermediate temporal outputs to dynamically update the input to the model at every time step for improved performance. This approach conforms to the feedback connections observed in the human visual + +cortex [9], providing a bio-plausible explanation for its efficacy. The ASR of the final encoder layer of the Spiking Transformer core is used as a temporal feedback to compute a dynamic saliency score with the input query enabling the design of a gating mechanism (Fig. 2a), allowing selective focusing on relevant segments of the video while minimizing computation on irrelevant segments. The saliency feedback gating mechanism is shown below, + +$$ +F _ {s} ^ {v _ {i}} [ t ] = \cos \left(\mathbf {a} _ {\mathbf {N} _ {\mathbf {v}}} ^ {\mathbf {i}} [ \mathbf {t} ], \mathbf {M}\right) := \frac {\mathbf {a} _ {\mathbf {N} _ {\mathbf {v}}} ^ {\mathbf {i}} [ \mathbf {t} ] \cdot \mathbf {M}}{\| \mathbf {a} _ {\mathbf {N} _ {\mathbf {v}}} ^ {\mathbf {i}} [ \mathbf {t} ] \| _ {2} \| \mathbf {M} \| _ {2}}, \tag {2} +$$ + +$$ +\bar {V} [ t + 1 ] = V [ t ] * \bar {F} _ {s} ^ {v} [ t ], +$$ + +$$ +\mathbf {S F G} (V, \mathbf {Q}, \mathbf {a} _ {\mathbf {N} _ {\mathbf {v}}}) = \bar {V} [ t + 1 ] \oplus \mathbf {Q}, +$$ + +where, using attentive pooling operation, sentence representation $\mathbf{M} = \mathbf{Q}^T\operatorname{Softmax}(\mathbf{Q}\mathbf{W}_{\mathbf{p}})$ , $\mathbf{M} \in \mathbb{R}^{D}$ , input textual query features $\mathbf{Q} \in \mathbb{R}^{L_q \times D}$ , input video features $V \in \mathbb{R}^{L_v \times D}$ and $\mathbf{W}_{\mathbf{p}} \in \mathbb{R}^{D \times 1}$ is a learnable embedding and $D$ is the hidden dimension. $F_{s}^{v_i}[t]$ is the dynamic saliency score, at time $t$ , for the $i$ -th segment of the video. We compute $\bar{F}_{s}^{v}$ by applying min-max normalization to $F_{s}^{v}$ , allowing us to obtain per-clip scores within the range [0, 1]. The ASR of the output of the spiking transformer core is given as $a_{N}[t] \in \mathbb{R}^{(L_v + L_q) \times D}$ . $a_{N_v}^{i}[t]$ is ASR of output of the spiking transformer core, corresponding to video segment $i$ , at time $t$ . The output of SFG module is the concatenation of saliency feedback gated video features and query features and serves as the input to the spiking transformer core at time $t + 1$ . + +To ensure compatibility between the two modalities during cosine similarity computation, both representations are projected into a shared latent space of dimensionality $D$ . The query representation is aggregated into a global sentence embedding via attention-based pooling with a learnable embedding vector. This adaptive mapping aligns the textual representation with the spatiotemporal semantics of the spiking video features, thereby facilitating effective cross-modal compatibility. + +SFG Computational Overhead: The SFG layer comprises $O(L_v \cdot D)$ floating-point multiplication operations; however, the computational overhead of this layer is significantly less than that of the transformer core which has a complexity of $O(L^2 \cdot D + L \cdot D^2)$ , where $L = L_v + L_q$ . + +# 4.3.1 Visualizing effect of SFG: + +The dynamic saliency score $(F_{s}^{v})$ achieves an equilibrium value $F_{s}^{*}$ following the convergence of ASR of the neuronal layers of the SpikingVTG model (Fig. 3). As shown in Fig. 2b, we empirically analyze the scores per clip at equilibrium to gain insights into the functioning of the SFG based multiplicative gating mechanism. Neighboring salient clips of the target clip exhibit higher $\bar{F}_{s}$ scores at equilibrium, while irrelevant clips show lower scores, highlighting the effectiveness of the SFG mechanism. + +The SFG mechanism not only results in better performance of our SpikingVTG architecture (Table 5) but also reduces overall neural activity by sparsifying input spikes. Empirical + +results (Fig. 3b) confirm that the model with the gating mechanism exhibits a lower neural activity, particularly in the input and spiking attention layers, compared to the model without SFG. + +![](images/ed67ea1fc9adaf6c625589c5dfec0b9264866a5a46e7c96a06a884343686cdba.jpg) +(a) + +![](images/c952b9dff065191d000953afcf1bd11694eaed329817a26b758ea4acc865b2cb.jpg) +(b) +Figure 2: (a) Overview of the internal operations of the saliency-feedback gating mechanism. The ASR of the output of the spiking transformer core at each time step is leveraged as the feedback signal (Fig. 1). (b) Heatmap showing the scores per clip $(\bar{F}_s)$ at equilibrium, with the target frame for highlight detection corresponding to clip index 28. + +# 4.4 Spiking Decoder Module + +The spiking decoder comprises of stacked 1-D convolutions followed by integrate-fire (IF) neuron layers $(\gamma = 1$ in Eqn. 1), for spike generation. The spiking decoder used for predicting foreground indicator $(f_{i})$ per clip, applies $n_1$ 1-D convolution operations with kernel size $k_{1}$ , each followed by an + +IF layer. The final layer consists of a single output channel, and its temporal mean is passed through a sigmoid activation to produce the prediction. The spiking decoder used for $d_{i}$ applies $n_2$ 1-D convolution operations with kernel size $k_{2}$ , each followed by an IF layer, and the final convolution layer has two output channels to predict $d_{i} = [d_{s_{i}}, d_{e_{i}}]$ , after which we compute $b_{i}$ . + +# 4.5 SpikingVTG: Convergence Dynamics + +The membrane potential $(u_{1})$ of the input LIF layer to the spiking transformer core, following the SFG mechanism can be formulated as below, + +$$ +u _ {1} [ t + 1 ] = \gamma u _ {1} [ t ] + \mathbf {S F G} (V, Q, a _ {N _ {v}} [ t ]) + b _ {1} - V _ {t h _ {1}} s _ {1} [ t + 1 ] \tag {3} +$$ + +where, $a^{N_v}[t]$ is the ASR of the final layer corresponding to the video features at time $t$ , $V$ is the input video features, $Q$ is the query features and SFG is defined in Eqn. 2. The layer-wise convergence dynamics of the SpikingVTG with SFG is demonstrated in Fig. 3. + +Following Eqn. 1, the layer-wise ASR is, $a_{i}[t + 1] = \frac{1}{V_{th_{i}}} (\hat{f} (a_{(i - 1)}[t + 1]) + b_{i} - \frac{u_{i}[t + 1]}{\sum_{j = 0}^{i}\gamma^{j}})$ where, $\hat{f}$ is operation of layer $i$ . Following empirical evidence (Fig. 3) and theoretical formulation [10, 26] as time $t\to \infty$ , the layer-wise ASRs converge to equilibrium, enabling the derivation of layer-wise steady-state equations given as, + +$$ +a _ {i} ^ {*} = \sigma \left(\frac {1}{V _ {t h _ {i}}} \left(\hat {f} \left(a _ {i - 1} ^ {*}\right) + b _ {i}\right)\right) \tag {4} +$$ + +The equation describing the steady-state ASR dynamics of the input layer is given as, + +$$ +a _ {1} ^ {*} = \sigma \left(\frac {1}{V _ {t h _ {1}}} \left(\mathbf {S F G} \left(V, Q, a _ {N _ {v}} ^ {*}\right) + b _ {1}\right)\right) \tag {5} +$$ + +where, clipping function $\sigma(x)$ clamps the values within $[-1, 1]$ . This is because we allow ternary spikes thus ASR must be with $[-1, 1]$ . Furthermore, the dynamic saliency score also achieves an equilibrium value $F_{s}^{*}$ since the ASR at the final layer achieves equilibrium state $a_{N_{v}}^{*}$ . Convergence dynamics of other layers are given in Appendix A. To analyze the overall layer-wise neural activity, which includes both positive and negative spiking event, we present the layer-wise dynamics of the absolute spiking activity events in Fig. 3b, i.e. $act_{i}[t] = \frac{\sum_{i=1}^{t} |s_{i}[t]|}{t}$ . + +Training: As described in the Section 4.4, the Spiking decoder is responsible for predicting $\tilde{f}_i$ and $\tilde{d}_i$ for individual video clip $i$ and $\tilde{s}_i$ is computed using the SFG module at equilibrium. Using these three predictions, we design a loss function that combines various components. The total loss over $N_{T}$ clips in the training set is given by $L = \frac{1}{N_T}\sum_{i = 1}^{N_T}(L_{f_i} + L_{d_i} + L_{c_i})$ , where $L_{f}$ is the binary cross-entropy loss for the indicator variable $f_{i}, L_{d}$ combines smooth L1 loss with IoU loss [31] for the predicted boundaries, and $L_{c}$ is an optional loss incorporating intra- and inter-video contrastive + +![](images/a1091df9fc97c0c295dce418dc046a05a28c248362dff24a3d458da35cfecce9.jpg) +Figure 3: Results from a random QVHighlights input passed through SpikingVTG models. (a) Layer-wise mean ASR convergence over operating time steps for a sample spiking transformer encoder layer (Fig. 1); note ASR can be negative due to ternary spikes. (b) Layer-wise mean spiking activity $(act_i[t]$ , averaged over neurons) versus time steps. Model with SFG exhibit reduced activity in both input and spiking attention layers, highlighting SFG's role in minimizing neural activity. + +![](images/e47c3747c5fd4152e7c81d7ea422ca02ce73417791544d9008d34c8bedeab2e9.jpg) + +learning [32]. Detailed formulation of the loss functions is in Appendix B. + +During training, leveraging implicit differentiation [33] at equilibrium, only ASR values at equilibrium are used, $\frac{\partial L(a^{*})}{\partial\theta} = -\frac{\partial L(a^{*})}{\partial a^{*}} (J_{g_{\theta}}^{-1}|_{a^{*}})\frac{\partial f_{\theta}(a^{*})}{\partial\theta}$ where, $\theta$ is the model parameters, $g_{\theta}(a) = f_{\theta}(a) - a$ + +$f$ is the steady-state equation of ASR, $J^{-1}$ is the inverse Jacobian of $g_{\theta}$ when $a = a^{*}$ , i.e., at equilibrium. Thus, unlike BPTT, we do not need to store the intermediate computational graph and the model parameters can be updated using a single backpropagation step. + +# 4.5.1 Cos-L2 Representation Matching (CLRM) + +We train the base SpikingVTG model on VTG tasks such as QVHighlights [1], achieving comparable performance to a similarly scaled non-spiking transformer-based model, UniVTG [6], as demonstrated in Table 1. The later to improve its performance further uses pre-training on large datasets such as Video-CC and Ego4D. Since, extensive pretraining of the SpikingVTG is considerably resource intensive we propose an effective knowledge transfer strategy. This mechanism enables SpikingVTG to inherit the generalization capabilities learned by the pre-trained detection transformer models like UniVTG model. Importantly, since the primary benefit of spiking architectures manifests during inference on resource-constrained edge devices, this knowledge transfer is a one-time process, which significantly improves performance. + +Hidden State Matching Loss: To align the hidden state (output of individual encoder layer) of the nonspiking multi-modal transformer model with the converged ASR (4.5) of the corresponding layer of the SpikingVTG model, we propose a hybrid loss function combining a + +Table 1: Ablation study of the effect of CLRM on SpikingVTG evaluated on the evaluation set of QVHighlights. + +
MethodQVHighlights-MRQVHighlights-HD
@0.5@0.7mAP@avgmAPHIT@1
UniVTG [6]59.7440.9036.1338.8361.81
SpikingVTG w/o CLRM60.1239.6836.2338.8462.49
UniVTG with PT [6]67.3552.6545.4441.3468.77
SpikingVTG w/ CLRM67.5850.8244.0740.8168.64
+ +squared cosine similarity based directional loss and a L2-norm based loss for minimizing the scale difference. For each layer, the student representation is projected into the feature space of the teacher via a learnable linear transformation $W_{d} \in \mathbb{R}^{d_{s} \times d_{t}}$ . The total loss is: + +$$ +\mathcal {L} _ {\mathrm {r e p}} = \frac {1}{B \times L} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {B} \sum_ {k = 1} ^ {L} \left[ \lambda_ {\cos} \cdot \left(1 - \cos \left(\theta_ {i, j, k} ^ {\mathrm {r e p}}\right)\right) ^ {2} + \lambda_ {\ell_ {2}} \cdot \left\| \mathbf {s} _ {i} ^ {(j, k)} - \mathbf {t} _ {i} ^ {(j, k)} \right\| _ {2} ^ {2} \right] \tag {6} +$$ + +where $N$ is total number of encoder layers, $B$ is batch size, $L$ is length of sequence, $\lambda_{\mathrm{cos}}$ , $\lambda_{l_2}$ are hyperparameters and the cosine similarity is computed as: $\cos (\theta_{i,j,k}^{\mathrm{rep}}) = \frac{\mathbf{s}_{i}^{(j,k)} \cdot \mathbf{t}_{i}^{(j,k)}}{\|\mathbf{s}_{i}^{(j,k)}\|_2 \cdot \|\mathbf{t}_{i}^{(j,k)}\|_2}$ , where, $\mathbf{s}_i^{(j,k)} = a_{r_i}^{*(j,k)} W_d \in \mathbb{R}^{d_t}$ is the projected student representation and $\mathbf{t}_i^{(j,k)} = T_{r_i}^{(j,k)} \in \mathbb{R}^{d_t}$ is the teacher representation at layer $i$ , sequence position $k$ , and batch index $j$ . $a_{r_i}^* \in \mathbb{R}^{B \times L \times d_s}$ and $T_{r_i} \in \mathbb{R}^{B \times L \times d_t}$ are the pre-activation outputs from student and teacher, respectively. + +Attention Score Matching Loss: We align the attention score map $(\mathbf{A}_{\mathbf{i}}^{\mathbf{T}} \in \mathbb{R}^{B \times L \times L})$ of encoder layer $i$ of the non-spiking model with the mean attention score $(\mathbf{A}_{\mathbf{i}}^{\mathbf{S}^{*}} = \frac{1}{T_{c}} \sum_{t=1}^{T_{c}} A_{i}^{S}[t] \in \mathbb{R}^{B \times L \times L})$ of our corresponding converged SpikingVTG model. The total attention map alignment loss $(\mathcal{L}_{\mathrm{att}})$ is then computed following Eqn. 6. This loss term encourages the student to learn cross-modal attention behavior consistent with the pre-trained teacher. + +The cumulative representation matching loss is given as $\mathcal{L}_{\mathrm{CLRM}} = \mathcal{L}_{\mathrm{rep}} + \mathcal{L}_{\mathrm{att}}$ . Optimizing this loss enables our SpikingVTG model to learn the generalability enabling better performance as demonstrated in Table 1. + +# 4.6 Normalization-Free and Quantized SpikingVTG + +We perform two key optimizations: removal of non-local operations and 1-bit weight quantization. Although SpikingVTG replaces floating-point MACs with ternary spikes, it retains softmax and layer normalization, which are inefficient for resource-constrained hardware. We eliminate these by introducing a Normalization-Free (NF) SpikingVTG. + +In NF-SpikingVTG, softmax in the spiking attention is replaced with a ReLU followed by scaling with $1 / L$ , reducing compute overhead. Given $d$ -dimensional queries, keys, and values $\{q_i[t], s_{k_i}[t], s_{v_i}[t]\}_{i=1}^L$ , at time $t$ , the attention weights $\alpha_{ij}$ are computed as follows: + +$$ +\alpha_ {i j} [ t ] = \phi \left(\left[ q _ {i} [ t ] ^ {\top} s _ {k _ {1}} [ t ], \dots , q _ {i} [ t ] ^ {\top} s _ {k _ {L}} [ t ] \right]\right) _ {j} \tag {7} +$$ + +where, $\phi$ is a custom kernel consisting of ReLU operation followed by scaling with $L^{-1}$ . We further remove all layer normalization layers to eliminate additional non-local operations. + +To reduce memory and computational cost, we introduce 1-bit NF-SpikingVTG by binarizing weights. Each weight matrix $W \in \mathbb{R}^{n \times m}$ is zero-centered and quantized as $W_{q} = \operatorname{sgn}\left(W - \frac{1}{nm}\sum_{i,j}W_{ij}\right)$ , $\beta = \frac{1}{nm}\sum_{i,j}|W_{ij}|$ , where $W_{q} \in \{-1, + 1\}$ . The output of each quantized linear layer is scaled by $\beta$ , yielding a binary-weight, ternary-activation model that supports efficient integer-ACC operations. 1-bit NF-SpikingVTG achieves competitive performance while drastically reducing memory and compute, making it well-suited for edge deployment. + +Table 2: Performance comparison of our SpikingVTG model against non-spiking VTG solutions on test sets of QVHighlights-MR and Charades-STA datasets for moment retrieval task. + +
MethodSNNQVHighlights-MRCharades-STA
@0.5@0.7mAP@0.5mAP@0.75@0.3@0.5@0.7mIoU
M-DETR [1]No52.8933.0254.8229.4065.8352.0730.5945.54
UMT [5]No56.2341.1853.8337.01-49.3526.16-
UniVTG [6]No58.8640.8657.6035.5970.8158.0135.6550.10
UniVTG+PT [6]No65.4350.0664.0645.0272.6360.1938.5552.17
UVCOM [34]No63.5547.4763.3742.67-56.6934.76-
SpikeMba [19]No64.1349.42-43.6771.2459.6536.1251.74
BAM-DETR [35]No62.7148.6464.5746.3372.9359.9539.3852.33
TR-DETR [36]No64.6648.9663.9843.73-57.6133.52-
CG-DETR [37]No65.4348.3864.5142.7770.4058.4036.3050.10
LLMEPET [38]No66.7349.9465.7643.9170.91-36.4950.25
SpikingVTGYes65.2948.1864.3142.2571.2058.7337.1650.62
+ +# 5 Experimentation + +We evaluate SpikingVTG variants on moment retrieval and highlight detection tasks using the QVHighlights, Charades-STA, TACoS and Youtube Highlight datasets. Since, to the best of our knowledge, our proposed model is the first spiking detection transformer evaluated on VTG tasks, we compare its performance against sota non-spiking detection transformers. Additional training, dataset, evaluation metric, hyperparameter and experimental details are provided + +in Appendix C & E. The experiments were run on a NVIDIA RTX A6000 GPU with 48GB memory. + +Table 3: Performance comparison on the test set of TACoS for moment retrieval task. + +
MethodSNNTACoS
@0.3@0.5@0.7mIoU
M-DETR [1]No37.9724.6711.9725.49
2D-TAN [6]No40.0127.9912.9227.22
UniVTG [6]No51.4434.9717.3533.60
QD-DETR [39]No52.3936.7721.0735.76
CG-DETR [37]No52.23-22.2336.48
UniVTG+PT [6]No56.1143.4424.2738.63
SpikeMba [19]No51.9839.3422.8335.81
SpikingVTGYes54.7139.2721.8436.02
+ +# 5.1 Results + +SpikingVTG establishes a baseline for spiking models on VTG tasks. The results are shown in Table 2, 3 & 4. Our model achieves competitive results compared to the current SOTA non-spiking models. + +Training & Inference Metrics: In the CLRM-based knowledge transfer stage, the memory requirement is 20GB when using a batch size of 32 on the QVHighlights dataset. In contrast, replacing our training method with BPTT would require over 100GB of memory for $T_{c} > 10$ , making BPTT computationally infeasible. Training on true labels with similar batch size requires 8GB of + +Table 4: Performance comparison of our SpikingVTG model against other non-spiking VTG solutions on the test set of the QVHighlights and Youtube Highlights for highlight detection task. + +
MethodSNNQVHighlights-HDYoutube Highlights
mAPHIT@1DogGym.SkatingSkiingParkingSurfingAvg.
UMT [5]No38.1859.9965.975.271.872.381.682.774.9
UniVTG [6]No38.2060.9671.876.573.373.273.982.275.2
UniVTG+PT [6]No40.5466.2874.379.084.975.174.483.978.6
QD-DETR [39]No38.9062.4072.277.472.772.871.080.674.4
SpikeMba [5]No--74.475.474.375.5--75.5
CG-DETR [37]No40.3066.2076.376.176.075.170.081.975.9
UVCOM [34]No39.9865.5873.877.176.075.175.782.776.4
LLMEPET [38]No38.1859.9973.673.275.374.072.582.575.3
SpikingVTGYes40.4665.8273.978.180.174.272.281.776.7
+ +memory. The clock time for 50 epochs of training on QVHighlights is $\approx 5$ HOURS. Inference latency on entire test set of QVHighlights ranges from 25 sec ( $T_{c} = 2$ ) to $\approx 4$ mins ( $T_{c} = 20$ ). + +Table 5: Ablation Study of SpikingVTG variants on the evaluation set of QVHighlights. + +
MethodQVHighlights-MRQVHighlights-HDOperationsLocalActivityEnergy
@0.5@0.7mAPHIT@1
Pre-trained UniVTG (sota)67.3552.6541.3468.77FP-MAC×1.023.92mJ
SpikingVTG without SFG64.9447.2140.4967.37FP-ACC×0.4115.2mJ
SpikingVTG with SFG67.5850.8240.8168.64FP-ACC×0.3413.8mJ
(NF)-SpikingVTG w/ SFG66.5948.3140.6167.73FP-ACC✓0.2510.1mJ
1-bit (NF)-SpikingVTG w/ SFG65.3147.4840.3567.30INT-ACC✓0.191.3mJ
1-bit (NF)-SpikingVTG w/ ReLU65.9147.0440.1667.07INT-ACC✓0.191.3mJ
+ +Ablation Study: As demonstrated in Table 5, the inclusion of the SFG mechanism enhances performance compared to the model without SFG. It results in reduced neuronal activity and overall less energy consumption. Thus, we enable SFG for the SpikingVTG variants we discuss next. Without non-local operations, the (NF)-SpikingVTG model achieves competitive performance even compared to other SOTA non-spiking VTG models. Although the 1-bit (NF)-SpikingVTG variant shows a slight reduction in performance, it is highly memory efficient and involves simpler INT-ACC computations, resulting in order of magnitude less energy consumption. Along with energy consumption ( $T_{c} = 10$ ) Table 5, also highlights the sparsity in the SpikingVTG variants particularly in the 1-bit version underscoring considerably reduced mean neural activity. For a more hardware friendly model, we train a variant of 1-bit (NF)-SpikingVTG by replacing all GELU layers with ReLU layer [40] and observe minimal degradation in performance. Extensive hyper-parameter study is done in Appendix E. + +Analysis of Energy Efficiency: We analyze the test-time energy efficiency of SpikingVTG variants compared to a non-spiking transformer-based model with same depth and hidden dimensions. The analysis considers arithmetic operation costs using a 45nm CMOS technology with 32-bit precision, where FP-MAC, FP-ACC, and INT-ACC ops. consume $4.6\mathrm{pJ}$ , $0.9\mathrm{pJ}$ , and $0.1\mathrm{pJ}$ respectively [41]. The energy cost for a non-spiking transformer encoder layer based on total FLOPs used in attention and linear layers, is $E_{A} = [(3LD^{2}) + (LD^{2} + L^{2}D) + (2LD^{2})]\times$ $4.6\mathrm{pJ}$ . Considering $L = 200$ , $D = 1024$ , we compute $E_{A} = 5.98\mathrm{mJ}$ . Since, UniVTG has 4 encoder layers, so total energy is $23.92mJ$ . + +In contrast, the 1-bit (NF)-SpikingVTG operates over $T_{c}$ time steps. Its per time-step cost is: $E_{S_t} = [(3\cdot IFR_{\mathrm{in}}\cdot LD^2) + (IFR_{\mathrm{k}}\cdot LD^2 + IFR_{\mathrm{v}}\cdot L^2 D) + (IFR_{\mathrm{attn}}\cdot LD^2) + (IFR_{\mathrm{int.}}\cdot$ + +![](images/8711a46c6d203fbfdadc67f22bbf551663d2f8e610db98e1e54b6999ebe2d269.jpg) +Figure 4: Graph showing tradeoff of energy efficiency $(E_{f})$ and HIT@1 on QVHighlights-HD for varying time steps. + +$[LD^{2})]\times 0.1\mathrm{pJ}$ , where $IFR_{l}$ denotes the mean activity of component $l$ . For our 1-bit (NF)-SpikingVTG model, which uses INT-ACC operations, we empirically measure: $IFR_{\mathrm{in}} = 0.40$ , $IFR_{\mathrm{k}} = 0.18$ , $IFR_{\mathrm{v}} = 0.19$ , $IFR_{\mathrm{attn}} = 0.03$ , and $IFR_{\mathrm{interm.}} = 0.09$ , resulting in $E_{S_t} = 0.03\mathrm{mJ}$ . With $T_{c} = 10$ , the total spiking energy is $E_{S} = T_{c}\cdot E_{S_{t}} = 0.3\mathrm{mJ}$ , yielding an energy efficiency of $E_{f} = E_{A} / E_{S} = 19.93$ . Model-wise energy comparison is shown in Table 5. + +Energy-Accuracy Tradeoff: Unlike conventional non-spiking VTG solutions, SpikingVTG enables a controllable trade-off between performance and energy consumption, as total energy usage scales with the number of operating time steps. In Fig. 4, we demonstrate this performance-energy tradeoff achieved using 1-bit (NF)-SpikingVTG model by running it for different number of time steps. Notably, when running the model for just 2 time steps, SpikingVTG attains a HIT@1 score of 66.4 (compared to the SOTA score of 68.77), while achieving an energy efficiency factor $E_{f} \approx 100$ . + +# 6 Conclusions + +In this paper we propose SpikingVTG, a bio-inspired computationally efficient solution for VTG tasks in resource constraint environment. We propose a saliency feedback gating mechanism, which leverages the temporal dynamics of the model to improve performance while lowering computational costs by reducing overall neuronal activity across the model. We empirically demonstrate the convergence dynamics of the SpikingVTG model and leverage the formulated steady-state equations to efficiently train our model using implicit differentiation at equilibrium. To improve generalization, we propose a Cos-L2 Representation Matching loss, enabling knowledge transfer from non-spiking VTG models and yielding substantial performance gains. We further optimize the model by removing non-local operations and applying extreme quantization, leading to the 1-bit NF-SpikingVTG. Our framework significantly improves computational efficiency and model compactness, enabling tunable energy-accuracy tradeoffs on low-resource devices. While this work presents the first spiking solution for VTG with competitive performance, there remains room for improvement in fully closing the performance gap. Future research can build upon our framework to further advance spiking models in this domain. + +# 7 Acknowledgments + +This work was supported in part by the United States Air Force and Defense Advanced Research Projects Agency (DARPA) under Contract No. FA8750-23-C-0519 and the U.S. Army Research Laboratory Cooperative Research Agreement W911NF-17-2-0196. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the Department of Defense or the United States Government. + +# References + +[1] Jie Lei, Tamara L Berg, and Mohit Bansal. Detecting moments and highlights in videos via natural language queries. Advances in Neural Information Processing Systems, 34:11846-11858, 2021. +[2] Songyang Zhang, Houwen Peng, Jianlong Fu, and Jiebo Luo. Learning 2d temporal adjacent networks for moment localization with natural language. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 12870-12877, 2020. +[3] Jonghwan Mun, Minsu Cho, and Bohyung Han. Local-global video-text interactions for temporal grounding. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10810-10819, 2020. +[4] Fa-Ting Hong, Xuanteng Huang, Wei-Hong Li, and Wei-Shi Zheng. Mini-net: Multiple instance ranking network for video highlight detection. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part XIII 16, pages 345-360. Springer, 2020. +[5] Ye Liu, Siyuan Li, Yang Wu, Chang-Wen Chen, Ying Shan, and Xiaohu Qie. Umt: Unified multimodal transformers for joint video moment retrieval and highlight detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3042-3051, 2022. +[6] Kevin Qinghong Lin, Pengchuan Zhang, Joya Chen, Shraman Pramanick, Difei Gao, Alex Jinpeng Wang, Rui Yan, and Mike Zheng Shou. Univtg: Towards unified video-language temporal grounding. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2794-2804, 2023. +[7] Siddharth Samsi, Dan Zhao, Joseph McDonald, Baolin Li, Adam Michaleas, Michael Jones, William Bergeron, Jeremy Kepner, Devesh Tiwari, and Vijay Gadepally. From words to watts: Benchmarking the energy costs of large language model inference. In 2023 IEEE High Performance Extreme Computing Conference (HPEC), pages 1-9. IEEE, 2023. +[8] Samanwoy Ghosh-Dastidar and Hojjat Adeli. Spiking neural networks. International journal of neural systems, 19(04):295-308, 2009. + +[9] Kohitij Kar, Jonas Kubilius, Kailyn Schmidt, Elias B Issa, and James J DiCarlo. 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Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need, 2017. +[14] Amar Shrestha, Haowen Fang, Zaidao Mei, Daniel Patrick Rider, Qing Wu, and Qinru Qiu. A survey on neuromorphic computing: Models and hardware. IEEE Circuits and Systems Magazine, 22(2):6-35, 2022. +[15] Soroush Abbasi Koohpayegani and Hamed Pirsiavash. Sima: Simple softmax-free attention for vision transformers. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 2607-2617, 2024. +[16] Boxun Xu, Hejia Geng, Yuxuan Yin, and Peng Li. Ds2ta: Denoising spiking transformer with attenuated spatiotemporal attention. arXiv preprint arXiv:2409.15375, 2024. +[17] Hongyu Wang, Shuming Ma, Li Dong, Shaohan Huang, Huaijie Wang, Lingxiao Ma, Fan Yang, Ruiping Wang, Yi Wu, and Furu Wei. Bitnet: Scaling 1-bit transformers for large language models. arXiv preprint arXiv:2310.11453, 2023. +[18] Malyaban Bal, Yi Jiang, and Abhronil Sengupta. Exploring extreme quantization in spiking language models. In 2024 International Conference on Neuromorphic Systems (ICONS), pages 272-276. IEEE, 2024. +[19] Wenrui Li, Xiaopeng Hong, and Xiaopeng Fan. Spikemba: Multi-modal spiking saliency mamba for temporal video grounding. arXiv preprint arXiv:2404.01174, 2024. +[20] Albert Gu and Tri Dao. Mamba: Linear-time sequence modeling with selective state spaces. arXiv preprint arXiv:2312.00752, 2023. +[21] Michael V DeBole, Brian Taba, Arnon Amir, Filipp Akopyan, Alexander Andreopoulos, William P Risk, Jeff Kusnitz, Carlos Ortega Otero, Tapan K Nayak, Rathinakumar Appuswamy, et al. Truenorth: Accelerating from zero to 64 million neurons in 10 years. Computer, 52(5):20-29, 2019. +[22] Mike Davies, Andreas Wild, Garrick Orchard, Yulia Sandamirskaya, Gabriel A Fonseca Guerra, Prasad Joshi, Philipp Plank, and Sumedh R Risbud. Advancing neuromorphic computing with loihi: A survey of results and outlook. 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In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 10998-11006, 2024. +[27] Rui-Jie Zhu, Qihang Zhao, and Jason K Eshraghian. Spikegpt: Generative pre-trained language model with spiking neural networks. arXiv preprint arXiv:2302.13939, 2023. +[28] Jan Hosang, Rodrigo Benenson, and Bernt Schiele. Learning non-maximum suppression. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4507-4515, 2017. +[29] Sangya Dutta, Vinay Kumar, Aditya Shukla, Nihar R Mohapatra, and Udayan Ganguly. Leaky integrate and fire neuron by charge-discharge dynamics in floating-body mosfet. Scientific reports, 7(1):8257, 2017. +[30] Yufei Guo, Yuanpei Chen, Xiaode Liu, Weihang Peng, Yuhan Zhang, Xuhui Huang, and Zhe Ma. Ternary spike: Learning ternary spikes for spiking neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 12244–12252, 2024. +[31] Hamid Rezatofighi, Nathan Tsoi, JunYoung Gwak, Amir Sadeghian, Ian Reid, and Silvio Savarese. Generalized intersection over union: A metric and a loss for bounding box regression. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 658-666, 2019. +[32] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pages 1597-1607. PMLR, 2020. +[33] Shaojie Bai, J Zico Kolter, and Vladlen Koltun. Deep equilibrium models. Advances in Neural Information Processing Systems, 32, 2019. +[34] Yicheng Xiao, Zhuoyan Luo, Yong Liu, Yue Ma, Hengwei Bian, Yatai Ji, Yujiu Yang, and Xiu Li. Bridging the gap: A unified video comprehension framework for moment retrieval and highlight detection, 2023. +[35] Pilhyeon Lee and Hyeran Byun. Bam-detr: Boundary-aligned moment detection transformer for temporal sentence grounding in videos, 2024. +[36] Hao Sun, Mingyao Zhou, Wenjing Chen, and Wei Xie. Tr-detr: Task-reciprocal transformer for joint moment retrieval and highlight detection, 2024. +[37] WonJun Moon, Sangeek Hyun, SuBeen Lee, and Jae-Pil Heo. Correlation-guided query-dependency calibration for video temporal grounding, 2024. +[38] Yiyang Jiang, Wengyu Zhang, Xulu Zhang, Xiao-Yong Wei, Chang Wen Chen, and Qing Li. Prior knowledge integration via llm encoding and pseudo event regulation for video moment retrieval. In Proceedings of the 32nd ACM International Conference on Multimedia, MM '24, page 7249-7258. ACM, October 2024. +[39] WonJun Moon, Sangeek Hyun, SangUk Park, Dongchan Park, and Jae-Pil Heo. Query-dependent video representation for moment retrieval and highlight detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 23023-23033, 2023. +[40] Jonathan Timcheck, Sumit Bam Shrestha, Daniel Ben Dayan Rubin, Adam Kupryjanow, Garrick Orchard, Lukasz Pindor, Timothy Shea, and Mike Davies. The intel neuromorphic dns challenge. Neuromorphic Computing and Engineering, 3(3):034005, 2023. +[41] Song Han, Jeff Pool, John Tran, and William J. Dally. Learning both weights and connections for efficient neural networks, 2015. +[42] Jiyang Gao, Chen Sun, Zhenheng Yang, and Ram Nevatia. Tall: Temporal activity localization via language query. In Proceedings of the IEEE international conference on computer vision, pages 5267-5275, 2017. + +[43] Michaela Regneri, Marcus Rohrbach, Dominikus Wetzel, Stefan Thater, Bernt Schiele, and Manfred Pinkal. Grounding action descriptions in videos. Transactions of the Association for Computational Linguistics, 1:25-36, 2013. +[44] Min Sun, Ali Farhadi, and Steve Seitz. Ranking domain-specific highlights by analyzing edited videos. In Computer Vision-ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part I 13, pages 787-802. Springer, 2014. + +# A Extended Architecture Overview + +The Spiking Transformer layer primarily consists of a spiking multi-head attention (MHA) block, followed by a spiking feedforward network comprising an intermediate layer and an output layer with both inter- and intra-layer communication happening using spikes. Details of the operations in each layer are provided below. + +Spiking Attention Block: In Spiking MHA, to enable computationally efficient accumulate based operations the input to the attention layer are spikes instead of real-valued data. The spiking attention mechanism [26] is given as follows, + +$$ +A t t n \left(X _ {s} [ t ], K _ {s} [ t ], V _ {s} [ t ]\right) = \phi \left(d * Q \left(X _ {s} [ t ]\right) \cdot \left(K _ {s} [ t ]\right) ^ {T}\right) \cdot V _ {s} (t) \tag {8} +$$ + +Here, $Q(X_{s}(t))$ represents the Query, obtained by passing the input spikes $X_{s}(t)$ at time $t$ through a linear layer $(W_{Q})$ . The spikes for the Key layer $(K_{s}(t))$ are generated by passing $X_{s}(t)$ through a linear mapping $(W_{K})$ , followed by an LIF neuron layer. Similarly, we generate spikes for Value. $d$ is a scaling constant. Since the input, key, and value matrices consist of spike trains rather than real-valued data, the primary computations in all matrix multiplications are floating-point accumulation operations rather than floating point multiplicative and accumulative operations. In the (NF)-SpikingVTG variant, as discussed in the paper, we use $\phi$ as the ReLU and scaling operation, significantly reducing the computational overhead compared to employing $\phi$ as the non-local Softmax operation. The output of the attention layer is fed to an LIF neuron, which outputs spikes. The convergence dynamics of the layer at equilibrium is given as, $a_{attn}^{*} = \sigma \left( \frac{1}{V_{th}} \left( Attn(a_{x}^{*}, a_{k}^{*}, a_{v}^{*}) + b_{attn} \right) \right)$ , where $a_{x}$ represents the ASR of the layer used to generate the Query, $a_{k}$ denotes the ASR of the Key, and $a_{v}^{*}$ corresponds to the ASR of the Value. $b_{attn}$ is a bias term. + +Intermediate Layer: The intermediate layer takes as input the spikes generated from the preceding layer and maps it to an intermediate dimension with a linear layer. The output is then passed through an LIF layer. The convergence dynamics of the layer at equilibrium is given as, $a_{interm}^{*} = \sigma\left(\frac{1}{V_{th}}(act(W_{interm}.a_{p}^{*}) + b_{interm.})\right)$ , where $W_{interm}$ is the linear weight and $\mathrm{gelu}$ is the activation used for the layer. $a_{p}^{*}$ is the ASR at equilibrium for the previous layer. $b_{interm.}$ is a bias term. In this paper, we have used explored different choices for act, such as GELU and ReLU. During inference, all matrix multiplications involve accumulative operations due to the nature of the input. + +Output Layer: The output layer takes as input the spikes generated from the preceding layers as shown in Fig. 1. The output is then passed through an LIF layer. The convergence dynamics of the layer at equilibrium is given as, $a_{output}^{*} = \sigma (\frac{1}{V_{th}} (\text{norm}(W_{output}a_{interm}^{*} + a_{p}^{*}) + b_{output}))$ , where $W_{output}$ is the linear weight and layer norm is used for normalization. $a_{interm}^{*}$ is the ASR at equilibrium for the previous intermediate layer. $b_{output}$ is a bias term. During inference, all matrix multiplications involve accumulative operations due to the nature of the input. In the (NF)-SpikingVTG model we further remove the layer normalization to improve on-chip deployability. + +# B Loss Function Details + +As described in the main paper, the total loss over $N$ clips in the training set is defined as $L = \frac{1}{N}\sum_{i=1}^{N}(L_{f_i} + L_{d_i} + L_{c_i})$ , where $L_f$ represents the binary cross-entropy loss for the indicator variable $f_i$ , $L_d$ combines the smooth L1 loss with the generalized IoU loss [31] for the predicted boundaries, and $L_c$ is an optional loss term incorporating intra- and inter-video contrastive learning [32]. We follow similar loss function construction as previous works on VTG [1, 6]. The loss for foreground parameter is given as follows, + +$$ +L _ {f} = - \lambda_ {f} \left[ f _ {i} \log \tilde {f} _ {i} + (1 - f _ {i}) \log \left(1 - \tilde {f} _ {i}\right) \right] \tag {9} +$$ + +where, $f_{i}$ is the true label and $\tilde{f}_{i}$ is the model prediction. The loss for predicted boundaries is given as follows, + +$$ +L _ {d} = \mathbf {1} _ {f _ {i} = 1} \left(\lambda_ {\mathrm {L} 1} L _ {\text {S m o o t h L} 1} \left(\tilde {d} _ {i}, d _ {i}\right) + \lambda_ {\mathrm {i o u}} L _ {\mathrm {i o u}} \left(\tilde {b} _ {i}, b _ {i}\right)\right) \tag {10} +$$ + +where, $d_{i}, b_{i}$ are the true label and $\tilde{d}_i,\tilde{b}_i$ is the model prediction. $L_{c} = \lambda_{\mathrm{inter}}L_{\mathrm{inter}} + \lambda_{\mathrm{intra}}L_{\mathrm{intra}}$ is used for inter-video and intra video contrastive learning [6]. For each video $V$ , we randomly select a clip $v_{i}$ with fore-ground indicator $= 1$ and positive saliency score. Clips from the same video, denoted as $v_{j}$ , with saliency scores $s_j < s_i$ are treated as negative samples. i.e., $A = \{j\mid s_j < s_i,1\leq j\leq L_v\}$ , and perform intra-video contrastive learning using the loss + +$$ +L _ {\text {i n t r a}} = - \log \frac {\exp \left(\tilde {s} _ {i} / \tau\right)}{\exp \left(\tilde {s} _ {i} / \tau\right) + \sum_ {j \in A} \exp \left(\tilde {s} _ {j} / \tau\right)} \tag {11} +$$ + +. Furthermore, we treat textual queries from other samples within the batch $(k\in S)$ as negative samples, enabling inter-video contrastive learning for cross-sample supervision: + +$$ +L _ {\text {i n t e r}} = - \log \frac {\exp \left(\tilde {s} _ {i} / \tau\right)}{\sum_ {k \in S} \exp \left(\tilde {s} _ {i} ^ {k} / \tau\right)} \tag {12} +$$ + +, where $S$ is the training batch, $\tilde{s}_i^k = \cos (v_i,M_k)$ and $M_{k}$ is the sentence representation (Eqn. 2) and cos is cosine similarity. + +# C Dataset Details + +QVHighlights: The QVHighlights dataset [1] stands out as the sole dataset providing annotations for both moment retrieval and highlight detection, making it an excellent resource for benchmarking on both the VTG tasks. Comprising 10,148 videos with an average duration of 150 seconds. It features a total of 10,310 queries linked to 18,367 moments, resulting in an average of 1.8 distinct moments per query within each video. The dataset spans a variety of scenarios, including daily vlogs, travel vlogs, and news events. + +Charades-STA: The Charades-STA [42] dataset comprises 16,128 indoor videos, each with an average duration of 30.6 seconds. It includes 12,408 query-interval pairs designated for training and 3,720 query-interval pairs reserved for testing. + +TACoS: TACoS [43] consists of 127 videos, each averaging 4.78 minutes in length. The dataset is split into 75 videos for training, 27 for validation, and 25 for testing. + +Youtube Highlights: YouTube Highlights [44] consists of 433 videos across 6 domains, using the domain names as text queries. + +# D Evaluation Metrics: + +For QVHighlights, following previous work [1] we use Recall@1 with IoU thresholds of 0.3, 0.5 and 0.7 and avg. mean average precision (mAP), mAP@0.5 and mAP@0.75 as the evaluation metric for moment retrieval tasks. For highlight detection, we use mAP and HIT@1 [1], where a clip is considered a true positive if it receives a score of "Very Good" [5]. For Charades-STA and TACoS, we employ Recall@1 with IoU thresholds of 0.3, 0.5, and 0.7, along with the mean IoU (mIoU). For Youtube Highlights we use mAP. + +# E Additional Experimental Details + +In this subsection, we provide a concise overview of the implementation details and provide additional experimental details. The GPU specifications for the experiments are detailed in the main paper, while the CPU utilized is an AMD Ryzen Threadripper 3960X 24-Core Processor. We have used Python and the PyTorch framework to write the code. The video and textual feature are developed following previous work [1, 6]. We have used the Adam optimizer to train our model. We list the hyper-parameters used in the work in Table 6. We used grid search to find optimal values. + +# E.1 Training Stages + +Training a multi-modal spiking architecture like SpikingVTG is resource-intensive. To enhance the efficiency of this process and develop computationally efficient variants of our model, we leverage a + +
Hyper-parametersRangeOptimal
N: Encoder Layers(2-6)4
D: Hidden Dimension(768-2048)1024
n1: f-decoder depth(1-5)3
k1: f-decoder kernel size(3-9)3
n2: d-decoder depth(1-5)3
k2: d-decoder kernel size(3-9)7
TCLRM: Timesteps for CLRM(5-100)50
Tf: Timesteps for Finetuning(5-50)16
Vth: Threshold Potential(0.5 - 2.0)1.0
γ: Leaky-factor(0.9 - 1.0)0.99
λf:Lf co-efficient(1 - 20)10
λL1:LSmoothL1 co-efficient(1 - 20)10
λintra:Lintra co-efficient(0 - 1.0)0.05
λinter:Inter-co-efficient(0 - 1.0)0.01
λiou:Liou co-efficient(1 - 20)10
λcos: Sq. cosine weight (CLRM)(0 - 1.0)0.2
λl2: L2 weight (CLRM)(0 - 1.0)0.8
lr: Learning Rate(1e-5 - 1e-6)8e-6
wd: weight decay(1e-5 - 1e-3)1e-4
Batch Size(8-64)32
Epochs: CLRM10-10050
Epochs: Finetuning20-200100
+ +Table 6: Hyper-parameters of our SpikingVTG model. Optimal values for QVHighlights dataset is also shown. + +multi-staged training framework. We utilize a transformer-based non-spiking VTG model (such as UniVTG) to perform CLRM loss optimization. After this initial stage, we fine-tune SpikingVTG using the true labels. Once the base SpikingVTG model is established, we modify its architecture to remove non-local operations and perform extreme quantization, followed by additional fine-tuning to create computationally efficient variants with minimal performance degradation. The resulting computationally efficient, lightweight models are well-suited for deployment on neuromorphic chips, enabling efficient inference. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. 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diff --git a/NeurIPS/2025/Spiral_ Semantic-Aware Progressive LiDAR Scene Generation and Understanding/full.md b/NeurIPS/2025/Spiral_ Semantic-Aware Progressive LiDAR Scene Generation and Understanding/full.md new file mode 100644 index 0000000000000000000000000000000000000000..e70a276e0b848f35704e776b468897ed2aea8092 --- /dev/null +++ b/NeurIPS/2025/Spiral_ Semantic-Aware Progressive LiDAR Scene Generation and Understanding/full.md @@ -0,0 +1,874 @@ +![](images/6ff86011a98653910b8afc7f456978c711f973d071280c895c90eeed18701c1b.jpg) + +# SPIRAL: Semantic-Aware Progressive LiDAR Scene Generation and Understanding + +Dekai Zhu $^{1,4,*}$ Yixuan Hu $^{1,*}$ Youquan Liu $^{2}$ Dongyue Lu $^{3}$ + +Lingdong Kong3 Slobodan Ilic1 + +$^{1}$ Technical University of Munich $^{2}$ Fudan University + +$^{3}$ National University of Singapore $^{4}$ Munich Center for Machine Learning + +https://dekai21.github.io/SPIRAL/ + +![](images/c5f258fa084892d2c881dc94cfd16bc27fd4ff7a9481d40228a5d9a90460767a.jpg) + +Figure 1: Visualization of LiDAR scenes and their semantic labels jointly generated by SPIRAL, exhibiting high geometric fidelity and semantic-geometric consistency. +![](images/5ab9ba0bfb7a7c2ae05030f27a8aafbda787dd345245710a089be39e9dc43f39.jpg) +car bicy moto truc o.veh ped b.list m.list road park walk o.gro build fenc veg trun pole sign + +![](images/1ae5ed12c253b86335811230df733080982535bbe93c1a7fdefcd64e88433364.jpg) + +![](images/0fb4b9b841be6502432119202f5bc41abf771cd6ed75499d01e9699257ea38b6.jpg) + +![](images/961c2711071b2fe7b73c4333243171381431fb753ac71f68538bc9a74b16256f.jpg) + +![](images/cae4542b54ad24d9d207f21ddb326cdfc4175f56ecdfe1fdd8037c647b701200.jpg) + +![](images/b8777861bf18c48bc3df95939af791187fbc712f1169c331f784fe4c7ac7cec6.jpg) + +![](images/0a2aadb686b9c0864749dce614ad573c2b37d3336a2c2d34a6d0cbd937e771c9.jpg) + +![](images/024e8e5d6cd7c7bbb21c38a8f5c9a12a2d00a59bcd953d4f7d3d5665f582deec.jpg) + +# Abstract + +Leveraging recent diffusion models, LiDAR-based large-scale 3D scene generation has achieved great success. While recent voxel-based approaches can generate both geometric structures and semantic labels, existing range-view methods are limited to producing unlabeled LiDAR scenes. Relying on pretrained segmentation models to predict the semantic maps often results in suboptimal cross-modal consistency. To address this limitation while preserving the advantages of range-view representations, such as computational efficiency and simplified network design, we propose SPIRAL, a novel range-view LiDAR diffusion model that simultaneously generates depth, reflectance images, and semantic maps. Furthermore, we introduce novel semantic-aware metrics to evaluate the quality of the generated labeled range-view data. Experiments on the SemanticKITTI and nuScenes datasets demonstrate that SPIRAL achieves state-of-the-art performance with the smallest parameter size, outperforming two-step methods that combine the generative and segmentation models. Additionally, we validate that range images generated by SPIRAL can be effectively used for synthetic data augmentation in the downstream segmentation training, significantly reducing the labeling effort on LiDAR data. + +# 1 Introduction + +By providing accurate distance measurements regardless of ambient illumination, LiDAR plays a crucial role in scene understanding and navigation for robotics and autonomous driving [11, 60, 10, 13, + +![](images/2fb5097745ddadf4e4d67d3bfe54e9e3966d0abbc47f8e89c8fbeec6f354cf12.jpg) +(a) Conventional Two-Step Pipeline + +![](images/f73055cfda3b55335d1df5c7a00ab204e00b023d0f8235507ea2840411f41d7c.jpg) +(b) SPIRAL: Unified One-Step & Closed-Loop Framework + +![](images/1100be347191d85b2c36181aa4883b48fea6b07d3f249a5c198773b948c61fe9.jpg) +(c) Quantitative Comparison +Figure 2: (a) Two-step methods: Existing range-view LiDAR generative models typically generate only depth and reflectance images, requiring an additional pre-trained segmentation model to predict semantic labels. (b) SPIRAL: In contrast, Spiral jointly generates depth, reflectance, and semantic maps. A closed-loop inference mechanism (highlighted in the dash arrow) further improves cross-modal consistency. (c) Results: Spiral achieves state-of-the-art performance with the smallest parameter size (61M) among the related methods. + +32, 26, 27, 58]. However, collecting and annotating large-scale LiDAR datasets is both expensive and time-consuming [12, 64, 41, 38, 63]. To address this issue, recent research has increasingly focused on using denoising diffusion probabilistic models (DDPMs) [14] for LiDAR generative modeling, aiming to create tools capable of generating unlimited LiDAR scenes [73, 48, 16, 42, 43, 37, 34]. + +Existing generative approaches can be categorized into voxel-based methods [43, 29, 49, 4] and range-view-based methods [73, 42, 48, 16]. The former divides the 3D space into regular volumetric grids (i.e., voxels) and captures detailed geometric structures with 3D convolutional networks [8]. However, they often suffer from high memory consumption and computational overhead [52]. Methods based on range-view, on the other hand, project LiDAR point clouds onto a 2D cylindrical image plane using the sensor azimuth and elevation angles [73, 48, 16, 33]. This leads to compact 2D representations of depth and reflectance, allowing for efficient processing via 2D convolutional networks [1, 23]. These methods are significantly more memory-efficient and computationally lightweight [62, 61]. + +In this work, we aim to address two limitations in existing range-view generative methods: + +1. While recent models such as LiDARGen [73] and R2DM [42] generate high-fidelity LiDAR scenes, their outputs are restricted to depth and reflectance images, without producing semantic labels. +2. Existing evaluations extract global features for each scene from three perspectives (range-view image, Cartesian point cloud, and BEV projection) to assess the distributional similarity between generated and real scenes. However, none of these evaluation methods consider semantic labels. + +For the first limitation, a straightforward solution is to adopt a two-step pipeline: first, produce unlabeled LiDAR scenes using generative models [73, 42, 48], and then apply a pretrained segmentation model (e.g., RangeNet++ [40]) to predict the corresponding semantic labels, as depicted in Figure 2 (a). However, this approach often results in suboptimal performance due to two key issues: (1) The generative and segmentation models are trained independently, which hinders shared representations between the two tasks and reduces training efficiency. (2) The semantic maps, being predicted post hoc, cannot serve as conditional guidance during generation, leading to limited consistency between semantics and other modalities, including depth and reflectance. + +Therefore, we propose a novel semantic-aware range-view LiDAR diffusion model, named SPIRAL, as depicted in Figure 2 (b), with the following key features: + +- Semantic-aware generation: Our framework aims to jointly generate depth, reflectance, and semantic maps from the Gaussian noise, different from existing models that lack semantic awareness and require separate segmentation models to obtain semantic labels. +- Progressive semantic prediction: At each denoising step, SPIRAL outputs an intermediate semantic map which is aggregated via exponential moving averaging (EMA) to suppress noise and produce stable, per-pixel confidence scores. The EMA trace serves as both the final semantic output and the basis for the closed-loop inference. + +- Closed-loop inference: Once the prediction confidence exceeds a threshold, the semantic map is fed back into the model as the condition to guide the generation of both depth and reflectance. By alternating between conditional and unconditional steps, SPIRAL enables joint refinement of geometry and semantics, thereby enhancing cross-modal consistency. + +For the second limitation, we extend all three types of metrics with semantic awareness, enabling a comprehensive assessment of geometric, physical, and semantic quality in the generated LiDAR scenes. (1) For the evaluation in range-view image and Cartesian point cloud perspectives that rely on pretrained models such as RangeNet++ [40] and PointNet [47] to extract learning-based global features, we integrate the semantic map conditional module from LiDM [48], which is originally designed for semantic-to-LiDAR generation, to encode semantic labels. The encoded semantic features are then concatenated with the original global features to form semantic-aware global features. (2) For the evaluation in BEV projection perspective that produces rule-based global features using 2D histograms over the xy-plane, we compute 2D histograms for each semantic category individually and then concatenate them into a unified semantic-aware histogram representation. + +Experiments on the SemanticKITTI [3] and nuScenes [5] datasets demonstrate that Spiral achieves state-of-the-art performance in labeled LiDAR scene generation with the smallest parameter size, as depicted in and Fig. 1 and Fig. 2 (c). Moreover, we demonstrate that Spiral-generated samples can be effectively used as synthetic data to augment downstream training, which is particularly valuable for autonomous driving tasks that require large-scale training data [39, 70, 71]. To summarize, the key contributions of this work are as follows: + +- We propose a novel state-of-the-art semantic-aware range-view LiDAR diffusion model, SPIRAL, which jointly produces depth and reflectance images along with semantic labels. +- We introduce unified evaluation metrics that comprehensively evaluate the geometric, physical, and semantic quality of generated labeled LiDAR scenes. +- We demonstrate the effectiveness of the generated LiDAR scenes for training segmentation models, highlighting Spiral's potential for generative data augmentation. + +# 2 Related Works + +LiDAR Generation from Range Images. LiDARGen [73] pioneers diffusion-based LiDAR generation by learning a score function [18] that models the log-likelihood gradient in range-view image space. Building on this foundation, LiDM [48] advances conditional generative modeling, enabling synthesis from multiple input modalities. R2DM [42] enhances unconditional LiDAR scene generation through diffusion models and provides in-depth analysis of crucial components that improve generation fidelity. RangeLDM [16] optimizes the computational efficiency to achieve real-time generation, while Text2LiDAR [59] develops text-guided generation capabilities for semantic control. However, for semantic segmentation applications, these methods still require additional dedicated models, increasing computational cost. + +LiDAR Generation from Voxel Grids. Several studies investigate the generation of complete LiDAR scenes in Cartesian space, aiming to preserve geometric reconstruction accuracy [29, 43, 4, 49, 30, 57, 67, 54, 56, 45]. SemCity [29] leverages a triplane representation to model outdoor LiDAR scenes by projecting the 3D space into three orthogonal 2D planes. XCube [49] utilizes a hierarchical voxel latent diffusion model to generate large-scale scenes. [43] proposes to generate semantic LiDAR scenes using the latent DDPM without relying on intermediate image projections or coarse-to-fine multi-resolution modeling. DynamicCity [4] utilizes a VAE [22] model and a DiT-based [44] DDPM to generate large-scale, high-quality dynamic 4D scenes. + +LiDAR Semantic Segmentation. Various approaches are proposed for LiDAR scene segmentation from different representations, including range-view [40, 1, 23, 62, 25, 7], BEV [65], voxel [8, 72, 24, 15], and multi-view fusion [19, 20, 36, 6, 46, 39, 31, 35, 17]. As a representative range-view segmentation model, RangeNet++ [40] offers a real-time and efficient solution for LiDAR semantic segmentation. Cylinder3D [72] proposes a cylindrical representation that is particularly well-suited for outdoor scenes, addressing the irregularity and sparsity issues in LiDAR point clouds. SPVCNN [55] serves as a point-voxel fusion framework that integrates point cloud and voxel representations to leverage their complementary advantages for improved segmentation performance. + +![](images/adbadf52b454e67c4dcb65c4b8d22412a7056d8b200e0ef4a03712d4cd05780e.jpg) +Figure 3: (a) Unconditional Step: Spiral takes noisy LiDAR scenes $x_{t}$ as input and predicts both the semantic map $\hat{y}_{t}$ and the noise $\hat{\epsilon}_{t}$ , where the switch $\mathcal{A}$ is off and $\mathcal{B}$ is on. (b) Conditional Step: Spiral predicts $\hat{\epsilon}_{t}$ conditioned on the given semantic map $y$ , where $\mathcal{A}$ is on and $\mathcal{B}$ is off. (c) During inference, Spiral begins in an open-loop mode with unconditional steps. Once the predicted semantic map smoothed by the progressive filter reaches high confidence, Spiral switches to a closed-loop mode that alternates between unconditional and conditional steps, enhancing cross-modal consistency. + +# 3 Methodology + +# 3.1 Preliminaries of Diffusion Models for Range-View LiDAR Generation + +The diffusion models [14] generate data by simulating a stochastic process. Starting from a real sample $x_0 \sim q(x_0)$ , a forward process gradually adds Gaussian noise over $T$ steps, such that the final sample approximates a standard normal distribution, ie $q(x_{T}) \approx \mathcal{N}(0,I)$ . For range-view LiDAR generation, the sample is represented as $x_0 \in \mathbb{R}^{H \times W \times d}$ , where $H$ and $W$ denote the height and width of the range image, and $d$ denotes the number of modalities. In LiDARGen [73] and R2DM [42], $d = 2$ as both depth and reflectance images are generated, while $d = 1$ in LiDM [48] since it only generates depth images. The model $\epsilon_{\theta}$ with parameters $\theta$ is trained to predict the noise $\epsilon$ added at an intermediate step $t \in \{1,\dots,T\}$ , by minimizing the following objective: + +$$ +\mathcal {L} = \mathbb {E} _ {t, x _ {0}, \epsilon} \left[ \| \epsilon_ {\theta} (x _ {t}, t, a) - \epsilon \| _ {2} ^ {2} \right], \tag {1} +$$ + +where $\epsilon \sim \mathcal{N}(0,I)$ is the random noise added during the forward process, and $a$ denotes the conditional input, such as textual description or semantic maps $y\in \mathbb{R}^{H\times W\times C}$ , where $C$ denotes the number of categories. During inference, the model starts from a random noise sample $x_{T}\sim \mathcal{N}(0,I)$ and iteratively denoises it to generate a novel sample $x_0^{\prime}$ . At each step, an additional noise $\eta \sim \mathcal{N}(0,I)$ is added on $x_{t}$ to diversify the final results. Building upon the vanilla DDPM, a more flexible diffusion framework [50] is proposed by introducing a continuous time variable $t\in [0,1]$ , which enables flexible control over the number of sampling steps during inference, allowing a trade-off between generation speed and quality. + +# 3.2 SPIRAL: Semantic-Aware Progressive LiDAR Generation + +As previously discussed, although existing range-view LiDAR generative models [73, 48, 42] have demonstrated impressive performance, they are limited to producing only depth and reflectance modalities. While LiDM [48] generates LiDAR scenes conditioned on semantic maps, it requires the semantic maps to be provided beforehand. Alternatively, two-step pipelines that first generate LiDAR scenes and then predict semantic labels suffer from low training efficiency and limited cross-modal consistency. Inspired by the insight that diffusion models can serve as powerful representation learners for various tasks such as classification and segmentation [2, 68, 28, 69], we propose a novel semantic-aware progressive range-view LiDAR diffusion model, named SPIRAL, as illustrated in Figure 2 (b), to address these limitations. Spiral contains three major innovations: + +Complete Semantic Awareness. The semantic awareness of Spiral contains two aspects: (1) In addition to generating depth and reflectance images, Spiral also predicts the corresponding semantic maps; (2) Spiral enables conditional generation of depth and reflectance images guided by a given semantic map. It alternates between two types of steps: unconditional and conditional. To control the switching between them, we introduce two control switches, $\mathcal{A}$ and $\mathcal{B}$ , as illustrated in Figure 3. + +Switches $\mathcal{A}$ and $\mathcal{B}$ follow an exclusive-or (XOR) relationship. Their on/off states are as follows: + +$$ +\left\{ \begin{array}{l l} \mathcal {A} = 0, \mathcal {B} = 1; & \text {f o r t h e u n c o n d i t i o n a l s t e p}, \\ \mathcal {A} = 1, \mathcal {B} = 0; & \text {f o r t h e c o n d i t i o n a l s t e p}, \end{array} \right. \tag {2} +$$ + +where 0 denotes the off state and 1 denotes the on state. During training, in the unconditional step, the Spiral with learnable parameters $\theta$ , $\epsilon_{\theta}$ , simultaneously predicts both the semantic map $\hat{y}_t$ and the noise $\hat{\epsilon}_t$ on the noisy LiDAR scene $x_t$ , i.e., + +$$ +\hat {\epsilon} _ {t}, \hat {y} _ {t} \leftarrow \epsilon_ {\theta} (x _ {t}), \tag {3} +$$ + +and the corresponding training loss $\mathcal{L}_u$ is calculated as: + +$$ +\mathcal {L} _ {u} = \mathcal {L} _ {\text {n o i s e}} + \mathcal {L} _ {\text {s e m}} = \operatorname {M S E} \left(\epsilon_ {t}, \hat {\epsilon} _ {t}\right) + H \left(\hat {y} _ {t}, y\right), \tag {4} +$$ + +where $\mathrm{MSE}(\cdot)$ denotes the mean squared error, and $H(\cdot)$ represents the cross-entropy loss. In the conditional step, $\epsilon_{\theta}$ takes the semantic map $y$ as conditional input and only predicts the denoising residual: + +$$ +\hat {\epsilon} _ {t} \leftarrow \epsilon_ {\theta} \left(x _ {t}, y\right), \tag {5} +$$ + +with the training loss $\mathcal{L}_c$ calculated as: + +$$ +\mathcal {L} _ {c} = \mathcal {L} _ {\text {n o i s e}} = \operatorname {M S E} \left(\epsilon_ {t}, \hat {\epsilon} _ {t}\right). \tag {6} +$$ + +We use a random variable $\psi \sim \mathrm{Uniform}(0,1)$ to determine the mode for each training step. Therefore, + +$$ +\mathcal {L} = \mathcal {L} _ {c} \cdot \mathbb {I} (\psi \leq \psi_ {c}) + \mathcal {L} _ {u} \cdot \mathbb {I} (\psi > \psi_ {c}), \tag {7} +$$ + +where $\mathbb{I}(\cdot)$ is the indicator function and $\psi_{c}$ is the ratio of training the conditional step. Empirically, we set $\psi_{c}$ as 0.5 to balance the training of these two step types. + +Progressive Semantic Predictions. During inference, Spiral predicts the semantic map $\hat{y}_t$ at each unconditional step. To mitigate the inherent stochasticity of the diffusion process and improve stability, we apply an exponential moving average (EMA) to obtain the smoothed predictions $\bar{y} \in \mathbb{R}^{H \times W \times C}$ . The EMA is initialized as $\bar{y}_T = \hat{y}_T$ , and updated recursively for $t = T - 1$ to 0 as: + +$$ +\bar {y} _ {t} \leftarrow \alpha \cdot \hat {y} _ {t} + (1 - \alpha) \cdot \bar {y} _ {t + 1}, \tag {8} +$$ + +where $\alpha$ denotes the EMA smoothing factor. At the end of inference, Spiral outputs not only the depth and reflectance images, but also the final smoothed semantic prediction $\bar{y}_0$ . + +Closed-Loop Inference. Two-step methods suffer from limited cross-modality consistency, since the predicted semantic map post hoc cannot guide the previous generation. To address this issue, Spiral introduces a novel closed-loop inference mechanism, where the semantic predictions $\bar{y}_t$ are continuously fed back into the model as conditional inputs during inference. Notably, to alleviate the potential artifacts in $\bar{y}_t$ that could degrade the generation quality, Spiral employs a confidence-based filtering strategy to select the reliable predictions as conditions. Specifically, during inference from step $t = T$ to 0, Spiral starts with an open-loop mode by default. However, if more than the proportion $\delta$ of the pixels in $\bar{y}_t$ have confidence scores exceeding $\delta$ , it switches to the closed-loop mode, as depicted in Figure 3 (c). For instance, setting $\delta$ to 0.8 requires that over $80\%$ of the pixels in $\bar{y}_t$ have prediction confidence scores exceeding 0.8. Once this condition is met, Spiral enters an alternating loop between unconditional and conditional steps: (1) In the unconditional step, Spiral predicts both $\hat{\epsilon}_t$ and $\hat{y}_t$ . (2) In the conditional step, Spiral predicts $\hat{\epsilon}_t$ conditioned on $\bar{y}_t$ , thereby improving the consistency between $\bar{y}_0$ and $x_0$ eventually. A quantitative study on the benefits of the closed-loop mode and the effect of the confidence threshold $\delta$ is presented in the experimental section. + +Overall Architecture. Spiral adopts a 4-layer Efficient U-Net [50] as the backbone and follows the default continuous DDPM framework. Spiral takes as input the perturbed depth and reflectance images $x_{t}$ , along with semantic maps $y$ encoded as RGB images. Notably, in unconditional step, the semantic maps are replaced with zero padding to disable semantic guidance. On the output side, Spiral utilizes two heads to separately predict the diffusion residuals $\hat{\epsilon}_{t}$ and the segmentation labels $\hat{y}_{t}$ . Each output branch consists of a 2D convolutional layer followed by a sequential MLP layer. More details about the architecture of Spiral are provided in the appendix. + +![](images/6574d260805b48fd7d3a39ec9f72489cddaf63a090d718667c56f82a8ebfa983.jpg) +(a) Range-View-Based Metrics + +![](images/da2c1826b16af56b08acefbe19e3fb281ef4efbc2be0147322233d064cc31441.jpg) +(b) BEV-Based Metrics +Figure 4: (a) Range-view based semantic-aware feature $f^{s}$ is constructed by concatenating the features extracted by the RangeNet++ [3] encoder and the LiDM [48] semantic encoder from the LiDAR scene $x$ and the semantic map $y$ , respectively. (b) BEV-based semantic-aware feature $h^{s}$ is constructed by aggregating per-category 2D histograms. + +# 3.3 Semantic-Aware Metrics for Better LiDAR Generation Evaluations + +Recent range-view LiDAR studies [73, 42, 48] assess the quality of generated scenes from three perspectives: range-view images, Cartesian point clouds, and BEV projections. (1) For the range image- and point cloud-based evaluations, they rely on pretrained models such as RangeNet++ [40] and PointNet [47] to extract learning-based features $f$ for each scene. The extracted features from the real and generated sets are then used to compute the Fréchet Range Distance (FRD) [73], Fréchet Point Cloud Distance (FPD) [42], and Maximum Mean Discrepancy (MMD), which quantify the distributional similarity between these two sets. (2) For the BEV-based evaluation, rule-based descriptions $h$ are computed for each scene using 2D histograms over the xy-plane. These features, extracted from both the real and generated sets, are then used to calculate the MMD and Jensen-Shannon Divergence (JSD). However, none of these evaluation methods take semantic labels into account. Since the "ground truth" for semantic labels on the generated scene is unavailable, standard metrics like mIoU cannot be directly applied. Thereby, we propose to assess the quality of the generated labeled LiDAR scenes based on semantic-aware features $f^s$ and $h^s$ . + +Learning-based Semantic Features. For the range-view based evaluation, following LiDAR-Gen [73], we use the RangeNet++ [40] as encoder $\mathcal{E}$ . Additionally, we propose to use a semantic map encoder $\mathcal{G}$ to extract the semantic latent features. For $\mathcal{G}$ , we use the semantic conditional module in LiDM [48]. Given a LiDAR scene $x$ with semantic labels $y$ , the semantic-aware features $f^s$ are obtained by the concatenation of these two features, as depicted in Figure 4 (a): + +$$ +f ^ {s} \leftarrow \mathcal {E} (x) \oplus \mathcal {G} (y). \tag {9} +$$ + +The extracted features from the real and generated sets, $\{f^s\}_r$ and $\{f^s\}_g$ , are ultimately used to compute the semantic-aware Fréchet Range Distance (S-FRD) and semantic-aware Maximum Mean Discrepancy (S-MMD), extending FRD and MMD to incorporate semantic information. For the Cartesian-based evaluation, we adopt the same procedure to extract $f^s$ , while the only difference is that the RangeNet++ [40] is replaced with PointNet [47]. Similarly, $\{f^s\}_r$ and $\{f^s\}_g$ are used to compute the semantic-aware Fréchet Point Cloud Distance (S-FPD) and S-MMD. + +Rule-based Semantic Features. The BEV-based evaluation in [73, 42] divides the xy-plane into a grid of $B \times B$ bins and computes a 2D BEV histogram as the rule-based feature $h \in \mathbb{R}^{B \times B}$ . However, the spatial distribution of points belonging to different categories is obviously distinct. For instance, "road" points are typically concentrated near the region along the x-axis, while "building" and "vegetation" points tend to appear farther away. The distribution of points across different categories within a scene encodes rich semantic information and effectively reflects the overall semantic structure of the scene. Thereby, we propose to compute histograms for each category individually and aggregate them into a semantic-aware histogram $h^s \in \mathbb{R}^{C \times B \times B}$ , as depicted in Figure 4 (b). Hence, only when a generated and a real scene share both similar point distributions and semantic classifications can their histograms be similar. The extracted descriptions from the real and generated sets, $\{h^s\}_r$ and $\{h^s\}_g$ , are used to compute the semantic-aware JSD (S-JSD) and S-MMD. + +# 4 Experiments + +# 4.1 Experimental Setup + +Datasets. We conduct an extensive experimental study on SemanticKITTI [3] and nuScenes [5] datasets and follow their official data splits. SemanticKITTI contains 23k annotated LiDAR scenes with 19 semantic classes, while nuScenes contains 28k LiDAR scenes with 16 semantic classes. During pre-processing, the LiDAR scenes are projected into range-view images of spatial resolutions $64 \times 1024$ and $32 \times 1024$ , respectively. To further assess robustness, we also evaluate Spiral-based generative data augmentation on the fog and wet-ground subsets of Robo3D [24], which simulate adverse weather conditions for out-of-distribution testing. + +Details on Training & Inference. We train SPIRAL on NVIDIA A6000 GPUs with 48 GB VRAM for 300k steps using the Adam optimizer [21] with a learning rate of 1e-4. The training process takes $\sim 36$ hours. For the generative models in two-step baseline methods, including LiDARN [73], LiDM [48], and R2DM [42], we follow the official training settings. To obtain semantic labels for the generated unlabeled LiDAR scenes, we use RangeNet++ [40] with official pretrained weights and SPVCNN++ [36], an improved implementation of SPVCNN [55] provided in UniSeg [36] codebase. During inference, the number of function evaluations (NFE) [42], i.e., the number of sampling steps, is set to 256 for both Spiral and R2DM. We also run LiDM using the DDIM [51] sampling method with 256 steps for fair comparison. LiDARN models the denoising process with 232 noise levels and requires 5 steps per level by default, resulting in a total NFE of 1160. Following [73], we generate 10k samples per method for evaluation. + +Evaluation Metrics. We follow previous works [73, 42] to assess the quality of generated LiDAR scenes from the perspectives of range images, point clouds, and BEV projections. Importantly, we report the evaluation of the generated labeled LiDAR scenes using our newly proposed semantic-aware metrics introduced in Section 3.3, including S-FRD, S-FPD, S-MMD, and S-JSD. For all the metrics, lower values indicate better generative quality. Note that the samples generated by LiDM [48] contain only depth images, which prevents their evaluation in range-view-based benchmarks where the RangeNet++ [16] encoder requires the reflectance channel as input. + +# 4.2 Experimental Results + +Evaluation on SemanticKITTI. We report the experimental results on the SemanticKITTI [3] dataset in Table 1. Despite having the smallest parameter size of only 61M, Spiral achieves the best performance across all semantic-aware metrics, outperforming the two-step method, R2DM [42] & SPVCNN++ [36], by $31.03\%$ , $56.33\%$ , and $50.94\%$ on S-FRD, S-FPD, and S-JSD, respectively. For the previous metrics that evaluate only the unlabeled LiDAR scenes, Spiral outperforms R2DM on most metrics, indicating that the additional semantic prediction task does not compromise the generation quality of depth and reflectance images. Surprisingly, the more advanced segmentation model SPVCNN++ performs worse than RangeNet++ on the unlabeled scenes generated by LiDAR-Gen [73] and LiDM [48], resulting in inferior performance on semantic-aware metrics. We attribute this drop to the higher sensitivity of larger models to noise, compounded by the greater noise present in the LiDAR scenes generated by LiDARGen and LiDM. Although the performance of SPVCNN++ improves after jittering-based fine-tuning, it still lags behind RangeNet++. Further discussion of this issue is provided in the appendix. The generated labeled LiDAR scenes from Spiral and other baseline methods, as shown in Figure 5, demonstrate the superior performance of Spiral in both geometric and semantic aspects. + +Evaluation on nuScenes. We report the experimental results on the nuScenes [5] dataset in Table 2. Spiral consistently outperforms the other baseline methods on all metrics with the smallest parameter size. Compared with the second best method (R2DM [42] & RangeNet++ [40]), Spiral achieves improvements of $49.03\%$ , $67.84\%$ , and $46.79\%$ on S-FRD, S-FPD, and S-JSD, respectively. Figure 6 presents the generated scenes from Spiral and the baseline methods for qualitative comparison, showing the superior performance of Spiral in both geometric and semantic aspects. + +Generative data augmentation. We evaluate the effectiveness of using Spiral's generated samples to augment the training set for segmentation learning on SemanticKITTI [3]. Using SPVCNN++ [36] as the segmentation backbone, we compare Spiral with R2DM [42] & RangeNet++ [40] under different ratios of available real labeled data. As shown in the first row of Table 3, the generated samples + +Table 1: Comparisons with state-of-the-art LiDAR generation models on the SemanticKITTI [3] dataset. We evaluate methods using the Range View, Cartesian, and BEV representations. Symbols † and ‡ denote the RangeNet++ [40] backbone and the SPVCNN++ [36] backbone, respectively. The parameter size includes both the generative and segmentation models. The best and second best scores under each metric are highlighted in bold and underline. + +
MethodParam (M)NFERange ViewCartesianBEV
FRD↓(×1)MMD↓(×10-2)S-FRD↓(×1)S-MMD↓(×10-2)FPD↓(×1)MMD↓(×10-1)S-FPD↓(×1)S-MMD↓(×10-1)JSD↓(×10-2)MMD↓(×10-3)S-JSD↓(×10-2)S-MMD↓(×10-3)
LiDARGen† [73]30+501160681.3747.871216.6135.65115.1746.37710.7952.7113.232.1928.6510.96
LiDARGen‡ [73]30+971708.0561.421366.17103.1254.8468.16
LiDM† [48]275+50256————108.7033.87458.3328.604.560.2916.695.59
LiDM‡ [48]275+97——522.4732.3921.036.23
R2DM† [42]31+50256192.814.93559.2611.5619.292.30363.1615.003.730.1618.133.91
R2DM‡ [42]31+97555.0911.50351.7314.0618.673.97
SPIRAL (Ours)61256170.184.81382.874.108.061.10153.613.203.760.159.161.41
+ +![](images/ffaea2419a9e37afe9dc78cc277df11acc2e83c31cbb74ec912cf9417886187f.jpg) +Figure 5: Visualizations of generated LiDAR scenes on SemanticKITTI [3]. For two-step methods, we use the labels produced by RangeNet++ [40] due to its superior performance over SPVCNN++ [36]. Artifacts are highlighted with dashed boxes. Examples of semantic artifacts are shown in $⑦$ , $⑧$ , $⑨$ , and $⑪$ , while geometric artifacts such as local distortion and large noise are illustrated in $⑩$ and $⑫$ . + +from Spiral consistently improve the performance of SPVCNN++ and outperform those from R2DM. Detailed per-category results are provided in the appendix. Additionally, we evaluate SPVCNN++ under the same settings on out-of-distribution subsets, fog and wet-ground, from Robo3D [24]. As shown in the second and third rows of Table 3, although Spiral is not fine-tuned for such extreme weather conditions, its generated data still enriches the training set and improves performance under these challenging scenarios. + +# 4.3 Experimental Analysis + +Impact of the NFE. Intuitively, increasing the number of function evaluations (NFE) improves the quality of the generated samples but results in a linear increase of inference cost. We evaluate the performance of Spiral under different NFE settings in $\{32,64,128,256,512,1024\}$ on SemanticKITTI [3]. The results shown in Figure 7 indicate that Spiral's performance improves significantly when $\mathrm{NFE} < 256$ , while further increases in NFE yield only marginal gains on most metrics. Therefore, we set $\mathrm{NFE} = 256$ as the default configuration. On an A6000 GPU, Spiral achieves an average inference speed of 5.7 seconds per sample. + +Closed-Loop vs. Open-Loop Inference. Closed-loop inference is a key innovation in Spiral that enhances cross-modality consistency. To quantify the impact of closed-loop inference, we disable the feedback of the predicted semantic map to Spiral as conditional input, maintaining the open-loop inference throughout the whole generative process. The experimental results on SemanticKITTI [3], as listed in the first row of Table 4, show that adopting the open-loop setting leads to a performance drop of Spiral on all metrics, e.g., a drop of $3.34\%$ and $8.68\%$ on S-FRD and S-FPD, respectively. + +Table 2: Comparisons with state-of-the-art LiDAR generation models on the nuScenes [5] dataset. We evaluate methods using the Range View, Cartesian, and BEV representations. Symbols † and ‡ denote the RangeNet++ [40] backbone and the SPVCNN++ [36] backbone, respectively. The parameter size includes both the generative and segmentation models. The best and second best scores under each metric are highlighted in bold and underline. + +
MethodParam (M)NFERange ViewCartesianBEV
FRD↓(×1)MMD↓(×10-3)S-FRD↓(×1)S-MMD↓(×10-3)FPD↓(×1)MMD↓(×10-3)S-FPD↓(×1)S-MMD↓(×10-3)JSD↓(×10-2)MMD↓(×10-3)S-JSD↓(×10-2)S-MMD↓(×10-3)
LiDARGen† [73]30+501160188.803.031381.4229.6326.5212.231395.8392.313.980.1537.3715.23
LiDARGen‡ [73]30+972133.0450.202370.18165.1754.4390.78
LiDM† [48]275+50256————124.9590.141472.81129.654.000.2927.6011.75
LiDM‡ [48]275+97——1823.77147.3632.4213.61
R2DM† [42]31+50256157.522.96871.2519.1416.865.61866.3375.363.120.0517.144.94
R2DM‡ [42]31+971108.4325.271198.16100.3835.8317.70
SPIRAL (Ours)61256153.402.95444.045.497.131.04278.6412.863.100.059.120.80
+ +![](images/f4cb123d7ffe1cab7a23b5f8909df9c93b123412da816e4ac4a636d5d9831e5e.jpg) +Figure 6: Visualizations of generated LiDAR scenes on nuScenes [5]. For two-step methods, we use the labels produced by RangeNet++ [40] due to its superior performance over SPVCNN++ [36]. Artifacts are highlighted with dashed boxes. Examples of semantic artifacts are shown in $⑧$ , $⑪$ , and $⑫$ , while geometric artifacts such as local distortion and large noise are illustrated in $⑦$ , $⑨$ , and $⑩$ . + +Impact of the Confidence Threshold in Closed-loop Inference. In the closed-loop mode, Spiral adopts a confidence-based filtering strategy to exclude unreliable semantic maps that frequently occur during the early stages of the denoising process. To quantify the effect of the confidence threshold $\delta$ , we evaluate the performance of Spiral under different $\delta$ settings in $\{0.3, 0.5, 0.6, 0.7, 0.8, 0.9\}$ . The results listed in Table 4 indicate that Spiral performs well when $\delta \in \{0.7, 0.8, 0.9\}$ and achieves slightly best performance at $\delta = 0.8$ . Therefore, we set $\delta = 0.8$ as the default configuration of Spiral. However, the performance of Spiral starts to deteriorate when $\delta < 0.6$ . With $\delta = 0.3$ , the performance of the closed-loop inference even falls behind that of the open-loop inference. + +Table 3: Generative Data Augmentation (GDA) for segmentation training. We assess GDA using synthetic samples from R2DM [42] and Spiral, under different ratios (1%, 10%, 20%) of real labeled data from SemanticKITTI [3], as well as fog and wet-ground scenes from Robo3D [24]. Symbol $\dagger$ denotes using RangeNet++ as the segmentation model. Results are reported in mIoU (\%). + +
Setting1%10%20%
GDAw/oR2DM† [42]SPIRALw/oR2DM† [42]SPIRALw/oR2DM† [42]SPIRAL
SemanticKITTI37.7644.1647.4159.0760.6261.1461.1661.2162.35
Robo3D (fog)32.0736.8244.0653.9354.2058.6155.8956.0761.24
Robo3D (wet-ground)34.2637.9644.1954.7055.9259.3156.8157.5362.02
+ +![](images/8ea3b6e6faae3df890c60067d6371f6e8d9b15b21b097d55b62b1c6cd784727b.jpg) +Figure 7: Impact of the number of function evaluations (NFE). Increasing the NFE (32, 64, 128, 256, 512, and 1024) improves the performance across all metrics. With fewer sampling steps, Spiral outperforms R2DM [42] using its default setting of $\mathrm{NFE} = 256$ , indicated by the dashed line. Note: The x-axis denotes NFE, and the y-axis denotes the evaluation metric (lower is better). + +![](images/c550ea61be833d3dcae3f35fc66a66ee979c10472ade005639d69f3b79bed431.jpg) + +![](images/6d38942db01f67802d99c32974817fc72d41b3aa31fa8020e9b2b939a370ee00.jpg) +(c) + +![](images/a77381878dde9ec773b108383a3fdfa16c044d5b0971654ff681fd46b6d3fbd6.jpg) +(d) + +![](images/66ca2971d04e9ece579cc9ab8137f55e0981d2e8354b488d7290ebe25b5b2ff6.jpg) +(e) + +![](images/bcf0fe22c6cffffa1c77612bbfc4328064d5461dd520e8c21f0442c65d8c0e1b.jpg) +(f) + +Table 4: Impact of the closed-loop inference and confidence threshold. The best and second best scores under each metric are highlighted in bold and underline. The highlighted row indicates the default configuration of Spiral. + +
Close-loopConfidence Threshold δRange ViewCartesianBEV
FRD↓(×1)MMD↓(×102)S-FRD↓(×1)S-MMD↓(×102)FPD↓(×1)MMD↓(×10)S-FPD↓(×1)S-MMD↓(×10)JSD↓(×102)MMD↓(×103)S-JSD↓(×102)S-MMD↓(×103)
x-173.934.98395.644.3510.611.82166.954.413.890.169.291.44
y0.3190.035.56444.316.0647.6717.97265.7917.654.680.1911.312.51
y0.5174.045.15396.714.6722.316.81187.908.513.960.169.551.71
y0.6173.255.02392.364.2411.142.59174.255.123.870.159.401.59
y0.7170.934.87385.054.178.321.60161.473.773.890.159.221.50
y0.8170.184.81382.874.108.061.10153.613.203.760.159.161.41
y0.9170.725.00384.424.168.361.22155.283.273.890.169.181.42
+ +Open-/Closed-Loop Mode Switching Point. We performed a statistical analysis on 1,000 samples each from SemanticKITTI and nuScenes. With the default 256 denoising steps, closed-loop inference is activated (i.e., once $\geq 80\%$ of semantic predictions exceed the confidence threshold) at an average step of $180 \pm 36$ on SemanticKITTI and $189 \pm 39$ on nuScenes. This indicates that switching typically occurs during the last $\sim 30\%$ of the generation process, when semantic predictions have stabilized, thereby avoiding early-stage noise and ensuring reliable semantic-geometric alignment. + +Inference Efficiency. We report the average sampling time per sample for LiDARGen [73], LiDM [48], R2DM [42], and SPIRAL on an A6000 GPU in Table 5. Additionally, the inference times per sample for RangeNet++ [40] and SPVCNN++ [36] are 0.08s and 0.05s respectively on the same hardware. Unlike the two-step methods, Spiral does not require a segmentation model to generate semantic labels. Spiral demonstrates superior inference efficiency compared to LiDM and LiDARGen. Although it is approximately 2.05 s slower than R2DM combined with SPVCNN++ when using the same number of generation steps, Spiral remains 0.75 s faster with 128 steps and achieves higher generative quality. + +Table 5: Average sampling time per sample of LiDARGen [73], LiDM [48], R2DM [42], and Spiral on an Nvidia A6000 GPU. + +
MethodNFEAverage Time (s)
LiDARGen [73]116072.0
LiDM [48]2567.2
R2DM [42]2563.6
SPIRAL2565.7
SPIRAL1282.9
+ +# 5 Conclusion + +We present SPIRAL, the first semantic-aware range-view LiDAR diffusion model that jointly generates depth, reflectance, and semantic labels in a unified framework. We further introduce novel semantic-aware evaluation metrics, enabling a holistic assessment of generative quality. Through extensive evaluations on SemanticKITTI and nuScenes, Spiral achieves state-of-the-art performance across multiple geometric and semantic-aware metrics with minimal model size. Additionally, Spiral demonstrates strong utility in downstream segmentation via generative data augmentation, reducing the reliance on manual annotations. We believe Spiral offers a new perspective on multi-modal LiDAR scene generation and opens up promising directions for scalable, label-efficient 3D perception. + +# Acknowledgments + +The authors would like to sincerely thank the Program Chairs, Area Chairs, and Reviewers for the time and effort devoted during the review process. + +# References + +[1] Angelika Ando, Spyros Gidaris, Andrei Bursuc, Gilles Puy, Alexandre Boulch, and Renaud Marlet. Rangevit: Towards vision transformers for 3D semantic segmentation in autonomous driving. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5240-5250, 2023. +[2] Dmitry Baranchuk, Ivan Rubachev, Andrey Voynov, Valentin Khrulkov, and Artem Babenko. Label-efficient semantic segmentation with diffusion models. arXiv preprint arXiv:2112.03126, 2021. +[3] Jens Behley, Martin Garbade, Andres Milioto, Jan Quenzel, Sven Behnke, Cyril Stachniss, and Juergen Gall. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# 6 Evaluation Metrics 22 + +6.1 Evaluation on Range View and Cartesian Point Clouds 22 +6.2 Evaluation on Bird's Eye View 23 + +# 7 Data Preprocessing 23 + +7.1 Rescaling of the Depth and Reflectance Remission Channels 23 +7.2 Semantic Encoding 23 + +# 8 Model Architecture Details 24 + +8.1 Spatial Feature Encoding 24 +8.2 Temporal Feature Encoding 24 + +# 9 Discussions of SoTA Segmentation Models in Two-Step LiDAR Generation 24 + +9.1 Experimental Setup 24 +9.2 Observations and Analyses 24 + +# 10 Additional Qualitative Results 24 + +10.1 Visualizations of Reflectance Remission Images 24 +10.2 Visualizations of Generated LiDAR Scenes 25 + +# 11 Generative Data Augmentation 25 + +11.1 Full-supervision Setup 25 +11.2 Semi-supervision Setup 25 + +# 12 Broader Impact & Limitations 29 + +12.1 Broader Impact 29 +12.2 Societal Influence 30 +12.3 Potential Limitations 30 + +# 13 Public Resource Used 30 + +13.1 Public Datasets Used 30 +13.2 Public Implementations Used 31 + +# 6 Evaluation Metrics + +In the main paper, we evaluate the generative quality of Spiral for the LiDAR scene $x$ with semantic map $y$ from three perspectives: range-view images, Cartesian point clouds, and BEV projections. Here, we elaborate on the details of evaluations on each representation. + +# 6.1 Evaluation on Range View and Cartesian Point Clouds + +For range-view image-based and point cloud-based evaluations, we extract a unified semantic-aware feature $f^s$ by concatenating the geometric feature $\mathcal{E}(x)$ , extracted by RangeNet++ [40] or + +PointNet [47], with the semantic feature $\mathcal{G}(y)$ from the conditional module in LiDM [48]: + +$$ +f ^ {s} \leftarrow \mathcal {E} (x) \oplus \mathcal {G} (y). \tag {10} +$$ + +The features from real and generated sets, $\{f^s\}_r$ and $\{f^s\}_g$ , are used to compute S-FRD, S-FPD, and S-MMD. The calculation formula for S-FRD is as follows: + +$$ +\mathrm {S} - \mathrm {F R D} = \| \mu_ {r} - \mu_ {g} \| _ {2} ^ {2} + \operatorname {T r} \left(\Sigma_ {r} + \Sigma_ {g} - 2 \left(\Sigma_ {r} \Sigma_ {g}\right) ^ {\frac {1}{2}}\right), \tag {11} +$$ + +where $\mu_r$ and $\mu_s$ are the mean of $\{f^s\}_r$ and $\{f^s\}_g$ , $\Sigma_r$ and $\Sigma_g$ are the covariance matrices, and $\mathrm{Tr}(\cdot)$ is the matrix trace. The calculation of S-FPD follows the same formula. For S-MMD, it can be measured through the kernel trick: + +$$ +\mathrm {S} - \mathrm {M M D} = \frac {1}{N ^ {2}} \sum_ {i} ^ {N} \sum_ {i ^ {\prime}} ^ {N} k \left(f _ {i} ^ {s}, f _ {i} ^ {s ^ {\prime}}\right) - \frac {2}{N M} \sum_ {i} ^ {N} \sum_ {j} ^ {M} k \left(f _ {i} ^ {s}, f _ {j} ^ {s}\right) + \frac {1}{M ^ {2}} \sum_ {j} ^ {M} \sum_ {j ^ {\prime}} ^ {M} k \left(f _ {j} ^ {s}, f _ {j} ^ {s ^ {\prime}}\right), \tag {12} +$$ + +where $N$ and $M$ are the number of samples in the real and generated sets, respectively. Following [42], we use 3rd-order polynomial kernel function: + +$$ +k \left(f ^ {s}, f ^ {s ^ {\prime}}\right) = \left(\frac {\left(f ^ {s}\right) ^ {T} \cdot f ^ {s ^ {\prime}}}{d _ {f}} + 1\right) ^ {3}, \tag {13} +$$ + +where $d_{f}$ is the dimension of $f^{s}$ . + +# 6.2 Evaluation on Bird's Eye View + +For the BEV-based evaluation, we first compute the semantic-aware histogram for the real and generated sets, $\{h^s\}_r$ and $\{h^s\}_g$ , and then compute the BEV-based S-JSD and S-MMD. The calculation of S-JSD is as follows: + +$$ +\operatorname {J S D} (P \mid \mid Q) = \frac {1}{2} D _ {K L} (P \mid \mid M) + \frac {1}{2} D _ {K L} (Q \mid \mid M), \tag {14} +$$ + +where $P$ and $Q$ are the approximation of Gaussian distribution on $\{h^s\}_r$ and $\{h^s\}_g$ , and $M$ is the mean of them: $M = \frac{1}{2} (P + Q)$ . + +# 7 Data Preprocessing + +In this section, we provide more details regarding data preprocessing. For both SemanticKITTI [3] and nuScenes [5], we first project the raw point cloud to range-view images including the depth, reflectance remission, and semantic channels. Then we rescale the depth and reflectance remission channels and encode the semantic maps to RGB images. + +# 7.1 Rescaling of the Depth and Reflectance Remission Channels + +For the depth channel, we first convert the range-view depth images $x^{d}$ to log-scale representation $x_{\log}^{d}$ as follows: + +$$ +x _ {\log} ^ {d} = \frac {\log \left(x ^ {d} + 1\right)}{\log \left(x _ {\max } ^ {d} + 1\right)}, \tag {15} +$$ + +where $x_{\mathrm{max}}^d$ is the maximum depth value. Then we linearly rescale the reflectance remission and log-scale depth images to $[-1, 1]$ . + +# 7.2 Semantic Encoding + +For the semantic channels, we use the official color scheme to convert the one-hot semantic map $y \in \mathbb{R}^{H \times W \times C}$ into 2D RGB images, where $C$ is the number of categories. Notably, Spiral is trained in both conditional and unconditional modes, where RGB images in the unconditional mode are replaced with zero padding. To distinguish between zero-padded images from the unconditional mode and the black regions in semantic images from the conditional mode, we add an additional channel to indicate whether a semantic map is provided. + +# 8 Model Architecture Details + +# 8.1 Spatial Feature Encoding + +We adopt a 4-layer Efficient U-Net [50] as the backbone of Spiral, with intermediate feature dimensions of 128, 256, 384, and 640, respectively. Each layer consists of three residual blocks and downsamples the spatial resolution by a factor of two along both the row and column dimensions. The prediction heads for generative residual and semantic predictions are composed of a 2D convolutional layer followed by a two-layer MLP. + +# 8.2 Temporal Feature Encoding + +For temporal and coordinate encoding, we follow the same strategy as R2DM [42]. For coordinate encoding, we map the per-pixel azimuth and elevation angles to 32-dimensional Fourier features [53], which are then concatenated with the input data. Specifically, the diffusion timestep is encoded to a 256-dimensional sinusoidal positional embedding, which is then integrated by the adaptive group normalization (AdaGN) [9] modules in each layer. + +# 9 Discussions of SoTA Segmentation Models in Two-Step LiDAR Generation + +In the two-step methods, we report results using both RangeNet++ [40] and SPVCNN++ [36] as segmentation backbones in the main paper. Although SPVCNN++ outperforms RangeNet++ on real-world datasets, it performs worse on the generated LiDAR scenes of LiDARGen [73] and LiDM [48]. In this section, we further present the performance of RangeNet++, SPVCNN++, and a state-of-the-art segmentation model, RangeViT [1], on the generated LiDAR scenes of LiDARGen. As we observe that larger-scale jittering can improve SPVCNN++'s robustness, we train all models with both the default jittering and an increased jittering scale. + +# 9.1 Experimental Setup + +We use the official implementation of all models. In the default setting, RangeNet++ and RangeViT are trained without jittering augmentation, while SPVCNN++ is trained with the Gaussian noise jittering, where $\sigma = 0.1$ . In the setting of larger-scale jittering, we increase the $\sigma$ of jittering to 0.3 across all models. + +# 9.2 Observations and Analyses + +Detailed results of RangeNet++, SPVCNN++, and RangeViT under different jittering scales are presented in Table 6. Although fine-tuning SPVCNN++ and RangeViT with stronger jittering $(\sigma = 0.3)$ improves its performance, they still lag behind RangeNet++, while RangeNet++ presents the best robustness on the generated samples. These experimental results indicate that segmentation models achieving high performance on "clean" real-world scenes do not necessarily perform well on generated scenes. This highlights the limitation of using predictions from state-of-the-art segmentation models as pseudo "ground truth" for evaluating generated scenes. Instead, measuring the distributional similarity of semantic-aware features between real and generated scenes provides a more reliable assessment of the quality of predicted semantic maps, as proposed in this work. + +# 10 Additional Qualitative Results + +In this section, we present additional visualizations of the generated reflectance remission images, along with more examples of generated LiDAR scenes and the corresponding semantic maps. + +# 10.1 Visualizations of Reflectance Remission Images + +We visualize the generated reflectance remission images on SemanticKITTI [3] in Figure 8. Since LiDM does not produce reflectance remission images, we only present results from Spiral, LiDARGen [73], and R2DM [42]. Among these, Spiral demonstrates strong cross-modal consistency across depth, reflectance intensity, and semantic labels. In contrast, scenes generated by LiDARGen + +Table 6: Performance of RangeNet++ [40], SPVCNN++ [36], and RangeViT [1] trained with different jittering scales on the SemanticKITTI [3] dataset. + +
Generative ModelSegmentation BackboneJittering ScaleRange ViewCartesianBEV
S-FRD↓(×1)S-MMD↓(×10-2)S-FPD↓(×1)S-MMD↓(×10-1)S-JSD↓(×10-2)S-MMD↓(×10-3)
LiDARGen [73]RangeNet++ [40]default1216.6135.65710.7952.7128.6510.96
large1202.4534.97702.3850.0328.4910.85
SPVCNN++ [36]default1978.1370.331826.54210.6756.4068.97
large1708.0561.421366.17103.1254.8468.16
RangeViT [1]default2034.1572.091726.54195.4455.2067.47
large1891.4270.171625.27187.0254.1365.09
+ +and R2DM often exhibit artifacts and inconsistencies across modalities, as highlighted by the red bounding boxes. + +# 10.2 Visualizations of Generated LiDAR Scenes + +We demonstrate more generated LiDAR scenes in SemanticKITTI [3] and nuScenes [5] in Figure 9 and Figure 10, respectively. The geometric and semantic artifacts of the generated LiDAR scenes are highlighted by dashed boxes. + +# 11 Generative Data Augmentation + +# 11.1 Full-supervision Setup + +In Table 7, we report the experimental results of using the samples generated by a two-step method (RangeNet++ [40] & R2DM [42]) and Spiral for generative data augmentation in the segmentation training of SPVCNN++ [36] on SemanticKITTI [3] dataset. We train SPVCNN++ using full-supervision method. Besides different ratios (1%, 10%, 20%) of available real labeled data, we extend the real data subsets with the generated samples (10k). The results show that the generated samples from Spiral consistently improve the performance of SPVCNN++ and outperform those from R2DM. + +# 11.2 Semi-supervision Setup + +Two switches in Spiral, $\mathcal{A}$ and $\mathcal{B}$ , provide high flexibility to alternate between the unconditional and conditional modes. In fully-supervised training, where all input samples have semantic labels, $\mathcal{A}$ and $\mathcal{B}$ operate in an exclusive-or (X-OR) manner: (1) in the unconditional mode, $\mathcal{A}$ is off and $\mathcal{B}$ is on; (2) in the conditional mode, $\mathcal{A}$ is on and $\mathcal{B}$ is off. When both switches are turned off ( $A = 0$ and $\mathcal{B} = 0$ ), Spiral degenerates into a normal unconditional LiDAR generative model, i.e., non-labeled mode. This design enables Spiral to support semi-supervised training. Specifically, when only a small fraction of training samples are labeled (e.g., $10\%$ ), Spiral is trained under labeled data using the conditional or unconditional modes described in the main paper and is trained with the remaining unlabeled samples under the non-labeled mode. + +We train Spiral on a training set where only $10\%$ of the samples are labeled, with the remaining samples unlabeled. For the two-step baseline methods, the generative model is trained on the full training set, while the segmentation models are trained solely on the labeled $10\%$ subset. The results presented in Table 8 show that the generative performance of Spiral consistently outperforms other baseline two-step methods in this case. + +Additionally, we leverage the samples generated by Spiral, trained under the semi-supervised setting, to augment the labeled training set within the state-of-the-art semi-supervised LiDAR segmentation framework, LaserMix [25]. We adopt MinkUNet [8] and FIDNet [66] as the segmentation backbones. The experimental results in Table 9 demonstrate that incorporating Spiral's generated samples further improves the performance of both MinkUNet and FIDNet within the LaserMix framework. + +![](images/7f1fcb1186592638c9e3f96aeee093f6fc056c5c2b4a1a711d8c5c27b7cd84b4.jpg) +Ground Truth + +![](images/26293707a65b623ba08723a4596b8935b416af10aac089a6bd183c37e506e679.jpg) +Spiral (Ours) + +![](images/9644872ae2afa287509958c35c1577a5512aa603d665a3e45d4ae85720b66422.jpg) + +![](images/4bb8b6a04d0b8a407a3cb603051882802b8a3371c1ceecffb55d5c72df8f74e7.jpg) + +![](images/f42f0f9268017d9ab2ad24f37a1f31db3fb410070d2043b7637366e7d3e28e30.jpg) +LiDARGen (ECCV'22) + +![](images/eaa7797f528ce98ff01f09f43c6daef38237ebbcee6569b0adf1e062cb7f91d7.jpg) + +![](images/47f03999f491209bb5b3b4306cb60432b27b9565acbeba53c6b40b3b3af4b458.jpg) +R2DM (ICRA'24) + +Figure 8: Visualizations of generated depth, reflectance remission, and semantic labels in range-view perspective on SemanticKITTI [3]. Generation artifacts and cross-modal inconsistencies are highlighted by the bounding boxes. +![](images/98826b57885c085a1099910867bf527aafc5b1f8fa2adda410ba5c219374eebd.jpg) +car bicy moto truc o.veh ped b.list m.list road park +walk o.gro build fenc veg trun terr pole sign + +![](images/a6e32e15f7c864236d079aca82b250a37400af9fd7d09b90deb479ff5248a998.jpg) +Figure 9: Visualizations of generated LiDAR scenes on SemanticKITTI [3]. + +![](images/b8fa1aba6ca781951cb8382c4325435f7516b79a03fab46ad73f173f955284eb.jpg) +Figure 10: Visualizations of generated LiDAR scenes on nuScenes [5]. + +Table 7: Generative Data Augmentation (GDA) for segmentation training. We evaluate the effectiveness of GDA using synthetic samples generated by R2DM [42] & RangeNet++ [40] and Spiral, under different ratios (1%, 10%, 20%) of available real labeled data. Symbol $\dagger$ indicates that RangeNet++ [40] is used as the segmentation backbone. + +
Setting(GDA)1%10%20%
None\(R2DM^†\)SPIRALNone\(R2DM^†\)SPIRALNone\(R2DM^†\)SPIRAL
mIoU37.7644.1647.4159.0760.6261.1461.1661.2162.35
per-class mIoUcar89.8287.6592.8794.5294.5395.1395.7894.5296.03
bicycle0.000.0012.6725.0443.2331.5725.9242.6234.07
motorcycle13.9823.3625.6862.7855.9955.4766.2258.9858.50
truck24.157.9347.9252.8949.4060.1950.5348.4457.82
other-vehicle15.5029.6033.7142.0652.3053.5855.8551.1955.44
person25.8228.5626.1868.3260.2561.8769.4865.9873.85
bicyclist0.0053.540.0082.7277.5586.0788.0283.3285.20
motorcyclist1.970.000.200.000.260.000.001.520.00
road78.3786.1492.7991.5992.5593.2392.9592.6493.42
parking15.4118.9951.0538.9943.4947.7343.6143.0347.94
sidewalk60.7870.5679.6177.8779.9080.6279.9679.7280.52
other-ground0.010.250.674.212.091.302.113.311.40
building79.2086.9388.3888.5890.4090.5289.8790.1390.03
fence32.6252.1453.1456.1362.2261.8259.8060.7760.45
vegetation83.6286.2888.0787.9789.4989.8288.3589.2188.69
trunk52.7846.1961.6266.1166.5868.2766.2966.6968.51
terrain69.3872.8477.6774.9778.3678.2875.6377.7475.34
pole54.3244.9654.2663.0361.8062.8164.2762.1263.85
traffic-sign19.7743.1214.2144.6151.4443.4447.3951.0153.66
+ +Table 8: Semi-supervised training of Spiral and the baseline two-step methods on the SemanticKITTI [3] dataset. We evaluate methods using the Range View, Cartesian, and BEV representations. Symbols $\dagger$ denotes the RangeNet++ [40] trained with $10\%$ labeled samples. The parameter size includes both the generative and segmentation models. + +
MethodParam (M)NFERange ViewCartesianBEV
S-FRD↓×1S-MMD↓×10-2S-FPD↓×1S-MMD↓×10-1S-JSD↓×10-2S-MMD↓×10-3
LiDARGen† [73]30+5011601285.5547.56750.6858.2634.4912.66
LiDM† [48]275+50256--486.2229.7816.916.33
R2DM† [42]31+50256528.827.82281.968.9516.132.99
SPIRAL61256508.865.77211.974.7911.472.65
+ +# 12 Broader Impact & Limitations + +In this section, we elaborate on the broader impact, societal influence, and potential limitations of the proposed approach. + +# 12.1 Broader Impact + +This work presents several significant implications for both academic research and practical applications in autonomous driving and computer vision. SPIRAL advances the field of LiDAR scene generation by introducing a unified framework for joint depth, reflectance, and semantic label generation. This breakthrough could substantially impact: + +# 1. Research Community + +Table 9: Generative Data Augmentation (GDA) for semi-supervised training in the LaserMix [25] framework. We evaluate the effectiveness of GDA using synthetic samples generated by Spiral in the semi-supervised training framework LaserMix. + +
BackboneMinkUNet [8]FIDNet [66]
Methodsup. onlyLaserMix [25]LaserMix +SPIRALsup. onlyLaserMix [25]LaserMix +SPIRAL
mIoU64.066.667.252.260.160.3
+ +- Establishes new benchmarks for semantic-aware LiDAR generation +- Provides a novel framework for multi-modal scene understanding +- Opens new research directions in 3D scene generation and understanding + +# 2. Industry Applications + +- Reduces the dependency on expensive and time-consuming manual data annotation +- Enables more efficient development of autonomous driving systems +- Provides a cost-effective solution for synthetic data generation + +# 12.2 Societal Influence + +The development of SPIRAL could lead to several positive societal outcomes: + +# 1. Transportation Safety + +- Enhanced autonomous vehicle perception capabilities through better training data +- Potential reduction in traffic accidents through improved scene understanding + +# 2. Economic Impact + +- Reduced development costs for data collection +- Accelerated deployment of self-driving vehicles + +# 12.3 Potential Limitations + +While SPIRAL demonstrates promising results, several limitations and challenges should be acknowledged: + +1. Technical Constraints: The quality of generated scenes may still have room for improvement. +2. Methodological Limitations: Generated data may not fully capture all real-world complexities. +3. Implementation Challenges: Real-time performance considerations for practical applications. + +This work represents a significant step forward in semantic-aware LiDAR scene generation while acknowledging the need for continued research to address these limitations and challenges. Future work could focus on enhancing the model's capabilities in handling complex scenarios and improving computational efficiency while maintaining generation quality. + +# 13 Public Resource Used + +In this section, we acknowledge the use of the public resources, during the course of this work: + +# 13.1 Public Datasets Used + +- nuScenes² CC BY-NC-SA 4.0 + +- nuScenes-DevKit3 Apache License 2.0 +- SemanticKITTI4 CC BY-NC-SA 4.0 +- SemanticKITTI-API5 MIT License +- Robo3D6 CC BY-NC-SA 4.0 + +# 13.2 Public Implementations Used + +- MMDetection7 Apache License 2.0 +- MMDetection3D8 Apache License 2.0 +- RangeNet++9 MIT License +- OpenPCSeg10 Apache License 2.0 +- RangeViT11 Apache License 2.0 +LiDARGen12 MIT License +$\bullet$ LiDM13 MIT License +$\cdot$ R2DM14 MIT License \ No newline at end of file diff --git a/NeurIPS/2025/Spiral_ Semantic-Aware Progressive LiDAR Scene Generation and Understanding/images.zip b/NeurIPS/2025/Spiral_ Semantic-Aware Progressive LiDAR Scene Generation 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0000000000000000000000000000000000000000..e3cb537ab4089a4e6e637477d621ca2e62d42b57 --- /dev/null +++ "b/NeurIPS/2025/SplashNet_ Split\342\200\221and\342\200\221Share Encoders for Accurate and Efficient Typing with Surface Electromyography/full.md" @@ -0,0 +1,654 @@ +# SplashNet: Split-and-Share Encoders for Accurate and Efficient Typing with Surface Electromyography + +Nima Hadidi $^{1,2}$ Jason Chan $^{3}$ Ebrahim Feghhi $^{1,2}$ Jonathan C. Kao $^{1,2,3}$ + +University of California, Los Angeles + +nhadidi@g.ucla.edu + +# Abstract + +Surface electromyography (sEMG) at the wrists could enable natural, keyboard-free text entry, yet the state-of-the-art emg2qwerty baseline still misrecognizes $51.8\%$ of characters zero-shot on unseen users and $7.0\%$ after user-specific fine-tuning. We trace much of these errors to mismatched cross-user signal statistics, fragile reliance on high-order feature dependencies, and the absence of architectural inductive biases aligned with the bilateral nature of typing. To address these issues, we introduce three simple modifications: (i) Rolling Time Normalization which adaptively aligns input distributions across users; (ii) Aggressive Channel Masking, which encourages reliance on low-order feature combinations more likely to generalize across users; and (iii) a Split-and-Share encoder that processes each hand independently with weight-shared streams to reflect the bilateral symmetry of the neuromuscular system. Combined with a five-fold reduction in spectral resolution $(33\rightarrow 6$ frequency bands), these components yield a compact Split-and-Share model, SplashNet-mini, which uses only $\frac{1}{4}$ the parameters and $0.6\times$ the FLOPs of the baseline while reducing character error rate (CER) to $36.4\%$ zero-shot and $5.9\%$ after fine-tuning. An upscaled variant, SplashNet ( $\frac{1}{2}$ parameters, $1.15\times$ FLOPs of the baseline), further lowers error to $35.7\%$ and $5.5\%$ , representing $31\%$ and $21\%$ relative improvements in the zero-shot and finetuned settings, respectively. SplashNet therefore establishes a new state-of-the-art without requiring additional data. + +# 1 Introduction + +Translating neuromuscular signals into typed text is a novel but rapidly developing area at the intersection of human-computer interaction and machine learning. The emg2qwerty dataset (14) is the first large-scale benchmark specifically for wrist EMG-based touch typing. It contains over 346 hours of data from 108 users typing sentences while wearing electrode bands on each wrist. This dataset was motivated by the promise of wrist EMG as an always-available input modality for AR/VR glasses and other scenarios where traditional keyboards are impractical. In parallel, related benchmarks like emg2pose have been developed for EMG-based hand pose estimation (13), reflecting a broad interest in neuromotor interfaces for controlling virtual objects, robots, or text entry. Early explorations of EMG for text input date back at least two decades: for example, Yu et al. (22) demonstrated an "EMG keyboard" that allowed a forearm amputee to input characters via muscle signals. These efforts were limited in scale and accuracy, but they proved the principle that muscle activity alone can convey typing information. Today, with much larger datasets and deep learning models, the accuracy of EMG-based typing has greatly improved (2; 14; 13), making it a viable research direction for assistive technology and next-generation user interfaces. + +![](images/2cd298fca66b271ea38f2aaac4ac79f6c051bda60d3d0838e36ca1993dc628fa.jpg) +a +User 1 +Without Rolling Time Normalization +Time relative to keystroke (s) + +![](images/110999b717a9152e84ee66859510831ffc7b25421261ce91911ab75f047c932e.jpg) +User 2 +b + +![](images/36395bff2f7e950d28d684004cd45bf5205b123abdcca6d5f5b2bcc18c1a6bb3.jpg) +With Rolling Time Normalization +Figure 1: a) Top row: Peri-stimulus time histograms (PSTHs) for the "e" key with (top) and without (bottom) RTN for two users. Each PSTH shows the spectral features derived from the left hand, with spectrograms from the 16 electrodes concatenated together. RTN mitigates the significant differences in across-user feature scale and bias. b) Top Row: PSTHs for the "e" key from 3 training users. Note that some of User 4's features show similar patterns to User 3, while others show similar patterns to User 5. ACM isolates small feature combinations, which are more often shared across users. + +![](images/592975777f98624e5a2434802686aa42fb203920caeeb7628ed11f8925b916a3.jpg) + +![](images/ddaeef75e9e1d8e412ca9d886b77199c60b514178527f50b6cf185002f2840ba.jpg) +User 3 +Original Unaugmented PSTHs +Augmented View 1 + +![](images/2e215f5218cd968891d6f5534079faeb113be5ebc8ee4096a8e32b0190fe9eef.jpg) +User 4 + +![](images/b726a2f0ff33ce57d62eb0913e42761a286cc9ad595fc33ee71de69bcbbd1e59.jpg) +User 5 + +![](images/b4b370fe6f90c02dd5945736110be4eb9d34916ae9cc6e91691477c85ce52adb.jpg) + +![](images/1e1c4824e881a28e320691300665ecd2beef2b1eed6dc51213487c7b98868290.jpg) + +![](images/5842167deef1fc0e3e2f6e7307eff1962510991bd5e6662c6e76cc8cc83b681f.jpg) +Augmented View 2 + +![](images/fc8acc6464cc251d93a47d043d05a93f3836114e50a320adb5d24c0b0b21df67.jpg) + +Although promising, EMG interfaces face critical challenges in generalization. EMG interfaces suffer from substantial domain shift across users and sessions: anatomy, electrode placement, fatigue, and day-to-day physiology all alter the signal for the same action (14). The emg2qwerty benchmark probes this shift with (i) a zero-shot test, where one trains on many users and tests on a new one, and (ii) a personalization test that finetunes on a few target-user samples. Their baseline ASR-style system (CNN on spectrograms + LM decoding) misrecognized $51.8\%$ of characters zero-shot, but finetuning cut errors to approximately $7\%$ , showing that most residual errors are systematic, user-specific variations. Yet even large training pools struggle: Sivakumar et al. (14) estimate $\mathcal{O}(10^{3})$ users are needed for robust out-of-the-box performance, a trend echoed by CTRL-Labs' gains from massive data (2). + +While scaling may significantly increase the performance of EMG interfaces, collecting these large-scale datasets is time-consuming and human-resource expensive. We instead present a simpler and complementary path towards improving sEMG generalization. Our key contribution is to first identify limitations in sEMG data that form obstacles to zero-shot generalization. We then propose simple, causal, and computationally inexpensive modifications to address these limitations, resulting in models that are more invariant to user/session differences. Our rationale for pursuing causal and computationally inexpensive modifications is to enable sEMG decoders to work on-device, an important consideration for future wrist EMG devices. + +# 2 Insights into improving sEMG decoding + +We summarily make three insights, and subsequently, three simple modifications to sEMG decoding that substantially improve zero-shot and finetuning performance while reducing computational costs. + +Insight 1: sEMG features should be causally normalized per session. sEMG amplitudes routinely differ by an order of magnitude across participants or sessions (3; 8). Figure 1a highlights this disparity: the same channels exhibit both higher baselines and greater variance for User 2 than for User 1. Hence, EMG signals should obviously be normalized between users. However, the method of normalization is critical. Sivakumar et al. (14) apply batch normalization that computes mean and variance over the mini-batch, time, and per-electrode frequency bins. Because each mini-batch mixes data from many users and sessions, these statistics fail to place features from different users/sections in a common space during training and remain fixed at inference. We instead employ Rolling Time Normalization (RTN), a causal z-scoring of every input feature that updates its mean and variance online. Though simple, this mitigates the scale and shift differences across users, visualized in Figure 1a. Further, RTN requires no calibration and adds virtually no latency for causal, real-time inference. We empirically find this significantly improves zero-shot and finetuned performance. + +![](images/84ff0934d7e2d21252615b55fb9fa8f117950e09683fae96302938c4f6bd9670.jpg) +Figure 2: a) The bilateral structure of keyboard typing. b) The Split-and-Share macro-architecture. + +Insight 2: sEMG feature subsets may be conserved across users. Through visualizing sEMG activity, we empirically find that while two typists rarely share identical high-dimensional EMG signatures for the same keystroke, they often share subsets of electrodes or frequency bands that behave similarly. A representative example is shown in Figure 1b, where we present the PSTHs of sEMG activity from three users. While the full set of features looks fairly different across users, randomly selected subsets of features have a greater chance of being similar between some users (Augmented View 1 and 2 in Figure 1b). Decoding strategies that hinge on these low-order combinations therefore stand a better chance of transferring across users. While there are many ways to encourage this, we focus on substantially modifying the hyperparameters of dataset augmentation to achieve this, incurring no test-time computational expense. In particular, we do Aggressive Channel Masking (ACM) that on average zeros out more than half of the electrodes and removes broad spectral chunks from the rest. ACM encourages the model to rely on smaller, more universal feature sets (Figure 1b), which empirically improves zero-shot generalization. We emphasize that this 'low-order similarity' perspective is a working hypothesis rather than a definitive statement about the structure of EMG representations: the empirical improvements we observe are consistent with this interpretation but do not prove it. + +Insight 3: an inductive bias for bilateral typing. Keyboard typing is fundamentally bilateral: each keystroke is driven solely by the muscles of its own wrist, while the neuromuscular mapping from activation to finger motion is almost mirror-symmetric across hands. We therefore propose a Split-and-Share architecture that accounts for these two facts in the model. Separate, but weight-shared, subnets encode left- and right-hand sEMG streams in parallel, converging only at the final linear layer where a single vocabulary-level softmax decodes the character (Figure 2). This enforces hand-specific locality, but prevents the encoder from entangling spurious cross-hand correlations. Further, re-use of parameters reduces resource demands: the compact Split-and-Share model uses $0.25 \times$ the parameters and approximately $60\%$ of the FLOPs of a conventional joint-hand encoder while matching zero-shot accuracy. + +Summarily, our new, lightweight architecture built on simple modifications from these three insights reduces zero-shot generalization by $31\%$ and finetuning performance by $21\%$ over the prior state-of-the-art in Sivakumar et al. (14). Together, we both increase performance while reducing computational cost, providing a more feasible path towards on-device, performant computation for sEMG interfaces. + +# 3 Related Works + +Per-Session Normalization in EMG: The EMG and brain-computer interface (BCI) literature commonly normalize input signals prior to decoding to account for variability. For instance, prior EMG studies expressed amplitudes as a percentage of a reference contraction (maximum voluntary contraction) or applied z-score normalization based on a calibration recording (15). These normalizations enable comparisons across muscles and subjects (3; 8), and several works have normalized EMG features per session for machine learning pipelines (15; 10; 7; 20; 16; 6). In BCI literature, Willett et al. (19) show that continuously updating z-score statistics on spiking features within each session is essential for their high-performance speech neuroprosthesis, and Jarosiewicz et al. (5) used online bias correction to improve a point-and-click BCI. As described in Section 2, an important insight is not whether sEMG activity should be normalized, but how it is normalized. We ultimately find RTN significantly outperforms batch normalization. + +Masking-Based Regularization for sEMG: SpecAugment, first introduced for speech recognition, has recently been ported to EMG decoding pipelines (12; 14). The version used by Sivakumar et al. (14) is relatively mild: on average only $\sim 6\%$ of inputs are masked per training sample, with a given electrode's spectrogram never being masked across more than $\sim 24\%$ of the frequencies. ACM corresponds to hyperparameters that are intentionally harsher, with at least half of the electrodes completely blanked in most mini-batches and $\sim 55\%$ of input features masked on average. This encourages the model to infer keystrokes from small, overlapping subsets of channels, discouraging brittle reliance on the full spatiotemporal feature pattern. + +Multi-stream architectures and weight sharing: The concept of processing multiple information streams independently before later integration has been established in Automatic Speech Recognition (ASR), where multi-stream systems effectively handle speech at various resolutions or from multiple arrays (9). Similarly, in Sign Language Recognition (SLR), previous works have implemented separate processing streams for each hand (11; 17). Leveraging anatomical symmetry through weight sharing has shown success in pose estimation (21) and SLR (17). In the domain of EMG-based handpose recognition, Salter et al. (13) combined data from both hands into a unified dataset for single-model training, though this addresses an inherently unimanual task. For bimanual keystroke decoding, past methods have processed signals from both hands jointly (14; 18; 1). Our Split-and-Share architecture keeps the two wrists separate to respect biomechanics, yet ties the weights, effectively training a single-hand encoder with data from both hands. To our knowledge, this has not been explored in EMG keyboard decoding. + +# 4 Methods + +# 4.1 Reduced Spectral Granularity + +Let $x_{c}(t)$ denote the raw 2 kHz EMG signal from electrode $c$ . Following Sivakumar et al. (14) we first form a log-power spectrogram + +$$ +S _ {c} (t, f) = \log_ {1 0} \Big (\big | \mathrm {S T F T} (x _ {c}) \big | _ {t, f} ^ {2} + 1 0 ^ {- 6} \Big), \qquad f = 1, \ldots , 3 3, +$$ + +using a 64-point FFT (33 linearly spaced bins up to $1\mathrm{kHz}$ ) and a hop size of 16. We instead aggregate the 33 bins into six broader, roughly log-spaced bands, using the same frequency ranges as in (2). We refer to this aggregation as reduced spectral granularity (RSG). + +$$ +\mathcal {B} _ {1} = [ 3 1. 2 5, 6 2. 5 ], \mathcal {B} _ {2} = [ 6 2. 5, 1 2 5 ], \mathcal {B} _ {3} = [ 1 2 5, 2 5 0 ], +$$ + +$$ +\mathcal {B} _ {4} = [ 2 5 0, 3 7 5 ], \mathcal {B} _ {5} = [ 3 7 5, 6 8 7. 5 ], \mathcal {B} _ {6} = [ 6 8 7. 5, 1 0 0 0 ] \mathrm {H z}. +$$ + +For each electrode the reduced spectrogram is obtained by + +$$ +R _ {c} (t, b) = \sum_ {f \in \mathcal {B} _ {b}} S _ {c} (t, f), \qquad b = 1, \dots , 6. +$$ + +Hence, we reduce the spectral dimensionality of each electrode from 33 to 6. This represents an over five-fold reduction in features, and empirically equals or improves performance. This also leads SpecAugment to mask significantly more frequencies, even prior to ACM. + +# 4.2 Rolling Time Normalization + +Sivakumar et al. (14) normalizes each electrode's spectrogram with batch-norm over the entire mini-batch, frequency, and time axes. This collapses all features recorded on an electrode into a single distribution and, crucially, re-uses statistics gathered during training at inference time—an issue when the test session (or user) drifts from the training distribution. RTN replaces this with a causal, per-feature normalizer that is computed independently for every sample, band, electrode channel and frequency bin. Let + +$$ +x _ {t, n, b, c, f} \in \mathbb {R}, \qquad t = 0 \dots T - 1, n = 0 \dots N - 1, +$$ + +denote the log-power value at time-step $t$ , mini-batch index $n$ , band $b$ , electrode $c$ and frequency bin $f$ . RTN maintains cumulative statistics + +$$ +\mu_ {t} = \frac {1}{t + 1} \sum_ {s = 0} ^ {t} x _ {s, n, b, c, f}, \qquad \sigma_ {t} = \sqrt {\left(\frac {1}{t + 1} \sum_ {s = 0} ^ {t} x _ {s , n , b , c , f} ^ {2}\right) - \mu_ {t} ^ {2} + \varepsilon}, +$$ + +which are cheap to update via running sums in streaming settings. During a warm-up period of $T_{w} = 125$ frames (the first 1 second) the statistics are frozen to those computed over the entire warm-up window. The normalized output is + +$$ +\hat {x} _ {t, n, b, c, f} = \frac {x _ {t , n , b , c , f} - \tilde {\mu} _ {t}}{\tilde {\sigma} _ {t}}, +$$ + +where $\tilde{\mu}_t$ and $\tilde{\sigma}_t$ denote the warm-up-frozen statistics for $t < T_w$ and the cumulative statistics above for $t \geq T_w$ . RTN adapts continuously to session-specific non-stationarities, avoids batch-level or training-level statistics, and is causal. + +# 4.3 Aggressive Channel Masking + +We apply ACM on the reduced spectral granularity features with $B = 6$ bands. We mask inputs in $2/3$ of mini-batches. For each mini-batch we draw the number of frequency masks, $n_f \sim \mathrm{Unif}\{0,1,2\}$ . Each mask is then sampled independently for every electrode and sample in the batch: + +1. Width $w \sim \mathrm{Unif}\{0, \dots, f_{\max} - 1\}$ , with $f_{\max} = 12$ . The implementation clamps the width to the number of bands, $w \gets \min(w, B)$ . Thus the probability that a single mask erases the entire electrode is $\Pr[w = B] = \frac{f_{\max} - B}{f_{\max}} = 0.5$ . +2. Start index $f_0 \sim \mathrm{Unif}\{0, \ldots, B - w\}$ . +3. Set $X_{t,c,f_0:f_0 + w - 1} \gets 0$ for all timesteps $t$ . + +An individual mask already removes a channel with probability 0.5, while two masks remove a channel with probability 0.75. Remaining electrodes lose large spectrally-coherent chunks. By forcing the network to succeed even when so many electrodes are fully masked, the augmentation discourages brittle high-order feature dependencies and promotes low-order motifs that transfer across users. + +# 4.4 EMG Encoder Architectures + +Following spectrogram normalization, all architectures apply a two-stage encoder composed of a Rotation-Invariant MLP and a Time-Depth Separable Convolution (TDSConv) stack, with a linear character prediction layer following the last TDSConv block. These architectures were first presented in Sivakumar et al. (14) and are reproduced here for completeness. + +Rotation Invariant MLP. Let $\mathbf{x} \in \mathbb{R}^{T \times N \times B \times C \times F}$ denote the input, where $T$ is the number of time steps, $N$ is the batch size, $B = 2$ denotes the number of bands (hands), $C = 16$ is the number of electrode channels per band, and $F$ is the number of spectral frequency bins. Each band is passed through a Rotation-Invariant MLP, which applies a multi-layer perceptron to rotated versions of the electrode channels to ensure robustness to cyclic spatial shifts: + +$$ +\mathbf {h} _ {b} = \frac {1}{| \mathcal {O} |} \sum_ {o \in \mathcal {O}} \operatorname {M L P} (\operatorname {r o l l} (\mathbf {x} _ {b}, o)), \quad \mathcal {O} = \{- 1, 0, 1 \}, +$$ + +where each rotated input is flattened across $C \times F$ before the MLP is applied. This results in an embedding $\mathbf{h}_b \in \mathbb{R}^{T \times N \times D}$ per band. Concatenating the left and right hand features yields a total input of shape $\mathbf{h} \in \mathbb{R}^{T \times N \times 2D}$ to the TDSCov blocks in the baseline model, whereas each hand is processed independently in the Split-only and Split-and-Share variants. + +TDSCov block architecture. The TDSCov stack (4) is composed of alternating temporal convolution and fully connected blocks. Each convolutional block first reshapes the input from $\mathbb{R}^{T\times N\times D}$ to $\mathbb{R}^{N\times K\times H\times T}$ , where $D = K\cdot H$ denotes the total feature dimension, $K$ is the number of convolution channels, and $H$ is the per-channel hidden width. A 2-D convolution with kernel size $1\times w$ is then applied along the time axis: + +$$ +\mathbf {z} [ n, k, h, t ] = \sum_ {i = 0} ^ {w - 1} \sum_ {k ^ {\prime} = 1} ^ {K} \theta_ {k, k ^ {\prime}, i} \mathbf {h} [ n, k ^ {\prime}, h, t - i ], \qquad \theta \in \mathbb {R} ^ {K \times K \times 1 \times w}. +$$ + +![](images/28b43885f55ae1d1a6dd9262db44c82649f53880b38f2d238f0e71dbb9a9400e.jpg) +Figure 3: EMG encoder architectures. Left (blue) and right (red) hand specific modules only process inputs from a single hand, with hand-specific weights. Joint (purple) modules jointly process inputs from both hands. Shared (gray) modules process inputs from either hand using identical weights. a) Joint-Hand baseline architecture of Sivakumar et al. (14), b) Split-only architecture, in which hand-specific modules process signals from each hand separately. c) Split-and-Share architecture, where shared-weight modules process signals from each hand separately. + +Because the kernel height is 1, the same weights are reused at every hidden-width position $h$ . The convolution still mixes all $K$ input channels to produce each of the $K$ output channels, giving the layer $K^2 w$ parameters. The output is passed through ReLU, summed with the residual, and normalized using LayerNorm. The subsequent fully connected block consists of two linear layers with a ReLU in between and a residual connection: + +$$ +\mathbf {z} _ {\mathrm {f c}} = \operatorname {L a y e r N o r m} \left(\mathrm {F C} _ {2} \left(\operatorname {R e L U} \left(\mathrm {F C} _ {1} \left(\mathbf {z} _ {\text {c o n v}}\right)\right)\right) + \mathbf {z} _ {\text {c o n v}}\right). +$$ + +Architecture variants. In the architecture of Sivakumar et al. (14), which we refer to as the Joint-Hand architecture, both hands are processed jointly following the Rotation-Invariant MLP, yielding a post-MLP embedding of $2D = 768$ . The TDSCov stack operates with this full dimensionality throughout. In contrast, both the Split-only and Split-and-Share models operate on embeddings of $D = 384$ per hand, and apply the TDSCov blocks separately to each hand's input. This reduces the width of all fully connected layers from $768 \times 768$ to $384 \times 384$ , leading to a four-fold reduction in parameters per FC layer, while preserving expressiveness by keeping the number of convolution channels $K = 24$ and kernel width $w = 32$ unchanged. Despite duplicating the encoder, the Split-only model has lower total FLOPs—approximately $60\%$ of the baseline—due to the narrower per-hand embeddings. The Split-and-Share model uses the same dual-stream structure but shares all encoder weights between hands. This model has roughly half the parameters of the Split-only model, since almost all parameters (excluding the final prediction layer) are shared between the two encoders. Notably, the Split-only and Split-and-Share models have the same FLOPs. + +To evaluate whether representational capacity might further enhance the strong performance of the Split-and-Share architecture, we also test an Upscaled Split-and-Share variant. This model increases the per-hand embedding size to $D = 528$ and expands the number of convolution channels in the final two TDS blocks from 24 to 48. Nonetheless, the total number of parameters remains about half that of the baseline, and the FLOPs are only modestly higher (about $15\%$ ). We refer to the Upscaled Split-and-Share model as SplashNet and the smaller Split-and-Share model as SplashNet-mini + +# 4.5 Backspace-aware beam search with 6-gram Character LM + +For decoding we adopt the backspace-aware beam search of Sivakumar et al. (14), which keeps the 50 most-probable CTC label prefixes at each frame. Blank transitions simply update the score of an unchanged prefix, while the highest-scoring non-blank characters extend each hypothesis and are re-ranked with a 6-gram character LM prior. When a backspace symbol appears, the algorithm retracts the last language-model contribution, letting deletions correct earlier mistakes without extra + +Table 1: Zero-shot CER (\%, mean ± s.d. across participants), GFLOPs, and parameters. Columns in gray correspond to training domain validation results, which are reported for transparency but not used as indicators of generalization. + +
MethodTrain domain valOther domain valTest domain valTest domain testGFLOPs (30 s)Params
Sivakumar et al. 202412.1472.0752.10 ± 5.5451.78 ± 4.6161.615.29M
+ RSG13.5267.4847.26 ± 5.2647.18 ± 5.1954.154.96M
+ RTN13.0961.9539.49 ± 7.4539.15 ± 6.2054.154.96M
+ ACM23.4763.0842.62 ± 7.1842.62 ± 7.1054.154.96M
+ RTN + ACM21.7158.8536.41 ± 7.2136.42 ± 7.1154.154.96M
+ Split23.9358.6437.28 ± 6.9137.37 ± 7.3436.842.68M
+ Share (SplashNet-mini)26.4458.2036.46 ± 7.0936.41 ± 7.3036.841.38M
+ Upscale (SplashNet)20.5956.9535.49 ± 7.5635.67 ± 6.7971.382.58M
+ +passes. After the final frame any open LM context is closed, and the best-scoring prefix provides the decoded keystroke sequence. All beam-search parameters are the same as in Sivakumar et al. (14) + +# 4.6 Dataset Splits + +The official emg2qwerty protocol introduced by Sivakumar et al. (14) assigns 100 participants to the training pool and eight to a held-out test pool. For the 96 training participants with $\geq 4$ recording sessions, every session except the final two is used to train generic models, while those last two sessions form the authors' validation set; models are selected on this set before being evaluated on the eight held-out participants. + +Because the validation data come from the same individuals seen during training, this procedure encourages models that memorize participant-specific idiosyncrasies rather than ones that generalize across users. To illustrate this effect while limiting computational cost, we evaluate training domain validation performance using the validation sessions from 18 of the 96 training users, which we term the training domain validation set. The same 18 users are also used in Appendix A.8, where they are fully held out from training to provide a more appropriate validation signal. + +To obtain a validation signal that better predicts zero-shot performance in the main experiments, we instead exploit the four training-pool participants who recorded $< 4$ sessions and were therefore excluded from model fitting in the original setup. We take the final session of each of these four participants and combine them into what we term the other domain validation set. These participants exhibit noisier EMG data than those in the official test pool, providing a harder proxy for real-world generalization. Nonetheless, they provide a way of validating zero-shot generalization without requiring a complete restructuring of the official dataset splits. + +# 5 Results + +# 5.1 Zero-shot model performance + +Before turning to the main cross-user generalization results, we first highlight a key discrepancy between training domain validation and held-out user evaluation. As shown in Table 1, the CERs on the training domain validation set are not only lower, but the relative ranking of models follows a markedly different pattern than on held-out user evaluation sets. This illustrates why training domain validation provides a misleading signal for cross-user generalization and should not be used for model selection. In Appendix A.8, we further show that when these same 18 users are held out entirely, their CERs rise substantially and align closely with the held-out evaluation trends shown here. + +Our analysis centers on beam-search decoding using the same 6-gram character LM as in Sivakumar et al. (14); corresponding greedy-decoding results appear in Appendix A.3. All reported p-values were obtained from one-tailed paired t-tests across participants. + +Table 2: Finetuned CER (%, mean ± s.d. across participants), GFLOPs, and parameters. + +
MethodTest domain valTest domain testGFLOPs (30 s)Params
Sivakumar et al. 20248.31 ± 3.196.95 ± 3.6161.615.29M
+ RSG6.70 ± 3.226.92 ± 3.7954.154.96M
+ RTN6.47 ± 2.926.63 ± 3.5854.154.96M
+ ACM7.18 ± 3.337.45 ± 3.8754.154.96M
+ RTN + ACM6.55 ± 2.916.53 ± 3.2754.154.96M
+ Split6.04 ± 2.816.10 ± 3.3236.842.68M
+ Share (SplashNet-mini)6.13 ± 2.965.87 ± 3.0436.841.38M
+ FT Unshare5.85 ± 2.835.96 ± 3.2836.842.68M
+ Upscale (SplashNet)5.46 ± 2.605.51 ± 2.8171.382.58M
+ FT Unshare5.57 ± 2.655.67 ± 2.9771.385.06M
+ +First, we apply RSG to replace the 33-bin spectrogram with six coarser frequency bands. RSG leads to a modest but consistent improvement in CER from $51.78\%$ to $47.18\%$ ( $p = 7e-5$ ). Part of this performance improvement may stem from the implicit amplification of frequency masking under SpecAugment (due to the lower spectral resolution). + +Second, we replace batch-level normalization with RTN. RTN significantly improves zero-shot generalization $(p = 6e - 4)$ , reducing the CER to $39.15\%$ . This suggests that input scale and shift differences across users are a primary obstacle to cross-user generalization in EMG decoding. + +Third, we apply ACM, which encourages the model to rely on lower-order combinations of input features. Without RTN, ACM reduces zero-shot CER to $42.6\%$ $(p = .02$ vs. $+\mathrm{RSG}$ only). Combined with RTN, ACM reduces zero-shot CER to $36.42\%$ $(p = 5e - 3$ vs. $+\mathrm{RSG} + \mathrm{RTN})$ , providing a strong Joint-Hand baseline. + +Fourth, we explore architectural modifications that better reflect the causal and bilateral structure of EMG typing. We evaluate a Split-only model that encodes each hand separately without parameter sharing, achieving a CER of $37.37\%$ . Despite having only half the parameters and $66\%$ of the FLOPs of our Joint-Hand baseline, this model performs competitively, though its lack of shared parameters may limit data efficiency. We then evaluate SPLashNet-mini, where both hands are encoded via identical weight-shared encoders. This model achieves a $36.41\%$ CER—on par with the Joint-Hand baseline—while using just a quarter of its parameters and $66\%$ of the FLOPs. Finally, by increasing the embedding width and expanding the final convolutional layers, we create SPLashNet, with similar FLOPs to the baseline of Sivakumar et al. (14) but still half the parameters. This yields a further improvement, reducing the CER to $35.67\%$ ( $p = .049$ vs. SPLashNet-mini). + +Together, these results demonstrate that a combination of architectural priors, per-session normalization and principled regularization can significantly improve zero-shot EMG decoding. + +# 5.2 Finetuned Model Performance + +We next evaluate the performance of models finetuned on user-specific data. As in Sivakumar et al. (14), we maintain identical training hyperparameters during both generic pretraining and finetuning. While this simplifies the analysis, we note that the optimal hyperparameters for finetuning likely differ from those used during pretraining, and further gains may be achievable through phase-specific hyperparameter tuning. We also note that, because each recording session involved doffing and redonning the wristbands, the finetuning experiments inherently probe generalization across electrode placements in different sessions. + +We report all results using beam search with an external character-level language model (LM), which remains the standard for achieving state-of-the-art performance in CTC-based decoding pipelines. While our models consistently outperform the baseline of Sivakumar et al. (14) under beam search, we observe slightly worse performance under greedy decoding. We attribute this in part to the more aggressive channel masking induced by our reduced spectral representation ( $\S 4$ ), which we discuss further in Appendix A.3. + +![](images/b051154923b3a7a83664ed0b74dfc039ce35d7de3453f68f901857c7fbcf5149.jpg) +Figure 4: Zero-shot and finetuned CER distribution across users. Each of the 8 test users are represented by a dot, with lines connecting the same user across models. Boxplots depict median and interquartile ranges. Our methods improve performance for all participants relative to the baseline of Sivakumar et al. (14), with some participants showing very large improvements: two users reach CER between $20 - 30\%$ in the zero-shot setting, and one user attains a CER below $2\%$ when finetuned. + +![](images/e4f3c0398bb81a05edea5b530c8e0069bf495742c19647727a4289a8281fa188.jpg) + +First, we assess whether the same methods that improved zero-shot generalization also enhance performance in the finetuned setting. Using RSG alone yields similar performance to the baseline of Sivakumar et al. (14), with a CER of $6.91\%$ , while adding RTN yields a significantly improved CER of $6.63\%$ ( $p = .028$ vs. +RSG). Interestingly, applying ACM in isolation—without RTN—leads to worse performance than the baseline of Sivakumar et al. (14), whereas the Joint-Hand baseline model, with both ACM and RTN, matches the model with RTN alone ( $6.53\%$ CER). This suggests that masking-induced variability may hinder learning when the model lacks an appropriate normalization strategy to stabilize the input feature space. Moreover, unlike the generic case, finetuned models may rely more on higher-order feature correlations that are relatively stable across sessions for the same user, diminishing the benefits of aggressive masking. + +We next examine architectural changes that explicitly encode inductive biases about the bilateral and causal structure of EMG typing. Both the Split-only and SplashNet-mini models yield substantial improvements over the Joint-Hand baseline, with CERs of $6.10\%$ and $5.87\%$ ( $p = 6e-3$ and $p = 2e-3$ , respectively). For the latter, we evaluate two strategies during finetuning: either maintaining shared weights or duplicating them to allow separate adaptation per hand. Interestingly, both strategies yield similar performance, suggesting that hand-specific encoder weights are unnecessary even in the finetuned setting, although it is possible that unsharing weights could become advantageous with more user-specific data for finetuning. + +Finally, we evaluate the SplashNet model. Again, we do not find any benefit from unsharing the weights during finetuning. With the weights kept shared, SplashNet achieves the best performance overall with a CER of $5.51\%$ ( $p = .02$ vs.SplashNet-mini), establishing a new state-of-the-art for user-specific EMG keystroke decoding in this benchmark. + +# 6 Discussion + +The central contribution of this work is instilling simple, well-motivated priors through preprocessing, augmentation and architecture that can close a surprising fraction of the generalization gap in wrist-EMG typing. SplashNet-mini and SplashNet both achieve large absolute and relative CER reductions $(-15.4\mathrm{pp} / -31.1\%$ zero-shot; $-1.44\mathrm{pp} / -20.7\%$ after fine-tuning) while cutting parameters to $1/4 - 1/2$ of the baseline and FLOPs to $0.6 - 1.15\times$ . These gains are on par with (and complementary to) the $\sim 25\%$ error reduction CTRL-Labs at Reality Labs et al. (2) reported from doubling dataset size in a handwriting task, suggesting that principled inductive biases are as potent as raw data scaling for sEMG. + +A practical ambition articulated in Sivakumar et al. (14) is to run the entire decoder on the wristbands themselves, thereby mitigating concerns around latency, privacy, and robustness to Bluetooth interference. Achieving on-band inference will require an architecture that is not only light-weight but also fully split—able to process and output keystrokes from each wrist independently, without any cross-hand coupling. SplashNet moves part-way toward this goal by duplicating (and sharing) the encoder streams, yet it still merges information at the final linear output layer, so each wrist must + +communicate its embeddings to the other. Removing this last dependency, or replacing it with a low-bandwidth handshake, remains an open engineering challenge and a fruitful direction for future model architecture work. Another promising direction is exploring hybrid cross-hand architectures that retain split-and-share components but introduce limited, structured interactions between the hands, potentially offering a middle ground between the Split-and-Share and Joint-Hand extremes. + +Beyond these concrete results, our study points to several avenues. Normalization: RTN is a single-pass causal z-score; richer adaptive schemes (momentum updates, learnable affine transforms, or brief self-supervised calibration) may yield further robustness. Structured masking: ACM already improves zero-shot CER, but spatially contiguous electrode "drop-blocks", global time masks, or anatomical adjacency priors could guide the network toward even more transferable features. Model scale and self-supervision: SplashNet shows that capacity can be reinvested profitably once good priors are in place; coupling larger split-and-share encoders with masked-prediction pre-training may unlock still-higher accuracy without additional labels. + +Two limitations deserve emphasis. First, our models are developed for the practical beam-search + LM pipeline; under pure greedy decoding, the Joint-Hand baseline scores the best on unseen users, and ACM dents finetuned accuracy (details and discussion in Appendix A.3). This likely reflects the fact that the Joint-Hand baseline can exploit cross-hand co-articulatory patterns, which tend to correlate strongly with character bi- and tri-grams. These weak statistical regularities effectively act as an implicit language model under greedy decoding. By contrast, Split-and-Share models do not have access to these bilateral dependencies, and ACM further suppresses within-user cues by enforcing lower-order feature reliance. Once an external 6-gram LM is applied, however, its much stronger prior readily compensates for both the lost co-articulatory signal of the Split-and-Share models and the reduced within-user discriminability induced by ACM, thereby revealing the underlying generalization advantage of our approach. Second, all experiments use healthy participants; the extent to which our priors transfer to populations with motor or limb differences (where electrode placement and muscle recruitment differ markedly) remains to be tested. + +In sum, we show that the long-standing vision of "keyboard-quality" EMG typing can be advanced not only by more data but also by better assumptions. By pairing causal normalization, aggressive yet structured regularization, and a symmetry-aware encoder, we make significant strides towards truly out-of-the-box wrist-sEMG typing—paving the way for real-world assistive and AR/VR interfaces. + +# 7 Acknowledgments + +This work was supported by the following awards to JCK: NSF CAREER 1943467, NIH DP2NS122037, NIH R01NS121097. The authors would also like to thank Andrea Ortone for carefully checking the FLOPs calculations and providing helpful feedback. + +# References + +[1] Michael S Crouch, Mingde Zheng, and Michael S Eggleston. Natural typing recognition via surface electromyography. arXiv preprint arXiv:2109.10743, 2021. +[2] CTRL-Labs at Reality Labs, David Sussillo, Patrick Kaifosh, and Thomas Reardon. A generic noninvasive neuromotor interface for human-computer interaction. bioRxiv, 2024. doi: 10. 1101/2024.02.23.581779. URL https://www.biorxiv.org/content/early/2024/02/ 28/2024.02.23.581779. +[3] Mark Halaki and Karen Ginn. 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URL https://onlinelibrary.wiley.com/doi/abs/10.3233/ABB-2003-9693530. + +# A Technical Appendices and Supplementary Material + +# A.1 Training Details + +All models were trained for 150 epochs with the hyperparameter configuration of Sivakumar et al. (14). we used the same optimizer (Adam), the same LR schedule (linear warmup from 1e-8 to 1e-3 for first 10 epochs, followed with cosine annealing to 1e-6 until epoch 150), and the same augmentations. These included a rotation augmentation, which shifts all electrodes on each band one electrode to the left or right for each training sample, and a temporal alignment jitter augmentation, which jitters the EMG signals from each hand by a maximum offset of $60~\mathrm{ms}$ . We only made two modifications to hyperparameters: + +1. Input window length. We employ 16 s training windows, rather than the 4 s clips used by Sivakumar et al. (14). This was done to speed up training. + +2. SpecAugment settings. + +- With ACM. We raise the maximum frequency mask width to 12 and disable time masking. +- Without ACM. We retain the original SpecAugment parameters of Sivakumar et al. (14) (maximum frequency mask width = 4; up to 3 time masks per electrode per sample, each masking out up to $200\mathrm{ms}$ ). + +# A.2 Evaluation Details + +All validation and test sessions were loaded as a single sample (i.e. with batch size of 1). By comparison, Sivakumar et al. (14) loaded validation (but not test) sessions as shortened chunks. Although this is unlikely to have affected their greedy decoding results as their architecture (and ours) has a receptive field of only 1 second, this might have affected the validation performance they reported with an external LM, which would lose the character history between each chunk and therefore make poorer predictions. This might explain why the validation CER they report with beam-search $(8.31\%$ CER) is conspicuously higher than their test performance $(6.95\%$ CER) compared to the gaps we see between validation and test sessions. + +# A.3 Greedy vs. Beam-Search Decoding + +Our main text focuses on beam-search decoding with an external 6-gram language model (LM), because this is the configuration most relevant to realistic, latency-constrained deployment. For completeness, we also report greedy CTC decoding results, i.e. decoding without any LM. + +Table 3: Zero-shot CER (\%, mean across participants). Columns in gray correspond to training domain validation results, which are reported for transparency but not used as indicators of generalization. Note that the training domain validation results shown here with greedy decoding were computed over all 96 training users; the beam search decoding results are computed on the 18 user subset. + +
MethodTrain domain val (greedy)Train domain val (beam)Other domain val (greedy)Other domain val (beam)Test domain val (greedy)Test domain val (beam)Test domain test (greedy)Test domain test (beam)
Sivakumar et al. 202422.5112.1472.4472.0755.5752.1055.3851.78
+ RSG23.5913.5268.5567.4852.2447.2652.2747.18
+ RTN23.4913.0964.1961.9546.3139.4946.0739.15
+ ACM34.7723.4765.6063.0848.8742.6249.2342.62
+ RTN + ACM32.8521.7161.8258.8543.7236.4143.8036.42
+ Split35.5923.9361.7458.6445.7337.2845.6937.37
+ Share (SplashNet-mini)37.7226.4461.0758.2045.3336.4645.2636.41
+ Upscale (SplashNet)33.5720.5960.1656.9544.7935.4944.7835.67
+ +Zero-shot generalization. Table 3 shows that, in the zero-shot regime (i.e. on non-training domains), the joint-hand architecture with RTN and ACM often surpasses the Split-and-Share architecture under greedy decoding, even though the ranking reverses once beam search is applied. The most plausible explanation is that the joint-hand encoder can observe both wrists simultaneously and therefore learns a stronger implicit LM, capturing bi- and tri-gram dependencies spread across hands. While that emergent linguistic prior is helpful when no external LM is available, it becomes redundant—and potentially counter-productive—once a more reliable 6-gram LM is introduced at inference time. + +Table 4: Finetuned CER (%) with and without beam search. + +
MethodTest domain val (greedy)Test domain val (beam)Test domain test (greedy)Test domain test (beam)
Sivakumar et al. 202411.398.3111.286.95
+ RSG12.116.7012.506.92
+ RTN11.836.4711.756.63
+ ACM14.247.1814.747.45
+ RTN + ACM12.706.5512.906.53
+ Split12.676.0412.896.10
+ Share13.016.1313.225.87
+ FT Unshare12.685.8513.075.96
+ Upscale (SplashNet)12.105.4712.395.51
+ FT Unshare11.805.5712.135.67
+ +Fine-tuned models. A different pattern emerges after user-specific fine-tuning (Table 4). None of our variants fully match the baseline of Sivakumar et al. (14) under greedy decoding, despite decisive gains once beam search is enabled. Because the only changes between the baseline and the "+" RSG" model are (i) four-fold longer clips during training, (ii) coarser spectral resolution, and (iii) a five-fold higher probability of channel masking, the degradation must stem from one (or a combination) of these factors. + +Two clues implicate aggressive channel masking (ACM). First, the model with RTN but not ACM achieves the best greedy scores, whereas the model with just ACM but not RTN yields the worst greedy performance. Second, the performance gap between the model with just RTN and the model with both RTN and ACM vanishes under beam search, indicating that the external LM compensates for information lost when ACM forces the network to rely on low-order feature combinations. We therefore hypothesize that ACM, while beneficial for cross-user generalization, removes within-user cues that help distinguish confusable keystrokes, a weakness that beam-search decoding can largely recover. + +To confirm that the implicitly increased channel masking in the "+" RSG" model (and all other models) is largely responsible for the uniformly worse greedy decoding performance we see compared to the baseline of Sivakumar et al. (14), we ran an additional experiment in which we trained a model similar to our +RSG model, but apply SpecAugment on the 33-bin spectrograms from each channel before the bin aggregation of RSG (rather than after). This keeps the extent of masking equivalent to that of Sivakumar et al. 2024 while also allowing us to use the RSG frontend. This model does not perform significantly worse than the baseline of Sivakumar et al. 2024 with greedy decoding, confirming our suspicion that the core reason for the worse greedy decoding results in the finetuned case is increased channel masking. + +Table 5: Finetuned CER (%) under greedy decoding. Values are mean $\pm$ standard deviation. + +
MethodTest domain val (greedy)Test domain test (greedy)
Sivakumar 202411.39 ± 4.2811.28 ± 4.45
+ RSG12.11 ± 4.6712.50 ± 4.97
+ RSG w/ pre-aggregation masking11.02 ± 4.3211.53 ± 4.70
+ +In summary, greedy decoding accentuates two complementary inductive biases: (1) joint-hand encoders learn a useful internal LM, an advantage that vanishes once an external LM is applied, and (2) aggressive masking of input channels trades cross-user generalization for within-user discriminability, a trade-off that RTN and beam-search decoding can effectively offset. + +# A.4 Calculation of FLOPs + +FLOPs were measured using FlopTensorDispatchMode in PyTorch with an arbitrary 30-second input. + +# A.5 Additional Ablations on ACM Masking, RTN Sliding Windows, and RSG + +Table 6: Zero-shot CER (\%, mean ± s.d. across participants) for ablations on ACM mask width and RTN sliding window. “-RSG” corresponds to applying ACM on the full-resolution spectrogram as described in the text. + +
MethodOther domain valTest domain valTest domain test
SplashNet-Mini58.20 ± 10.5036.46 ± 7.0936.41 ± 7.30
+ 4s SW inference60.43 ± 7.8637.57 ± 7.1837.71 ± 7.47
+ 4s SW train + inference60.43 ± 7.7036.95 ± 7.8936.69 ± 7.80
+ 16s SW inference57.24 ± 11.0036.43 ± 7.3036.73 ± 7.40
+ mask width = 858.12 ± 11.7236.57 ± 7.0236.05 ± 6.74
+ mask width = 1657.65 ± 10.4736.66 ± 7.2136.92 ± 7.40
- RSG60.70 ± 11.2337.90 ± 6.5837.19 ± 6.33
Joint-Hand baseline (+ RSG + RTN + ACM)58.85 ± 10.5036.41 ± 7.2136.42 ± 7.11
+ mask width = 859.41 ± 10.6337.55 ± 6.9737.30 ± 6.90
+ mask width = 1657.17 ± 11.3236.95 ± 7.6236.47 ± 7.96
+ +Table 7: Finetuned CER (\%, mean ± s.d. across participants) for ablations on ACM mask width and RTN sliding window. + +
MethodTest domain valTest domain test
SplashNet-Mini6.13 ± 2.965.87 ± 3.04
+ 4s SW inference6.10 ± 2.916.28 ± 3.09
+ 4s SW train + inference6.34 ± 2.866.37 ± 3.10
+ 16s SW inference5.76 ± 2.645.74 ± 2.86
+ mask width = 85.85 ± 2.726.02 ± 3.11
+ mask width = 166.09 ± 2.885.72 ± 2.96
- RSG5.87 ± 2.735.75 ± 3.06
Joint-Hand baseline (+ RSG + RTN + ACM)6.55 ± 2.916.53 ± 3.27
+ mask width = 86.14 ± 2.856.56 ± 3.60
+ mask width = 166.55 ± 3.096.45 ± 3.18
+ +We conducted additional ablations to examine the sensitivity of model performance to the strength of ACM, the presence of RSG, and the temporal window used for RTN normalization. Unless otherwise noted, none of the differences reported here reached significance under a two-tailed paired $t$ -test across participants. + +ACM mask width. In our main experiments, ACM uses a maximum frequency mask width of 12 bins, while models without ACM use a maximum width of 4. Here, we trained SplashNet-mini and the Joint-Hand baseline with maximum mask widths of 8 and 16. As shown in Tables 6 and 7, these variations produced small, nonsignificant changes in CER in both the zero-shot and finetuned settings, indicating that performance is not particularly sensitive to the precise mask width within this range. + +-RSG. We also evaluated a variant without RSG, in which ACM was applied to the full-resolution spectrogram by masking a 6-bin dummy vector and inverting the 33-to-6 bin mapping used in RSG to obtain a 33-bin mask. Although performance differences relative to the standard RSG configuration were small and nonsignificant, this variant considerably increases both compute and memory requirements since it has more than sixfold greater input feature dimensionality. + +RTN sliding window. Our original RTN normalization uses all past time points within a sample to compute normalization statistics. We additionally evaluated inference-time sliding-window variants with 4-second and 16-second windows. Using a 4-second window led to a small but statistically significant degradation in zero-shot performance ( $p < 0.05$ , two-tailed), while a 16-second window yielded performance comparable to the default setting. This likely reflects the fact that models were trained on 16-second samples and benefited from matching normalization context at test time. When models were also trained with a 4-second RTN sliding window, this degradation disappeared, and differences were no longer significant in either the zero-shot or finetuned settings. + +Summary. Across all tested configurations, performance differences were small and generally nonsignificant. ACM is robust to maximum frequency mask width changes in the range of 8-16 bins. Removing RSG does not significantly degrade performance but incurs higher compute and memory costs. RTN performance remains stable across temporal windows when training and inference are matched, with only a modest increase in CER when using 4 second windows at inference time alone. + +# A.6 Aggressive channel masking with mean imputation + +Table 8: CER (%, mean ± s.d. across participants) for different ACM masking configurations. + +
MethodTest domain valTest domain test
Sivakumar et al. 202452.10 ± 5.5451.78 ± 4.61
+ RSG47.26 ± 5.2647.18 ± 5.19
+ ACM42.62 ± 7.1842.62 ± 7.10
+ ACM (mean impute)47.59 ± 7.1648.43 ± 6.24
+ RTN + ACM36.41 ± 7.2136.42 ± 7.11
+ +We performed an ablation to test whether RTN might interact with ACM by stabilizing input features such that the default masking value of 0 corresponds to their per-session mean. When using ACM with RTN, the default mask value of 0 corresponds to the per-sample mean for each feature, whereas under BatchNorm this same value can be far out of distribution for a given feature from a given sample. To test whether it is important that the masking value is the per-sample feature mean, we replaced RTN with standard BatchNorm while setting the ACM masking value to the per-sample mean of each feature, training a Joint-Hand model with RSG, BatchNorm, and ACM using per-channel sample mean imputation. + +Somewhat surprisingly, this variant performed on par with the model with RSG alone and substantially worse than the RSG + BatchNorm + ACM configuration with standard zero imputation. This suggests that simply matching the mask value to the per-session mean is not sufficient to reproduce the performance benefits obtained with RTN + ACM, let alone to obtain the benefits of ACM in the absence of RTN. More generally, these results indicate that the choice of masking value being stable across samples (rather than corresponding to the per-sample mean) appears to be the more important factor for ACM's effectiveness in this setting. + +# A.7 UMAP analyses on early intermediate representations + +![](images/06f8aec2e0d0efdf812f73da288ab1021a6c64ad136d409fdc94baedcd7a48e9.jpg) ++RSG + +![](images/5f6db56111512495a0fe5ba5262b979216a63406e5a1f062d8510da0aa3875e4.jpg) ++RSG+ACM + +![](images/d1e56c1fe41f5d75ab29bf8aea0311b7b0238f8a8fab0eb17bfbda10fd8ddb5a.jpg) +Figure 5: UMAP visualization of model activations after the first TDSConv block for four models (+RSG, +RSG+ACM, +RSG+ACM+RTN, and SplashNet-mini). We extracted activations from every 100th timestep from one session of each of the 8 held-out users. Colors indicate the user identity of each point. In the models without RTN, some users' representations occupy largely disjoint regions of the activation manifold, whereas models with RTN (+RSG+ACM+RTN and SplashNet-mini) produce markedly more overlapping per-user representations, indicating improved cross-user alignment in the learned feature space. + +![](images/6f0592b9c735943667f4e1fd348712f5d6b642ff233548d3c5226d04b1201f3e.jpg) ++RSG+ACM+RTN +SplashNet-mini + +# A.8 Alternative train-test split analyses + +Table 9: CER (\%, mean ± s.d. across participants) with a new 78/18 train/validation split. The first column shows performance on the same 18 users when they were included in the original training set (i.e., the previous training domain validation performance), while the remaining columns show results when these 18 users are held out for validation. The test set remains the same 8 held-out users. + +
MethodTrain domain val (prev. split)Train domain val (18 held-out)Test domain val (18 held-out)Test domain test (18 held-out)
+ RSG13.52 ± 7.5958.42 ± 10.4347.89 ± 6.3649.01 ± 5.35
+ RTN13.09 ± 6.3253.99 ± 11.1442.81 ± 6.7542.62 ± 6.89
+ ACM23.47 ± 9.7455.95 ± 10.4644.59 ± 6.4745.10 ± 7.26
+ RTN + ACM21.71 ± 9.6751.25 ± 11.4938.72 ± 6.6339.05 ± 7.22
+ Split23.93 ± 10.9951.57 ± 11.7839.81 ± 6.7240.14 ± 7.16
+ Share26.44 ± 10.6449.84 ± 12.5738.60 ± 7.5238.45 ± 7.86
+ +# A.9 Analyses on emg2pose dataset + +Table 10: Mean angular error (degrees) and landmark distance (mm) under different generalization regimes. All results are in the "tracking" setting. + +
GeneralizationModelAngular ErrorLandmark Dist
UserRTN+ACM7.595110.1567
StageRTN+ACM11.124315.2404
User, StageRTN+ACM10.843315.3644
UserRTN7.600710.1641
StageRTN11.162715.2241
User, StageRTN10.908515.3851
UserACM7.638810.2394
StageACM11.260115.4489
User, StageACM11.034315.6413
UserBaseline7.654910.2585
StageBaseline11.289215.4360
User, StageBaseline11.122215.7388
+ +Although the focus of this work is on EMG keystroke decoding, we also performed preliminary experiments to assess how RTN and ACM transfer to the emg2pose benchmark (13). A key consideration is that the baseline model of Salter et al. (13) uses a learned convolutional featurizer that immediately mixes signals across all electrodes, rather than spectrogram-based per-electrode features as in our keystroke decoding setting. This architectural difference makes a direct application of RTN and ACM less straightforward. + +To adapt these methods, we applied RTN and ACM to the outputs of the first convolutional layer in the featurizer. Since these intermediate features are not spectrograms, ACM was implemented by randomly zeroing out $50\%$ of the features in one-third of training samples and $75\%$ in another third. The resulting performance, summarized in Table 10, shows minimal differences relative to the baseline in both mean angular error and landmark distance across user, stage, and combined generalization regimes. + +These findings are somewhat unsurprising: applying ACM after early feature mixing likely limits its ability to enforce robust low-order structure, and RTN is less meaningful without clear per-electrode feature boundaries. Future work should explore applying these methods directly to the raw EMG signal before featurization, or using architectures with explicit per-electrode feature streams (e.g., spectrogram-based or structured learned featurizers), where RTN and ACM may have stronger effects. + +# A.10 Compute Resources + +All experiments were run on a single RTX 4090 (with 24GB VRAM) GPU. Each generic model took roughly 24 hours to train. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: LLMs were not used in any of our methods. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git "a/NeurIPS/2025/SplashNet_ Split\342\200\221and\342\200\221Share Encoders for Accurate and Efficient Typing with Surface Electromyography/images.zip" "b/NeurIPS/2025/SplashNet_ Split\342\200\221and\342\200\221Share Encoders for Accurate and Efficient Typing with Surface Electromyography/images.zip" new file mode 100644 index 0000000000000000000000000000000000000000..3820e39bc2d7a20554764480e219e6ee72ac1041 --- /dev/null +++ "b/NeurIPS/2025/SplashNet_ Split\342\200\221and\342\200\221Share Encoders for Accurate and Efficient Typing with Surface Electromyography/images.zip" @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dee22492d3edc1b76116fcec6623c906761800875f64359a8ee6ef9352554ba6 +size 890716 diff --git "a/NeurIPS/2025/SplashNet_ Split\342\200\221and\342\200\221Share Encoders for Accurate and Efficient Typing with Surface Electromyography/layout.json" "b/NeurIPS/2025/SplashNet_ Split\342\200\221and\342\200\221Share Encoders for Accurate and Efficient Typing with Surface Electromyography/layout.json" new file mode 100644 index 0000000000000000000000000000000000000000..fa3104311f0dc75ae8a54c375460018831e5b544 --- /dev/null +++ "b/NeurIPS/2025/SplashNet_ Split\342\200\221and\342\200\221Share Encoders for Accurate and Efficient Typing with Surface Electromyography/layout.json" @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bfab480499ba92853f234d39fc33b7b4ecc33bb7196c2ea02e5af294fef4a181 +size 723196 diff --git a/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_content_list.json b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..94c86003e1d93f9b6ddafbb7a86d600b465fbe21 --- /dev/null +++ b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:09804aa85d58acbfb991c4de06cea2f14e08dfef84e463484b500126c379d2f0 +size 209623 diff --git a/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_model.json b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_model.json new file mode 100644 index 0000000000000000000000000000000000000000..2833f68fcb6d2469a38b00fba3baa0ca204f92e4 --- /dev/null +++ b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f243cbdf49d84c8f1ffdcc3faf4cea6217bffdce279ba65d658d96bb7842a77c +size 259975 diff --git a/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_origin.pdf b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..68ff400c6bf7a0d1d6ddb0b23098ae767dc51435 --- /dev/null +++ b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/6273db03-8159-4f40-be33-59fc1c917133_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8c22054750e83406c7457ad27f91e84aa40805e1afce0fc7aa1e4241ac2753e6 +size 3183404 diff --git a/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/full.md b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/full.md new file mode 100644 index 0000000000000000000000000000000000000000..f2f45192e99910a28d9b078bcd6fff90779a9167 --- /dev/null +++ b/NeurIPS/2025/Split Gibbs Discrete Diffusion Posterior Sampling/full.md @@ -0,0 +1,1071 @@ +# Split Gibbs Discrete Diffusion Posterior Sampling + +Wenda Chu1 Zihui Wu1 Yifan Chen2 Yang Song3 Yisong Yue1 +1California Institute of Technology New York University OpenAI + +# Abstract + +We study the problem of posterior sampling in discrete-state spaces using discrete diffusion models. While posterior sampling methods for continuous diffusion models have achieved remarkable progress, analogous methods for discrete diffusion models remain challenging. In this work, we introduce a principled plug-and-play discrete diffusion posterior sampling algorithm based on split Gibbs sampling, which we call SGDD. Our algorithm enables reward-guided generation and solving inverse problems in discrete-state spaces. We demonstrate the convergence of SGDD to the target posterior distribution and verify this through controlled experiments on synthetic benchmarks. Our method enjoys state-of-the-art posterior sampling performance on a range of benchmarks for discrete data, including DNA sequence design, discrete image inverse problems, and music infilling, achieving more than $30\%$ improved performance compared to existing baselines. + +# 1 Introduction + +We study the problem of posterior sampling in finite spaces with discrete diffusion models. Given a pretrained discrete diffusion model [35, 1, 4, 29] that models the prior distribution $p(\mathbf{x})$ , our goal is to sample from the posterior $p(\mathbf{x}|\mathbf{y})$ in a plug-and-play fashion, where $\mathbf{y}$ represents a guiding signal of interest. This setting encompasses both solving inverse problems (e.g., if $\mathbf{y}$ is an incomplete measurement of $\mathbf{x}$ ), and conditional or guided generation (e.g., sampling from $p(\mathbf{x})\exp (\beta r(\mathbf{x})) / Z$ according to a reward function $r(\cdot)$ ). + +Existing works on diffusion posterior sampling [7, 37, 30, 51] primarily focus on continuous diffusion models operating in the Euclidean space for $\mathbf{x}$ . These methods have achieved remarkable success in applications including natural image restorations [24, 7, 57], medical imaging [20, 42, 6], and various scientific inverse problems [14, 51, 55]. However, posterior sampling with discrete diffusion models in discrete-state spaces remains a challenging problem, mainly due to not having well-defined gradients of the likelihood function $p(\mathbf{y}|\mathbf{x})$ . Initial work to address this gap for discrete diffusion models relies on approximating value functions (similar to reinforcement learning) for derivative-free sampling [27], which can be difficult to tune for good performance, particularly when the guidance $\mathbf{y}$ is complicated. + +In this paper, we develop a rigorous plug-and-play method based on the principle of split Gibbs sampling [47], which we refer to as SGDD. Our method addresses the posterior sampling problem $\mathbf{x} \sim p(\mathbf{x}|\mathbf{y})$ by introducing an auxiliary variable $\mathbf{z}$ along with a regularization potential function $D(\mathbf{x},\mathbf{z};\eta)$ to relax the original problem. The relaxed formulation enables efficient sampling via Gibbs sampling, which alternates between a likelihood sampling step and a prior sampling step using a pretrained discrete diffusion model. As regularization increases, both variables $\mathbf{x}$ and $\mathbf{z}$ converge to the target posterior distribution. Our method generalizes split Gibbs samplers with $\ell_2$ regularizers to any potential functions with certain limiting conditions, making it compatible with discrete diffusion models by carefully designing the potential function. We provide a rigorous stationary guarantee on the convergence of SGDD, with error bounds that account for imperfect score function and discretization errors. + +We validate SGDD on diverse inverse problems and reward-guided generation problems that involve discrete data. We demonstrate that SGDD converges to the true posterior distribution on synthetic data, that SGDD solves discrete inverse problems accurately and efficiently on discretized image data and monophonic music data, and that SGDD excels at reward-guided sampling on guided DNA generation. Our method demonstrates strong sampling quality in all experiments. For instance, we achieve a $42\%$ higher median activity in enhancer DNA design, an 8.36 dB improvement of PSNR values in solving the XOR inverse problem on MNIST, and over $2\times$ smaller Hellinger distance in music infilling, compared to existing methods. + +# 2 Preliminaries + +# 2.1 Discrete Diffusion Models + +Diffusion models [39, 40, 18, 41, 22, 23] generate data $\mathbf{x} \in \mathcal{X}$ by reversing a predefined forward process. When $\mathcal{X} \in \mathbb{R}^n$ is a Euclidean space, the forward process is typically defined to be a stochastic differential equation [22]: $\mathrm{d}\mathbf{x}_t = \sqrt{2\dot{\sigma}_t\sigma_t}\mathrm{d}\mathbf{w}_t$ , where $\mathbf{w}_t$ is a standard Wiener process and $\{\sigma_t\}_{t=0}^T$ is a predefined noise schedule. This diffusion process is reversible once the score function $s(\mathbf{x}_t,t) \coloneqq \nabla_{\mathbf{x}_t}\log p(\mathbf{x}_t;\sigma_t)$ is learned. Starting from $\mathbf{x}_T \sim \mathcal{N}(0,\sigma_T^2\mathbf{I})$ , we generate data following the reverse SDE: + +$$ +\mathrm {d} \mathbf {x} _ {t} = - 2 \dot {\sigma} _ {t} \sigma_ {t} \nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {x} _ {t}; \sigma_ {t}) \mathrm {d} t + \sqrt {2 \dot {\sigma} _ {t} \sigma_ {t}} \mathrm {d} \mathbf {w} _ {t}. \tag {1} +$$ + +Recent works on discrete diffusion models [35, 1, 4, 29] extend score-based generative methods from modeling continuous distributions in Euclidean spaces to categorical distributions in discrete-state spaces. Specifically, when the data distribution lies in a finite support $\mathcal{X} = \{1,\dots ,N\}$ , one can evolve a family of categorical distributions $p_t$ over $\mathcal{X}$ following a continuous-time Markov chain over the discrete elements, + +$$ +\partial_ {t} p _ {t} = Q _ {t} ^ {\mathrm {f w}} p _ {t}, \tag {2} +$$ + +where $p_0 = p_{\mathrm{data}}$ and $\mathbf{Q}_t^{\mathrm{fw}} \in \mathbb{R}^{N \times N}$ are diffusion matrices with a simple stationary distribution. To reverse this continuous-time Markov chain, it suffices to learn the concrete score $s(\mathbf{x}; t) := \left[\frac{p_t(\tilde{\mathbf{x}})}{p_t(\mathbf{x})}\right]_{\tilde{\mathbf{x}} \neq \mathbf{x}}$ , as the reverse process is given by + +$$ +\partial_ {t} p _ {T - t} = \boldsymbol {Q} _ {t} p _ {T - t} \tag {3} +$$ + +with $Q_{t}^{[i,j]} = \frac{p_{T - t}(\mathbf{x}_{i})}{p_{T - t}(\mathbf{x}_{j})} Q_{T - t}^{\mathrm{fw}[j,i]}$ and $Q_{t}^{[i,i]} = -\sum_{\mathbf{x}_{j}\neq \mathbf{x}_{i}}Q_{t}^{[j,i]}$ . + +For sequential data $\mathbf{x} \in \mathcal{X}^D$ , there are $N^D$ states in total. Instead of constructing an exponentially large diffusion matrix, we use a sparse matrix $Q_{t}^{\mathrm{fw}}$ that perturbs tokens independently in each dimension [29]. + +Example: uniform kernel. An example of such diffusion matrices is + +$$ +\boldsymbol {Q} _ {t} ^ {\mathrm {f w}} = \dot {\sigma} _ {t} \boldsymbol {Q} ^ {\text {u n i f o r m}} = \dot {\sigma} _ {t} \left(\mathbf {1 1} ^ {T} / N - \boldsymbol {I}\right), \tag {4} +$$ + +where $\sigma_{t}$ is a predefined noise schedule with $\sigma_0 = 0$ and $\sigma_T = \sigma_{\mathrm{max}}$ . This uniform kernel transfers any distribution to a uniform distribution as $\sigma \rightarrow \infty$ . Moreover, + +$$ +p _ {t} = \exp \left(\int_ {0} ^ {t} Q _ {\tau} ^ {\mathrm {f w}} \mathrm {d} \tau\right) p _ {0} = \exp \left(\sigma_ {t} Q ^ {\text {u n i f o r m}}\right) p _ {0} = \left(e ^ {- \sigma_ {t}} I + \left(1 - e ^ {- \sigma_ {t}}\right) \frac {\mathbf {1 1} ^ {T}}{N}\right) p _ {0}. \tag {5} +$$ + +When $\mathbf{x} \in \mathcal{X}^D$ is a sequence of length $D$ , we have $p_t(\mathbf{x}_t | \mathbf{x}_0) \propto \beta_t^{d(\mathbf{x}_t, \mathbf{x}_0)} (1 - \beta_t)^{D - d(\mathbf{x}_t, \mathbf{x}_0)}$ , where $d(\cdot, \cdot)$ is the Hamming distance between two sequences, and $\beta_t = \frac{N - 1}{N} (1 - e^{-\sigma_t})$ . + +# 2.2 Diffusion Posterior Sampling Methods + +Diffusion posterior sampling is a class of methods that draw samples from $p(\mathbf{x}|\mathbf{y}) \propto p(\mathbf{x})p(\mathbf{y}|\mathbf{x})$ using a diffusion model trained on the prior $p(\mathbf{x})$ . This technique has been applied in two settings with slightly different formulations: solving inverse problems and reward guided generation. + +The goal of inverse problems is to reconstruct the underlying data $\mathbf{x}_0$ from its measurements, which are modeled as $\mathbf{y} = \pmb {G}(\mathbf{x}_0) + \mathbf{n}$ , where $\pmb{G}$ is the forward model, and $\mathbf{n}$ denotes the measurement noise. In this case, the likelihood function is given by $p(\mathbf{y}|\mathbf{x}) = p_{\mathbf{n}}(\mathbf{y} - \pmb {G}(\mathbf{x}))$ , where $p_{\mathbf{n}}$ denotes the noise distribution. + +![](images/2e1639a40a055cbab1c8c90ec8dc02db3de9fcf518d504721d81cfcf25ae31a3.jpg) +Figure 1: An illustration of our method on inpainting discretized MNIST. Our method implements the Split Gibbs sampler, which alternates between likelihood sampling steps and prior sampling steps. Likelihood sampling steps enforce data consistency with regard to measurement $\mathbf{y}$ , while prior sampling steps involve denoising uniform noise from $z^{(k)}$ by a pretrained discrete diffusion model starting from noise level $\eta_{k}$ . Two variables converge to a sample from the posterior distribution as the noise level $\eta_{k}$ reduces to zero. + +On the other hand, reward-guided generation [38, 19, 32] focuses on generating samples that achieve higher rewards according to a given reward function $r(\cdot)$ . Here, the likelihood is modeled as $p(\mathbf{y}|\mathbf{x})\propto \exp (\beta r(\mathbf{x}))$ , where $\beta >0$ is a parameter that controls the strength of reward weighting. + +Existing diffusion posterior sampling methods leverage (continuous) diffusion models as plug-and-play priors and modify Eq. (1) to generate samples from the posterior distribution $p(\mathbf{x}|\mathbf{y}) \propto p(\mathbf{x})p(\mathbf{y}|\mathbf{x})$ . These methods can be broadly classified into four categories [9, 56]. + +Guidance-based methods [7, 38, 37, 24, 49, 55] perform posterior sampling by adding an additional likelihood score term $\nabla_{\mathbf{x}_t}\log p(\mathbf{y}|\mathbf{x}_t)$ to the score function in Eq. (1), and modifies the reverse SDE as: $\mathrm{d}\mathbf{x}_t = -2\dot{\sigma}_t\sigma_t(\nabla_{\mathbf{x}_t}\log p(\mathbf{x}_t;\sigma_t) + \nabla_{\mathbf{x}_t}\log p(\mathbf{y}|\mathbf{x}_t))\mathrm{d}t + \sqrt{2\dot{\sigma}_t\sigma_t}\mathrm{d}\mathbf{w}_t$ . The intractable likelihood score term is approximated with various assumptions. However, these methods cannot be directly applied to discrete posterior sampling problems, where $\log p(\mathbf{y}|\mathbf{x})$ is defined only on a finite support and thus does not have such gradient information. + +Sequential Monte Carlo (SMC) methods sample batches of particles iteratively from a series of distributions, which converge to the posterior distribution in the limit. This class of methods [45, 12, 26, 50, 33, 34, 5] exploits the sequential diffusion sampling process to sample from the posterior distribution. Similar ideas have been applied in Li et al. [27] for reward-guided generation using discrete diffusion models, which rely on approximating a soft value function at each iteration. + +Variational methods [30, 14] solve inverse problems by approximating the posterior distribution with a parameterized distribution $q_{\theta}$ and optimizing its Kullback-Leibler (KL) divergence with respect to the posterior. However, how to extend these methods to discrete diffusion models remains largely unexplored. + +Variable splitting methods [51, 54, 57, 36, 52] decompose posterior sampling into two simpler alternating steps. A representative of this category is the split Gibbs sampler (see Section 3), where the variable $\mathbf{x}_t$ is split into two random variables that both eventually converge to the posterior distribution. These methods are primarily designed for continuous diffusion models with Gaussian kernels, leveraging Langevin dynamics or optimization in Euclidean space as subroutines. Murata et al. [31] extends this category to VQ-diffusion models in the discrete latent space using the Gumbel-softmax reparameterization trick. However, this approach is limited to discrete latent spaces with an underlying continuous embedding due to its reliance on softmax dequantization. In this work, we generalize the split Gibbs sampler to posterior sampling in categorical data by further exploiting the unique properties of discrete diffusion models. + +# 3 Method + +# 3.1 Split Gibbs for plug-and-play posterior sampling + +Suppose we observe $\mathbf{y}$ from a likelihood function $p(\mathbf{y}|\mathbf{x})$ and assume that $\mathbf{x}$ satisfies a prior distribution $p(\mathbf{x})$ . Our goal is to sample from the posterior distribution: + +$$ +p (\mathbf {x} | \mathbf {y}) \propto p (\mathbf {y} | \mathbf {x}) p (\mathbf {x}) = \exp (- f (\mathbf {x}; \mathbf {y}) - g (\mathbf {x})), \tag {6} +$$ + +where $f(\mathbf{x};\mathbf{y}) = -\log p(\mathbf{y}|\mathbf{x})$ and $g(\mathbf{x}) = -\log p(\mathbf{x})$ . The challenge in sampling from this distribution arises from the interplay between the prior distribution, modeled by a diffusion model, and the likelihood function. + +The Split Gibbs sampler (SGS) relaxes this sampling problem by introducing an auxiliary variable $\mathbf{z}$ , allowing sampling from an augmented distribution: + +$$ +\pi (\mathbf {x}, \mathbf {z}; \eta) \propto \exp (- f (\mathbf {z}; \mathbf {y}) - g (\mathbf {x}) - D (\mathbf {x}, \mathbf {z}; \eta)), \tag {7} +$$ + +where $D(\mathbf{x}, \mathbf{z}; \eta)$ measures the distance between $\mathbf{x}$ and $\mathbf{z}$ , and $\eta > 0$ is a parameter that controls the strength of regularization. While prior works [8, 51, 52] consider $D(\mathbf{x}, \mathbf{z}; \eta) = \frac{\|\mathbf{x} - \mathbf{z}\|_2^2}{2\eta^2}$ , the potential function can be generalized to any continuous function $D$ , such that $D(\mathbf{x}, \mathbf{z}; \eta) \to \infty$ as $\eta \to 0$ for any $\mathbf{x} \neq \mathbf{z}$ . As shown in Appendix A.3, this ensures that both marginal distributions, + +$$ +\pi^ {X} (\mathbf {x}; \eta) := \int \pi (\mathbf {x}, \mathbf {z}; \eta) d \mathbf {z}, \text {a n d} \pi^ {Z} (\mathbf {z}; \eta) := \int \pi (\mathbf {x}, \mathbf {z}; \eta) d \mathbf {x} +$$ + +converges to the posterior distribution $p(\mathbf{x}|\mathbf{y})$ . The decoupling of the prior and likelihood in Eq. (7) enables split Gibbs sampling, which alternates between two steps: + +# 1. Likelihood sampling step: + +$$ +\mathbf {z} ^ {(k)} \sim \pi (\mathbf {x} = \mathbf {x} ^ {(k)}, \mathbf {z}; \eta) \propto \exp (- f (\mathbf {z}; \mathbf {y}) - D (\mathbf {x} ^ {(k)}, \mathbf {z}; \eta)), \tag {8} +$$ + +# 2. Prior sampling step: + +$$ +\mathbf {x} ^ {(k + 1)} \sim \pi (\mathbf {x}, \mathbf {z} = \mathbf {z} ^ {(k)}; \eta) \propto \exp (- g (\mathbf {x}) - D (\mathbf {x}, \mathbf {z} ^ {(k)}; \eta)). \tag {9} +$$ + +A key feature of split Gibbs sampling is that it does not rely on knowing gradients of the guidance term $f(\mathbf{z};\mathbf{y})$ , which is highly desirable in our setting with discrete data. In this way, split Gibbs is arguably the most natural and simplest framework for developing principled posterior sampling algorithms for discrete diffusion models. + +Prior work has studied split Gibbs posterior sampling for continuous diffusion models [8, 51, 52]. These methods specify the regularization potential as $D(\mathbf{x}, \mathbf{z}; \eta) = \frac{\|\mathbf{x} - \mathbf{z}\|_2^2}{2\eta^2}$ , which transforms Eq. (9) into a Gaussian denoising problem solvable by a diffusion model. A key challenge, which we tackle, is developing a performant approach for the discrete diffusion setting that is easy to implement, and enjoys rigorous guarantees (i.e., sampling from the correct posterior distribution). + +# 3.2 Prior Step with Discrete Diffusion Models + +Suppose $p(\mathbf{x})$ is a discrete-state distribution over $\mathcal{X}^D$ modeled by a diffusion process. We consider a discrete diffusion model with uniform transition kernel $Q_{t}^{\mathrm{fw}} = \frac{1}{N}\mathbf{11}^{T} - \mathbf{I}$ . To connect split Gibbs sampling to discrete diffusion models, we specify the potential function $D(\mathbf{x},\mathbf{z};\eta)$ as: + +$$ +D (\mathbf {x}, \mathbf {z}; \eta) := d (\mathbf {x}, \mathbf {z}) \log \frac {1 + (N - 1) e ^ {- \eta}}{(N - 1) (1 - e ^ {- \eta})} \tag {10} +$$ + +where $d(\mathbf{x},\mathbf{z})$ denotes the Hamming distance between $\mathbf{x}$ and $\mathbf{z}$ . When $\eta \to 0^{+}$ , the regularization potential $D(\mathbf{x},\mathbf{z};\eta)$ goes to infinity unless $d(\mathbf{x},\mathbf{z}) = 0$ , ensuring the convergence of marginal distributions to $p(\mathbf{x}|\mathbf{y})$ . Given Eq. (10), the prior step from Eq. (9) can be written as + +$$ +\mathbf {x} ^ {(k + 1)} \sim \pi (\mathbf {x}, \mathbf {z} = \mathbf {z} ^ {(k)}; \eta) \propto p _ {0} (\mathbf {x}) \left(\tilde {\beta} / (1 - \tilde {\beta})\right) ^ {d (\mathbf {z} ^ {(k)}, \mathbf {x})} \tag {11} +$$ + +Algorithm 1 Split Gibbs Discrete Diffusion Posterior Sampling (SGDD) +Require:Concrete score model $\mathbf{s}_{\theta}$ , measurement y, noise schedule $\{\eta_k\}_{k = 0}^{K - 1}$ Initialize $\mathbf{x}_0\in \mathcal{X}^d$ +for $k = 0,\dots ,K - 1$ do Likelihood step following Eq. (13): $\mathbf{z}^{(k)}\sim \pi (\mathbf{x} = \mathbf{x}^{(k)},\mathbf{z};\eta_k)$ Prior step following Eq. (11): $\mathbf{x}^{(k + 1)}\sim \pi (\mathbf{x},\mathbf{z} = \mathbf{z}^{(k)};\eta_k)$ +end for +Return $\mathbf{x}^{(K)}$ + +where $\tilde{\beta} = \frac{N - 1}{N} (1 - e^{-\eta})$ . On the other hand, the distribution of clean data $\mathbf{x}_0$ conditioned on $\mathbf{x}_t$ is given by: + +$$ +p \left(\mathbf {x} _ {0} \mid \mathbf {x} _ {t}\right) \propto p _ {0} \left(\mathbf {x} _ {0}\right) p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {0}\right) \propto p _ {0} \left(\mathbf {x} _ {0}\right) \beta_ {t} ^ {d \left(\mathbf {x} _ {t}, \mathbf {x} _ {0}\right)} \left(1 - \beta_ {t}\right) ^ {D - d \left(\mathbf {x} _ {t}, \mathbf {x} _ {0}\right)} \tag {12} +$$ + +where $\beta_{t} = \frac{N - 1}{N} (1 - e^{-\sigma_{t}})$ . Thus, sampling from Eq. (11) is equivalent to unconditional generation from $p(\mathbf{x}_0|\mathbf{x}_t)$ when $\tilde{\beta} = \beta_{t}$ , i.e., $\eta = \sigma_t$ . Therefore, we can solve the prior sampling problem by simulating a partial discrete diffusion sampler starting from $\sigma_{t} = \eta$ and $\mathbf{x}_t = \mathbf{z}^{(k)}$ . + +# 3.3 Likelihood Sampling Step + +With the potential function $D(\mathbf{x}, \mathbf{z}; \eta)$ specified in Section 3.2, at iteration $k$ , the likelihood sampling step from Eq. (8) can be written as: + +$$ +\mathbf {z} ^ {(k)} \sim \pi (\mathbf {x} = \mathbf {x} ^ {(k)}, \mathbf {z}; \eta) \propto \exp \left(- f (\mathbf {z}; \mathbf {y}) - d (\mathbf {x}, \mathbf {z}) \log \frac {1 + (N - 1) e ^ {- \eta}}{(N - 1) (1 - e ^ {- \eta})}\right). \tag {13} +$$ + +Since the unnormalized probability density function of $\pi (\mathbf{x} = \mathbf{x}^{(k)},\mathbf{z};\eta)$ is accessible, we can efficiently sample from Eq. (13) using Markov Chain Monte Carlo methods. + +# 3.4 Overall Algorithm + +We now summarize the complete SGDD algorithm. Like in Wu et al. [51], our algorithm alternates between likelihood steps and prior steps while employing an annealing schedule $\{\eta_k\}$ , which starts at a large $\eta_0$ and gradually decays to $\eta_K \to 0$ . This annealing scheme accelerates the mixing time of the Markov chain, which we further provide some theoretical intuition in Appendix B.1; and ensures the convergence of $\pi^X(\mathbf{x}; \eta)$ and $\pi^Z(\mathbf{z}; \eta)$ to $p(\mathbf{x}|\mathbf{y})$ as $\eta \to 0$ . We present the complete pseudocode of our method in Algorithm 1 and further explain the choice of the noise schedule in Appendix B.1. + +# 3.5 Convergence of Split Gibbs Diffusion Posterior Sampling + +We provide theoretical guarantees on the convergence of SGDD. For two probability measures in a finite $\mathcal{X}$ , we consider the Kullback-Leibler (KL) divergence and the relative Fisher information (a.k.a. Fisher divergence), respectively, as (where we define $f\coloneqq \mu /\pi$ ) + +$$ +\mathsf {K L} (\mu \| \pi) := \mathbb {E} _ {\mathbf {x} \sim \mu} \left[ \ln \frac {\mu}{\pi} (\mathbf {x}) \right], \mathsf {F l} _ {Q} (\mu \| \pi) := \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j} \in \mathcal {X}} \pi (\mathbf {x} _ {i}) Q ^ {[ j, i ]} \left(f (\mathbf {x} _ {j}) - f (\mathbf {x} _ {i}) - f (\mathbf {x} _ {i}) \log \frac {f (\mathbf {x} _ {j})}{f (\mathbf {x} _ {i})}\right). +$$ + +Both divergences are nonnegative and equal to zero if and only if $\mu = \pi$ , when $Q$ is irreducible. Fisher information has been widely used to analyze the convergence of sampling algorithms in continuous spaces due to its inherent connection to KL divergence [2, 43, 51]. In the continuous case, the Fisher information can be written as a quadratic form $\mathsf{Fl}(\mu \| \pi) = \mathbb{E}_{\mu}\| \nabla \log (\mu /\pi)\| ^2$ . However, it does not have an analogous form in finite spaces, which poses additional challenges to the analysis of SGDD. To address this challenge, we adopt a generalized definition of Fisher information [3, 17] and analyze the convergence of SGDD to the stationary distribution using a more general technique that encompasses both continuous and discrete settings. + +We define a distribution $\mu_t$ over $\mathcal{X}$ that evolves according to likelihood steps and prior steps alternatively. We assume each likelihood step is implemented with the Metropolis-Hastings algorithm, and that each prior step is solved by the Euler method with an approximated score function. We compare $\mu_t$ to the continuous-time stationary distribution $\pi_t$ , which alternates between $\pi^X$ and $\pi^Z$ . + +Theorem 1. Consider running $K$ iterations of SGDD with a fixed $\eta > 0$ and an estimated concrete score $s_{\theta}(\mathbf{x};t)$ , and suppose that each prior step is solved by an $H$ step Euler method. Let $t^{*} > 0$ with $\sigma(t^{*}) = \eta$ . Define $\pi_t$ and $\mu_t$ as stationary and non-stationary distributions. Over $K$ iterations of SGDD, the average relative Fisher information between $\mu_t$ and $\pi_t$ has + +$$ +\frac {1}{K} \sum_ {k = 0} ^ {K - 1} \underbrace {\frac {1}{t ^ {*}} \int_ {k (t ^ {*} + 1) + 1} ^ {(k + 1) (t ^ {*} + 1)} \mathsf {F I} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t} \| \mu_ {t}\right) \mathrm {d} t} _ {\text {a v e r a g e F i s h e r d i v e g e n c e i n t h e k - t h p r i o r s t e p}} \leqslant \underbrace {\frac {2 \mathrm {K L} \left(\pi_ {0} \| \mu_ {0}\right)}{K t ^ {*}}} _ {\text {c o n v e r g e n c e b y O (1 / K)}} + \underbrace {\frac {4 M \epsilon}{c}} _ {\text {s c o r e e r r o r}} + \underbrace {\frac {2 M L t ^ {*}}{c H}} _ {\text {d i s c r e t i z a t i o n e r r o r}}. \tag {14} +$$ + +where $\left\| \frac{s_{\theta}(\cdot;t) - s(\cdot;t)}{s(\cdot;t)} \right\|_{\infty} \leqslant \epsilon \leqslant 1$ , and $L, M, c$ are positive constants defined in Appendix A. + +We include the proof in Appendix A. Theorem 1 states that the average Fisher divergence of the non-stationary process to the stationary process in all prior steps converges at the rate of $O(1 / K)$ , up to a constant error term. This result extends our theoretical understanding of diffusion posterior sampling by generalizing the analysis [43, 51] on SDEs to general Markov processes using the free-energy-rate-functional-relative-Fisher-information (FIR) inequality [16]. Moreover, compared to existing analyses for continuous diffusion models, our analysis accounts not only for the imperfect score function but also for the discretization error in solving CTMCs as well. + +Unlike some previous methods [27, 50] that rely on assumptions of surrogate value functions and an infinite number of particles, SGDD does not rely on such approximations and eventually converges to the posterior distribution with robustness to score approximation and discretization error. Consequently, SGDD can be viewed as a natural and principled approach for posterior sampling using discrete diffusion models, leading to empirical benefits as we demonstrate in Section 4. + +# 4 Experiments + +We conduct experiments to demonstrate the effectiveness of our algorithm on various posterior sampling tasks in discrete-state spaces, including discrete inverse problems and conditional generation guided by a reward function. + +# 4.1 Experimental Setup + +We evaluate our method across multiple domains, including synthetic data, DNA sequences, discrete image prior, and monophonic music, as listed in Table 1. We define the likelihood function for all inverse problems as $p(\mathbf{y}|\mathbf{x}) \propto \exp(-\| G(\mathbf{x}) - \mathbf{y} \| / \sigma_{\mathbf{y}})$ , while the likelihood functions for reward guiding tasks are $p(\mathbf{y}|\mathbf{x}) \propto \exp(\beta r(\mathbf{x}))$ . + +Pretrained models. For synthetic data, we use a closed-form concrete score function. For real datasets, we train a SEDD [29] model with the uniform transition kernel (details in Appendix C). + +Baselines. We compare our methods to existing approaches for discrete diffusion posterior sampling: DPS [7], SVDD-PM [27], and SMC [50]. The original version of DPS relies on gradient backpropagation through both the likelihood function and the pretrained diffusion model, making it unsuitable for discrete-domain posterior sampling. In our experiments, we consider its discrete analogy by replacing the gradient with a guidance rate matrix. SVDD-PM is originally designed to sample from a reward-guided distribution $p^{\beta}(\mathbf{x}) \propto p(\mathbf{x}) \exp (\beta r(\mathbf{x}))$ , but it is also adaptable to solving inverse problems. More implementation details are elaborated in Appendix C.2. + +SGDD implementation details. We implement our method with a total of $K$ iterations. In each prior sampling step, we simulate the reverse continuous-time Markov chain with $H = 20$ steps. Additional details on hyperparameter choices are provided in Appendix C.3. + +Table 1: A summary of posterior sampling tasks considered in our experiments. + +
DomainSyntheticDNA enhancersImagesMonophonic music
Task typeInverse problemReward guidanceInverse problemInverse problem
Dimensionality5010420021024129256
+ +Table 2: Quantitative evaluation of posterior sampling on synthetic dataset for sequence lengths of $D = 2,5$ and 10. The Hellinger distance and total variation are computed between the empirical distribution of 10k samples and the true posterior distribution on the first two dimensions. + +
D=2D=5D=10
Hellinger ↓TV ↓Hellinger ↓TV ↓Hellinger ↓TV ↓
SVDD-PM (M=20)0.2970.3200.4130.4580.4550.503
SVDD-PM (M=100)0.2750.2920.4010.4450.4480.494
SMC0.2380.2480.3530.3910.4120.460
DPS0.1820.1760.3450.3830.4100.453
SGDD0.1490.1250.2140.2220.3340.365
+ +# 4.2 Controlled Study using Synthetic Data + +To evaluate the accuracy of SGDD in posterior sampling, we first conduct a case study on a synthetic task where the true posterior distribution is accessible. Let $\mathbf{x} \in \mathcal{X}^D$ follow a prior distribution $p(\mathbf{x})$ , obtained by discretizing a Gaussian distribution $\mathcal{N}(0, \sigma^2 I_D)$ over an $D$ -dimensional grid. We define a forward model by $\boldsymbol{G}(\mathbf{x}) = \| f(\mathbf{x})\|_1$ where $f(\mathbf{x})$ maps $\mathbf{x}$ to $\mathbb{R}^D$ . + +We draw 10k independent samples with each algorithm and compute the frequency map for the first two dimensions of $\mathbf{x}$ . We measure the Hellinger distance and total variation distance between the empirical distribution and the ground truth posterior distribution. As shown in Table 2, our method achieves substantially more accurate posterior sampling compared to baseline methods. We include visualization of the generated samples in Appendix D.3 (Fig. 8) by projecting samples onto the first two dimensions and computing their heatmaps. When the number of dimensions is small, e.g., $D = 2$ , all methods approximate the true posterior distribution well. However, as the dimensionality increases, DPS, SMC, and SVDD-PM struggle to capture the posterior and tend to degenerate to the prior distribution. While SGDD consistently outperforms baseline methods in terms of Hellinger distance and total variation distance, it also shows a slight deviation from the posterior distribution as the dimensionality increases. + +# 4.3 Guided DNAs Generation + +We evaluate our method for generating regulatory DNA sequences that drive gene expression in specific cell types, which is an important task for cell and gene therapy and commonly studied in the literature [27, 44, 48]. + +We train a discrete diffusion model [29] on a publicly available dataset from Gosai et al. [15] that consists of $\sim 700\mathrm{k}$ DNA sequences. The expression driven by each sequence in human cell lines is measured by massively parallel reporter assays (MPRAs). Following Wang et al. [48], we split the dataset into two subsets and train surrogate reward oracles to predict the activity level in the HepG2 cell line. We use the oracle trained on the training set, $r(\cdot)$ , to guide our diffusion model, and evaluate its performance using the other one trained on the test dataset. The likelihood function is therefore defined as $p(\mathbf{y}|\mathbf{x})\propto \exp (\beta r(\mathbf{x}))$ + +Evaluation metrics. We evaluate the quality of generated samples using the following metrics: + +- Predicted activity (Pred-Activity). We evaluate the activity level of generated sequences using the oracle trained on the test dataset. This corresponds to the reward function that we aim to optimize. +- Binary classification on chromatin accessibility (ATAC-Acc). Following [48], we use a binary classifier trained on chromatin accessibility data in the HepG2 cell line. As active enhancers should have accessible chromatin, ATAC-Acc can be used to validate the generated sequences. +- Log-likelihood. We estimate the log-likelihood of the generated samples using the ELBO of a pretrained masked diffusion model [48]. This regulates the generated samples to stay close to the prior distribution of natural DNA sequences. + +We include more baseline methods that require additional training for this task, including a discrete diffusion fine-tuning method, DRAKES [48], classifier guidance (CG) for discrete diffusion models [32], and twisted diffusion sampler (TDS) [50] which uses CG as the proposal function for sequential Monte Carlo sampling. We further include a Feynman-Kac diffusion sampler [33] with MAX potential function as well. + +Results. We generate 640 DNA samples using different algorithms and compute the median and average Pred-Activity of the generated sequences. As shown in Table 3, our method achieves the + +Table 3: Quantitative results for guided DNA (enhancers) sampling. We generate 640 independent samples with different algorithms according to the reward-shaped posterior distribution. We report the mean and median of the enhancer's activity computed by the surrogate reward. + +
Pred-Activity (median) ↑Pred-Activity (average) ↑ATAC Acc % ↑Log-likelihood (average) ↑
SVDD-PM5.415.08±1.5149.9-241±47
SMC4.153.99±1.4939.9-259±13
CG2.902.76±0.940.0-265±5
TDS4.644.62±1.0945.3-257±9
FK-MAX4.904.79±1.0247.1-243±9
DRAKES w/o KL6.446.28±0.7682.5-281±6
DRAKES w/ KL5.615.24±1.3992.5-264±9
SGDD (β = 30)8.828.69±1.1291.7-246±41
SGDD (β = 50)9.148.96±1.1793.0-261±29
+ +![](images/2aeccac6c146180dfcbe29172ceceba31cf41dc649bf4eb639faa451662e00fc.jpg) +Figure 2: Generating DNA sequences involves trade-offs between targeted objectives and consistency with the prior distribution. SGDD is plotted in blue and darker colors mean higher $\beta$ . + +![](images/f3692ef64cb41a0ed0ab853320ab72044b9b218aee657caffaf5dab401b5c575.jpg) +Figure 3: Diversified samples when the measurement $y$ is sparse. Samples are generated by SGDD when solving an MNIST inpainting task. + +highest Pred-Activity (reward values) compared to existing baselines, including $42\%$ higher median reward compared to the previous state-of-the-art method. + +Moreover, as SGDD samples from the posterior distribution, $p(\mathbf{x})\exp (\beta r(\mathbf{x})) / Z$ , the guidance strength $\beta$ controls the strength of guidance imposed in the sampling algorithm. We instantiate SGDD with $\beta$ ranging from 3 to 100 and calculate the Pred-Activity and Log-likelihood of the generated samples. We plot the distribution of the samples' Pred-Activity with various $\beta$ in Fig. 2. In general, using a larger $\beta$ results in higher Pred-Activity (reward), at the expense of a lower likelihood as the target distribution deviates further from the prior distribution. + +# 4.4 Discrete Image Inverse Problems + +Setup. We evaluate on a discretized image domain. Specifically, we convert the MNIST dataset [25] to binary strings by discretizing the images, and train a discrete diffusion prior on 60k training data. As examples of linear and nonlinear forward models on binary strings, we consider AND and XOR operators. We randomly pick $\gamma D$ pairs of positions $(i_p,j_p)$ over $\{1,\dots ,D\}$ , and compute: + +$$ +\boldsymbol {G} _ {\text {A N D}} (\mathbf {x}) = \left[ \mathbf {x} _ {i _ {p}} \wedge \mathbf {x} _ {j _ {p}} \right] _ {p = 1, \dots , \gamma D}, \quad \boldsymbol {G} _ {\text {X O R}} (\mathbf {x}) = \left[ \mathbf {x} _ {i _ {p}} \oplus \mathbf {x} _ {j _ {p}} \right] _ {p = 1, \dots , \gamma D}. \tag {15} +$$ + +These logical operators are challenging to approximate using surrogate functions to guide sampling. + +Evaluation metrics. We use 1k binary images from the test set of MNIST and calculate the peak signal-to-noise ratio (PSNR) of the reconstructed image. Furthermore, we train a simple convolutional neural network on MNIST as a surrogate and report the classifier accuracy of the generated samples. + +Results. As shown in Table 4, SGDD outperforms baseline methods by a large margin in both XOR and AND tasks. We present the samples generated by SGDD for the XOR task in Appendix D.2. The reconstructed samples are visually consistent with the underlying ground truth signal, which explains the high class accuracy of our method shown in Table 4. + +Sample diversity. Furthermore, we demonstrate that SGDD generates diversified samples when the measurement $y$ is sparse. For example, when a digit is masked with a large box, as shown + +Table 4: Quantitative results for XOR and AND problems on discretized MNIST. We report the mean and standard deviation of PSNR and class accuracy across 1k generated samples. SGDD demonstrates superior performance on both tasks. + +
XORAND
PSNR ↑Accuracy (%) ↑PSNR ↑Accuracy (%) ↑
SVDD-PM (M=20)11.81±2.5451.410.04±1.4933.7
SMC10.05±1.5427.810.25±1.6324.4
DPS9.04±1.2130.08.67±0.9124.5
SGDD20.17±3.4791.217.25±3.8279.4
+ +Table 5: Quantitative evaluation of infilling monophonic music sequences. + +
γ = 40%γ = 60%
Hellinger ↓Outliers ↓Hellinger ↓Outliers ↓
SVDD-PM0.465±0.1290.112±0.0980.529±0.1370.114±0.119
SMC0.492±0.1430.152±0.1380.552±0.1310.164±0.140
SGDD0.126±0.0980.003±0.0020.244±0.1640.060±0.138
+ +in Fig. 3, the measurement lacks sufficient information to fully recover the original digit. In this scenario, SGDD generate samples from multiple plausible modes, including digits 1,4,7 and 9. This highlights the ability of our method to produce diverse samples from the posterior distribution while preserving consistency with the measurement. + +Higher dimensional images. To demonstrate the scalability of SGDD, we further evaluate SGDD on a higher-dimensional dataset, FFHQ 256 [21] in Appendix D.1, where we solve the super-resolution task by operating in the discrete latent space of a pretrained vector-quantized diffusion model. + +# 4.5 Monophonic Music + +We conduct experiments on monophonic songs from the Lakh pianoroll dataset [11]. Following the preprocessing steps in [4], we obtain monophonic musical sequences of length $D = 256$ and vocabulary size $N = 129$ . The ordering of musical notes is scrambled to destroy any ordinal structure. Therefore, the likelihood function for this task is $p(\mathbf{y}|\mathbf{x}) \propto \exp(-\| G(\mathbf{x}) - \mathbf{y}\|_0 / \sigma_{\mathbf{y}})$ . This $\ell_0$ norm makes this inverse problem ill-posed, in stark contrast to most reward-guided generation tasks, where the reward functions are much smoother. We evaluate our method on an inpainting task, where the forward model $G(\cdot)$ randomly masks $\gamma = 40\%$ , $60\%$ of the notes. As in Campbell et al. [4], we use the Hellinger distance of histograms and the proportion of outlier notes as metrics to evaluate our method. We run experiments on 100 samples in the test set and report the quantitative results in Table 5. SGDD successfully completes monophonic music sequences in a style more consistent with the given conditions, achieving a 2x reduction in Hellinger distance compared to SVDD-PM. + +# 4.6 More Ablation Studies + +We conduct more ablation studies on our method to investigate how specific design choices may affect its performance. We discuss the effectiveness of using an annealing noise schedule $\eta_{k}$ over a fixed noise $\eta_{k} \equiv \eta$ in Appendix B.1, including empirical discovery and theoretical justification. We include runtime analysis of SGDD in Appendix B.2, showing the tradeoff between sample quality and computing budgets regarding NFEs. Finally, we highlight the value of using an informative discrete diffusion prior in solving inverse problems by comparing SGDD to MCMC methods without a prior (Appendix B.3). + +# 5 Conclusion + +We introduce SGDD, a principled discrete diffusion posterior sampling algorithm based on the split Gibbs sampler, extending plug-and-play diffusion methods from Euclidean spaces to discrete-state spaces. Our algorithm alternates between prior and likelihood sampling steps with decreasing regularization level $\eta_{k}$ . By carefully designing the regularization potential function, we establish a connection between prior sampling steps and discrete diffusion samplers, integrating discrete diffusion models into the split Gibbs sampling framework. We prove the convergence of SGDD to the targeted distribution. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: We only used LLMs for editing and grammar checks. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Theory + +Notations. We consider a linear stochastic process whose forward Kolmogorov equation can be written as $\partial_t\pi_t = Q_t\pi_t$ with boundary condition $\pi_{t=0} = \pi_0$ , where $\pi_t \sim \mathcal{P}(\mathcal{X})$ . For the simplicity of notation, we let $Q_t$ to generate reverse continuous-time Markov chain, which means $Q_t^{[ij]} = s(\mathbf{x}_j; T - t)_{\mathbf{x}_i} Q_{\mathrm{uniform}}^{[ji]}$ ; and time $t$ flows with the sampling algorithm, which means $\pi_0$ is a uniform distribution and $\pi_T$ is the data distribution. + +For two probability mass functions $\mu$ and $\pi$ , we discuss their KL divergence, + +$$ +\mathsf {K L} (\mu \| \pi) := \mathbb {E} _ {\mathbf {x} \sim \mu} \left[ \log \frac {\mu}{\pi} (\mathbf {x}) \right]. \tag {16} +$$ + +We define the relative Fisher information between $\mu$ and $\pi$ as + +$$ +\operatorname {F l} _ {\boldsymbol {Q}} (\mu \| \pi) := \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j} \in \mathcal {X}} \pi (\mathbf {x} _ {i}) \boldsymbol {Q} ^ {[ j, i ]} \left(f (\mathbf {x} _ {j}) - f (\mathbf {x} _ {i}) - f (\mathbf {x} _ {i}) \log \frac {f (\mathbf {x} _ {j})}{f (\mathbf {x} _ {i})}\right), f := \mu / \pi . \tag {17} +$$ + +Note that when $Q$ is irreducible, the relative Fisher information $\mathsf{F}\mathsf{I}_Q(\mu \| \pi)\geqslant 0$ , and $\mathsf{F}\mathsf{I}_Q(\mu \| \pi) = 0$ if and only if $\mu = \pi$ . + +In continuous state spaces, the relative Fisher information criterion has been used to derive general first-order guarantees for non-log-concave sampling [2]. This line of analysis has been adapted to provide theoretical insights for posterior sampling methods using diffusion models [43, 51]. The formula in Eq. (17) represents a discrete state space analogue of Fisher information. We refer interested readers to [3, 10, 17] for more discussions on this topic and the relation between KL and Fisher information in the discrete state space. Our analysis in this paper adapts these techniques and provides general first-order guarantees for posterior sampling with discrete diffusion models. + +Time interpolation of SGDD. SGDD alternates between likelihood steps and prior steps. Let $t^* > 0$ such that $\sigma_{t^*} = \eta$ . We define $\{\pi_t\}$ as the distributions at time $t$ of the stationary process, and $\{\mu_t\}$ as the distributions of the non-stationary process. + +- In time intervals $\tau \in [k(t^{*} + 1) + 1, (k + 1)(t^{*} + 1)]$ . The stationary distribution is initialized with $\pi_{k(t^{*} + 1) + 1}(\mathbf{x}) = \pi_{\eta}^{Z}(\mathbf{x})$ . We run a prior step on $\mu_{\tau}$ with the learned concrete score function for $H$ steps, while $\pi_{\tau}$ evolves in continuous time with the true concrete score function. +- In time intervals $\tau \in [k(t^{*} + 1), k(t^{*} + 1) + 1]$ , we run a Metropolis-Hastings sampling algorithm on both $\pi_{\tau}$ and $\mu_{\tau}$ . + +Assumptions. Our analysis relies on the following assumptions: + +(i) Concrete score is well estimated: $\left\| \frac{s_{\theta}(\cdot;t) - s(\cdot;t)}{s(\cdot;t)}\right\|_{\infty}\leqslant \epsilon < 1.$ +(ii) Smoothness of score function in $t$ : $\| \frac{s(\cdot ;t + \Delta t) - s(\cdot ;t)}{s(\cdot ;t)}\|_{\infty}\leqslant L\cdot \Delta t$ +(iii) Strong irreducibility: $Q_{t}^{[i,j]} > 0$ for $i\neq j$ +(iv) Bounded probability ratio: $\sup_t\| \log \frac{\mu_t(\mathbf{x})}{\pi_t(\mathbf{x})}\|_\infty \leqslant B.$ +(v) The entry-wise absolute of the reverse-time transition matrix is bounded: $\sup_t\| |Q_t\| _1\leqslant M$ + +# A.1 Lemmas + +Lemma 1 (Data processing inequality of Metropolis Hasting). Running Metropolis Hasting on two distributions $\pi_{\tau}$ and $\mu_{\tau}$ does not increase their KL divergence, i.e., + +$$ +\mathsf {K L} \left(\pi_ {k (t * + 1)} \| \mu_ {k (t * + 1)}\right) \geqslant \mathsf {K L} \left(\pi_ {k (t * + 1) + 1} \| \mu_ {k (t * + 1) + 1}\right). \tag {18} +$$ + +Lemma 2 (Free-energy-rate-functional-relative-Fisher-information (FIR) inequality (from Theorem 6.2.3. in [16])). Consider two continuous time Markov chains: $\partial_t\pi_t = Q_t\pi_t$ and $\partial_t\mu_t = \tilde{Q}_t\mu_t$ . Suppose Assumption (iv) holds, then there exists a constant $c > 0$ , such that + +$$ +\partial_ {t} \mathrm {K L} \left(\pi_ {t} \| \mu_ {t}\right) \leqslant - \frac {1}{2} \mathrm {F I} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t} \| \mu_ {t}\right) + \frac {2}{c} \mathcal {L} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t}, \boldsymbol {Q} _ {t} \pi_ {t}\right), \tag {19} +$$ + +where $\mathcal{L}_{\tilde{\mathbf{Q}}}(\pi_t,\partial_t\pi_t)\geqslant 0$ is the Lagrangian defined as + +$$ +\mathcal {L} _ {\tilde {Q}} \left(\pi_ {t}, \partial_ {t} \pi_ {t}\right) = \sup _ {\varphi \in \mathcal {C} _ {b} (\mathcal {X})} \left[ \langle \varphi , \partial_ {t} \pi_ {t} \rangle - \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j} \in \mathcal {X}} \pi_ {t} \left(\mathbf {x} _ {i}\right) \tilde {Q} ^ {[ j, i ]} \exp \left(\varphi \left(\mathbf {x} _ {j}\right) - \varphi \left(\mathbf {x} _ {i}\right)\right) \right]. \tag {20} +$$ + +Proof Sketch. This lemma follows from Theorem 6.2.3. in Hilder et al. [16]. For completeness, we provide a sketch of the proof here. + +By a direct calculation of derivatives, we have + +$$ +\begin{array}{l} \partial_ {t} \mathrm {K L} (\pi_ {t} \| \mu_ {t}) = \partial_ {t} \sum_ {\mathbf {x} _ {i}} \pi_ {t} (\mathbf {x} _ {i}) \log \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} = \sum_ {\mathbf {x} _ {i}} \left(\partial_ {t} \pi_ {t} (\mathbf {x} _ {i}) \log \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} - \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} \partial_ {t} \mu_ {t} (\mathbf {x} _ {i})\right) \\ = \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j}} \left(Q _ {t} ^ {[ i, j ]} \pi_ {t} (\mathbf {x} _ {j}) \log \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} - \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} \tilde {Q} _ {t} ^ {[ i, j ]} \mu_ {t} (\mathbf {x} _ {j})\right) \\ = \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j}} \tilde {Q} _ {t} ^ {[ i, j ]} \mu_ {t} (\mathbf {x} _ {j}) \left(\frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} \log \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} - \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})}\right) - \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j}} (\tilde {Q} _ {t} ^ {[ i, j ]} - Q _ {t} ^ {[ i, j ]}) \pi_ {t} (\mathbf {x} _ {j}) \log \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} \\ \end{array} +$$ + +Using the fact that $\sum_{\mathbf{x}_i} \tilde{\mathbf{Q}}_t^{[i,j]} = 0$ and the definition in Eq. (17), we have the equality + +$$ +\sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j}} \tilde {\boldsymbol {Q}} _ {t} ^ {[ i, j ]} \mu_ {t} (\mathbf {x} _ {j}) \left(\frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} \log \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})} - \frac {\pi_ {t} (\mathbf {x} _ {i})}{\mu_ {t} (\mathbf {x} _ {i})}\right) = - \mathrm {F I} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t} \| \mu_ {t}\right) +$$ + +Using the relation $\partial_t\pi_t = Q_t\pi_t$ this leads to + +$$ +\partial_ {t} \mathrm {K L} \left(\pi_ {t} \| \mu_ {t}\right) = - \operatorname {F I} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t} \| \mu_ {t}\right) - \underbrace {\left(\log \frac {\pi_ {t}}{\mu_ {t}}\right) ^ {T} \left(\tilde {\boldsymbol {Q}} _ {t} - \partial_ {t}\right) \pi_ {t}} _ {\text {e r r o r t e r m}}. \tag {21} +$$ + +This formula is similar to Lemma 1 in Yau [53]. + +We define + +$$ +\begin{array}{l} \mathcal {L} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t}, \partial_ {t} \pi_ {t}\right) = \sup _ {\varphi \in \mathcal {C} _ {b} (\mathcal {X})} \left[ \left\langle \varphi , \partial_ {t} \pi_ {t} \right\rangle - \left(e ^ {- \varphi} \pi_ {t}\right) ^ {T} \tilde {\boldsymbol {Q}} _ {t} ^ {T} e ^ {\varphi} \right] \\ = \sup _ {\varphi \in \mathcal {C} _ {b} (\mathcal {X})} \left[ \left\langle \varphi , \partial_ {t} \pi_ {t} - \tilde {\boldsymbol {Q}} _ {t} \pi_ {t} \right\rangle - \underbrace {\pi_ {t} ^ {T} \left(e ^ {- \varphi} \tilde {\boldsymbol {Q}} _ {t} ^ {T} e ^ {\varphi} - \tilde {\boldsymbol {Q}} _ {t} ^ {T} \varphi\right)} _ {\text {d e n o t e d a s} \tilde {\mathcal {H}} (\pi_ {t}, \varphi)} \right] \tag {22} \\ \end{array} +$$ + +By the variational characterization of the Lagrangian, for any continuous, bounded $\varphi$ + +$$ +\langle \varphi , \partial_ {t} \pi_ {t} - \tilde {\mathbf {Q}} _ {t} \pi_ {t} \rangle \leqslant \mathcal {L} _ {\tilde {\mathbf {Q}} _ {t}} \left(\pi_ {t}, \partial_ {t} \pi_ {t}\right) + \tilde {\mathcal {H}} \left(\pi_ {t}, \varphi\right). \tag {23} +$$ + +If we choose $\varphi = \log \frac{\pi_t}{\mu_t}$ , the inequality above gives a bound to the error term in Eq. (21). Moreover, it is easy to verify that + +$$ +\tilde {\mathcal {H}} \left(\pi_ {t}, \log \frac {\pi_ {t}}{\mu_ {t}}\right) = \mathrm {F I} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t} \| \mu_ {t}\right). \tag {24} +$$ + +In fact, when the space $\mathcal{X}$ is continuous, choosing $\varphi (\mathbf{x}) = \log \frac{\pi_t}{\mu_t} (\mathbf{x})$ exactly recovers Lemma A.4 in [51]. However, as pointed out in [16], it is necessary to consider a rescaled $\varphi = \lambda \log \frac{\pi_t}{\mu_t}$ with $\lambda \in (0,1)$ to derive a bound in finite space $\mathcal{X}$ . + +Fortunately, according to Lemma 6.2.2. in [16], for any $\varphi \in \mathcal{C}_b(\mathcal{X})$ with $\| \varphi \|_{\infty}\leqslant B$ , there exists a positive constant $c = c(B)\in (0,1)$ , such that + +$$ +\tilde {\mathcal {H}} \left(\pi_ {t}, \lambda \varphi\right) \leqslant \frac {\lambda^ {2}}{c} \tilde {\mathcal {H}} \left(\pi_ {t}, \varphi\right). \tag {25} +$$ + +Plugging in $\varphi = \lambda \log \frac{\pi_t}{\mu_t}$ in Eq. (23), we have + +$$ +\begin{array}{l} \left\langle \lambda \log \frac {\pi_ {t}}{\mu_ {t}}, \partial_ {t} \pi_ {t} - \tilde {\boldsymbol {Q}} _ {t} \pi_ {t} \right\rangle \leqslant \mathcal {L} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t}, \partial_ {t} \pi_ {t}\right) + \tilde {\mathcal {H}} \left(\pi_ {t}, \lambda \log \frac {\pi_ {t}}{\mu_ {t}}\right) (26) \\ \leqslant \mathcal {L} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t}, \partial_ {t} \pi_ {t}\right) + \frac {\lambda^ {2}}{c} \tilde {\mathcal {H}} \left(\pi_ {t}, \log \frac {\pi_ {t}}{\mu_ {t}}\right) (27) \\ \end{array} +$$ + +Combine this with Eq. (21), + +$$ +\begin{array}{l} \partial_ {t} \mathrm {K L} \left(\pi_ {t} \| \mu_ {t}\right) \leqslant - \mathrm {F I} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t} \| \mu_ {t}\right) + \frac {1}{\lambda} \mathcal {L} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t}, \partial_ {t} \pi_ {t}\right) + \frac {\lambda}{c} \tilde {\mathcal {H}} \left(\pi_ {t}, \log \frac {\pi_ {t}}{\mu_ {t}}\right) \\ = - \left(1 - \frac {\lambda}{c}\right) \mathsf {F I} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t} \| \mu_ {t}\right) + \frac {1}{\lambda} \mathcal {L} _ {\tilde {\boldsymbol {Q}} _ {t}} \left(\pi_ {t}, \partial_ {t} \pi_ {t}\right). \tag {28} \\ \end{array} +$$ + +Finally, choosing $\lambda = \frac{c}{2} \in (0,1)$ recovers the statement of Lemma 2. + +Lemma 3. When the learned score function $s_{\theta}$ satisfies Assumption (i) and (v), + +$$ +\mathcal {L} _ {\boldsymbol {Q} _ {t}} \left(\mu_ {t}, \tilde {\boldsymbol {Q}} _ {t} \mu_ {t}\right) \leqslant M \epsilon . \tag {29} +$$ + +Proof. By definition, + +$$ +\begin{array}{l} \mathcal {L} _ {\boldsymbol {Q} _ {t}} \left(\mu_ {t}, \tilde {\boldsymbol {Q}} _ {t} \mu_ {t}\right) = \sup _ {\varphi \in \mathcal {C} _ {b} (\mathcal {X})} \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j} \in \mathcal {X}} \left[ \mu_ {t} (\mathbf {x} _ {i}) \tilde {\boldsymbol {Q}} _ {t} ^ {[ j, i ]} \varphi (\mathbf {x} _ {j}) - \mu_ {t} (\mathbf {x} _ {i}) \boldsymbol {Q} _ {t} ^ {[ j, i ]} e ^ {\varphi (\mathbf {x} _ {j}) - \varphi (\mathbf {x} _ {i})} \right] \\ = \sup _ {\varphi \in \mathcal {C} _ {b} (\mathcal {X})} \sum_ {\mathbf {x} _ {i}, \mathbf {x} _ {j} \in \mathcal {X}} \left[ \mu_ {t} (\mathbf {x} _ {i}) \tilde {\boldsymbol {Q}} _ {t} ^ {[ j, i ]} (\varphi (\mathbf {x} _ {j}) - \varphi (\mathbf {x} _ {i})) - \mu_ {t} (\mathbf {x} _ {i}) \boldsymbol {Q} _ {t} ^ {[ j, i ]} e ^ {\varphi (\mathbf {x} _ {j}) - \varphi (\mathbf {x} _ {i})} \right] \\ \left(\mathbf {1} ^ {T} \tilde {\boldsymbol {Q}} _ {t} \mathbf {u} = 0 \text {f o r a n y u}\right) \\ = \sup _ {\varphi \in \mathcal {C} _ {b} (\mathcal {X})} \sum_ {\mathbf {x} _ {i} \neq \mathbf {x} _ {j}} \left[ \mu_ {t} (\mathbf {x} _ {i}) Q _ {t} ^ {[ j, i ]} \left(- \frac {\tilde {Q} _ {t} ^ {[ j , i ]}}{Q _ {t} ^ {[ j , i ]}} z _ {i j} - e ^ {- z _ {i j}}\right) \right] + \sum_ {\mathbf {x} _ {i}} \mu_ {t} (\mathbf {x} _ {i}) \left(\tilde {Q} _ {t} ^ {[ i, i ]} - Q _ {t} ^ {[ i, i ]}\right) \\ \left(z _ {i j} = \varphi (\mathbf {x} _ {i}) - \varphi (\mathbf {x} _ {j})\right) \\ \leqslant \sum_ {\mathbf {x} _ {i} \neq \mathbf {x} _ {j}} \left[ \mu_ {t} (\mathbf {x} _ {i}) Q _ {t} ^ {[ j, i ]} \cdot \sup _ {z _ {i j} \in \mathbb {R}} \left(- \frac {\tilde {Q} _ {t} ^ {[ j , i ]}}{Q _ {t} ^ {[ j , i ]}} z _ {i j} - e ^ {- z _ {i j}}\right) \right] + \epsilon \sum_ {\mathbf {x} _ {i}} \mu_ {t} (\mathbf {x} _ {i}) | Q _ {t} ^ {[ i, i ]} |. \tag {30} \\ \end{array} +$$ + +where the inequality is due to swapping sup and summation, and that $\mu_t(\mathbf{x}_i) \mathbf{Q}_t^{[j,i]} \geqslant 0$ for $i \neq j$ . + +Consider the function $g(z) = -uz - e^{-z}$ . When $u \geqslant 0$ , this function is maximized when $z = -\log u$ . Using $u = Q_t^{[j,i]} / \tilde{Q}_t^{[j,i]}$ , + +$$ +\sup _ {z _ {i j}} \left(- \frac {\mathbf {Q} _ {t} ^ {[ j , i ]}}{\tilde {\mathbf {Q}} _ {t} ^ {[ j , i ]}} z _ {i j} - e ^ {- z _ {i j}}\right) = (u - 1) \log u \leqslant | u - 1 | \leqslant \epsilon , \tag {31} +$$ + +for $\epsilon \leqslant 1$ . Combining with Eq. (30), we have + +$$ +\mathcal {L} _ {\boldsymbol {Q} _ {t}} \left(\mu_ {t}, \tilde {\boldsymbol {Q}} _ {t} \mu_ {t}\right) \leqslant \epsilon \mathbf {1} ^ {T} | \boldsymbol {Q} _ {t} | \mu_ {t} \leqslant \epsilon \| | \boldsymbol {Q} _ {t} \| _ {1} \| \mu_ {t} \| _ {1} \leqslant \epsilon M. \tag {32} +$$ + +# Additional Notes on Lemma 3. + +With a slightly different derivation, assumptions (i) and (v) can potentially be replaced by the following assumption that links directly to the score entropy loss defined in Definition 3.1 of Lou et al. [29]. + +# Alternative assumption. + +(i') Score function is well estimated, i.e., the score entropy loss for distributions on the stationary process is bounded: + +$$ +\mathbb{E}_{\mathbf{x}_{i}\sim \pi_{t}}\mathsf{SE}_{Q^{fw}}(\mathbf{x}_{i};\theta ,t):= \mathbb{E}_{\mathbf{x}_{i}\sim \pi_{t}}\sum_{\mathbf{x}_{j}\neq \mathbf{x}_{i}}K\left(\frac{s_{\theta}(\mathbf{x}_{i};t)_{\mathbf{x}_{j}}}{s(\mathbf{x}_{i};t)_{\mathbf{x}_{j}}}\right)Q_{t}^{fw}[i,j]s(\mathbf{x}_{i};t)_{\mathbf{x}_{j}}\leqslant \epsilon_{\mathsf{SE}}, +$$ + +for any $t \in [0,1]$ , where $K(a) = a - 1 - \log a$ . + +Lemma 4. When the learned score function $s_{\theta}$ satisfies Assumption (i'), + +$$ +\mathcal {L} _ {\tilde {Q} _ {t}} \left(\pi_ {t}, Q _ {t} \pi_ {t}\right) \leqslant \epsilon_ {\mathrm {S E}}. \tag {33} +$$ + +Proof. + +$$ +\begin{array}{l} \mathcal {L} _ {\tilde {\boldsymbol {Q}} _ {t}} (\pi_ {t}, \boldsymbol {Q} _ {t} \pi_ {t}) = \sup _ {\varphi \in \mathcal {C} _ {b} (\mathcal {X})} \sum_ {\mathbf {x} _ {i} \neq \mathbf {x} _ {j}} \left[ \pi_ {t} (\mathbf {x} _ {i}) \tilde {\boldsymbol {Q}} _ {t} ^ {[ j, i ]} \left(\frac {\boldsymbol {Q} _ {t} ^ {[ j , i ]}}{\tilde {\boldsymbol {Q}} _ {t} ^ {[ j , i ]}} (1 - z _ {i j}) - e ^ {- z _ {i j}}\right) \right] + \sum_ {\mathbf {x} _ {i}} \pi_ {t} (\mathbf {x} _ {i}) (\boldsymbol {Q} _ {t} ^ {[ i, i ]} - \tilde {\boldsymbol {Q}} _ {t} ^ {[ i, i ]}) \\ \leqslant \sum_ {\mathbf {x} _ {i} \neq \mathbf {x} _ {j}} \left[ \pi_ {t} (\mathbf {x} _ {i}) \tilde {\boldsymbol {Q}} _ {t} ^ {[ j, i ]} \frac {s (\mathbf {x} _ {i} ; t)}{s _ {\theta} (\mathbf {x} _ {i} ; t)} \log \frac {s (\mathbf {x} _ {i} ; t)}{s _ {\theta} (\mathbf {x} _ {i} ; t)} \right] - \sum_ {\mathbf {x} _ {i}} \pi_ {t} (\mathbf {x} _ {i}) \sum_ {j \neq i} \left(\boldsymbol {Q} _ {t} ^ {[ j, i ]} - \tilde {\boldsymbol {Q}} _ {t} ^ {[ j, i ]}\right) (34) \\ = \sum_ {\mathbf {x} _ {i} \neq \mathbf {x} _ {j}} \left[ \pi_ {t} (\mathbf {x} _ {i}) Q _ {t} ^ {[ j, i ]} \log \frac {s (\mathbf {x} _ {i} ; t)}{s _ {\theta} (\mathbf {x} _ {i} ; t)} \right] - \sum_ {\mathbf {x} _ {i}} \pi_ {t} (\mathbf {x} _ {i}) \sum_ {j \neq i} (Q _ {t} ^ {[ j, i ]} - \tilde {Q} _ {t} ^ {[ j, i ]}) \\ = \sum_ {\mathbf {x} _ {i} \neq \mathbf {x} _ {j}} \left[ \pi_ {t} (\mathbf {x} _ {i}) Q _ {t} ^ {\mathrm {f w} [ i, j ]} \left(s _ {\theta} (\mathbf {x} _ {i}; t) _ {\mathbf {x} _ {j}} \log \frac {s (\mathbf {x} _ {i} ; t)}{s _ {\theta} (\mathbf {x} _ {i} ; t)} - s (\mathbf {x} _ {i}; t) + s _ {\theta} (\mathbf {x} _ {i}; t)\right) \right] \\ = \mathbb {E} _ {\mathbf {x} _ {i} \sim \pi_ {t}} \sum_ {\mathbf {x} _ {j} \neq \mathbf {x} _ {i}} \left[ K \left(\frac {s _ {\theta} (\mathbf {x} _ {i} ; t)}{s (\mathbf {x} _ {i} ; t)}\right) Q _ {t} ^ {\mathrm {f w} [ i, j ]} s \left(\mathbf {x} _ {i}; t\right) _ {\mathbf {x} _ {j}} \right] \leqslant \epsilon_ {\mathrm {S E}}. (35) \\ \end{array} +$$ + +where Eq. (34) comes from the previous Lemma 3. + +![](images/edcc949434e7eccdac19ffe864f25ed1c547b1c77ec6ee1477a8b91523dbedfd.jpg) + +# A.2 Proof of Theorem 1 + +Proof. Consider discretizing a prior step $[0, t^{*}]$ uniformly into $H$ intervals, $[h\delta, (h + 1)\delta]$ , for $h = 0, \ldots, H - 1$ , where $\delta = t^{*} / H$ . In the $h$ -th interval, + +$$ +\partial_ {t} \mu_ {t} = \hat {\mathbf {Q}} _ {h \delta} \mu_ {t}. \tag {36} +$$ + +Applying Lemma 2 on $\mu_t$ and $\pi_t$ in the $h$ -th interval, + +$$ +\partial_ {t} \mathrm {K L} \left(\pi_ {t} \| \mu_ {t}\right) \leqslant - \frac {1}{2} \mathrm {F I} _ {\tilde {\mathbf {Q}} _ {h \delta}} \left(\pi_ {t} \| \mu_ {t}\right) + \frac {2}{c} \mathbb {E} _ {\mathbf {x} _ {i} \sim \pi_ {t}} \mathcal {L} _ {\tilde {\mathbf {Q}} _ {h \delta}} \left(\pi_ {t}, \mathbf {Q} _ {t} \pi_ {t}\right). \tag {37} +$$ + +After integrating both sides by $t$ and rearranging, (for simplicity of notation, we denote $\tau_{k} = k(t^{*} + 1) + 1$ the starting time of the $k$ -th prior step) + +$$ +\begin{array}{l} \int_ {\tau_ {k} + h \delta} ^ {\tau_ {k} + (h + 1) \delta} \mathsf {F I} _ {\boldsymbol {Q} _ {t}} (\mu_ {t} \| \pi_ {t}) \mathrm {d} t \leqslant \int_ {k (t ^ {*} + 1) + 1 + h \delta} ^ {k (t ^ {*} + 1) + 1 + (h + 1) \delta} \left[ - 2 \partial_ {t} \mathsf {K L} (\mu_ {t} \| \pi_ {t}) + \frac {4}{c} \mathcal {L} _ {\boldsymbol {Q} _ {t}} (\mu_ {t}, \tilde {\boldsymbol {Q}} _ {\tau_ {k} + h \delta} \mu_ {t}) \right] \mathrm {d} t \\ = 2 \left(\mathrm {K L} \left(\mu_ {\tau_ {k} + h \delta} \| \pi_ {\tau_ {k} + h \delta}\right) - \mathrm {K L} \left(\mu_ {\tau_ {k} + (h + 1) \delta} \| \pi_ {\tau_ {k} + (h + 1) \delta}\right)\right) + \frac {4}{c} \int_ {\tau_ {k} + h \delta} ^ {\tau_ {k} + (h + 1) \delta} \mathcal {L} _ {\boldsymbol {Q} _ {t}} \left(\mu_ {t}, \tilde {\boldsymbol {Q}} _ {\tau_ {k} + h \delta} \mu_ {t}\right) \mathrm {d} t. \tag {38} \\ \end{array} +$$ + +Taking a summation over $h$ + +$$ +\begin{array}{l} \int_ {k (t ^ {*} + 1) + 1} ^ {(k + 1) (t ^ {*} + 1)} \mathsf {F I} _ {\boldsymbol {Q} _ {t}} (\mu_ {t} \| \pi_ {t}) \leqslant 2 \left(\mathsf {K L} \left(\mu_ {k (t ^ {*} + 1) + 1} \| \pi_ {k (t ^ {*} + 1) + 1}\right) - \mathsf {K L} \left(\mu_ {(k + 1) (t ^ {*} + 1)} \| \pi_ {(k + 1) (t ^ {*} + 1)}\right)\right) \\ + \sum_ {h = 0} ^ {H - 1} \int_ {\tau_ {k} + h \delta} ^ {\tau_ {k} + (h + 1) \delta} \frac {4}{c} \mathcal {L} _ {\boldsymbol {Q} _ {t}} \left(\mu_ {t}, \tilde {\boldsymbol {Q}} _ {\tau_ {k} + h \delta} \mu_ {t}\right) d t \tag {39} \\ \end{array} +$$ + +Since $\| \frac{s_{\theta}(\cdot ;t + s) - s(\cdot ;t)}{s(\cdot ;t)}\|_{\infty}\leqslant \epsilon +Ls.$ According to Lemma 3, and $\delta = t^{*} / H$ + +$$ +\int_ {\tau_ {k} + h \delta} ^ {\tau_ {k} + (h + 1) \delta} \frac {4}{c} \mathcal {L} _ {\boldsymbol {Q} _ {t}} \left(\mu_ {t}, \tilde {\boldsymbol {Q}} _ {\tau_ {k} + h \delta} \mu_ {t}\right) \mathrm {d} t \leqslant \int_ {0} ^ {\delta} \frac {4}{c} M (\epsilon + L s) \mathrm {d} s = \frac {4 M}{c} \left(\frac {\epsilon t ^ {*}}{H} + \frac {L t ^ {* 2}}{2 H ^ {2}}\right) \tag {40} +$$ + +Thus, + +$$ +\int_ {k (t ^ {*} + 1) + 1} ^ {(k + 1) (t ^ {*} + 1)} \mathsf {F I} _ {\boldsymbol {Q} _ {t}} (\mu_ {t} \| \pi_ {t}) \leqslant 2 \Big (\mathsf {K L} (\tilde {p} _ {k (t ^ {*} + 1) + 1} \| p _ {k (t ^ {*} + 1) + 1}) - \mathsf {K L} (\tilde {p} _ {(k + 1) (t ^ {*} + 1)} \| p _ {(k + 1) (t ^ {*} + 1)}) \Big) + \frac {4 M}{c} (\epsilon t ^ {*} + \frac {L t ^ {* 2}}{2 H}) +$$ + +Finally, taking summation over $k = 0,\ldots ,K - 1$ , and applying Lemma 1 for each likelihood step, + +$$ +\sum_ {k = 0} ^ {K - 1} \int_ {k (t ^ {*} + 1) + 1} ^ {(k + 1) (t ^ {*} + 1)} \mathsf {F l} _ {\boldsymbol {Q} _ {t}} (\mu_ {t} \| \pi_ {t}) \leqslant 2 \mathsf {K L} (\mu_ {0} \| \pi_ {0}) + \frac {4 M}{c} (\epsilon K t ^ {*} + \frac {L K t ^ {* 2}}{2 H}). +$$ + +Dividing by $Kt^{*}$ on both sides, + +$$ +\frac {1}{K} \sum_ {k = 0} ^ {K - 1} \frac {1}{t ^ {*}} \int_ {k (t ^ {*} + 1) + 1} ^ {(k + 1) (t ^ {*} + 1)} \mathsf {F l} _ {\boldsymbol {Q} _ {t}} \left(\mu_ {t} \| \pi_ {t}\right) \leqslant \frac {2 \mathsf {K L} \left(\mu_ {0} \| \pi_ {0}\right)}{K t ^ {*}} + \frac {4 M \epsilon}{c} + \frac {2 M L t ^ {*}}{c H}. \tag {41} +$$ + +![](images/047855364a0653daa6699a47ea9417642c3723d84f78ffc3bff8f7c386aea772.jpg) + +# A.3 Potential function of split Gibbs samplers + +As defined in Eq. (7), split Gibbs samplers draws samples from the augmented distribution + +$$ +\pi (\mathbf {x}, \mathbf {z}; \eta) \propto \exp (- f (\mathbf {z}; \mathbf {y}) - g (\mathbf {x}) - D (\mathbf {x}, \mathbf {z}; \eta)). +$$ + +The key requirement of split Gibbs samplers is that the potential function $D(\mathbf{x},\mathbf{z};\eta)$ satisfies + +$$ +\lim _ {\eta \rightarrow 0 ^ {+}} D (\mathbf {x}, \mathbf {z}; \eta) = \infty , \forall \mathbf {x} \neq \mathbf {z}. \tag {42} +$$ + +Or more precisely, + +$$ +\lim _ {\eta \rightarrow 0 ^ {+}} \frac {\exp (- D (\mathbf {x} , \mathbf {z} ; \eta))}{\int \exp (- D (\mathbf {x} , \mathbf {z} ; \eta)) \mathrm {d} \mathbf {z}} = \delta_ {\mathbf {x}} (\mathbf {z}). \tag {43} +$$ + +Therefore, + +$$ +\begin{array}{l} \lim _ {\eta \rightarrow 0 ^ {+}} \pi^ {X} (\mathbf {x}; \eta) \propto \lim _ {\eta \rightarrow 0 ^ {+}} \int p (\mathbf {x}) p (\mathbf {y} | \mathbf {z}) \exp (- D (\mathbf {x}, \mathbf {z}; \eta)) d \mathbf {z} \\ = \int p (\mathbf {x}) p (\mathbf {y} | \mathbf {z}) \delta (\mathbf {x} - \mathbf {z}) \mathrm {d} z = p (\mathbf {x}) p (\mathbf {y} | \mathbf {x}). \tag {44} \\ \end{array} +$$ + +Also, the similar derivation holds for $\pi^Z (\mathbf{z};\eta)$ . This result has also been shown in [46]. Combining this with Theorem 1, SGDD is guaranteed to sample from the true posterior distribution when $\eta$ goes to zero. + +# B Ablation studies + +![](images/525d14f06685106993a45cd93bf3e5befab459a9ef5a7e04ebc430202e8f9da9.jpg) +Figure 4: Convergence speed of different noise schedules. We compare our noise annealing scheduler $\{\eta_k\}_{k=1}^K$ (orange) to fixed noise schedules of various noise levels on the MNIST AND task. $y$ -axis is the PSNR of variables $\mathbf{x}^{(k)}$ with respect to the ground truth. Our noise schedule converges faster than schedulers with fixed noises. + +![](images/b4d63e2ed3b936cf8d2f9e5404553a0dcf957466623679c1d5d58c4e29d344fb.jpg) +Figure 5: Sampling quality vs. computing budget. The $x$ -axis indicates the number of function evaluations of the discrete diffusion, while the $y$ -axis shows the PSNR metric. Experiments are done with 10 discretized MNIST samples on XOR and AND tasks. + +# B.1 Effectiveness of the annealing noise schedule. + +We investigate how the choice of noise schedule $\{\eta_k\}_{k=0}^K$ affects the performance of SGDD. In our main experiments, we use a geometric noise schedule from [29], defined as $\eta_k = \eta_{\min}^{k/K} \eta_{\max}^{1-k/K}$ . We argue that an annealing noise schedule $\{\eta_k\}_{k=0}^K$ that gradually decays to $\eta_{\min} \approx 0$ allows SGDD to converge faster and more accurately to the posterior distribution. + +To validate this hypothesis, we conduct an ablation study on the MNIST AND task. We run our method for $K = 50$ iterations and compare the annealing noise schedule with fixed noise schedules at different noise levels, ranging from $10^{-3}$ to 0.5. We then compute the PSNR between $\mathbf{x}^{(k)}$ and the true underlying signal. + +As shown in Fig. 4, running SGDD with the annealing noise schedule achieves the highest PSNR and also the fastest convergence speed. Moreover, when $\eta$ is large, SGDD converges quickly to a poor solution, whereas when $\eta$ is small, it converges slowly to an accurate solution. This finding aligns with our intuition that SGDD is most accurate with $\eta \rightarrow 0$ , while sampling with $\eta > 0$ regularizes and simplifies the sampling problem. + +This trade-off between convergence speed and accuracy results in a non-monotonic relationship between PSNR and fixed noise levels: PSNR first increases and then decreases with $\eta$ . Therefore, using an annealing noise schedule helps provide both fast initial convergence and accurate posterior sampling as $\eta_{k}$ approaches 0. + +Theoretical insights. Since Eq. (44) holds when $\eta \to 0$ , a natural question is why we cannot fix a small $\eta$ to begin with. + +The answer to this question is contained in Theorem 1. When fixing a small $\eta$ , we also fix a small $t^*$ as $\sigma(t^*) = \eta$ . Then the transition matrix $Q_{\tau}$ ( $\tau \in [0, t^*]$ ) will be ill-posed, in the sense that $Q_{\tau}^{[i,j]}$ is either very large or very close to 0. In this case, the relative Fisher information $\mathsf{Fl}_{Q_{\tau}}(\mu_{\tau} \| \pi_{\tau})$ becomes a poor criterion of measuring the divergence between $\mu_{\tau}$ and $\pi_{\tau}$ . To see this, we note that the spectral gap $\gamma(Q_{\tau})$ has + +$$ +\gamma \left(\boldsymbol {Q} _ {\tau}\right) = \min _ {\lambda \neq 0} - \operatorname {R e} \lambda \left(\boldsymbol {Q} _ {\tau}\right) \geqslant N \epsilon , \tag {45} +$$ + +where $N = |\mathcal{X}|$ , if $\pmb{Q}_{\tau}^{i,j} > \epsilon \geqslant 0$ for all $i \neq j$ . By the logarithmic Sobolev inequality [3], we have that + +$$ +\mathrm {F I} _ {\boldsymbol {Q} _ {\tau}} \left(\mu_ {\tau} \| \pi_ {\tau}\right) \geqslant \frac {n \min _ {i \neq j} \boldsymbol {Q} _ {\tau} ^ {[ i , j ]}}{2} \cdot \mathrm {K L} \left(\mu_ {\tau} \| \pi_ {\tau}\right) \tag {46} +$$ + +Ignoring the errors in the score function and discretization, the KL divergence converges linearly, i.e., + +$$ +\mathrm {K L} \left(\mu_ {t} \| \pi_ {t}\right) \leqslant \exp \left(- \frac {n}{4} \int_ {0} ^ {t} \min _ {i \neq j} Q _ {\tau} ^ {[ i, j ]} \mathrm {d} \tau\right) \mathrm {K L} \left(\mu_ {0} \| \pi_ {0}\right). \tag {47} +$$ + +When $\eta_{k}$ is large, $\int_0^{t^*} \min_{i \neq j} Q_{\tau}^{[i,j]} \mathrm{d}\tau$ is considerably large, because the probability ratio $p_t(\mathbf{x})$ is being smoothed by the uniform transition kernel, which makes $Q_{t}^{[i,j]}$ large. However, as discussed above, $\int_0^{t^*} Q_{\tau}^{[i,j]} \mathrm{d}\tau$ can be very close to 0 when $\eta_{k}$ is small, which makes the convergence slow. + +# B.2 Time efficiency. + +We run SGDD with different configurations to evaluate its performance under different computing budgets. We control the number of iterations, $K$ , and the number of unconditional sampling steps $T$ in each prior sampling step, so the number of function evaluations (NFE) of the discrete diffusion model is $KT$ . We evaluate SGDD with NFEs ranging from 100 to 2k. The detailed configuration is specified in Table 6. + +We present a plot in Fig. 5 showing the tradeoff between sample quality and computing budgets in terms of NFEs. SGDD is able to generate high-quality samples with a reasonable budget comparable to unconditional generation. Note that we calculate the NFEs for SVDD and SMC with the number of Monte Carlo samples $\mathrm{mc} = 1$ and the number of Euler steps $H = 1$ , while some results in our experiments are done with larger mc and $H$ for better performance. We report both the number of total NFEs and the number of sequential NFEs that decide the cost of the algorithm when it is fully parallelized. + +Table 6: Configurations of SGDD and baseline methods. For DPS and SVDD-PM, we calculate both the total NFE and the sequential NFE. The numbers are collected from the MNIST XOR task as an example. The runtime is amortized on a batch size of 10 samples with one NVIDIA A100 GPU. + +
Configuration# IterationsPrior sampling stepsNFE (total/sequential)Runtime (sec/sample)
DPS100-102400/100820
SVDD-PM (M=20)128-2560/12816
SMC (M=20)128-5120/25621
SGDD-50252504
SGDD-1002541006
SGDD-2004052008
SGDD-400401040010
SGDD-1k5020100013
SGDD-2k10020200020
+ +# B.3 Comparison to sampling without prior. + +We evaluate the value of discrete diffusion priors in solving discrete inverse problems by comparing our method to the Metropolis-Hastings algorithm without a diffusion prior. We pick the AND problem on discretized MNIST as an example, in which we choose $\gamma D$ random pairs of pixels and compute their AND values as measurement $\mathbf{y}$ . Fig. 6 shows the reconstructed samples of SGDD and direct MCMC sampling under different levels of measurement sparsity. As demonstrated in the figure, the discrete diffusion model complements the missing information when the measurement is sparse (e.g., $\gamma = 2, 4, 8$ ), underscoring the importance of discrete diffusion priors in SGDD for solving discrete inverse problems. + +![](images/79e3bee5a64775311f240f18f83afb03730ebc7940086fdaf2054f07dae0a4ab.jpg) + +![](images/b97df768473a3b3a1691297a139b1490314367a4a84f3fa509ccbe874e21081f.jpg) + +![](images/fa912a118cc8990ff546a057ab9765290812a765b149e216d6864293a8f9298c.jpg) + +![](images/0fd1511ba792f60428da34005f7e91a3ed10f2538c2deeeff00a1f409aeb4d6a.jpg) + +![](images/a696b10ab592e421b0b0ebb4b7ce7e51b3c07072580c241737d4d246afb761b7.jpg) + +![](images/086118c7ea86a61be127f5040f044bbe1608d2fab21954481f559b711cf6a5b4.jpg) +Figure 6: Comparison of SGDD and direct sampling without data prior. The PSNR values are computed with 10 test samples. SGDD replenishes the missing information when the measurement is sparse. + +![](images/bfee870bb326302f190821e5bc4bf13d0fc68c37d4f7b9b052403671cfcf6b32.jpg) + +![](images/fa7ab17f17bee5a614391ad05c2616a04bb1fdb15956f1e158db3a6202f71eab.jpg) + +![](images/99aba1785ca471736296f4644d4c350263b58fdca5056c20db1dde369b464227.jpg) + +![](images/6c5e72499874dd57912f1a8ce8962bf99d59f3a20ca870c9776c3d5e32d29098.jpg) + +# C Experimental details + +# C.1 Pretrained Models + +We learn prior distributions for each dataset using SEDD [29] discrete diffusion models. We use the SEDD small architecture with around 90M parameters for all experiments, and the models are trained with AdamW [28] with batch size 32 and a learning rate of $3 \times 10^{-4}$ . + +# C.2 Baseline Methods + +# C.2.1 DPS + +DPS [7] is designed to solve general inverse problems with a pretrained (continuous) diffusion model. It performs posterior sampling from $p(\mathbf{x}|\mathbf{y})$ by modifying the reverse SDE + +$$ +\begin{array}{l} \mathrm {d} \mathbf {x} _ {t} = - 2 \dot {\sigma} _ {t} \sigma_ {t} \nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {x} _ {t} | \mathbf {y}) \mathrm {d} t + \sqrt {2 \dot {\sigma} _ {t} \sigma_ {t}} \mathrm {d} t (48) \\ = - 2 \dot {\sigma} _ {t} \sigma_ {t} \left(\nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {x} _ {t}; \sigma_ {t})\right) + \nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {y} | \mathbf {x} _ {t})) \mathrm {d} t + \sqrt {2 \dot {\sigma} _ {t} \sigma_ {t}} \mathrm {d} t. (49) \\ \end{array} +$$ + +To estimate the intractable guidance term $\nabla_{\mathbf{x}_t}\log p(\mathbf{y}|\mathbf{x}_t)\mathrm{d}t$ , DPS proposes to approximate it with $p(\mathbf{y}|\mathbf{x}_t)\approx p(\mathbf{y}|\mathbb{E}[\mathbf{x}_0|\mathbf{x}_t])$ . The guidance term is thus approximated by + +$$ +\nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {y} | \mathbf {x} _ {t}) \approx - \nabla_ {\mathbf {x} _ {t}} \frac {\| G \left(\boldsymbol {D} _ {\theta} \left(\mathbf {x} _ {t} , \sigma_ {t}\right)\right) - \mathbf {y} \| _ {2} ^ {2}}{2 \sigma_ {\mathbf {y}} ^ {2}}, \tag {50} +$$ + +where $D_{\theta}$ is a one-step denoiser using the pretrained diffusion model, and the measurement is assumed to be $\mathbf{y} = \mathbf{G}(\mathbf{x}) + \mathbf{n}$ with $\mathbf{n} \sim \mathcal{N}(0, \sigma_{\mathbf{y}}^2 \mathbf{I})$ . + +However, DPS is not directly applicable to inverse problems in discrete-state spaces since propagating gradients through $\pmb{G}$ and $\pmb{D}_{\theta}$ in Eq. (50) is impossible. Therefore, we consider the counterpart of DPS in discrete spaces. We modify the continuous-time Markov chain of the discrete diffusion model by + +$$ +\frac {\mathrm {d} p _ {T - t}}{\mathrm {d} t} = Q _ {t} ^ {\mathbf {y}} p _ {T - t}, \tag {51} +$$ + +in which + +$$ +Q _ {t} ^ {\mathbf {y} [ i, j ]} = \frac {p _ {T - t} (\mathbf {x} _ {i} | \mathbf {y})}{p _ {T - t} (\mathbf {x} _ {j} | \mathbf {y})} = \frac {p _ {T - t} (\mathbf {x} _ {i})}{p _ {T - t} (\mathbf {x} _ {j})} \frac {p _ {T - t} (\mathbf {y} | \mathbf {x} _ {i})}{p _ {T - t} (\mathbf {y} | \mathbf {x} _ {j})} = Q _ {t} ^ {[ i, j ]} \cdot \frac {p _ {T - t} (\mathbf {y} | \mathbf {x} _ {i})}{p _ {T - t} (\mathbf {y} | \mathbf {x} _ {j})} \tag {52} +$$ + +Similar ideas are applied to classifier guidance for discrete diffusion models [32], where the matrix $\mathbf{R}_t^{\mathbf{y}} = \left[\frac{p_{T - t}(\mathbf{y}|\mathbf{x}_i)}{p_{T - t}(\mathbf{y}|\mathbf{x}_j)}\right]_{i,j}$ is called a guidance rate matrix. We compute $\mathbf{R}_t^{\mathbf{y}}$ at $\mathbf{x}_j$ -column by enumerating every neighboring $\mathbf{x}_i$ and calculating $\frac{p_t(\mathbf{y}|\mathbf{x}_i)}{p_t(\mathbf{y}|\mathbf{x}_j)}$ for each $\mathbf{x}_i$ . The discrete version of DPS can be summarized by + +$$ +\frac {\mathrm {d} p _ {T - t}}{\mathrm {d} t} = \boldsymbol {Q} _ {t} \boldsymbol {R} _ {t} ^ {\mathbf {y}} p _ {T - t}. \tag {53} +$$ + +However, the discrete version of DPS is very time-consuming, especially when the vocabulary size is large, for it enumerates $(N - 1) \times D$ number of neighboring $\mathbf{x}$ when computing $\mathbf{R}_t^{\mathbf{y}}$ . We find it slow for binary MNIST $(N = 2)$ and DNA generation tasks $(N = 4)$ but unaffordable for monophonic music generation $(N = 129)$ . + +# C.2.2 SVDD + +SVDD [27] aims to sample from the distribution $p^{\beta}(\mathbf{x}_0) \propto p(\mathbf{x}_0) \exp (\beta r(\mathbf{x}_0))$ , which is equivalent to the regularized MDP problem: + +$$ +p ^ {\beta} \left(\mathbf {x} _ {0}\right) = \underset {\pi} {\arg \max } \mathbb {E} _ {\mathbf {x} _ {0} \sim \pi} r \left(\mathbf {x} _ {0}\right) - \operatorname {K L} (\pi \| p) / \beta \tag {54} +$$ + +They calculate the soft value function as + +$$ +v _ {t} \left(\mathbf {x} _ {t}\right) = \log \mathbb {E} _ {\mathbf {x} _ {0} \sim p \left(\mathbf {x} _ {0} \mid \mathbf {x} _ {t}\right)} \left[ \exp \left(\beta r \left(\mathbf {x} _ {0}\right)\right) \right] / \beta \tag {55} +$$ + +and propose to sample from the optimal policy + +$$ +p _ {t} ^ {\star} \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t + 1}\right) \propto p _ {t} \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t + 1}\right) \exp \left(\beta v _ {t} \left(\mathbf {x} _ {t}\right)\right). \tag {56} +$$ + +In time step $t$ , SVDD samples a batch of $M$ particles from the unconditional distribution $p_t(\mathbf{x}_t|\mathbf{x}_{t+1})$ , and conduct importance sampling according to $\exp(\beta v_t(\mathbf{x}_t))$ . + +Although [27] is initially designed for guided diffusion generation; it also applies to solving inverse problems by carefully choosing reward functions. We consider $r(\mathbf{x}) = -\| \pmb{G}(\mathbf{x}) - \mathbf{y}\|_0 / \sigma_{\mathbf{y}}$ and $\beta = 1$ , so that it samples from the posterior distribution $p(\mathbf{x}|\mathbf{y})$ . As recommended in [27], we choose $\beta = \infty$ ( $\alpha = 0$ in their notation) in practice, so the importance sampling reduces to finding the particle with the maximal value in each iteration. We use SVDD-PM, a training-free method provided by [27] in our experiments. It approximates the value function by $v_t(\mathbf{x}_t) = r(\hat{\mathbf{x}}_0(\mathbf{x}_t))$ , where $\hat{\mathbf{x}}_0(\mathbf{x}_t)$ is an approximation of $\mathbb{E}[\mathbf{x}_0|\mathbf{x}_t]$ . In practice, we find that approximating $\hat{\mathbf{x}}_0(\mathbf{x}_t)$ by Monte Carlo sampling with a few-step Euler sampler achieves slightly better results. We use 3 Monte Carlo samples to estimate $v_t(\mathbf{x}_t)$ in our experiments. + +# C.2.3 SMC + +Sequential Monte Carlo (SMC) methods evolve multiple particles to approximate a series of distributions, eventually converging to the target distribution. Specifically, in our experiments, we implement the SMC method to sample from $p(\mathbf{x}_t|\mathbf{y})$ for $t = T,T - 1\ldots ,0$ , using the unconditional discrete diffusion sampler $p(\mathbf{x}_t|\mathbf{x}_{t + 1})$ as the proposal function. + +We maintain a batch of $M = 20$ particles $\{\mathbf{x}^{[m]}\}$ . At time $t$ , we sample $\mathbf{x}_t^{[m]} \sim p(\mathbf{x}_t|\mathbf{x}_{t+1}^{[m]})$ by the pretrained discrete diffusion model and estimate the likelihood $p(\mathbf{y}|\mathbf{x}_t^{[m]}) = \mathbb{E}_{\mathbf{x}_0 \sim p(\mathbf{x}_0|\mathbf{x}_t^{[m]})} p(\mathbf{y}|\mathbf{x}_0)$ by Monte Carlo sampling. We then resample the particles $\{\mathbf{x}_t^{[m]}\}$ according to their weights $w_t^{[m]} = p(\mathbf{y}|\mathbf{x}_t^{[m]}) / p(\mathbf{y}|\mathbf{x}_{t+1}^{[m]})$ . In practice, we find that we have to carefully tune a hyperparameter $\beta$ , where $w_t^\beta$ , or otherwise the resampling step can easily degenerate to arg max or uniform random sampling. + +# C.3 Hyperparameters + +We use an annealing noise schedule of $\eta_{k} = \eta_{\mathrm{min}}^{k / K}\eta_{\mathrm{max}}^{1 - k / K}$ with $\eta_{\mathrm{min}} = 10^{-4}$ and $\eta_{\mathrm{max}} = 20$ . We run SGDD for $K$ iterations. In each likelihood sampling step, we run Metropolis-Hastings for $T$ steps, while in each prior sampling step, we run a few-step Euler discrete diffusion sampler with $H$ steps. The hyperparameters used for each experiment are listed in Table 7. We also include the dimension of data spaces $\mathcal{X}^D$ for each experiment in Table 7, where $|\mathcal{X}| = N$ . + +Table 7: Hyperparameters used in each experiment. + +
SyntheticDNA designMNIST XORMNIST ANDMusic infilling
Metropolis-Hastings T10200200050005000
SGDD iterations K105050100100
Euler sampler H2020202020
Sequence length D2/5/1020010241024256
Vocab size N50422129
+ +# D Additional Experimental Results + +# D.1 Solving image restoration problems in discrete latent space. + +To demonstrate the performance of SGDD on higher-dimensional datasets, we conduct an additional experiment on FFHQ 256 with a vector-quantized diffusion model. Specifically, we use a pretrained VQVAE model from [13] and train a discrete diffusion model in the discrete latent space. We test our algorithm on the super-resolution task $(\times 4)$ using 100 images from the test dataset. While existing methods fail to solve this high-dimensional task with a complicated forward model that includes a latent space decoder (the cost of DPS is unaffordable), our method is able to reconstruct the underlying images with high PSNR values. + +While the latent codes of VQVAE lie in a discrete-state space, they are embedded in an underlying Euclidean space, which makes this problem different from solving inverse problems in categorical distributions without ordinal information. A potential way to speed up and boost the performance of SGDD in solving image restoration problems with VQ-diffusion is to replace the Metropolis-Hastings in discrete latent codes with Langevin dynamics in the continuous embedding space of latent codes. We leave the exploration of this idea as a future direction. + +Table 8: Quantitative results for the superresolution $(\times 4)$ task on FFHQ 256 dataset. + +
PSNR ↑LPIPS ↓
SVDD-PM12.08±0.390.594±0.061
SGDD25.85±0.390.288±0.039
+ +Table 9: Class-conditioned generation on MNIST dataset. + +
MSID ↓Accuracy %
SMC39.1345.0
SGDD14.5998.5
+ +# D.2 More results for inverse problems on discretized image. + +Qualitative results. Here we present the visualization of the reconstructed samples of MNIST from their XOR measurements. + +![](images/b5ba71a55fd7323422c70042735392defe69d93747694fd430474459091bd3e1.jpg) +Figure 7: Sampling results of the XOR task on the discretized MNIST dataset. SGDD faithfully recovers the structural information of the ground truth signal. + +Approximate distribution divergence on class-conditioned generation. Since a classifier on MNIST data can also be viewed as a forward model, SGDD can also be directly applied to sample digit-conditioned samples. Moreover, while the posterior distribution for a general inverse problem is inaccessible, we can approximate the posterior distribution of class-conditioned generation by a digit-specific prior distribution. + +Specifically, we define the posterior distribution as proportional to $p(\mathbf{x})p(-\beta \ell (G(\mathbf{x}),\mathbf{y}))$ , where $\ell$ is the cross-entropy loss between the predicted class probability and target class label $\mathbf{y}$ . We let $\mathbf{y} = 0$ to guide diffusion models to generate digits 0, and extract all digits 0 in the dataset as the approximated true posterior distribution. We train another classifier with a different random seed to evaluate the generated samples. To measure the divergence of the generated samples from the posterior distribution, we extract the features of MNIST before the last layer of a CNN classification network, and compute the multi-scale intrinsic distance (MSID) score. + +Experimental results in Table 9 show that SGDD effectively samples from the true posterior distribution, as it achieves a low MSID score and a high class accuracy of $98.5\%$ . + +# D.3 More experimental results on the synthetic dataset + +![](images/27c80992593efabf1e28b6a74890fbdeb2892ae6aa49849081e6cb8b51db6546.jpg) +Figure 8: Empirical distributions of various sampling algorithms on the synthetic dataset. Heatmaps are drawn with 10k samples for sequence lengths of $D = 2,5$ and 10. We project the sequences to the first two dimensions for ease of visualization. + +# E Discussions + +# E.1 Limitations and future directions + +Our method is built on discrete diffusion models with a uniform transition matrix. It is unclear how our method can be applied to other types of discrete diffusion models, such as masked diffusion models. Moreover, our method calls the forward model (for inverse problems) or the reward function multiple times while running Metropolis-Hastings in likelihood steps, which potentially increases the computational overhead when the forward model or reward function is complicated. We leave these issues as possible future extensions. + +# E.2 Boarder Impacts + +We anticipate that SGDD offers a clean and principled framework for posterior sampling using discrete diffusion models. SGDD tackles these problems by leveraging a discrete diffusion model as a general denoiser in the finite space, which models the prior distribution with rich data. However, we note that it could be the case that SGDD will generate biased samples, if the discrete diffusion model is itself trained on biased data. 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Existing methods provide remarkably strong performance guarantees with minimal computational costs. However, they require the use calibration samples composed by labeled examples different to those used for training. This requirement can be highly inconvenient, as it prevents the use of all labeled examples for training and may require acquiring additional labels solely for calibration. This paper presents an effective methodology for split conformal prediction with unsupervised calibration for classification tasks. In the proposed approach, set-prediction rules are obtained using unsupervised calibration samples together with supervised training samples previously used to learn the classification rule. Theoretical and experimental results show that the presented methods can achieve performance comparable to that with supervised calibration, at the expenses of a moderate degradation in performance guarantees and computational efficiency. + +# 1 Introduction + +Conformal prediction methods can significantly increase the reliability of machine learning techniques by providing prediction sets that contain the true response (class) with a coverage probability that can be chosen in advance (see e.g., [1-3]). In particular, methods for split (aka inductive) conformal prediction [2, 4] are particularly convenient since they can transform any prediction rule (possibly highly unreliable) into a set-prediction rule that complies with a target coverage probability. In these methods, the set-prediction rule is obtained using a fresh set of examples referred to as calibration samples. Existing methods for split conformal prediction provide remarkably strong performance guarantees with minimal computational costs. For general data distributions, they ensure a small coverage gap between the actual and target coverage probabilities. In addition, these methods only require to compute an empirical quantile from the calibration samples, which can be carried out with quasilinear complexity and negligible running time in practice. While conformal prediction has been developed for both classification and regression tasks, this paper focuses exclusively on classification problems, in line with other works for conformal prediction using weakly-supervised data [5, 6]. + +Conventional methods for split conformal prediction require the use supervised calibration samples composed by accurately labeled examples. This requirement forces to either leave aside a portion of labeled examples during the training phase or to acquire new labels solely for calibration. Both cases are problematic, the use of fewer samples for training can increase classification error, and the acquisition of new labeled examples is often costly. It is worth highlighting that obtaining new labels after the training phase may be even unfeasible in certain practical scenarios, including cases + +![](images/cb2a21e90e29838effcb0621a683a558313c3cd1dbd3318c0296569e4db936b4.jpg) +Figure 1: Coverage probabilities and prediction set sizes over 400 random partitions of 'CIFAR10' dataset with target coverage $1 - \alpha = 0.9$ and number of calibration samples ranging from 100 to 2,000. The presented methods with unsupervised calibration achieve performance that is only slightly inferior to that of conventional methods with supervised calibration (coverage probabilities slightly less centered around 0.9). With both approaches, the coverage gap between the actual and target coverage probabilities decreases with more calibration samples and remains below 0.02 with high probability using a few thousand calibration samples. + +![](images/ece00ee92000776a9817f3993f50d2fb1c6f3475c4e6cd1848e09eab5be598a4.jpg) + +where labels are protected by privacy regulations or correspond to future events that are not yet observable. Several methods have been recently proposed for situations with weakly-supervised calibration samples [5-8]. These methods utilize calibration samples with noisy or partial labels, and show that small coverage gaps can be achieved with a moderated amount of incorrect or missing labels. However, existing methods cannot handle cases with unsupervised calibration where all the calibration samples are unlabeled. In particular, it has been shown that set-prediction rules obtained using only unsupervised samples for calibration provide either uninformative or unreliable predictions (see Def. 2 and Thm. 3 in [6]). + +This paper presents effective methods for split conformal prediction with unsupervised calibration samples. In particular, the proposed approach avoids the negative result for weakly-supervised calibration in [6] because the set-prediction rule is obtained leveraging also supervised training samples previously used to learn the classification rule. Theoretical and experimental results show that the presented methods can achieve performance comparable to that with supervised calibration (see Figure 1), at the expenses of a moderate weakening of the performance guarantees and an affordable increase in computational complexity. The main contributions presented in the paper are as follows. + +- We present split conformal methods that assign label weights to unsupervised calibration samples so that they are statistically indistinguishable from training data. In particular, the methods proposed minimize a two-sample nonparametric test statistic given by a family of functions. +- We provide performance guarantees for the methods presented in terms of high-confidence bounds for coverage probabilities. In addition, the coverage gap achieved is characterized in terms of a bias-variance trade-off corresponding to the family of functions used. +- We detail the methods' implementation using families of functions given by reproducing kernel Hilbert spaces (RKHSs), which result in particularly advantageous performance guarantees that also serve to select the most adequate kernel. +- We experimentally show that the proposed methods can achieve performance comparable to that with supervised calibration. + +Notations: Vectors and matrices are denoted by bold lower and upper case letters, respectively (e.g., $\mathbf{v}$ and $\mathbf{M}$ ); random variables (RVs) are distinguished from instantiations by using capital upright symbols (e.g., Z); $\preceq$ and $\succ$ denote vector (component-wise) inequalities; $\mathbb{I}\{\cdot\}$ denotes the indicator function; for an integer $t$ , $[t]$ denotes the set $[t] = \{1, 2, \dots, t\}$ ; $\mathbf{1}_t$ denotes the vector of size $t$ with all components equal to 1; and $\mathbf{I}_t$ denotes the identity matrix of size $t \times t$ . + +# 2 Preliminaries and conventional approach for split conformal prediction + +Let $\mathcal{Z} = \mathcal{X}\times \mathcal{Y}$ be the set of instance-label pairs, with $\mathcal{X}\subset \mathbb{R}^d$ the set of instances (features) and $\mathcal{V} = \{1,2,\dots ,c\}$ the set of labels (classes), $(\mathrm{X},\mathrm{Y})$ be an RV of instance-label pairs in $\mathcal{Z}$ , and $\mathcal{D}$ be a dataset composed by i.i.d. copies of $(\mathrm{X},\mathrm{Y})$ . Conformal prediction methods use the dataset $\mathcal{D}$ to find a set-prediction rule $\mathcal{C}:\mathcal{X}\to 2^{\mathcal{V}}$ , so that prediction sets $(\mathcal{C}(\mathrm{X})\subset \mathcal{V})$ have reduced cardinality and often cover the true label $(\mathrm{Y}\in \mathcal{C}(\mathrm{X}))$ . In particular, such methods aim to ensure a coverage probability near a target value $1 - \alpha$ (e.g., $\alpha = 0.1$ ). + +The most common approach for conformal prediction is known as split conformal prediction. In this approach, the dataset is randomly split into two groups $\mathcal{D} = \mathcal{D}_{\mathrm{train}} \cup \mathcal{D}_{\mathrm{cal}}$ : the $m$ samples $\mathcal{D}_{\mathrm{train}} = \{(\widetilde{\mathrm{X}}_i, \widetilde{\mathrm{Y}}_i)\}_{i \in [m]}$ are referred to as training (or pretraining) samples, and the remaining $n$ samples $\mathcal{D}_{\mathrm{cal}} = \{(\mathrm{X}_i, \mathrm{Y}_i)\}_{i \in [n]}$ are referred to as calibration samples. Then, prediction sets are obtained as shown in Algorithm 1 below (see e.g., [2]). + +Algorithm 1 Supervised split conformal prediction +Input: Training samples $\{(\widetilde{X}_i,\widetilde{Y}_i)\}_{i\in [m]}$ , labeled calibration samples $\{(X_i,Y_i)\}_{i\in [n]}$ , and coverage $1 - \alpha$ +1: Using the training samples, construct a conformal score function $S:\mathcal{X}\times \mathcal{Y}\to \mathbb{R}$ +2: Compute the conformal scores of the calibration samples $\{S(X_i,Y_i)\}_{i\in [n]}$ +3: Compute the conformal quantile $\widehat{q} = \mathrm{Quantile}\big(\{(S(X_i,Y_i),1 / n)\}_{i\in [n]};(1 - \alpha)(1 + 1 / n)\big)$ +4: For any test instance $X$ , return the prediction set $\mathcal{C}(X) = \{y\in \mathcal{V}:S(X,y)\leq \widehat{q}\}$ + +The $m$ training samples are used to learn a classification rule that determines a conformal score $S:\mathcal{X}\times \mathcal{Y}\to \mathbb{R}$ assessing the plausibility of different instance-label pairs. For instance, the conformal score can be defined as $S(X,Y) = 1 - \widehat{p} (Y|X)$ where $\widehat{p} (Y|X)$ is the probability estimate (e.g., softmax output) assigned to label $Y$ by the classification rule at instance $X$ (see [2,9] for other types of conformal scores for classification tasks). Subsequently, the $n$ calibration samples are used to obtain score values and the corresponding conformal quantile $\widehat{q}$ in Step 3 of Algorithm 1. Specifically, the expression Quantile $\big(\{(S_i,p_i)\}_{i\in [t]};\beta \big)$ denotes the $\beta$ -quantile of values $S_{1},S_{2},\ldots ,S_{t}\in \mathbb{R}$ with probabilities $p_1,p_2,\ldots ,p_t\in [0,1],\sum_{i\in [t]}p_i = 1$ , that is + +$$ +\text {Q u a n t i l e} \left(\left\{\left(S _ {i}, p _ {i}\right) \right\} _ {i \in [ t ]}; \beta\right) = \inf \left\{q \in \mathbb {R}: \sum_ {i \in [ t ]} p _ {i} \mathbb {I} \left\{S _ {i} \leq q \right\} \geq \beta \right\}. \tag {1} +$$ + +Then, the set-prediction rule is directly obtained using the score $S$ and the conformal quantile $\widehat{q}$ . + +The conventional approach for split conformal prediction provides remarkably strong performance guarantees for the resulting prediction sets [1-3, 10]. Specifically, for any data distribution, type of conformal score, and target coverage $1 - \alpha$ , the set-prediction rule $\mathcal{C}$ from Algorithm 1 satisfies + +$$ +1 - \alpha \leq \mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) \right\} \leq 1 - \alpha + \frac {1}{n + 1} \tag {2} +$$ + +$$ +1 - \alpha - \sqrt {\frac {\log (2 / \delta)}{2 n}} \leq \mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D} \right\} \leq 1 - \alpha + \frac {1}{n + 1} + \sqrt {\frac {\log (2 / \delta)}{2 n}}, \quad \mathrm {w . p .} \geq 1 - \delta . \tag {3} +$$ + +These performance guarantees are satisfied with wide generality. The lower bounds for the marginal coverage and the (training) conditional coverage in (2) and (3), respectively, do not require additional assumptions. As described above, we assume that training, calibration, and test samples are randomly drawn from the same data distribution. The lower bound in (2) does not even require the independence of samples and is valid also if samples are exchangeable. The upper bounds in (2) and (3) additionally require that the score values are distinct with probability one, which can be easily achieved for instance by adding a negligible amount of noise [2]. The bounds for training-conditional coverage in (3) are arguably more relevant in a practical sense: in a given application, there is only one dataset $\mathcal{D}$ , and the achieved coverage corresponds to a specific realization of $\mathbb{P}\{\mathrm{Y} \in \mathcal{C}(\mathrm{X}) | \mathcal{D}\}$ , rather than its expected value $\mathbb{P}\{\mathrm{Y} \in \mathcal{C}(\mathrm{X})\}$ . + +Split conformal prediction has been shown to be extremely useful in practice since it allows to obtain reliable set-prediction rules from general prediction rules (possibly highly unreliable) by using an additional set of calibration samples. However, existing methods require the use supervised calibration samples, leading to additional labeling costs. In the following, we describe the proposed approach for split conformal prediction that uses unsupervised calibration samples. + +# 3 Split conformal prediction with unsupervised calibration samples + +In the proposed approach, the labels of the calibration samples are not available, and prediction sets are obtained as shown in Algorithm 2 below. + +# Algorithm 2 Split conformal prediction with unsupervised calibration + +Input: Training samples $\{(\widetilde{X}_i,\widetilde{Y}_i)\}_{i\in [m]}$ , unlabeled calibration samples $\{X_{i}\}_{i\in [n]}$ , and coverage $1 - \alpha$ + +1: Using the training samples, construct a conformal score function $S:\mathcal{X}\times \mathcal{Y}\to \mathbb{R}$ +2: Compute the conformal scores of the calibration instances for all possible labels $\{S(X_i,y)\}_{(i,y) \in [n] \times \mathcal{Y}}$ +3: Obtain label weights $\{w_{i}(y)\}_{(i,y)\in [n]\times \mathcal{V}}$ for the calibration instances +4: Compute the conformal quantile $\widehat{q} =$ Quantile $\left(\left\{(S(X_i,y),w_i(y) / n)\right\}_{(i,y)\in [n]\times y};(1 - \alpha)(1 + 1 / n)\right)$ +5: For any test instance $X$ , return the prediction set $\mathcal{C}(X) = \{y \in \mathcal{Y} : S(X, y) \leq \widehat{q}\}$ + +The main difference in the proposed approach with respect to conventional split conformal prediction consists of Step 3 in Algorithm 2 that obtains label weights for unsupervised calibration samples. Indeed, the conventional approach can be seen as a particular case of Algorithm 2 in which the label weights correspond to the actual calibration labels, i.e., $w_{i}(y)$ is taken as $w_{i}^{*}(y) = \mathbb{I}\{y = Y_{i}\}$ for any $(i,y)\in [n]\times \mathcal{V}$ . + +In the presented methods, the label weights for calibration instances are obtained using the training samples. A naive approach would obtain such weights using the labels predicted by a classification rule $\mathbf{h}$ learned using the training samples, i.e., $w_{i}(y)$ taken as $\widehat{w}_{i}(y) = \mathbb{I}\{y = \mathrm{h}(X_{i})\}$ for any $(i,y)\in [n]\times \mathcal{V}$ . However, this naive approach only provides acceptable performance in cases where the classification rule is already highly accurate, making conformation prediction somewhat unnecessary. The following sections present methods that avoid the limitations of such naive approaches and obtain the label weights by solving a tractable optimization problem determined by a family of functions. The proposed approach can be seen as obtaining labels that result in calibration and training samples that are statistically indistinguishable by a two-sample test based on an integral probability metric (IPM). This type of non-parametric tests can reliably detect statistical differences between two sets of samples without requiring any distributional assumption (see e.g., [11, 12]). + +# 3.1 Label weights minimizing IPMs + +Let $\mathbf{w} = [w_{1}(1), w_{1}(2), \ldots, w_{1}(c), w_{2}(1), \ldots, w_{n}(c)]^{\top} \in \mathbb{R}^{nc}$ denote label weights for the calibration samples, the Step 3 in Algorithm 2 can be carried out by solving the optimization problem + +$$ +\min _ {\mathbf {w}} \sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {(i, y) \in [ n ] \times \mathcal {Y}} w _ {i} (y) f \left(X _ {i}, y\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {X} _ {i}, \widetilde {Y} _ {i}\right) \right| \tag {4} +$$ + +$$ +\begin{array}{l} \text {s . t .} \quad \mathbf {w} \succeq \mathbf {0}, \mathbf {A w} = \mathbf {1} _ {n}, \mathbf {B w} \preceq \mathbf {b} \end{array} +$$ + +where $\mathcal{F}$ is a family of functions $f:\mathcal{X}\times \mathcal{Y}\to \mathbb{R}$ . The optimization in (4) is convex for any family $\mathcal{F}$ since the supremum of convex functions is a convex function [13], and is tractable for common and general families of functions. In particular, using functions in RKHSs, optimization (4) becomes a quadratic program of size $\mathcal{O}(nc)$ (see details in Section 4 below), while using Lipschitz functions, optimization (4) becomes a linear program of size $\mathcal{O}(ncm)$ , as a consequence of the Kantorovich-Rubinstein duality theorem [14]. + +Label weights aim to describe the probabilities of different labels, and the linear feasible set in (4) encodes quite general prior knowledge for label probabilities. In particular, the matrix $\mathbf{A} \in \mathbb{R}^{n \times nc}$ just encodes the constraints $\sum_{y \in \mathcal{Y}} w_i(y) = 1$ for $i \in [n]$ , i.e., $\mathbf{A} = \mathbf{I}_n \otimes \mathbf{1}_c^\top$ for $\otimes$ the matrix Kronecker product. On the other hand, the matrix $\mathbf{B} \in \mathbb{R}^{r \times nc}$ and vector $\mathbf{b} \in \mathbb{R}^r$ can be used to encode other types of prior knowledge about label probabilities. For instance, an upper bound $L$ for the expected loss $\mathbb{E}\{\ell(\mathrm{h}(\mathrm{X}), \mathrm{Y})\}$ of a classifier $\mathrm{h}$ (e.g., its cross-entropy loss) can be used to have the constraint given by $\mathbf{b} = nL \in \mathbb{R}$ and + +$$ +\mathbf {B} = \left[ \ell (\mathrm {h} \left(X _ {1}\right), 1), \ell (\mathrm {h} \left(X _ {1}\right), 2), \dots , \ell (\mathrm {h} \left(X _ {1}\right), c), \ell (\mathrm {h} \left(X _ {2}\right), 1), \dots , \ell (\mathrm {h} \left(X _ {n}\right), c) \right] \in \mathbb {R} ^ {1 \times n c}. \tag {5} +$$ + +Such feasible sets often contain the label weights corresponding to the actual calibration labels, $w_{i}^{*}(y) = \mathbb{I}\{y = Y_{i}\}$ for $i\in [n]$ , $y\in \mathcal{V}$ , because $\mathbf{w}^* \succeq \mathbf{0}$ and $\mathbf{A}\mathbf{w}^{*} = \mathbf{1}_{n}$ are clearly achieved, and we have $\mathbf{B}\mathbf{w}^{*}\preceq \mathbf{b}$ as long as the prior knowledge encoded by $\mathbf{B}$ and $\mathbf{b}$ is reliable. + +As shown in the following, Algorithm 2 with label weights obtained by (4) can provide theoretical guarantees and practical performance comparable to those achieved by conventional split conformal prediction with supervised calibration samples. The optimization in (4) can be seen as finding the label weights for calibration samples that minimize the IPM between probability distributions corresponding to training and calibration samples, where for a family of functions $\mathcal{F}$ + +$$ +\operatorname {IPM}(P,Q) = \sup_{f\in \mathcal{F}}| \mathbb{E}_{P}\{f\} -\mathbb{E}_{Q}\{f\} |. +$$ + +In particular, for families given by RKHSs, IPMs correspond to maximum mean discrepancy metrics while for families given by Lipschitz functions, IPMs correspond to L1-Wasserstein metrics (see e.g., [12, 15]). + +The objective function in (4) denoted as $\Phi_{n,m}^{\mathcal{F}}(\mathbf{w})$ corresponds to the IPM between the distribution $\mathbf{w} / n$ supported on the $nc$ possible instance-label calibration pairs, and the empirical distribution of the $m$ training samples. IPMs between empirical distributions have been used to design non-parametric two-sample tests in which two sets of samples are considered to be statistically different (draw from different distributions) if the IPM between the corresponding empirical distributions is large enough [11, 12]. Therefore, the label weights obtained from (4) can be interpreted as those making calibration and training samples to be statistically indistinguishable by a non-parametric two-sample test. In particular, at the weights $\mathbf{w}^{*}$ corresponding to the actual calibration samples the value of (4) is given by the test statistic achieved by two sets of i.i.d. samples that are actually drawn from the same distribution. + +A key methodological difference in the proposed approach is that the training samples are not only used to learn the score function but also to obtain label weights for calibration samples. In particular, the methods presented bypass the negative results in [6] for weakly-supervised calibration because the set prediction function is obtained using unlabeled calibration samples and also labeled training samples. This reuse of the training samples may seem doomed to fail since scores and label weights for calibration samples are no longer independent (both depend on the training samples). The following briefly describes the rationale that allows the proposed approach to avoid such concerns, Theorem 1 below provides the precise performance guarantees achieved by the methods proposed. + +Let $\mathfrak{X}_{\mathcal{C}}:\mathcal{X}\times \mathcal{Y}\to \{0,1\} \subset \mathbb{R}$ be the function + +$$ +\mathfrak {X} _ {\mathcal {C}} (x, y) := \mathbb {I} \{y \in \mathcal {C} (x) \} = \mathbb {I} \{S (x, y) \leq \widehat {q} \} \in \{0, 1 \}. \tag {6} +$$ + +The coverage gap of the set-prediction rule from Algorithm 2 is small as long as the weighted average of function $\mathfrak{X}_{\mathcal{C}}$ at the calibration instances is near the expectation of $\mathfrak{X}_{\mathcal{C}}$ because + +$$ +\mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) \right\} = \mathbb {E} \left\{\mathfrak {X} _ {\mathcal {C}} \right\} \text {a n d} \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right) \approx 1 - \alpha +$$ + +by definition of $\mathfrak{X}_{\mathcal{C}}$ in (6) and $\widehat{q}$ in the step 4 of Algorithm 2. Since the minimization of (4) ensures that + +$$ +\frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) f (\mathrm {X} _ {i}, y) \approx \frac {1}{m} \sum_ {i \in [ m ]} f (\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}), \forall f \in \mathcal {F} +$$ + +prediction sets given by label weights from (4) have a small coverage gap as long as the family $\mathcal{F}$ utilized: 1) can accurately approximate the function $\mathfrak{X}_{\mathcal{C}}$ at the calibration instances (e.g., $\mathcal{F}$ can interpolate finite sets of points), and 2) ensures the uniform convergence of i.i.d. averages (e.g., $\mathcal{F}$ has reduced Rademacher complexity). This type of analysis for conformal prediction methods is novel and can be of independent interest beyond split conformal prediction with unsupervised calibration. + +# 3.2 Performance guarantees + +The following provides performance bounds for the coverage probability of set-prediction rules obtained using the proposed approach in Algorithm 2. In particular, such performance guarantees are given in terms of the Rademacher complexity of the family $\mathcal{F}$ , and the quantity + +$$ +D \left(\mathfrak {X} _ {\mathcal {C}}, \mathcal {F}\right) = \inf _ {f \in \mathcal {F}} \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} \left| \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right) - f \left(\mathrm {X} _ {i}, y\right) \right| \tag {7} +$$ + +that describes the smallest error achieved approximating the function $\mathfrak{X}_{\mathcal{C}}$ with functions in $\mathcal{F}$ at the calibration instances. For the next result we denote by $\mathcal{R}_t(\mathcal{F})$ the Rademacher complexity of $\mathcal{F}$ for $t$ i.i.d. samples, and consider general families of real-valued functions with bounded differences, (i.e., $|f(z) - f(z')| \leq B$ , for any $f \in \mathcal{F}$ and $z, z' \in \mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ ). The requirement of bounded differences is satisfied by common families including functions in RKHSs given by bounded kernels or Lipschitz functions defined in bounded sets. + +Theorem 1. Let $\mathbf{w}$ be label weights with objective value in (4) upper bounded by $V_{\mathrm{opt}}$ (i.e., $\Phi_{n,m}^{\mathcal{F}}(\mathbf{w}) \leq V_{\mathrm{opt}}$ ). For any data distribution, type of conformal score, and target coverage $1 - \alpha$ , the set-prediction rule $\mathcal{C}$ from Algorithm 2 satisfies + +$$ +\mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D} \right\} \geq 1 - \alpha - G _ {n, m} ^ {\mathcal {F}} - \sqrt {\frac {\log (2 / \delta)}{2 n}} \tag {8} +$$ + +with probability at least $1 - \delta$ , for $G_{n,m}^{\mathcal{F}}$ given by + +$$ +G _ {n, m} ^ {\mathcal {F}} = V _ {\text {o p t}} + D \left(\mathfrak {X} _ {\mathcal {C}}, \mathcal {F}\right) + 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) + B \sqrt {\left(\frac {1}{n} + \frac {1}{m}\right) \frac {\log (2 / \delta)}{2}}. \tag {9} +$$ + +In addition, if the values $\{S(\mathrm{X}_i,y)\}_{(i,y)\in [n]\times \mathcal{Y}}$ are distinct with probability one, we have + +$$ +\mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D} \right\} \leq 1 - \alpha + \frac {2}{n + 1} + G _ {n, m} ^ {\mathcal {F}} + \sqrt {\frac {\log (2 / \delta)}{2 n}} \tag {10} +$$ + +with probability at least $1 - \delta$ + +Proof. See Appendix B. + +![](images/3b73b07f048cc4e69a4b439f2e9156b4188ccaeee7e21c0e670f9158d5767a3c.jpg) + +The result above presents performance guarantees for the coverage probability of the methods proposed comparable to those in (3) for the conventional approach with supervised calibration. The excess coverage gap $G_{n,m}^{\mathcal{F}}$ in the bounds of Theorem 1 describes the effectiveness of Step 3 in Algorithm 2. In particular, the term $V_{\mathrm{opt}}$ accounts for the objective value obtained by $\mathbf{w}$ in (4), and is small using appropriate optimization algorithms and number of calibration and training samples. For instance, if the feasible set in optimization (4) includes the label weights $\mathbf{w}^{*}$ corresponding to the actual calibration labels, the objective value $\Phi_{n,m}^{\mathcal{F}}(\mathbf{w})$ satisfies + +$$ +\Phi_ {n, m} ^ {\mathcal {F}} (\mathbf {w}) \leq \varepsilon_ {\text {o p t}} + 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) + B \sqrt {\left(\frac {1}{n} + \frac {1}{m}\right) \frac {\log (1 / \delta)}{2}}, \quad \mathrm {w . p .} \geq 1 - \delta \tag {11} +$$ + +for $\varepsilon_{\mathrm{opt}}$ the optimization error achieved by the algorithm used to solve (4) (see Appendix D for a detailed proof). Hence, the term in (9) due to the optimization value is generally of the same order as the Rademacher complexity of the family. In addition, Theorem 1 shows that the methods proposed do not require to solve the optimization problem (4) with a high accuracy, as approximate optimization algorithms increase the bound only by their optimization error $\varepsilon_{\mathrm{opt}}$ . For instance optimization errors on the order of $10^{-3}$ do not significantly affect the final coverage probability. + +The excess in coverage gap achieved using unsupervised calibration is given by the terms $D(\mathfrak{X}_{\mathcal{C}}, \mathcal{F})$ and $\mathcal{R}_n(\mathcal{F}) + \mathcal{R}_m(\mathcal{F})$ that describe a bias-variance trade-off for the family $\mathcal{F}$ . The bias term $D(\mathfrak{X}_{\mathcal{C}}, \mathcal{F})$ corresponds to the distance between function $\mathfrak{X}_{\mathcal{C}}$ and the family $\mathcal{F}$ at the calibration instances, while the variance term $\mathcal{R}_n(\mathcal{F}) + \mathcal{R}_m(\mathcal{F})$ accounts for the usage of finite sample sizes. Both terms are low using an appropriate family $\mathcal{F}$ and a sufficient number of samples. The bias term $D(\mathfrak{X}_{\mathcal{C}}, \mathcal{F})$ can be made arbitrarily small using functions that are Lipschitz or belong to an RKHS given by a strictly positive kernel (e.g., Gaussian, Laplacian, or polynomial) since both types of functions can interpolate any finite set of points. The variance term $\mathcal{R}_n(\mathcal{F}) + \mathcal{R}_m(\mathcal{F})$ decreases with the sample sizes at a rate $O(Rn^{-1/2} + Rm^{-1/2})$ using functions in an RKHS with norm bounded by $R$ (see e.g., Chapter 6 in [16]). On the other hand, using $R$ -Lipschitz functions such a rate becomes $O(Rn^{-1/d} + Rm^{-1/d})$ [17, 18] that is significantly affected by the instances' dimensionality $d$ . + +Theorem 1 shows that the coverage gap decreases with both the number of calibration samples $n$ and training samples $m$ . Although the total number of training samples available is commonly much larger than that of calibration samples, there is not a significant benefit in using $m > n$ for calibration, and label weights can be obtained using in (4) only a subset of the training samples available. The results in the theorem also expose an important limitation in the methods presented since the performance guarantees deteriorate for an increased number of classes $|\mathcal{V}| = c$ . Specifically, both + +bias and variance terms increase with the number of classes, the bias term $D(\mathfrak{X}_{\mathcal{C}}, \mathcal{F})$ is given by the average of approximation errors multiplied by $|\mathcal{Y}|$ , and the variance term $\mathcal{R}_n(\mathcal{F}) + \mathcal{R}_m(\mathcal{F})$ is given by the complexity of functions over $\mathcal{X} \times \mathcal{Y}$ that generally increases with $|\mathcal{Y}|$ . + +The performance guarantees shown in Theorem 1 can also be compared with those of the naive approach that labels calibration instances using a classification rule. It is easy to show that for such a naive approach the coverage gap is $\mathcal{O}(\mathrm{Err(h)})$ for $\mathrm{Err(h)}$ the error probability of the classification rule $h$ used to label the calibration instances. In particular, such a bound for the coverage gap can be obtained by using the results in [19] since $\mathrm{Err(h)}$ coincides with the total variation distance between $(\mathrm{X},\mathrm{Y})$ and $(\mathrm{X},\mathrm{h}(\mathrm{X}))$ . A coverage gap near $\mathrm{Err(h)}$ is not acceptable in common scenarios because it would only be small in cases where a highly accurate classifier is available, which would commonly defeat the purpose of conformal prediction. + +In this section we have described the general approach presented together with the corresponding performance guarantees. As indicated above, kernel-based methods offer an advantageous balance for optimization efficiency and bias-variance trade-off in general scenarios, so that the rest of the paper focuses in such type of approaches. In addition, the kernel-based methods provide enhanced performance guarantees that enable effective model selection, as described in the following. + +# 4 Kernel-based implementation + +Let $\mathcal{H}$ be an RKHS of real-valued functions over $\mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ given by kernel $K: \mathcal{Z} \times \mathcal{Z} \to \mathbb{R}$ . The optimization problem in (4) using family $\mathcal{F} = \{f \in \mathcal{H}: \| f \|_{\mathcal{H}} \leq R\}$ with $R = 1$ is equivalent to + +$$ +\min _ {\mathbf {w}} \left(\frac {1}{n ^ {2}} \mathbf {w} ^ {\top} \mathbf {K} \mathbf {w} - \frac {2}{n m} \mathbf {v} ^ {\top} \mathbf {w} + \frac {1}{m ^ {2}} \mathbf {1} _ {m} ^ {\top} \widetilde {\mathbf {K}} \mathbf {1} _ {m}\right) ^ {\frac {1}{2}} \tag {12} +$$ + +$$ +\begin{array}{l} \text {s . t .} \quad \mathbf {w} \succeq \mathbf {0}, \mathbf {A w} = \mathbf {1} _ {n}, \mathbf {B w} \preceq \mathbf {b} \end{array} +$$ + +where $\mathbf{K} \in \mathbb{R}^{nc \times nc}$ is the kernel matrix corresponding to the $nc$ possible instance-label calibration pairs $Z_{1}, Z_{2}, \ldots, Z_{nc}$ with $Z_{(i-1)c+y} = (X_{i}, y)$ for $i \in [n], y \in \mathcal{Y}$ , that is, the $(i,j)$ -th component of $\mathbf{K}$ is given by $K_{i,j} = K(Z_{i}, Z_{j})$ for $i,j \in [nc]$ . Similarly, $\widetilde{\mathbf{K}} \in \mathbb{R}^{m \times m}$ is the kernel matrix corresponding to the training samples $\widetilde{Z}_{1}, \widetilde{Z}_{2}, \ldots, \widetilde{Z}_{m}$ with $\widetilde{Z}_{i} = (\widetilde{X}_{i}, \widetilde{Y}_{i})$ for $i \in [m]$ . In addition, $\mathbf{v} \in \mathbb{R}^{nc}$ is the vector with $i$ -th component given by $v_{i} = \sum_{j \in [m]} K(Z_{i}, \widetilde{Z}_{j})$ for $i \in [nc]$ . The equivalence between (4) and (12) follows as a consequence of the dual characterization of the norm in $\mathcal{H}$ (see the detailed derivation in Appendix D). The kernel $K$ can be any joint kernel over $\mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ , for instance $K$ can be given by a kernel $K_{0}$ over $\mathcal{X}$ as the separable kernel $K\big((x,y),(x',y')\big) = K_{0}(x,x')\mathbb{I}\{y = y'\}$ (see [20] for other types of joint kernels). Separable kernels are simple and convenient computationally since they lead to highly sparse kernel matrices. + +The usage of a family with other maximum norm $R \neq 1$ results in an optimization problem equivalent to (12), namely, a problem that has the same solution with optimum value multiplied by $R$ . This fact has important consequences because for families of functions $\mathcal{F} = \{f \in \mathcal{H} : \| f \|_{\mathcal{H}} \leq R\}$ the bias/variance trade-off described in Section 3.2 is given by the value of the norm $R$ . Increased values of $R$ result in more general families of functions at the expenses of an increased Rademacher complexity. Therefore, kernel-based methods provide an automatic capacity/complexity control: the trade-off for the value of $R$ does not need to be explicitly addressed because the optimization problem solved does not really change with $R$ . Therefore, a solution of (12) enjoys the performance guarantees of Theorem 1 corresponding to the value of $R$ that achieves the best bias/variance tradeoff. In particular, kernel-based methods result in the following enhanced performance guarantees. + +Theorem 2. Let $\mathcal{H}$ be an RKHS given by kernel $K$ over $\mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ with $0 \leq K(z, z') \leq \kappa^2 \forall z, z' \in \mathcal{Z}$ , and $\mathbf{w}$ be a solution of optimization problem (12) with feasible set that includes the label weights $\mathbf{w}^*$ corresponding to the actual calibration labels. For any data distribution, type of conformal score, and target coverage $1 - \alpha$ , the set-prediction rule $\mathcal{C}$ from Algorithm 2 satisfies + +$$ +\mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D} \right\} \geq 1 - \alpha - G _ {n, m} ^ {K} - \sqrt {\frac {\log (2 / \delta)}{2 n}} \tag {13} +$$ + +with probability at least $1 - \delta$ , for $G_{n,m}^{K}$ given by + +$$ +G _ {n, m} ^ {K} = \min _ {f \in \mathcal {H}} \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} | \mathfrak {X} _ {\mathcal {C}} (\mathrm {X} _ {i}, y) - f (\mathrm {X} _ {i}, y) | + 2 \kappa (1 + \sqrt {\log (2 / \delta)}) \sqrt {\frac {1}{n} + \frac {1}{m}} \| f \| _ {\mathcal {H}}. \tag {14} +$$ + +In addition, if the values $\{S(\mathrm{X}_i,y)\}_{(i,y)\in [n]\times \mathcal{Y}}$ are distinct with probability one, we have + +$$ +\mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D} \right\} \leq 1 - \alpha + \frac {2}{n + 1} + G _ {n, m} ^ {K} + \sqrt {\frac {\log (2 / \delta)}{2 n}} \tag {15} +$$ + +with probability at least $1 - \delta$ + +Proof. See Appendix C. + +![](images/62d865cbc86410c4505de675810225ea63095270c9aee321f2fadbb579611ad4.jpg) + +The result above shows that the kernel-based approach in (12) yields an excess coverage gap given by the best bias-variance trade-off in the corresponding RKHS. This best trade-off is described by the term $G_{n,m}^{K}$ in (14) as a combination of the approximation error and the RKHS norm weighted by the number of samples used. For simplicity, Theorem 2 is stated for the case where $\mathbf{w}$ is an exact solution of (12) and $\mathbf{w}^*$ is included in the feasible set. A similar result holds in general cases, by including the term $V_{\mathrm{opt}}$ in the expression of $G_{n,m}^{K}$ to account for the optimization value attained, as in Theorem 1. + +For RKHSs given by strictly positive definite kernels, Theorem 2 shows that the coverage gap is $\mathcal{O}(M\sqrt{1 / n + 1 / m})$ where $M$ denotes the minimum norm of interpolators of $\mathfrak{X}_{\mathcal{C}}$ at the calibration instances. More generally, the excess coverage gap can be bounded as $G_{n,m}^{K}\leq c\varepsilon +2\kappa (1 + \sqrt{\log(2 / \delta)})M_{\varepsilon}\sqrt{1 / n + 1 / m}$ where $M_{\varepsilon}$ denotes the minimum norm of $\varepsilon$ -approximations of $\mathfrak{X}_{\mathcal{C}}$ at the calibration instances, that is + +$$ +M _ {\varepsilon} = \min _ {f \in \mathcal {H}} \left\{\| f \| _ {\mathcal {H}}: \left| \mathfrak {X} _ {\mathcal {C}} \left(X _ {i}, y\right) - f \left(X _ {i}, y\right) \right| \leq \varepsilon , i \in [ n ], y \in \mathcal {Y} \right\}. \tag {16} +$$ + +The magnitude of the minimum norm $M$ (or $M_{\varepsilon}$ ) depends on the smoothness of the function $\mathfrak{X}_{\mathcal{C}}$ at the calibration instances and on how well the selected kernel is adapted to such values. The smoothness of $\mathfrak{X}_{\mathcal{C}}$ is determined by that of the conformal score function $S(X,Y)$ defined by the classification rule used. Such a function may be highly non-smooth at the training instances used to learn the classification rule. However, for classifiers that generalize well, the conformal score is expected to be fairly smooth at the calibration instances, which are drawn independently of the training samples. Cases in which the conformal score is highly non-smooth across the instance space are possible but of reduced practical relevance, as such conformal scores would be largely uninformative. + +The bounds in Theorem 2 not only provide theoretical guarantees but can also be used in practice for kernel selection. In particular, a kernel can be chosen from a candidate set (e.g., multiple choices of bandwidth for a Gaussian kernel) as the one achieving the smallest minimum norm interpolation (or $\varepsilon$ -approximation). The minimum norm of interpolating functions can be efficiently computed using standard properties of RKHSs (see e.g., [21]). Specifically, for kernel $K$ the minimum norm of interpolators for $\mathfrak{X}_{\mathcal{C}}$ at the calibration instances is given by $\mathbf{u}^{\top}\gamma$ for $\gamma$ that solves the linear system $\mathbf{K}\gamma = \mathbf{u}$ , where the kernel matrix $\mathbf{K}$ and vector $\mathbf{u}$ are given by $K_{i,j} = K(Z_i,Z_j)$ and $u_{i} = \mathfrak{X}_{\mathcal{C}}(Z_{i})$ for $i,j\in [nc]$ . Notice also that the preprocessing step that selects the most suitable kernel over $s$ candidates only changes the term $\log (2 / \delta)$ in the bounds of Theorem 2 to the term $\log (2s / \delta)$ , as a direct consequence of the union bound. + +Algorithm 3 details the implementation of the proposed kernel-based methods to obtain label weights for the calibration instances. The kernel selection process requires to solve $s$ linear systems of equations of size $nc$ so that it has time complexity $\mathcal{O}(sn^3c^3)$ . Then, label weights are obtained by solving a quadratic program with $nc$ variables and $n + r$ linear constraints. Highly accurate solutions of such problems can be obtained using interior point methods with complexity per iteration $\mathcal{O}(n^3c^3)$ due to solving a linear system (see Chapter 16 in [22]). These methods allow to efficiently solve the quadratic program for a number classes $c$ and unlabeled calibration instances $n$ in the order of tens and thousands, respectively. For cases with a higher number of classes or calibration samples, first-order methods can result in efficient implementations since their complexity per iteration is $\mathcal{O}(n^2c^2)$ due to a matrix-vector multiplication (see Chapter 16 in [22]). The memory complexity of Algorithm 3 is $\mathcal{O}(n^2c^2)$ corresponding to the storage of a kernel matrix. Furthermore, the computational complexity of Algorithm 3 can be significantly reduced taking into account the common sparsity of the matrices involved. The matrix $\mathbf{A}$ corresponding to the $n$ linear equality constraints is always highly sparse (ratio of nonzero values $1/n$ ), and the kernel matrix $\mathbf{K}$ is also highly sparse using separable kernels (ratio of nonzero values $1/c^2$ ). Appendix E shows that Algorithm 3 + +using interior point methods achieves running times ranging from tens to hundreds of seconds for cases with up to 10 classes and up to 3,000 calibration samples. + +Algorithm 3 Kernel-based method to obtain label weights for calibration instances +Input: Unsupervised calibration samples $\{X_i\}_{i\in [n]}$ training samples $\{(\widetilde{X}_i,\widetilde{Y}_i)\}_{i\in [m]}$ Kernels $K^{(1)},K^{(2)},\ldots ,K^{(s)}$ matrix and vector for inequality constraints B and b Output: Label weights $w_{i}(y),i\in [n],y\in \mathcal{V}$ +1: $Z_{(i - 1)c + y}\gets (X_i,y)$ for $i\in [n],y\in \mathcal{V},\widetilde{Z}_i\gets (\widetilde{X}_i,\widetilde{Y}_i)$ for $i\in [m]$ Kernel selection +2: Compute a rough approximate $\widehat{q}_0$ of the quantile using Algorithm 2 with the naive approach given by label weights $\{\widehat{w}_i(y) = \mathbb{I}\{y = h(X_i)\} \}$ +3: Compute vector $\mathbf{u}\in \mathbb{R}^{nc}$ given by $u_{i} = \mathfrak{X}_{0}(Z_{i}) = \mathbb{I}\{S(Z_{i})\leq \widehat{q}_{0}\}$ for $i\in [nc]$ +4: for $t\in [s]$ do +5: Compute kernel matrix $\mathbf{K}^{(t)}\in \mathbb{R}^{nc\times nc}$ given by $K_{i,j}^{(t)} = K^{(t)}(Z_i,Z_j)$ for $i,j\in [nc]$ +6: $\pmb{\gamma}^{(t)}\gets$ solution of $\mathbf{K}^{(t)}\pmb{\gamma}^{(t)} = \mathbf{u}$ +7: end for +8: $t^* \gets \arg \min_{t\in [s]}\{\mathbf{u}^\top \pmb{\gamma}^{(t)}\}$ Label weights computation +9: $\mathbf{K}\gets \mathbf{K}^{(t^*)},\mathbf{A}\gets \mathbf{I}_n\otimes \mathbf{1}_c^\top$ +10: Compute vector $\mathbf{v}\in \mathbb{R}^{nc}$ given by $v_{i} = \sum_{j\in [m]}K^{(t^{*})}(Z_{i},\widetilde{Z}_{j})$ for $i\in [nc]$ +11: w $\leftarrow$ solution of min $\frac{1}{n}\mathbf{w}^{\top}\mathbf{K}\mathbf{w}-\frac{2}{m}\mathbf{v}^{\top}\mathbf{w}$ s.t. w $\succeq 0,\mathbf{A}\mathbf{w} = 1_n,\mathbf{B}\mathbf{w}\preceq \mathbf{b}$ + +# 5 Numerical results + +The experiments assess the performance obtained by the proposed approach for split conformal prediction with unsupervised classification using 9 common benchmark datasets. The coverage probabilities and prediction set sizes are compared with those provided by the conventional approach that uses supervised calibration samples and by the naive approach that uses label predictions for calibration samples. Due to the amount of theoretical results presented, this section is necessarily concise. The code implementing the methods presented and reproducing the experiments can be found at https://github.com/MachineLearningBCAM/Unsupervised-conformal-prediction-NeurIPS2025. The supplementary materials provide implementation details and additional results in Appendix E, including running time assessments, as well as results for different target coverages and types of conformal scores. + +For each dataset, 400 random realizations are generated by randomly partitioning the samples in training, calibration, and test sets. The training samples are used to learn classification rules: random forests for tabular datasets and neural networks for image datasets (fine-tuned Resnet50 for CIFAR10' and ImageNet10' datasets). The results in the main paper use the adaptive conformal score proposed in [2,9], and the Appendix E reports results with the conformal score directly given by probability estimates [1]. Set-prediction rules are obtained from the scoring functions using $n$ calibration samples for multiple values of $n$ , and coverage probabilities and prediction sets sizes are then computed using the test samples. The presented methods are implemented using Algorithm 2 with label weights obtained by Algorithm 3 using $m = n$ training samples randomly drawn from the samples used to learn the classification rule, and a Gaussian kernel selected from a set given by 10 candidate bandwidth (scale) parameters. Full implementation details are provided in Appendix E. + +Table 1 shows the coverage probability and prediction set size obtained in the 9 datasets using $n = 1,000$ calibration samples for a target coverage of $1 - \alpha = 0.9$ . The results in the table indicate that the performance of the methods presented is only slightly inferior to that of conventional methods with supervised calibration. In accordance with the discussion in Section 3.2 regarding the weakened guarantees in scenarios with many classes, 'Letter' – which includes 26 classes – is the only dataset showing significantly poorer performance. The table also shows that the naive approach + +Table 1: Coverage probability and prediction set size for conventional approach with supervised calibration ('Supervised'), proposed approach with unsupervised calibration ('Unsupervised'), and naive unsupervised approach ('Unsup. Naive'). Results are shown as: mean (interquartile interval), and correspond to $1 - \alpha = 0.9$ . + +
DatasetCoverage probabilityPrediction set size
SupervisedUnsupervisedUnsup. NaiveSupervisedUnsupervised
Drybean0.90 (0.89,0.91)0.91 (0.90,0.91)0.84 (0.83,0.85)1.23 (1.20,1.25)1.25 (1.22,1.27)
Forestcov0.90 (0.89,0.91)0.89 (0.89,0.90)0.62 (0.61,0.63)1.90 (1.87,1.94)1.87 (1.85,1.89)
Satellite0.90 (0.89,0.91)0.90 (0.89,0.91)0.84 (0.83,0.85)1.45 (1.42,1.48)1.44 (1.41,1.48)
USPS0.90 (0.89,0.91)0.90 (0.89,0.91)0.88 (0.87,0.89)1.11 (1.09,1.13)1.11 (1.09,1.14)
MNIST0.90 (0.89,0.91)0.90 (0.90,0.91)0.86 (0.85,0.87)1.19 (1.16,1.21)1.20 (1.18,1.23)
Fashion0.90 (0.89,0.91)0.91 (0.90,0.91)0.80 (0.79,0.81)1.43 (1.39,1.47)1.45 (1.42,1.48)
CIFAR100.90 (0.89,0.91)0.91 (0.90,0.92)0.88 (0.87,0.89)1.00 (0.98,1.01)1.01 (0.99,1.02)
Imagnet100.90 (0.89,0.91)0.90 (0.89,0.91)0.86 (0.85,0.87)1.65 (1.66,1.70)1.62 (1.57,1.68)
Letter0.90 (0.89,0.91)0.85 (0.84,0.87)0.69 (0.68,0.71)5.58 (5.24,5.90)4.14 (3.80,4.48)
+ +![](images/40cf794361f4a4adaf0a3de2abf6165e2432e6b13e8db093037009d7f5f325ea.jpg) +(a) CIFAR10' Dataset + +![](images/ea65ef631a04ab189270ca8c9b4e5301a1b3269a4edb8a0fa8f5d4ff58eba05c.jpg) +(b) 'Drybean' Dataset +Figure 2: Decrease of average coverage gap with the number of calibration samples. + +can only provide acceptable performance in cases where the learned classification rule is highly accurate (e.g., the classification error in 'CIFAR10' is $4\%$ ). Figure 2 shows the average coverage gap $(\mathbb{E}\{\left|1 - \alpha - \mathbb{P}\{\mathrm{Y} \in \mathcal{C}(\mathrm{X})|\mathcal{D}\}\right|\}$ with $\alpha = 0.1$ ) obtained with different number of calibration samples. The results in the figure exhibit that coverage gaps smaller than 0.015 can be obtained with few thousand unsupervised calibration samples. In accordance with the theoretical results in Section 3.2, the coverage gap decreases with the number of unsupervised calibration samples comparably to cases with supervised calibration. + +# 6 Conclusion + +The paper presents methods for split conformal classification with unsupervised calibration. In the proposed approach, label weights for calibration samples are obtained so that training and calibration samples become statistically indistinguishable by a non-parametric two-sample test. The theoretical and experimental results show that the methods presented can achieve performance comparable to that with supervised calibration, at the expenses of a moderate degradation in performance guarantees and computational efficiency. The methodology presented can lead to methods that remove the need to acquire labeled examples at the calibration stage, thus making conformal prediction more cost-efficient in general and viable even when new labels cannot be obtained after training. + +Limitations: The proposed methodology is broadly effective but does not scale well with the number of classes, as discussed in the theoretical and experimental results. As shown in Sections 3 and 4, the excess coverage gap in the methods presented decreases at a rate $\mathcal{O}(1 / \sqrt{n} + 1 / \sqrt{m})$ in general scenarios, but the constants can be large if the conformal score used is highly non-smooth at the calibration samples. Such cases are rare since the score function is learned without calibration samples, and globally non-smooth scores are largely uninformative in the first place. The other main limitation of the methods is a significantly higher computational cost compared to conventional split conformal methods, which only require computing one quantile. This drawback is of limited relevance in practice since the computational cost remains small relative to typical learning processes. + +# Acknowledgements + +Funding in direct support of this work has been provided by project PID2022-137063NB-I00 funded by MCIN/AEI/10.13039/501100011033 and the European Union "NextGenerationEU"/PRTR, BCAM Severo Ochoa accreditation CEX2021-001142-S/MICIN/AEI/10.13039/501100011033 funded by the Ministry of Science and Innovation (Spain), and program BERC-2022-2025 funded by the Basque Government. + +# References + +[1] Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world. Springer, New York, NY, 2005. +[2] Anastasios N. Angelopoulos and Stephen Bates. Conformal prediction: A gentle introduction. 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TensorFlow Datasets: A collection of ready-to-use datasets. https://www.tensorflow.org/datasets, 2022. + +# Appendices + +# A Auxiliary lemmas + +Lemma 1. For $i \in [n]$ , let $(X_i, Y_i)$ be an instance-label pair in $\mathcal{X} \times \mathcal{Y}$ , and $w_i(1), w_i(2), \ldots, w_i(c)$ be positive label weights that add to one. If $\widehat{q}$ is the quantile + +$$ +\widehat {q} = \text {Q u a n t i l e} \left( \right.\left\{ \right.\left(S \left(X _ {i}, y\right), w _ {i} (y) / n\right)\left. \right)\left. \right\} _ {(i, y) \in [ n ] \times \mathcal {Y}}; (1 - \alpha) (1 + 1 / n)\left. \right) \tag {17} +$$ + +for some $\alpha \in [0,1]$ , and $E$ is given by + +$$ +E = \frac {1}{n} \sum_ {i \in [ n ]} \mathfrak {X} _ {\mathcal {C}} (X _ {i}, Y _ {i}) - \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} (X _ {i}, y) +$$ + +for the function $\mathfrak{X}_{\mathcal{C}}$ defined in (6). Then, we have + +$$ +\widehat {q} \geq \text {Q u a n t i l e} \left(\left\{\left(S \left(X _ {i}, Y _ {i}\right), 1 / n\right) \right\} _ {i \in [ n ]}; \left(1 - \alpha + n E / (n + 1)\right) \left(1 + 1 / n\right)\right). \tag {18} +$$ + +In addition, if the values $\{S(X_i,y)\}_{(i,y)\in [n]\times \mathcal{Y}}$ are distinct, we have + +$$ +\hat {q} \leq \text {Q u a n t i l e} \left(\left\{\left(S \left(X _ {i}, Y _ {i}\right), 1 / n\right) \right\} _ {i \in [ n ]}; (1 - \alpha + 1 / (n + 1) + n E / (n + 1)) (1 + 1 / n)\right). \tag {19} +$$ + +Proof. From the definition of $\widehat{q}$ in (17), we have + +$$ +\frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} \left(X _ {i}, y\right) = \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathbb {I} \{S \left(X _ {i}, y\right) \leq \widehat {q} \} \geq \frac {(n + 1) (1 - \alpha)}{n} \tag {20} +$$ + +then, we directly get + +$$ +\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathfrak {X} _ {\mathcal {C}} \left(X _ {i}, Y _ {i}\right) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {I} \left\{S \left(X _ {i}, Y _ {i}\right) \leq \widehat {q} \right\} \\ = \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathbb {I} \left\{S \left(X _ {i}, y\right) \leq \widehat {q} \right\} + E \geq \frac {(n + 1) (1 - \alpha + \frac {n E}{n + 1})}{n} \tag {21} \\ \end{array} +$$ + +that gives (18). + +If the values $\{S(X_i, y)\}_{(i, y) \in [n] \times \mathcal{Y}}$ are distinct, by the definition of $\widehat{q}$ in (17) we have + +$$ +\frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} \left(X _ {i}, y\right) = \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathbb {I} \{S \left(X _ {i}, y\right) \geq \widehat {q} \} \leq \frac {(n + 1) (1 - \alpha)}{n} + \frac {1}{n} \tag {22} +$$ + +because $0 \leq w_{i}(y) \leq 1$ for any $y \in \mathcal{V}, i \in [n]$ . Then, analogously to the previous case we get + +$$ +\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathfrak {X} _ {\mathcal {C}} \left(X _ {i}, Y _ {i}\right) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {I} \{S \left(X _ {i}, Y _ {i}\right) \leq \widehat {q} \} (23) \\ \leq \frac {(n + 1) (1 - \alpha + \frac {n E}{n + 1})}{n} + \frac {1}{n} = \frac {(n + 1) (1 - \alpha + \frac {1}{n + 1} + \frac {n E}{n + 1})}{n} (24) \\ \end{array} +$$ + +that together with (21) gives (19). + +Lemma 2. Let $\mathbf{w} = [w_{1}(1), w_{1}(2), \ldots, w_{1}(c), w_{2}(1), \ldots, w_{n}(c)]^{\top} \in \mathbb{R}^{nc}$ be label weights for the calibration samples that satisfy $\Phi_{n,m}^{\mathcal{F}}(\mathbf{w}) \leq V_{\mathrm{opt}}$ , and $\mathcal{F}$ be a family of real-valued functions with bounded differences, i.e., $|f(z) - f(z')| \leq B$ , for any $f \in \mathcal{F}$ and $z, z' \in \mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ . + +If E is the RV + +$$ +\mathrm {E} = \frac {1}{n} \sum_ {i \in [ n ]} \mathfrak {X} _ {\mathcal {C}} (\mathrm {X} _ {i}, \mathrm {Y} _ {i}) - \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} (\mathrm {X} _ {i}, y). +$$ + +Then, with probability at least $1 - \delta$ we have + +$$ +\left| \mathrm {E} \right| \leq V _ {\text {o p t}} + D \left(\mathfrak {X} _ {C}, \mathcal {F}\right) + 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) + B \sqrt {\left(\frac {1}{n} + \frac {1}{m}\right) \frac {\log (1 / \delta)}{2}}. \tag {25} +$$ + +Let $\mathcal{H}$ be the RKHS given by kernel $K$ that satisfies $0\leq K(z,z^{\prime})\leq \kappa^{2}$ for any $z,z^{\prime}\in \mathcal{Z}$ . If $\mathbf{w}$ is a solution of (4) using the family $\mathcal{F}_R = \{f\in \mathcal{H}:\| f\|_{\mathcal{H}}\leq R\}$ , and the weights $\mathbf{w}^*$ corresponding to the actual calibration labels are included in the feasible set of (4), we have + +$$ +| \mathrm {E} | \leq D \left(\mathfrak {X} _ {\mathcal {C}}, \mathcal {F} _ {R}\right) + 2 \kappa \left(1 + \sqrt {\log (1 / \delta)}\right) R \sqrt {\frac {1}{n} + \frac {1}{m}} \text {, f o r a l l} R > 0 \tag {26} +$$ + +with probability at least $1 - \delta$ + +Proof. For any $f \in \mathcal{F}$ we have + +$$ +\begin{array}{l} | \mathrm {E} | = \left| \frac {1}{n} \sum_ {i \in [ n ]} \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right) \right| \\ = \left| \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} \left(\mathbb {I} \{y = Y _ {i} \} - w _ {i} (y)\right) \mathfrak {X} _ {\mathcal {C}} \left(X _ {i}, y\right) \right\rbrace \\ \leq \left| \right. \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} \left(\mathbb {I} \{y = Y _ {i} \} - w _ {i} (y)\right)\left( \right.\mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right)\left. \right\rbrace - f \left(\mathrm {X} _ {i}, y\right)\left. \right)\left. \right| (27) \\ + \left| \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} \left(\mathbb {I} \{y = \mathrm {Y} _ {i} \} - w _ {i} (y)\right) f \left(\mathrm {X} _ {i}, y\right) \right|. (28) \\ \end{array} +$$ + +We bound separately the two terms in (27) and (28). For (27) we directly have + +$$ +\left| \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} \left(\mathbb {I} \{y = \mathrm {Y} _ {i} \} - w _ {i} (y)\right) \left(\mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right) - f \left(\mathrm {X} _ {i}, y\right)\right) \right| \leq \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} \left| \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right) - f \left(\mathrm {X} _ {i}, y\right) \right| \tag {29} +$$ + +because $0 \leq w_{i}(y) \leq 1$ for any $i \in [n]$ and $y \in \mathcal{V}$ . For (28), we have + +$$ +\begin{array}{l} \left| \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} \left(\mathbb {I} \{y = \mathrm {Y} _ {i} \} - w _ {i} (y)\right) f (\mathrm {X} _ {i}, y) \right| \\ = \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) + \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) - \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) f \left(\mathrm {X} _ {i}, y\right) \right| \\ \leq \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right| + \left| \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) - \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) f \left(\mathrm {X} _ {i}, y\right) \right| \\ \leq \sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right| + \Phi_ {n, m} ^ {\mathcal {F}} (\mathbf {w}) \tag {30} \\ \end{array} +$$ + +Therefore, we have + +$$ +| \mathrm {E} | \leq \Phi_ {n, m} ^ {\mathcal {F}} (\mathbf {w}) + D \left(\mathfrak {X} _ {\mathcal {C}}, \mathcal {F}\right) + \sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right|. \tag {31} +$$ + +Then, the bound in (25) follows using that $\Phi_{n,m}^{\mathcal{F}}(\mathbf{w})\leq V_{\mathrm{opt}}$ , and upper bounding the 'sup' in (31). Specifically, to bound the expectation of the 'sup' in (31) we use a usual argument based on symmetrization. If $(\overline{\mathrm{X}}_1,\overline{\mathrm{Y}}_1),(\overline{\mathrm{X}}_2,\overline{\mathrm{Y}}_2),\ldots ,(\overline{\mathrm{X}}_t,\overline{\mathrm{Y}}_t)$ for $t = \max (n,m)$ form a different set of i.i.d. + +samples, we have + +$$ +\begin{array}{l} \mathbb {E} \Big \{\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f (\mathrm {X} _ {i}, \mathrm {Y} _ {i}) - \frac {1}{m} \sum_ {i \in [ m ]} f (\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}) \right| \Big \} \\ = \mathbb {E} \Big \{\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \mathbb {E} \Big \{\frac {1}{n} \sum_ {i \in [ n ]} f \left(\bar {\mathrm {X}} _ {i}, \bar {\mathrm {Y}} _ {i}\right) \Big \} \right. \\ + \mathbb {E} \Bigl \{\frac {1}{m} \sum_ {i \in [ m ]} f (\bar {\mathrm {X}} _ {i}, \bar {\mathrm {Y}} _ {i}) \Bigr \} - \frac {1}{m} \sum_ {i \in [ m ]} f (\tilde {\mathrm {X}} _ {i}, \tilde {\mathrm {Y}} _ {i}) \Big | \Big \} \\ \leq \mathbb {E} \left\{\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - f \left(\bar {\mathrm {X}} _ {i}, \bar {\mathrm {Y}} _ {i}\right) \right| \right\} + \mathbb {E} \left\{\sup _ {f \in \mathcal {F}} \left| \frac {1}{m} \sum_ {i \in [ m ]} f \left(\bar {\mathrm {X}} _ {i}, \bar {\mathrm {Y}} _ {i}\right) - f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) \right| \right\} (32) \\ = \mathbb {E} \left\{\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} \sigma_ {i} \left(f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - f \left(\bar {\mathrm {X}} _ {i}, \bar {\mathrm {Y}} _ {i}\right)\right) \right| \right\} \\ + \mathbb {E} \left\{\sup _ {f \in \mathcal {F}} \left| \frac {1}{m} \sum_ {i \in [ m ]} \sigma_ {i} \left(f \left(\bar {\mathrm {X}} _ {i}, \bar {\mathrm {Y}} _ {i}\right) - f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right)\right) \right| \right\} \\ \leq 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) (33) \\ \end{array} +$$ + +where in (32) we have used Jensen's and triangular inequalities, and $\sigma_1,\sigma_2,\ldots ,\sigma_k$ are i.i.d. Rademacher variables $\sigma_{i}\in \{-1,1\}$ . Then, using McDiarmid's inequality (see e.g., [16]), with probability at least $1 - \delta$ we get + +$$ +\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right| \leq 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) + B \sqrt {\frac {1}{2} \left(\frac {1}{n} + \frac {1}{m}\right) \log \frac {1}{\delta}} \tag {34} +$$ + +because changing any $(\mathrm{X}_i,\mathrm{Y}_i)$ or $(\widetilde{\mathrm{X}}_i,\widetilde{\mathrm{Y}}_i)$ by other sample in the left hand side of (34), changes the 'sup' at most $B / n$ or $B / m$ , since $|f(x,y) - f(x^{\prime},y^{\prime})|\leq B$ , for any $f\in \mathcal{F}$ and pairs $(x,y),(x^{\prime},y^{\prime})$ . Therefore, the bound in (25) is directly obtained from (31) and (34). + +The bound in (26) for families in RKHSs is obtained similarly from (30). The term corresponding to maximum deviations of empirical averages can be bounded more tightly in the RKHS case, without resorting to Rademacher complexities, as follows. Firstly, for the case $R = 1$ we have + +$$ +\begin{array}{l} \sup _ {f \in \mathcal {F} _ {1}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right| \\ = \sup _ {f \in \mathcal {H}, \| f \| _ {\mathcal {H}} \leq 1} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right| \\ = \left\| \frac {1}{n} \sum_ {i \in [ n ]} K \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}, \cdot\right) - \frac {1}{m} \sum_ {i \in [ m ]} K \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}, \cdot\right) \right\| _ {\mathcal {H}} \tag {35} \\ \end{array} +$$ + +as a consequence of the dual characterization of the norm in $\mathcal{H}$ . In addition, the expression in (35) also bounds $\Phi_{n,m}^{\mathcal{F}_1}(\mathbf{w})$ since $\mathbf{w}$ is solution of (12) and $\mathbf{w}^*$ is feasible in (12). + +The expectation of (35) can be bounded using Jensen's inequality $(\mathbb{E}\{s\} \leq \sqrt{\mathbb{E}\{s^2\}})$ and the bound + +$$ +\begin{array}{l} \mathbb {E} \Big \{\left\| \frac {1}{n} \sum_ {i \in [ n ]} K (\mathrm {X} _ {i}, \mathrm {Y} _ {i}, \cdot) - \frac {1}{m} \sum_ {i \in [ m ]} K (\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}, \cdot) \right\| _ {\mathcal {H}} ^ {2} \Big \} \\ = \mathbb {E} \left\{\frac {1}{n ^ {2}} \sum_ {i _ {1}, i _ {2} \in [ n ]} K \left(\mathrm {X} _ {i _ {1}}, \mathrm {Y} _ {i _ {1}}, \mathrm {X} _ {i _ {2}}, \mathrm {Y} _ {i _ {2}}\right) + \frac {1}{m ^ {2}} \sum_ {j _ {1}, j _ {2} \in [ m ]} K \left(\widetilde {\mathrm {X}} _ {j _ {1}}, \widetilde {\mathrm {Y}} _ {j _ {1}}, \widetilde {\mathrm {X}} _ {j _ {2}}, \widetilde {\mathrm {Y}} _ {j _ {2}}\right) \right. \\ \left. - \frac {2}{n m} \sum_ {i \in [ n ], j \in [ m ]} K \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}, \widetilde {\mathrm {X}} _ {j}, \widetilde {\mathrm {Y}} _ {j}\right) \right\} \\ = \left(\frac {1}{n} + \frac {1}{m}\right) \mathbb {E} \{K (\mathrm {Z}, \mathrm {Z}) - K (\mathrm {Z}, \mathrm {Z} ^ {\prime}) \} \leq \left(\frac {1}{n} + \frac {1}{m}\right) \kappa^ {2} \tag {36} \\ \end{array} +$$ + +where $Z$ and $Z'$ denote two independent copies of the RV (X, Y). + +Therefore, the bound in (26) follows using McDiarmid's inequality. Specifically, with probability at least $1 - \delta$ we get + +$$ +\begin{array}{l} \sup _ {f \in \mathcal {F} _ {1}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f (\mathrm {X} _ {i}, \mathrm {Y} _ {i}) - \frac {1}{m} \sum_ {i \in [ m ]} f (\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}) \right| \\ = \left\| \frac {1}{n} \sum_ {i \in [ n ]} K \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}, \cdot\right) - \frac {1}{m} \sum_ {i \in [ m ]} K \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}, \cdot\right) \right\| _ {\mathcal {H}} (37) \\ \leq \kappa \sqrt {\frac {1}{n} + \frac {1}{m}} + R \kappa \sqrt {\left(\frac {1}{n} + \frac {1}{m}\right) \log \frac {1}{\delta}} (38) \\ \end{array} +$$ + +because changing any $(\mathrm{X}_i,\mathrm{Y}_i)$ or $(\widetilde{\mathrm{X}}_i,\widetilde{\mathrm{Y}}_i)$ by other sample in (37), changes the norm at most $\sqrt{2}\kappa /n$ or $\sqrt{2}\kappa /m$ , as a consequence of the triangular inequality. Moreover, the bound in (26) is satisfied simultaneously for all $R > 0$ with probability at least $1 - \delta$ because + +$$ +\sup _ {f \in \mathcal {F} _ {R}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f (\mathrm {X} _ {i}, \mathrm {Y} _ {i}) - \frac {1}{m} \sum_ {i \in [ m ]} f (\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}) \right| = R \sup _ {f \in \mathcal {F} _ {1}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f (\mathrm {X} _ {i}, \mathrm {Y} _ {i}) - \frac {1}{m} \sum_ {i \in [ m ]} f (\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}) \right| +$$ + +for any $R > 0$ + +![](images/08e07008406534c79983a285d4520f107d5c4ebc38c497f87bd323a2dcdad2b1.jpg) + +# B Proof of Theorem 1 + +Let $\widehat{q}$ be the quantile obtained in Step 4 of Algorithm 2. Using (18) in Lemma 1 we have + +$$ +\widehat {q} \geq \text {Q u a n t i l e} \left(\left\{\left(S \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right), 1 / n\right) \right\} _ {i \in [ n ]}; (1 - \alpha + n \mathrm {E} / (n + 1)) (1 + 1 / n)\right) +$$ + +for + +$$ +\mathrm {E} = \frac {1}{n} \sum_ {i \in [ n ]} \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right). +$$ + +In addition, using (25) in Lemma 2 we have + +$$ +| \mathrm {E} | \leq V _ {\text {o p t}} + D \left(\mathfrak {X} _ {\mathcal {C}}, \mathcal {F}\right) + 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) + B \sqrt {\left(\frac {1}{n} + \frac {1}{m}\right) \log \frac {\log (2 / \delta)}{2}} = G _ {n, m} ^ {\mathcal {F}} \tag {39} +$$ + +with probability at least $1 - \delta /2$ . Therefore, with probability at least $1 - \delta /2$ we have + +$$ +\widehat {q} \geq \text {Q u a n t i l e} \left(\left\{\left(S \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right), 1 / n\right) \right\} _ {i \in [ n ]}; \left(1 - \alpha - n G _ {n, m} ^ {\mathcal {F}} / (n + 1)\right) \left(1 + 1 / n\right)\right) +$$ + +due to the monotonicity of the quantile function. + +Therefore, with probability at least $1 - \delta /2$ we have + +$$ +\begin{array}{l} \mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D} \right\} = \mathbb {P} \left\{\mathrm {S} (\mathrm {X}, \mathrm {Y}) \leq \widehat {q} | \mathcal {D} \right\} \\ \geq \mathbb {P} \left\{ \right.\mathrm {S} (\mathrm {X}, \mathrm {Y}) \leq \text {Q u a n t i l e} \left( \right.\left\{ \right.\mathrm {S} \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right), 1 / n\left. \right)\left. \right\} _ {i \in [ n ]}; (1 - \alpha - n G _ {n, m} ^ {\mathcal {F}} / (n + 1)) (1 + 1 / n)\left. \right) | \mathcal {D} \left. \right\}. \tag {40} \\ \end{array} +$$ + +Using standard results for training-conditional coverage probabilities in conventional split conformal prediction (see e.g., [2, 10]), in cases with $0 \leq 1 - \alpha - G_{n,m}^{\mathcal{F}} n / (n + 1) \leq 1$ , the probability in (40) is lower bounded by + +$$ +1 - \alpha - \frac {n}{n + 1} G _ {n, m} ^ {\mathcal {F}} - \sqrt {\frac {\log (2 / \delta)}{2 n}} \geq 1 - \alpha - G _ {n, m} ^ {\mathcal {F}} - \sqrt {\frac {\log (2 / \delta)}{2 n}} \tag {41} +$$ + +with probability at least $1 - \delta / 2$ . In addition, the lower bound in (41) is also satisfied in other cases because if $1 - \alpha - G_{n,m}^{\mathcal{F}} n / (n + 1) < 0$ the lower bound is trivially valid and the case $1 - \alpha - G_{n,m}^{\mathcal{F}} n / (n + 1) > 1$ is not possible because both $\alpha$ and $G_{n,m}^{\mathcal{F}}$ are positive. Therefore, the inequality in (8) follows directly from the lower bound in (41) using the union bound. + +In case the values $\{S(X_i, y)\}_{(i, y) \in [n] \times \mathcal{Y}}$ are distinct with probability one, using (19) in Lemma 1 we have + +$$ +\widehat {q} \leq \operatorname {Q u a n t i l e} \left(\left\{\left(S \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right), 1 / n\right) \right\} _ {i \in [ n ]}; \left(1 - \alpha + n G _ {n, m} ^ {\mathcal {F}} (n + 1) + 1 / (n + 1)\right) \left(1 + 1 / n\right)\right) +$$ + +with probability at least $1 - \delta /2$ + +Therefore, with probability at least $1 - \delta /2$ we have + +$$ +\begin{array}{l} \mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D}\right) = \mathbb {P} \left\{\mathrm {S} (\mathrm {X}, \mathrm {Y}) \leq \hat {q} | \mathcal {D} \right\} \\ \leq \mathbb {P} \left\{ \right.\mathrm {S} (\mathrm {X}, \mathrm {Y}) \leq \text {Q u a n t i l e} \left( \right.\left\{ \right.\mathrm {S} \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right), 1 / n\left. \right)\left. \right\} _ {i \in [ n ]}; (1 - A) (1 + 1 / n)\left. \right) | \mathcal {D} \left. \right\} \tag {42} \\ \end{array} +$$ + +with $A = \alpha - n G_{n,m}^{\mathcal{F}} / (n + 1) - 1 / (n + 1)$ . + +Using standard results for training-conditional coverage probabilities in conventional split conformal prediction (see e.g., [2, 10]), in cases with $0 \leq 1 - \alpha + G_{n,m}^{\mathcal{F}} n / (n + 1) + 1 / (n + 1) \leq 1$ , the probability in (42) follows a Beta distribution $\operatorname{Beta}(n + 1 - \lfloor (n + 1)A \rfloor, \lfloor (n + 1)A \rfloor)$ where $\lfloor \cdot \rfloor$ denotes the largest integer that lower bounds the argument. Such a Beta distribution has mean + +$$ +\frac {n + 1 - \lfloor (n + 1) A \rfloor}{n + 1} \leq 1 - A + \frac {1}{n + 1} = 1 - \alpha + \frac {2}{n + 1} + \frac {n}{n + 1} G _ {n, m} ^ {\mathcal {F}} +$$ + +and is sub-Gaussian with parameter at most $1 / 4n$ . Hence, using a Chernoff bound the probability in (42) can be upper bounded by + +$$ +1 - \alpha + \frac {2}{n + 1} + \frac {n}{n + 1} G _ {n, m} ^ {\mathcal {F}} + \sqrt {\frac {\log (2 / \delta)}{2 n}} \leq 1 - \alpha + \frac {2}{n + 1} + G _ {n, m} ^ {\mathcal {F}} + \sqrt {\frac {\log (2 / \delta)}{2 n}} \tag {43} +$$ + +with probability at least $1 - \delta / 2$ . In addition, the upper bound in (43) is also satisfied in other cases because if $1 - \alpha + G_{n,m}^{\mathcal{F}} n / (n + 1) + 1 / (n + 1) > 1$ the upper bound is trivially valid and the case $1 - \alpha + G_{n,m}^{\mathcal{F}} n / (n + 1) + 1 / (n + 1) < 0$ is not possible because $\alpha \leq 1$ and $G_{n,m}^{\mathcal{F}} \geq 0$ . Therefore, the inequality in (10) follows directly from the upper bound in (43) using the union bound. + +# C Proof of Theorem 2 + +The proof is analogous to that of Theorem 1 above. The main difference lies in the high-probability bound of $|\mathrm{E}|$ for + +$$ +\mathrm {E} = \frac {1}{n} \sum_ {i \in [ n ]} \mathfrak {X} _ {\mathcal {C}} (\mathrm {X} _ {i}, \mathrm {Y} _ {i}) - \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} w _ {i} (y) \mathfrak {X} _ {\mathcal {C}} (\mathrm {X} _ {i}, y). +$$ + +If $\mathcal{F}_R = \{f\in \mathcal{H}:\| f\|_{\mathcal{H}}\leq R\}$ , using (26) in Lemma 2, we have + +$$ +| \mathrm {E} | \leq D (\mathfrak {X} _ {\mathcal {C}}, \mathcal {F} _ {R}) + 2 \kappa R \sqrt {\frac {1}{n} + \frac {1}{m}} \left(1 + \sqrt {\log (2 / \delta)}\right), \text {f o r a l l} R > 0 +$$ + +with probability at least $1 - \delta /2$ . Therefore, + +$$ +\begin{array}{l} | \mathrm {E} | \leq \inf _ {R > 0} D (\mathfrak {X} _ {\mathcal {C}}, \mathcal {F} _ {R}) + 2 \kappa R \sqrt {\frac {1}{n} + \frac {1}{m}} \left(1 + \sqrt {\log (2 / \delta)}\right) \\ = \min _ {f \in \mathcal {H}} \frac {1}{n} \sum_ {i \in [ n ], y \in \mathcal {Y}} | \mathfrak {X} _ {\mathcal {C}} \left(\mathrm {X} _ {i}, y\right) - f \left(\mathrm {X} _ {i}, y\right) | + 2 \kappa \left(1 + \sqrt {\log (2 / \delta)}\right) \sqrt {\frac {1}{n} + \frac {1}{m}} \| f \| _ {\mathcal {H}} = G _ {n, m} ^ {K} \tag {44} \\ \end{array} +$$ + +with probability at least $1 - \delta /2$ + +The rest of the proof is totally analogous to that for Theorem 1 (see Appendix B for the details). In particular, the bound for $|\mathbf{E}|$ in (44) results in + +$$ +\widehat {q} \geq \operatorname {Q u a n t i l e} \big (\{(S (\mathrm {X} _ {i}, \mathrm {Y} _ {i}), 1 / n) \} _ {i \in [ n ]}; (1 - \alpha - n G _ {n, m} ^ {K} / (n + 1)) (1 + 1 / n) \big), \text {w . p .} \geq 1 - \frac {\delta}{2} +$$ + +using (18) in Lemma 1, and + +$$ +\widehat {q} \leq \text {Q u a n t i l e} \left(\left\{\left(S \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right), 1 / n\right) \right\} _ {i \in [ n ]}; \left(1 - \alpha + n G _ {n, m} ^ {K} (n + 1) + 1 / (n + 1)\right) (1 + 1 / n)\right), \text {w . p .} \geq 1 - \frac {\delta}{2} +$$ + +if $\{S(\mathrm{X}_i,y)\}_{(i,y)\in [n]\times \mathcal{Y}}$ are distinct with probability one, using (19) in Lemma 1. + +Therefore, (13) and (15) are obtained taking into account that + +$$ +\mathbb {P} \left\{\mathrm {Y} \in \mathcal {C} (\mathrm {X}) | \mathcal {D} \right\} = \mathbb {P} \left\{\mathrm {S} (\mathrm {X}, \mathrm {Y}) \leq \hat {q} | \mathcal {D} \right\} +$$ + +and that, for any $0 \leq \alpha' \leq 1$ , the RV + +$$ +\mathbb {P} \left\{\mathrm {S} (\mathrm {X}, \mathrm {Y}) \leq \text {Q u a n t i l e} \left(\left\{\left(S \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right), 1 / n\right) \right\} _ {i \in [ n ]}; \left(1 - \alpha^ {\prime}\right) (1 + 1 / n)\right) | \mathcal {D} \right\} +$$ + +follows a distribution that is dominated by a $\mathrm{Beta}(n + 1 - \lfloor (n + 1)\alpha^{\prime}\rfloor ,\lfloor (n + 1)\alpha^{\prime}\rfloor)$ , and is equal to such a Beta distribution in the case where $\{S(\mathrm{X}_i,y)\}_{(i,y)\in [n]\times \mathcal{Y}}$ are distinct with probability one. + +# D Additional results + +The following provides the detailed derivations leading to (11) and (12) in the main paper. + +Proposition 1. Let $\varepsilon_{\mathrm{opt}}$ be an upper bound for the difference between $\Phi_{\mathcal{F}}(\mathbf{w})$ and the optimum value of (4). If the feasible set in (4) includes the label weights $\mathbf{w}^*$ corresponding to the actual calibration labels, with probability at least $1 - \delta$ we have + +$$ +\Phi_ {n, m} ^ {\mathcal {F}} (\mathbf {w}) \leq \varepsilon_ {\text {o p t}} + 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + R _ {m} (\mathcal {F})\right) + B \sqrt {\frac {1}{2} \left(\frac {1}{n} + \frac {1}{m}\right) \log \frac {1}{\delta}}. \tag {45} +$$ + +Proof. If $\Phi^{*}$ denotes the optimum value of (4), we have + +$$ +\Phi_ {n, m} ^ {\mathcal {F}} (\mathbf {w}) = \left(\Phi_ {n, m} ^ {\mathcal {F}} (\mathbf {w}) - \Phi^ {*}\right) + \Phi^ {*} \leq \varepsilon_ {\text {o p t}} + \Phi^ {*} \tag {46} +$$ + +and since $\mathbf{w}^*$ is included in the feasible set of (4), we have + +$$ +\Phi^ {*} \leq \sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right|. \tag {47} +$$ + +Using McDiarmid's inequality (see e.g., [16]) with probability at least $1 - \delta$ we have + +$$ +\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right| \leq 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) + B \sqrt {\frac {1}{2} \left(\frac {1}{n} + \frac {1}{m}\right) \log \frac {1}{\delta}} \tag {48} +$$ + +because changing any $(\mathrm{X}_i,\mathrm{Y}_i)$ or $(\widetilde{\mathrm{X}}_i,\widetilde{\mathrm{Y}}_i)$ by other sample in the left hand side of (48), changes the 'sup' at most $B / n$ or $B / m$ , and we have + +$$ +\mathbb {E} \left\{\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i \in [ n ]} f \left(\mathrm {X} _ {i}, \mathrm {Y} _ {i}\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {\mathrm {X}} _ {i}, \widetilde {\mathrm {Y}} _ {i}\right) \right| \right\} \leq 2 \left(\mathcal {R} _ {n} (\mathcal {F}) + \mathcal {R} _ {m} (\mathcal {F})\right) \tag {49} +$$ + +using a symmetrization argument, as in the proof of Lemma 2. + +Therefore, the result in (45) is obtained by combining the inequalities in (46), (47), and (48). + +![](images/a2f8538d7db03fca8495b9eb7e4234b3e121aad38745d19da850044275dec7e8.jpg) + +Proposition 2. Let $\mathcal{H}$ be an RKHS given by kernel $K$ over $\mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ . If $\mathcal{F}$ is the family of real-valued functions $\mathcal{F} = \{f \in \mathcal{H} : \| f\|_{\mathcal{H}} \leq 1\}$ , we have + +$$ +\sup _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {(i, y) \in [ n ] \times \mathcal {Y}} w _ {i} (y) f \left(X _ {i}, y\right) - \frac {1}{m} \sum_ {i \in [ m ]} f \left(\widetilde {X} _ {i}, \widetilde {Y} _ {i}\right) \right| = \left(\frac {1}{n ^ {2}} \mathbf {w} ^ {\top} \mathbf {K} \mathbf {w} - \frac {2}{n m} \mathbf {v} ^ {\top} \mathbf {w} + \frac {1}{m ^ {2}} \mathbf {1} ^ {\top} \widetilde {\mathbf {K}} \mathbf {1}\right) ^ {\frac {1}{2}}. \tag {50} +$$ + +Proof. Using the reproducing property of RKHSs and the dual characterization of the norm in $\mathcal{H}$ , we have + +$$ +\begin{array}{l} \sup _ {f: \| f \| _ {\mathcal {H}} \leq 1} \left| \frac {1}{n} \sum_ {(i, y) \in [ n ] \times \mathcal {Y}} w _ {i} (y) f (X _ {i}, y) - \frac {1}{m} \sum_ {i \in [ m ]} f (\widetilde {X} _ {i}, \widetilde {Y} _ {i}) \right| ^ {2} \\ = \sup _ {f: \| f \| _ {\mathcal {H}} \leq 1} \left| \frac {1}{n} \sum_ {(i, y) \in [ n ] \times \mathcal {Y}} w _ {i} (y) < f, K ((X _ {i}, y), \cdot) > _ {\mathcal {H}} - \frac {1}{m} \sum_ {i \in [ m ]} < f, K ((\widetilde {X} _ {i}, \widetilde {Y} _ {i}), \cdot) > _ {\mathcal {H}} \right| ^ {2} \\ = \sup _ {f: \| f \| _ {\mathcal {H}} \leq 1} \left| < f, \frac {1}{n} \sum_ {(i, y) \in [ n ] \times \mathcal {Y}} w _ {i} (y) K ((X _ {i}, y), \cdot) > _ {\mathcal {H}} - < f, \frac {1}{m} \sum_ {i \in [ m ]} K ((\widetilde {X} _ {i}, \widetilde {Y} _ {i}), \cdot) > _ {\mathcal {H}} \right| ^ {2} \tag {51} \\ \end{array} +$$ + +$$ +\begin{array}{l} = \sup _ {f: \| f \| _ {\mathcal {H}} \leq 1} \left| < f, \frac {1}{n} \sum_ {(i, y) \in [ n ] \times \mathcal {Y}} w _ {i} (y) K ((X _ {i}, y), \cdot) - \frac {1}{m} \sum_ {i \in [ m ]} K ((\widetilde {X} _ {i}, \widetilde {Y} _ {i}), \cdot) > _ {\mathcal {H}} \right| ^ {2} \\ = \left\| \frac {1}{n} \sum_ {(i, y) \in [ n ] \times \mathcal {Y}} w _ {i} (y) K ((X _ {i}, y), \cdot) - \frac {1}{m} \sum_ {i \in [ m ]} K ((\widetilde {X} _ {i}, \widetilde {Y} _ {i}), \cdot) \right\| _ {\mathcal {H}} ^ {2} \\ = \frac {1}{n ^ {2}} \sum_ {(i _ {1}, y _ {1}), (i _ {2}, y _ {2}) \in [ n ] \times \mathcal {Y}} w _ {i _ {1}} (y _ {1}) w _ {i _ {2}} (y _ {2}) K ((X _ {i _ {1}}, y _ {1}), (X _ {i _ {2}}, y _ {2})) \\ - \frac {2}{n m} \sum_ {\left(i _ {1}, y _ {1}\right) \in [ n ] \times \mathcal {Y}, i _ {2} \in [ m ]} w _ {i _ {1}} \left(y _ {1}\right) K \left(\left(X _ {i _ {1}}, y _ {1}\right), \left(\widetilde {X} _ {i _ {2}}, \widetilde {Y} _ {j}\right)\right) \tag {52} \\ + \frac {1}{m ^ {2}} \sum_ {i _ {1}, i _ {2} \in [ m ]} K \left(\left(\tilde {X} _ {i _ {1}}, \tilde {Y} _ {i _ {1}}\right), \left(\tilde {X} _ {i _ {2}}, \tilde {Y} _ {i _ {2}}\right)\right) \\ \end{array} +$$ + +where $< \cdot ,\cdot >_{\mathcal{H}}$ denotes the inner product in $\mathcal{H}$ . Then, the result in (50) is directly obtained using the definition of $\mathbf{K}\in \mathbb{R}^{nc\times nc}$ , $\widetilde{\mathbf{K}}\in \mathbb{R}^{m\times m}$ , and $\mathbf{v}\in \mathbb{R}^{nc}$ as + +$$ +K _ {i, j} = K \left(Z _ {i}, Z _ {j}\right), \text {f o r} i, j \in [ n c ], \widetilde {K} _ {i, j} = K \left(\widetilde {Z} _ {i}, \widetilde {Z} _ {j}\right), \text {f o r} i, j \in [ m ] +$$ + +$$ +v _ {i} = \sum_ {j \in [ m ]} K \left(Z _ {i}, \widetilde {Z} _ {j}\right), \text {f o r} i \in [ n c ] +$$ + +for $Z_{(i - 1)c + y} = (X_i,y)$ for $i\in [n],y\in \mathcal{V}$ , and $\widetilde{Z}_i = (\widetilde{X}_i,\widetilde{Y}_i)$ for $i\in [m]$ . + +![](images/5aea2f8f0c1b13774f1abc48f56bc0fbe6168986b069411c579191c9547d31ea.jpg) + +# E Implementation details and additional experimental results + +In the following we provide further implementation details and describe the datasets used in the experimental results. Then, we complement the results in the main paper by including the results for the running times of the methods presented as well as additional results with other target coverages and conformal scores. In addition, the Github https://github.com/MachineLearningBCAM/Unsupervised-conformal-prediction-NeurIPS2025 and the folder Implementation_Unsupervised_Conformal in the supplementary materials provide Matlab code corresponding to the presented methods with the setting used in the numerical results. + +# E.1 Implementation and datasets details + +We utilize 9 publicly available and common benchmark datasets for classification tasks: 'Drybean', 'Forestcov', 'Satellite', 'USPS', 'MNIST', 'FashionMNIST', 'CIFAR10', 'ImageNet10' (first 10 classes of ImageNet), and 'Letter'. These datasets can be found in the UCI repository [23], Tensor Flow datasets [24], and Kaggle website https://www.kaggle.com. The main characteristics of the datasets used is provided in Table 2. + +Table 2: Key characteristics of benchmark datasets used in the experiments. + +
Dataset# Samples# ClassesInput Dimensionality
Drybean13,611716
Forestcov581,012754
Satellite6,435636
Letter20,0002616
USPS9,29810256
MNIST70,00010784
FashionMNIST70,00010784
CIFAR1060,000103,072
ImageNet1013,00010150,528
+ +In each random realization, the datasets are randomly partitioned in training, calibration, and test sets. The sizes of the training and test sets are 3,000 samples and 1,000, and that of the calibration set is varied from 10 to 3,000. The classification rules are obtained using random forests in the tabular datasets ('Drybean', 'Forestcov', 'Satellite', and 'Letter') and using neural networks in + +![](images/eecc841f9bcb3e8366e24b1320d368c665db16b11bacca016d1beffe0c65abce.jpg) +Figure 3: Running times of the methods presented as the number of calibration samples increases. For datasets with up to 10 classes, the running times range from tens to hundreds of seconds using few thousands of samples. + +the image datasets ('USPS', 'MNIST', 'FashionMNIST', 'CIFAR10', and 'ImageNet10'). Specifically, the random forests are given by 200 decision trees and learning is carried out with 20 maximum number of splits and 10 minimum leaf size. The neural networks for 'USPS', 'MNIST', and 'FashionMNIST' datasets have two hidden layers of sizes 128 and 64 and learning is carried out with regularization parameter 0.001. The implementation for 'CIFAR10' and 'ImageNet10' datasets is slightly different due their higher complexity. In particular, the classification rule for those datasets is learned only once by fine-tuning a Resnet50 until a validation error of $4\%$ in 'CIFAR10' and $11\%$ in 'ImageNet10' datasets, using SGD with momentum, initial learning rate of 0.001, minibatch size of 64, and regularization parameter 0.001. + +Set-prediction rules are obtained using the adaptive conformal score proposed in [2,9] and also using the conformal score directly given by probability estimates [1]. Such scores are computed from the probability estimates provided by the classification rules (proportion of votes from trees in random forest methods, and soft-max outputs in neural networks methods). The conventional approach for split conformal prediction with supervised calibration is implemented using Algorithm 1 (see also, e.g., [2]). The naive approach with unsupervised calibration is implemented using Algorithm 1 with labels corresponding to the predictions obtained by the classification rules. + +The presented method is implemented using Algorithm 2 with label weights obtained by Algorithm 3 using a Gaussian kernel selected from a set given by 10 candidate bandwidth (scale) parameters $\sigma = \sigma_0\sqrt{d / 2}$ , for $\sigma_0\in \{10^{-1 + t / 3},t = 0,1,\ldots ,9\}$ , and $m = n$ training samples randomly drawn from the samples used to learn the classification rule. The inequality constraints in (12) given by $\mathbf{B}$ and $\mathbf{b}$ correspond to estimates for the cross-entropy loss of the classification rule obtained in the training stage. Specifically, $\mathbf{B}\in \mathbb{R}^{1\times nc}$ is given as in (5) and $\mathbf{b} = nL\in \mathbb{R}$ with $L$ the estimate of the cross-entropy plus its sample standard deviation obtained using validation samples (10-fold cross validation in all datasets but in 'CIFAR10' and 'ImageNet10' datasets for which 1,000 validation samples were used for those estimates). In addition, for 'CIFAR10' and 'ImageNet10' datasets the instances $\{\widetilde{X}_i\}_{i\in [m]}$ and $\{X_{i}\}_{i\in [m]}$ used in Algorithm 3 are given by the 512-dimensional feature representations given by the penultimate layer of the pre-trained Resnet18 network. + +# E.2 Running times of the methods presented + +The main computational burden of the methods presented corresponds to the Step 11 in Algorithm 3 that solves a quadratic optimization problem. In the experimental results carried out, such optimization problem is solved using interior point methods with Mosek solver https://www.mosek.com. Figure 3 shows the running time achieved by Algorithm 3 using a regular desktop machine in the datasets 'Drybean', 'CIFAR10', and 'Letter'. These datasets are representative of the overall running time because they have assorted characteristics. In particular the number of classes in those datasets is 7 ('Drybean'), 10 ('CIFAR10'), and 26 ('Letter'). As shown in the figure, the pre + +sented methods have running times ranging from tens to hundreds of seconds for cases with up to 10 classes and up to 3,000 calibration samples. In line with the discussion in Section 4 of the paper, the only problematic cases are those composed by a sizable number of classes since the computational complexity of the methods presented increases cubically with the number of classes. + +# E.3 Additional results with the adaptive conformal score + +The main paper shows experimental results using the adaptive conformal score [2, 9] given by + +$$ +S (X, Y) = \sum_ {i \in [ c ]} \widehat {p} (i | X) \mathbb {I} \{\widehat {p} (i | X) > \widehat {p} (Y | X) \} + \mathrm {U} \widehat {p} (Y | X) +$$ + +for U a random value from Unif(0, 1) and $\widehat{p}$ the probability estimate provided by the classification rule used. This section complements the results in the main paper with other target probabilities and with other datasets for figures such as those in Fig. 1 and Fig. 2. Specifically, Tables 3 and 4 show coverage probabilities and prediction set sizes analogous to those in Table 1 in the main paper but using target probabilities $1 - \alpha = 0.95$ and $1 - \alpha = 0.85$ . In addition, Figures 4, 5, and 6 show violin plots such as those in Fig. 1 using 'ImageNet10', 'FashionMNIST', and datasets, and Figure 7 shows the decrease of coverage gap with the number of calibration samples using 'ImageNet10' and 'MNIST' datasets. + +Table 3: Coverage probability and prediction set size for conventional approach with supervised calibration ('Supervised'), proposed approach with unsupervised calibration ('Unsupervised'), and naive unsupervised approach ('Unsup. Naive'). Results are shown as: mean (interquartile interval), and correspond to $1 - \alpha = .95$ . + +
DatasetCoverage probabilityPrediction set size
SupervisedUnsupervisedUnsup. NaiveSupervisedUnsupervised
Drybean0.95 (0.94,0.96)0.95 (0.94,0.96)0.91 (0.91,0.92)1.52 (1.47,1.56)1.51 (1.47,1.55)
Forestcov0.95 (0.94,0.96)0.94 (0.93,0.95)0.68 (0.67,0.69)2.24 (2.18,2.29)2.15 (2.11,2.19)
Satellite0.95 (0.94,0.96)0.95 (0.94,0.96)0.91 (0.90,0.92)1.77 (1.71,1.82)1.79 (1.73,1.84)
USPS0.95 (0.94,0.96)0.95 (0.95,0.96)0.94 (0.93,0.94)1.32 (1.28,1.35)1.33 (1.29,1.36)
MNIST0.95 (0.94,0.95)0.95 (0.95,0.96)0.93 (0.92,0.93)1.43 (1.38,1.48)1.45 (1.41,1.50)
Fashion0.95 (0.94,0.96)0.95 (0.95,0.96)0.87 (0.86,0.88)1.84 (1.76,1.90)1.84 (1.79,1.89)
CIFAR100.95 (0.94,0.96)0.95 (0.95,0.96)0.93 (0.93,0.94)1.10 (1.08,1.12)1.11 (1.10,1.13)
ImagNet100.95 (0.94,0.96)0.94 (0.94,0.95)0.93 (0.92,0.94)2.20 (2.13,2.27)2.07 (2.00,2.14)
Letter0.95 (0.94,0.96)0.90 (0.89,0.92)0.80 (0.79,0.82)9.06 (8.38,9.64)5.63 (4.99,6.33)
+ +Table 4: Coverage probability and prediction set size for conventional approach with supervised calibration ('Supervised'), proposed approach with unsupervised calibration ('Unsupervised'), and naive unsupervised approach ('Unsup. Naive'). Results are shown as: mean (interquartile interval), and correspond to $1 - \alpha = .85$ . + +
DatasetCoverage probabilityPrediction set size
SupervisedUnsupervisedUnsup. NaiveSupervisedUnsupervised
Drybean0.85 (0.84,0.86)0.86 (0.85,0.87)0.78 (0.76,0.79)1.09 (1.06,1.11)1.11 (1.09,1.13)
Forestcov0.85 (0.84,0.86)0.86 (0.85,0.87)0.57 (0.56,0.58)1.65 (1.61,1.69)1.70 (1.67,1.72)
Satellite0.85 (0.84,0.86)0.85 (0.84,0.86)0.78 (0.76,0.79)1.27 (1.24,1.30)1.26 (1.23,1.29)
USPS0.85 (0.84,0.86)0.85 (0.84,0.86)0.83 (0.81,0.84)1.00 (0.97,1.01)1.01 (0.98,1.03)
MNIST0.85 (0.84,0.86)0.86 (0.84,0.87)0.81 (0.80,0.82)1.06 (1.03,1.08)1.07 (1.05,1.09)
Fashion0.85 (0.84,0.86)0.86 (0.85,0.87)0.74 (0.72,0.75)1.24 (1.21,1.27)1.26 (1.24,1.29)
CIFAR100.85 (0.84,0.86)0.86 (0.85,0.87)0.82 (0.81,0.83)0.93 (0.91,0.94)0.94 (0.92,0.95)
ImagNet100.85 (0.84,0.86)0.85 (0.84,0.87)0.80 (0.79,0.81)1.37 (1.32,1.41)1.38 (1.33,1.43)
Letter0.85 (0.84,0.86)0.81 (0.80,0.83)0.60 (0.58,0.62)4.07 (3.84,4.28)3.38 (3.18,3.58)
+ +![](images/dbf4fcdfebb5896a4bf19e5118a3a4138512aeb408ee554bc4232737298afdca.jpg) +Figure 4: Coverage probabilities and prediction set sizes over 400 random partitions of 'ImageNet10' dataset with target coverage $1 - \alpha = 0.9$ and number of calibration samples ranging from 100 to 2,000. + +![](images/64bfb30b71673a0d57667f00a66cd2c5874e52e1b64bcaafde5bf64aa5ffeea5.jpg) + +![](images/ba4d8937462e0924c30f5afd5dfc53ac116e72948874f2b25fe4439aff620a87.jpg) +Figure 5: Coverage probabilities and prediction set sizes over 400 random partitions of 'FashionMNIST' dataset with target coverage $1 - \alpha = 0.9$ and number of calibration samples ranging from 100 to 2,000. + +![](images/c71a783565fd1531ce8e99ec4f4b4c7d1795d24e2e6bb0235b53628e1c8e93f4.jpg) + +![](images/a856ff18b10e5fd8b73e632175edc20978dc5396373a2b7b4a9ab0a40c9de0b3.jpg) +Figure 6: Coverage probabilities and prediction set sizes over 400 random partitions of 'Forestcov' dataset with target coverage $1 - \alpha = 0.9$ and number of calibration samples ranging from 100 to 2,000. + +![](images/c24771002d759a6fe5ad07400b5552b6ec884a37b413b531eb4cab2d8cac8e8b.jpg) + +![](images/3906baa79b006605a247268d9561d5124e343b40b90e8c6d631ee8efc269cb96.jpg) +(a) 'ImageNet10' Dataset +Figure 7: Decrease of average coverage gap with the number of calibration samples. + +![](images/613e0cba4c4552c1c7e1dfedb46661971e822a8e8f1770bfcf5fdcc7006ad556.jpg) +(b) 'MNIST' Dataset + +# E.4 Additional results with the conformal score given by probability estimates + +This section complements the results provided above by using the conformal score given by probability estimates, i.e., $S(X,Y) = 1 - \widehat{p}(Y|X)$ for $\widehat{p}$ the probability estimate provided by the classification rule used. Specifically, Tables 5, 6, and 7 show coverage probabilities and prediction set for target coverages $1 - \alpha \in \{0.9, 0.95, 0.85\}$ , Figures 8, 9, and 10 show violin plots using 'CIFAR10', 'FashionMNIST', and 'Forestcov' datasets, and Figures 11 and 11 show decrease of coverage gap with the number of calibration samples using 'CIFAR10', 'Drybean', 'FashionMNIST', and 'MNIST' datasets. + +Table 5: Coverage probability and prediction set size for conventional approach with supervised calibration ('Supervised'), proposed approach with unsupervised calibration ('Unsupervised'), and naive unsupervised approach ('Unsup. Naive'). Results are shown as: mean (interquartile interval), and correspond to the conformal score given by probability estimates with $1 - \alpha = 0.9$ . + +
DatasetCoverage probabilityPrediction set size
SupervisedUnsupervisedUnsup. NaiveSupervisedUnsupervised
Drybean0.90 (0.89,0.91)0.89 (0.89,0.90)0.84 (0.83,0.85)1.02 (1.01,1.04)1.01 (1.00,1.02)
Forestcov0.90 (0.89,0.91)0.90 (0.89,0.91)0.64 (0.63,0.65)1.74 (1.69,1.79)1.75 (1.72,1.77)
Satellite0.90 (0.89,0.91)0.90 (0.89,0.91)0.84 (0.83,0.86)1.44 (1.40,1.48)1.44 (1.41,1.48)
USPS0.90 (0.89,0.91)0.91 (0.90,0.92)0.89 (0.88,0.90)0.91 (0.90,0.92)0.93 (0.92,0.94)
MNIST0.90 (0.89,0.91)0.92 (0.91,0.92)0.87 (0.86,0.88)0.94 (0.93,0.95)0.97 (0.96,0.98)
Fashion0.90 (0.89,0.91)0.91 (0.90,0.92)0.79 (0.78,0.80)1.19 (1.16,1.22)1.24 (1.23,1.26)
CIFAR100.90 (0.89,0.91)0.91 (0.90,0.92)0.89 (0.88,0.90)0.91 (0.90,0.92)0.92 (0.91,0.93)
Letter0.90 (0.89,0.91)0.86 (0.85,0.87)0.72 (0.70,0.73)5.22 (4.85,5.56)3.91 (3.72,4.09)
+ +Table 6: Coverage probability and prediction set size for conventional approach with supervised calibration ('Supervised'), proposed approach with unsupervised calibration ('Unsupervised'), and naive unsupervised approach ('Unsup. Naive'). Results are shown as: mean (interquartile interval), and correspond to the conformal score given by probability estimates with $1 - \alpha = 0.95$ . + +
DatasetCoverage probabilityPrediction set size
SupervisedUnsupervisedUnsup. NaiveSupervisedUnsupervised
Drybean0.95 (0.94,0.96)0.95 (0.94,0.96)0.86 (0.86,0.87)1.16 (1.13,1.18)1.16 (1.14,1.18)
Forestcov0.95 (0.94,0.96)0.94 (0.93,0.94)0.67 (0.66,0.68)2.10 (2.06,2.14)2.00 (1.98,2.02)
Satellite0.95 (0.94,0.96)0.95 (0.94,0.96)0.91 (0.90,0.92)1.78 (1.72,1.83)1.79 (1.71,1.84)
USPS0.95 (0.94,0.96)0.96 (0.95,0.97)0.93 (0.92,0.93)0.99 (0.98,1.00)1.01 (1.00,1.02)
MNIST0.95 (0.94,0.96)0.96 (0.95,0.96)0.90 (0.90,0.91)1.07 (1.05,1.09)1.11 (1.09,1.13)
Fashion0.95 (0.94,0.96)0.95 (0.94,0.96)0.82 (0.81,0.82)1.55 (1.49,1.63)1.56 (1.53,1.60)
CIFAR100.95 (0.94,0.96)0.96 (0.96,0.97)0.93 (0.92,0.94)0.98 (0.98,0.99)1.01 (1.00,1.02)
Letter0.95 (0.94,0.96)0.90 (0.90,0.91)0.75 (0.74,0.76)8.66 (8.18,9.12)5.46 (5.18,5.72)
+ +Table 7: Coverage probability and prediction set size for conventional approach with supervised calibration ('Supervised'), proposed approach with unsupervised calibration ('Unsupervised'), and naive unsupervised approach ('Unsup. Naive'). Results are shown as: mean (interquartile interval), and correspond to the conformal score given by probability estimates with $1 - \alpha = 0.85$ . + +
DatasetCoverage probabilityPrediction set size
SupervisedUnsupervisedUnsup. NaiveSupervisedUnsupervised
Drybean0.85 (0.84,0.86)0.85 (0.84,0.86)0.81 (0.80,0.82)0.92 (0.90,0.93)0.92 (0.91,0.93)
Forestcov0.85 (0.84,0.86)0.86 (0.85,0.87)0.62 (0.61,0.63)1.49 (1.46,1.52)1.54 (1.51,1.56)
Satellite0.85 (0.84,0.86)0.86 (0.85,0.87)0.62 (0.61,0.63)1.49 (1.46,1.52)1.54 (1.51,1.56)
USPS0.85 (0.84,0.86)0.86 (0.85,0.87)0.84 (0.83,0.85)0.86 (0.85,0.87)0.87 (0.86,0.88)
MNIST0.85 (0.84,0.86)0.86 (0.85,0.87)0.83 (0.82,0.84)0.87 (0.86,0.88)0.89 (0.88,0.90)
Fashion0.85 (0.84,0.86)-0.87 (0.86,0.88)-0.76 (0.75,0.78)1.02 (1.00,1.04)1.08 (1.06,1.09)
-CIFAR100.85 (0.84,0.86)0.85 (0.84,0.86)0.84 (0.83,0.85)0.86 (0.84,0.87)0.86 (0.85,0.87)
Letter0.85 (0.84,0.86)0.82 (0.81,0.83)0.69 (0.68,0.70)3.59 (3.34,3.82)3.04 (2.92,3.17)
+ +![](images/210f9d83b4573b301d734afa9913f1ca2610028ec9b58964c24dcc397e1489d5.jpg) +Figure 8: Coverage probabilities and prediction set sizes over 400 random partitions of 'CIFAR10' dataset with conformal score given by probability estimates, target coverage $1 - \alpha = 0.9$ and number of calibration samples ranging from 100 to 2,000. + +![](images/feb986b9e852a8385a38cfbfa44610066fed297abc99e177e9dd436bdcfc941b.jpg) + +![](images/74481f45f72e8955fdd844de73e5b7b7601c0a7ab0592b625cc2e394c5933970.jpg) +Figure 9: Coverage probabilities and prediction set sizes over 400 random partitions of 'FashionMNIST' dataset with conformal score given by probability estimates, target coverage $1 - \alpha = 0.9$ and number of calibration samples ranging from 100 to 2,000. + +![](images/99ac2733cff71eb878ecd901878e4db328da7318f15a832697bc725058fb1a25.jpg) + +![](images/7675f06bbc98a124e366efcc46ba7dc434fa411037e69abf80c2a5c6deb00a77.jpg) +Figure 10: Coverage probabilities and prediction set sizes over 400 random partitions of 'Forestcov' dataset with conformal score given by probability estimates, target coverage $1 - \alpha = 0.9$ and number of calibration samples ranging from 100 to 2,000. + +![](images/31f1519dfd0a033f57ad50e4e44a97aa1c0f00754d7fefb0aa6c8e1f168d7da8.jpg) + +![](images/c3c632a773000ee353c67b314d7efbb36fc066d1f88804225e5ef81ba56cde59.jpg) +(a) 'CIFAR10' Dataset + +![](images/a9cf25a3c1dcaae0a176ec0d20d535a461ed3048f1cf61a9e3b3f552ef7ded6c.jpg) +(b) 'Drybean' Dataset + +![](images/dc1b794d640bbc7b1666106e8a236b2fc308c1370a84f91c4946df49fd6ab998.jpg) +Figure 11: Decrease of average coverage gap with the number of calibration samples with conformal score given by probability estimates. +(a) 'FashionMNIST' Dataset +Figure 12: Decrease of average coverage gap with the number of calibration samples with conformal score given by probability estimates. + +![](images/defe7931ed8898ade549b92c2cca427f0fbd5df3e5db21de3e53fae0f9b33162.jpg) +(b) 'MNIST' Dataset + +# NeurIPS Paper Checklist + +# 1. 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Despite their strong generative capabilities, they face critical limitations in image editing tasks: inaccurate inversion processes for mapping real images back into the latent space, and gradient entanglement issues during editing often result in outputs that do not faithfully reflect the target prompt. Recent efforts have attempted to directly map source and target distributions via ODE-based approaches without inversion; however, these methods still yield suboptimal editing quality. In this work, we propose a flow decomposition-and-aggregation framework built upon an inversion-free formulation to address these limitations. Specifically, we semantically decompose the target prompt into multiple sub-prompts, compute an independent flow for each, and aggregate them to form a unified editing trajectory. While we empirically observe that decomposing the original flow enhances diversity in the target space, generating semantically aligned outputs still requires consistent guidance toward the full target prompt. To this end, we design a projection and soft-aggregation mechanism for flow, inspired by gradient conflict resolution in multi-task learning. This approach adaptively weights the sub-target velocity fields, suppressing semantic redundancy while emphasizing distinct directions, thereby preserving both diversity and consistency in the final edited output. Experimental results demonstrate that our method outperforms existing zero-shot editing approaches in terms of semantic fidelity and attribute disentanglement. The code is available at https://github.com/Harvard-AI-and-Robotics-Lab/SplitFlow. + +# 1 Introduction + +Flow-based generative models have demonstrated superiority in synthesizing images and also in text-to-image generation task. Building on these advances, recent research has actively explored image editing methods that modify a given image to align with a target prompt. Due to the nature of diffusion [20, 7, 23] and flow-based generative models [1, 13, 15], which generate samples from noise through iterative refinement, editing typically requires an inversion step to estimate the corresponding noisy latent representation. However, this inversion process is often imperfect and fails to recover an exact latent that reconstructs the original image. As a result, the editing process may suffer from semantic drift, visual distortion, or inconsistent attribute manipulation. Even + +when a precise inversion allows near-perfect reconstruction, an empirical trade-off arises between reconstructability and editability. Latents that are highly optimized to match the input image tend to be rigid and entangled, meaning that adjusting a single attribute often leads to unintended changes in unrelated features [2]. + +Recently, some efforts have been made to improve the editing process based on rectified flow models [19, 26, 3, 29, 31] and also inversion-free method that directly maps the source and target distribution [11], but these methods show limited semantic fidelity due to gradient entanglement and prompt-latent misalignment. In particular, when a complex target prompt contains multiple semantic attributes (e.g., "a german shepherd dog with black sunglasses jumping on the grass with mouth opened"), a single editing trajectory guided by the full prompt often leads to entangled gradients and conflicting directions in the semantic space. This makes it difficult to isolate and control the influence of individual attributes, resulting in either under-edited or overly distorted outputs. + +Motivated by these issues, we propose SplitFlow, which decomposes the flow induced by the target prompt into independent sub-target flows derived from semantically decomposed sub-prompts. By computing independent editing flows for each sub-prompt and aggregating them into a unified trajectory while mitigating flow conflict, our method achieves improvements in both fidelity and editability. Performance on the PIE-Bench benchmark [10] demonstrates that the proposed approach outperforms prior methods. + +The contributions of this paper are threefold: + +- We demonstrate that decomposing the editing flow improves the fidelity of the edited image, especially in preserving background consistency. +- We propose SplitFlow, which progressively approximates the target latent through flow decomposition followed by flow aggregation. +- We introduce a projection-and-aggregation method that aligns sub-target flows with the target direction, while preserving their semantic diversity and alleviating potential conflicts velocity field that arise during integration. + +# 2 Related Work + +Diffusion and Rectified Flow. Diffusion models [21, 18] synthesize images by gradually reversing a noising process that transforms data into Gaussian noise. Starting from random noise, they iteratively denoise through learned score functions, generating high-quality outputs over many timesteps. Rectified Flow (RF) [13, 15] introduces a more stable and efficient generation process via a straight trajectory in latent space. + +Text-to-Image Inversion and Editing. Text-guided image editing aims to modify a given image to match a target text prompt. Most diffusion-based editing pipelines rely on first mapping the image back into the model's latent space, typically in the form of Gaussian noise. DDIM [21] enables approximate inversion via a deterministic trajectory, but suffers from cumulative errors due to its linear approximation. To improve inversion accuracy, optimization-based approaches [16, 25, 8] have been proposed, though at a high computational cost. On the editing side, early methods achieve localized edits by fine-tuning diffusion models [5, 12] or manipulating cross-attention maps [6, 24]. While effective, these methods still operate within the diffusion framework and depend heavily on either inversion or fine-tuning, which are often inefficient for practical use. + +Recent efforts have explored RF models for editing, which offer more stable and efficient sampling trajectories than diffusion models. RF-Inversion [19] formulates editing as an optimal control problem to balance editability and fidelity, RF-Solver [26] incorporates attention injection to guide editing, and FireFlow [3] employs higher-order ODE solvers to improve inversion accuracy. Despite their advantages, these methods still depend on iterative re-noising procedures, making them prone to cumulative error. FTEdit [29] reduces errors by performing iterative average at each inversion timestep, which is basically equivalent to increasing the number of sampling steps. + +Inversion-Free Editing. Unlike traditional inversion-based approaches, which require recovering the noise latent that originally generated the image, inversion-free methods bypass this step and directly optimize in the image or latent space. InfEdit [30] introduces an early inversion-free + +framework based on the consistency models [22]. FlowEdit [11] extends this idea to RF models by directly manipulating image-space velocity differences. However, it lacks directional selectivity and often struggles with global edits. These works highlight that explicitly modeling semantic flow differences enhances editing control and efficiency. + +Multi-Task and Flow Decomposition. Our method builds upon insights from gradient conflict resolution in multi-task learning [32, 17, 14], where conflicting objectives are resolved via adaptive re-weighting. Similarly, we propose to semantically decompose the target prompt into multiple sub-flows and adaptively aggregate them for consistent and diverse edits. To our knowledge, our work is the first to incorporate flow decomposition and aggregation into text-based image editing framework. + +# 3 Preliminaries + +# 3.1 Flow Matching and Rectified Flow + +Let $\mathbb{R}^d$ denote the data space, and let $x_0\in \mathbb{R}^d$ be an initial data point sampled from a source distribution $p_0$ . Flow Matching (FM) methods [15, 13, 1] aim to learn a time-dependent vector field $v_{\theta}(x_{t},t):[0,1]\times \mathbb{R}^{d}\to \mathbb{R}^{d}$ , such that the solution to the following ODE transports $x_0\sim p_0$ to $x_{1}\sim p_{1}$ , where $p_1$ denotes the target distribution: + +$$ +d x _ {t} = v _ {\theta} \left(x _ {t}, t\right) d t \tag {1} +$$ + +The solution $x_{t}$ describes a continuous trajectory defined by the ODE, starting from the initial point $x_{0}$ and reaching the target point $x_{1}$ . The ground-truth vector field $v^{*}(x_{t}, t)$ governs this trajectory and induces a distributional path $x_{t}$ that satisfies the boundary conditions $x_{t=0} = x_{0}$ and $x_{t=1} = x_{1}$ . The objective of FM is to learn a parameterized vector field $v_{\theta}(x_{t}, t)$ that approximates $v^{*}(x_{t}, t)$ by minimizing the following regression loss: + +$$ +\mathcal {L} (\theta) = \mathbb {E} _ {x _ {t}, t} [ \| v _ {\theta} (x _ {t}, t) - v ^ {*} (x _ {t}, t) \| _ {2} ^ {2} ], \tag {2} +$$ + +where $||\cdot ||_2^2$ denotes mean square error. Here, FM enables deterministic sampling via ODEs, avoiding the stochastic noise accumulation seen in diffusion models. By leveraging known couplings between distributions, it ensures both interpretability and fidelity. This leads to faster, more stable sampling and a principled connection to optimal transport, making FM a strong candidate for generative modeling. Rectified Flow (RF) [15], a special case of FM, aims to learn an ODE whose solution closely follows straight-line trajectories between pairs of points sampled from $x_0$ and $x_{1}$ . The ground-truth vector field is defined as $v^{*}(x_{t},t) = x_{1} - x_{0}$ , where $x_{t}$ is the linear interpolation between $x_0$ and $x_{1}$ . The training objective can thus be formulated as: + +$$ +\mathcal {L} (\theta) = \mathbb {E} _ {x _ {t}, t} \| v _ {\theta} (x _ {t}, t) - (x _ {1} - x _ {0}) \| _ {2} ^ {2}, \quad \text {w h e r e} \quad x _ {t} = (1 - t) x _ {0} + t x _ {1}. \tag {3} +$$ + +By eliminating the need for stochastic sampling and instead following deterministic linear paths, RF enables faster and more stable performance in downstream tasks such as ODE-based generative modeling and image editing. + +# 3.2 Inversion-free Image Editing + +Image editing aims to transform a source image $x_0^{src}$ , guided by a source text prompt $\varphi^{src}$ , into an edited image $x_0^{tgt}$ using a target text prompt $\varphi^{tgt}$ . In RF, the trajectory between noise and image is assumed to be linear, meaning that the noisy latent at any timestep can be obtained via linear interpolation. Leveraging this property, inversion-free image editing method [11] bypass the need for latent inversion by estimating the vector field difference between the source and target images at each timestep. This enables the construction of a transition path in the clean image space, gradually mapping the source image toward the target. Specifically, the target-aligned latent at timestep $t$ , denoted as $x_t^{\mathrm{FE}}$ , can be approximated by: + +$$ +x _ {t} ^ {\mathrm {F E}} = x _ {0} ^ {s r c} + x _ {t} ^ {t g t} - x _ {t} ^ {s r c}. \tag {4} +$$ + +The evolution of this trajectory follows the ODE: + +$$ +d x _ {t} ^ {\mathrm {F E}} = v _ {\theta} ^ {\Delta} \left(x _ {t} ^ {t g t}, x _ {t} ^ {s r c}\right) d t \approx v _ {\theta} ^ {\Delta} \left(x _ {t - 1} ^ {\mathrm {F E}} + x _ {t} ^ {s r c} - x _ {0} ^ {s r c}, x _ {t} ^ {s r c}\right) d t, \tag {5} +$$ + +![](images/5ac9687abf9786c449c44be7bf4be8b6519f73fc3930507b777154917b916778.jpg) +Source caption: + +A german shepherd dog stands on the grass with mouth [closed] + +A german shepherd dog [with black sunglasses] [jumping] on the grass with mouth [opened] + +![](images/72102aef8bab76cf421595077656a8f3bbb2bee1833f529a9ec6e77dce6cf462.jpg) +Target caption: +LLM (or etc) + +"Write semantic captions that split the target caption." + +![](images/e0cb32e94ac671b2435e5d2268ae968c1e56a62746f4f23de5c62fa004a4950e.jpg) +Figure 1: Simplified visual illustration of the proposed SplitFlow. Purple, Blue, Orange line indicates independent sub-target flow. The aggregation is done on a certain timestep. After the aggregation, we use a single, unified flow. The notation in this figure follows the paper. + +![](images/8c724a9d53b8cb79997e3809a76c1bafbdbcccffe121e9696fb7c438b1c5ff33.jpg) + +![](images/442911a667d69b06c6095ec654e223b0731341dbb9df382e709218673042bc5d.jpg) + +![](images/0421d6a451e3f4da84767975f797109294fbc4d5e56d521f0402fcab4b74db1d.jpg) +$\varphi^{tgt(1)}$ A black sunglasses dog jumps grass +$\varphi^{tgt(2)}$ A dog with open mouth jump grass +$\varphi^{tgt(3)}$ A german Shepard dog with black sunglass + +where the difference in velocity fields guided by the source and target prompts is defined as: $v_{t}^{\Delta}(x_{t}^{tgt}, x_{t}^{src}) = v_{\theta}(x_{t}^{tgt}, t, \varphi^{tgt}) - v_{\theta}(x_{t}^{src}, t, \varphi^{src})$ . Since the source image is known, $x_{t}^{\mathrm{src}}$ can be directly obtained by linearly interpolating between it and a randomly sampled noise. In contrast, the target image is unknown and cannot be directly interpolated. Therefore, $x_{t}^{\mathrm{tgt}}$ is approximated using the previous timestep $x_{t-1}^{\mathrm{FE}}$ and Eq. (4) as follows: $x_{t}^{\mathrm{tgt}} \approx x_{t-1}^{\mathrm{FE}} + x_{t}^{\mathrm{src}} - x_{0}^{\mathrm{src}}$ . + +Based on the estimated velocity field difference, the entire editing process can be implemented as an iterative trajectory in the clean image space. Starting from the source image, we initialize the path with $x_0^{\mathrm{FE}} = x_0^{\mathrm{src}}$ . At each timestep $t$ , the edited latent is updated using the approximated velocity difference: $x_{t-1}^{\mathrm{FE}} = x_t^{\mathrm{FE}} + \Delta_t v_\theta^\Delta(x_t^{tgt}, x_t^{src})$ . This iterative procedure continues until the trajectory converges to the desired target image. + +# 4 Method + +Long target prompts often contain multiple attributes and complex semantics, resulting in a large semantic gap between the source and target in the latent space. This gap makes direct editing challenging, as entangled gradients can degrade edit quality. Providing simultaneous guidance for all attributes may lead to conflicting flows, which can cause semantic drift or even failure in the editing process. To address this challenge, we propose SplitFlow, an editing framework that progressively approximates the target latent through flow decomposition (Sec. 4.1) followed by flow composition (Sec. 4.2). Specifically, we first decompose the semantic complexity of the target prompt into a set of sub-target prompts and compute an independent flow for each, enabling latent directional components to be isolated and manipulated separately. These sub-flows are then aggregated into a unified flow that semantically aligns with the original target prompt. The editing process proceeds along this unified trajectory, resulting in more stable, diverse edits. + +# 4.1 Progressive Target Approximation with Flow Decomposition + +In image editing tasks, the semantic gap between a source prompt $\varphi^{src}$ and a target prompt $\varphi^{tgt}$ is often complex and high-dimensional. Direct transformation from source to target may lead to unstable or imprecise editing results. To address this, we propose to decompose the overall semantic transition into a sequence of intermediate sub-target prompts $\{\varphi^{tgt(i)}\}_{i=1}^N$ , where $N$ is the number of sub-target prompts. This decomposition simplifies the editing task into semantically controllable and progressively guided transformations. + +To perform the decomposition, we leverage a Large Language Model (LLM) as a prompt reasoning engine, represented as a function $f_{\mathrm{LLM}}$ . We construct a composite input sequence by concatenating an instruction prompt $\psi$ , the source prompt $\varphi^{src}$ , and the target prompt $\varphi^{tgt}$ . Then we feed the input + +to $f_{\mathrm{LLM}}$ to generate a sequence of sub-target prompts that represent incremental semantic transitions: + +$$ +\left\{\varphi^ {t g t (i)} \right\} _ {i = 1} ^ {N} = f _ {\mathrm {L L M}} (\Phi), \quad \Phi = \operatorname {c o n c a t} [ \psi , \varphi^ {s r c}, \varphi^ {t g t} ]. \tag {6} +$$ + +Each sub-target prompt \(\varphi^{tgt(i)}\) captures a localized semantic component that contributes to the overall transformation from \(\varphi^{src}\) to \(\varphi^{tgt}\). For example, as shown in Fig. 1, given a source prompt \(\varphi^{src} = "A\ german shepherd dog stands on the grass with mouth closed" and a target prompt \(\varphi^{tgt} = "A\ german shepherd dog with black sunglasses jumping on the grass with mouth opened", and instruction prompt \(\psi = "Write semantic captions that split the target caption.", the resulting sub-target prompts may be: \(\{\varphi^{tgt(i)}\}_{i=1}^N = \{\text{"A black sunglasses dog jumps grass", "A dog with open mouth jump grass", "A german shepherd with black sunglasses"}\}\). The number of sub-target prompts \(N\) is adaptively determined by \(f_{\mathrm{LLM}}\) based on the degree and complexity of the semantic difference. In most cases, \(N \leq 3\), which yields a compact yet effective semantic trajectory for guided editing. + +Following the formulation of the baseline (Eq. (4)-(5)), we extend the framework to handle each sub-target prompt individually. For each sub-target prompt $\varphi^{tgt(i)}$ , we define a corresponding flow $x_{t}^{\mathrm{FE}(i)}$ governed by an independent velocity field. Specifically, the sub-target flow is expressed as follows: + +$$ +x _ {t} ^ {\mathrm {F E} (i)} = x _ {0} ^ {s r c} + x _ {t} ^ {t g t (i)} - x _ {t} ^ {s r c}, \tag {7} +$$ + +where $x_0^{src}$ denotes the initial latent of the source image, $x_{t}^{src}$ is the interpolated latents at timestep $t$ between the source image and a Gaussian noise. Accordingly, the ODE governing each decomposed sub-target flow is given by the sub-target relative velocity field $v_{t}^{\Delta}(x_{t}^{tgt(i)},x_{t}^{src})$ : + +$$ +d x _ {t} ^ {\mathrm {F E} (i)} = v _ {t} ^ {\Delta} (x _ {t} ^ {t g t (i)}, x _ {t} ^ {s r c}) \cdot d t, \quad v _ {t} ^ {\Delta} (x _ {t} ^ {t g t (i)}, x _ {t} ^ {s r c}) = v _ {\theta} (x _ {t} ^ {t g t (i)}, t, \varphi^ {t g t (i)}) - v _ {\theta} (x _ {t} ^ {s r c}, t, \varphi^ {s r c}). +$$ + +Since $x_{t}^{tgt(i)}$ is not directly observable during inference, we approximate it based on the previously updated latent in Eq. (7) as: $x_{t}^{tgt(i)} \approx x_{t-1}^{\mathrm{FE}(i)} + x_{t}^{src} - x_{0}^{src}$ . The decomposition phase starts from $\eta_{max}$ and proceeds until $\eta_{dec}$ . + +# 4.2 Flow Composition + +While decomposing the flow, as described in Sec. 4.1, allows us to disentangle the gradients and achieve independent transformations, the ultimate objective in the image editing task is to perform editing aligned with the target prompt faithfully. Therefore, we devise a method to compose the previously generated sub-target flows into a unified flow that adheres to the full semantics of the target prompt. Considering the formulation of the ODE as shown in Eq. 1, the velocity field computed for each sub-target prompt can be interpreted as a gradient-like vector in latent space that guides the latent representation toward the target state. In multi-task learning (MTL), it is well known that gradients from different tasks can conflict, leading to unstable optimization or degraded performance on each tasks. To address this issue, various strategies such as gradient projection and orthogonality constraints have been proposed. Inspired by gradient conflict resolution methods [17, 14] in MTL, we propose a method that mitigates interference between sub-target velocity fields while effectively achieving the desired image editing objectives. + +# 4.2.1 Latent Trajectory Projection (LTP). + +To enforce global semantic consistency with the target flow while maintaining the diversity of each sub-target flow, we project each sub-target latent (conditioned on $\varphi^{tgt(i)}$ ) onto the target latent (conditioned on $\varphi^{tgt}$ ): $x_{t}^{\mathrm{FE}}$ . To perform this projection, we normalize the target latent as $\hat{x}_{t}^{\mathrm{FE}} = x_{t}^{\mathrm{FE}} / \| x_{t}^{\mathrm{FE}}\|_{2}$ . Here, the projection of each sub-target latent $\{x_{t}^{\mathrm{FE}(i)}\}_{i=1}^{N}$ onto $\hat{x}_{t}^{\mathrm{FE}}$ is computed as follows: + +$$ +x _ {t} ^ {\operatorname {p r o j} (i)} = \left(\langle x _ {t} ^ {\mathrm {F E} (i)}, \hat {x} _ {t} ^ {\mathrm {F E}} \rangle\right) \hat {x} _ {t} ^ {\mathrm {F E}}, \tag {8} +$$ + +where the inner product $(\langle \cdot ,\cdot \rangle)$ is computed along the channel dimension of the latent. By projecting the sub-target latent onto the target latent, we ensure that the overall editing process remains consistent with the intended semantic shift. We then aggregate the projected sub-target latent to form the projected latent: + +$$ +x _ {t} ^ {\text {p r o j}} = \frac {1}{N} \sum_ {i = 1} ^ {N} x _ {t} ^ {\text {p r o j} (i)}. \tag {9} +$$ + +While both the target latent $x_{t}^{tgt}$ and the projected sub-target latents $\{x_{t}^{\mathrm{proj(i)}}\}_{i=1}^{N}$ are aligned along the same semantic direction, their origins differ fundamentally: the former stems from a unified trajectory conditioned on the full target prompt, whereas the latter are partial trajectories derived from sub-prompts and aligned through projection. This approach retains the global coherence of the target trajectory while preserving localized variations introduced by sub-target prompts, enhancing both semantic consistency and editing diversity. + +# 4.2.2 Velocity Field Aggregation (VFA). + +After projection, to further enhance flow diversity, we introduce Velocity Field Aggregation (VFA), which combines the velocity fields of sub-target flows. To quantify the directional consistency among sub-target flows, we compute the cosine similarity between their relative velocity fields with respect to the source latent. Specifically, we first compute the relative velocity vector between $x_{t}^{\mathrm{proj}(i)}$ and $x_{t}^{src}$ as follows: + +$$ +\mathbf {g} _ {i} := v _ {t} ^ {\Delta} \left(x _ {t} ^ {\operatorname {p r o j} (i)}, x _ {t} ^ {s r c}\right) = v _ {\theta} \left(x _ {t} ^ {\operatorname {p r o j} (i)}, t, \varphi^ {t g t (i)}\right) - v _ {\theta} \left(x _ {t} ^ {s r c}, t, \varphi^ {s r c}\right) \tag {10} +$$ + +The cosine similarity $\mathrm{S}_{ij}$ between the $i$ -th and $j$ -th sub-target prompt is defined as: + +$$ +\mathbf {S} _ {i j} = \left\langle \hat {v} _ {t} ^ {\Delta (i)}, \hat {v} _ {t} ^ {\Delta (j)} \right\rangle , \quad \hat {v} _ {t} ^ {\Delta (i)} = \frac {v _ {t} ^ {\Delta} \left(x _ {t} ^ {\operatorname* {p r o j} (i)} , x _ {t} ^ {s r c}\right)}{\| v _ {t} ^ {\Delta} \left(x _ {t} ^ {\operatorname* {p r o j} (i)} , x _ {t} ^ {s r c}\right) \|}. \tag {11} +$$ + +Here, $\hat{v}_t^{\Delta(i)}$ is the normalized from $v_t^{\Delta}(x_t^{proj(i)}, x_t^{src})$ to compute cosine similarity. This metric captures the angular agreement between projected velocity directions, thereby reflecting the consistency of semantic changes introduced by each sub-target prompt. By applying the softmax operation to the cosine similarity map $S \in \mathbb{R}^{N \times N \times H \times W}$ , we obtain a weight map $w \in \mathbb{R}^{N \times H \times W}$ that determines the relative contribution of each velocity field at every spatial location $(h, w)$ in the latent grid: + +$$ +\bar {v} _ {t} ^ {\Delta} (h, w) = \sum_ {i = 1} ^ {N} w _ {i} (h, w) \cdot v _ {t} ^ {\Delta (i)} (h, w), \quad w _ {i} (h, w) = \frac {\exp \left(\sum_ {j \neq i} \mathbf {S} _ {i j} (h , w)\right)}{\sum_ {k} \exp \left(\sum_ {j \neq k} \mathbf {S} _ {k j} (h , w)\right)}. \tag {12} +$$ + +Combining the projected latent from Eq. (9) with the aggregated velocity field $\bar{v}_{\theta}^{\Delta}$ , the latent after aggregation can be updated as follows: + +$$ +x _ {t} ^ {\mathrm {F E}} \leftarrow x _ {t} ^ {\operatorname {p r o j}} + \bar {v} _ {t} ^ {\Delta} \cdot d t. \tag {13} +$$ + +Since the proposed VFA adaptively weights the sub-target velocity fields based on their semantic alignment, it not only suppresses the influence of redundant flows but also emphasizes those with distinct semantic directions. This enables the model to preserve editing diversity while maintaining coherent alignment with the target prompt. Also, note that the LTP and VFA are applied at the end of the decomposition phase ( $\eta_{dec}$ ). + +Mathematical Justification of VFA. Here, we mathematically verify why VFA improves both fidelity and editability over mere averaging by showing: + +$$ +\langle \bar {\mathbf {g}}, \mathbf {g} _ {\text {a v g}} \rangle \geq \| \mathbf {g} _ {\text {a v g}} \| ^ {2}, \tag {14} +$$ + +where $\bar{\mathbf{g}} = \sum_{i=1}^{K} w_i \mathbf{g}_i$ and $\mathbf{g}_{\mathrm{avg}} = \frac{1}{K} \sum_{i=1}^{K} \mathbf{g}_i$ . Here, $S_{kj}$ is denoted as $\langle \hat{\mathbf{g}}_k, \hat{\mathbf{g}}_j \rangle$ and $a_k = \sum_j S_{kj}$ . We first reformulate Eq. (14) in terms of the scores $a_i$ , where the LHS is $\langle \bar{g}, g_{\mathrm{avg}} \rangle = \frac{1}{K} \sum_{i=1}^{K} w_i a_i$ and the RHS is $\| g_{\mathrm{avg}} \|^2 = \frac{1}{K^2} \sum_{i=1}^{K} a_i$ . Thus, the proposition is equivalent to proving $\sum_i w_i a_i \geq \frac{1}{K} \sum_i a_i$ . The proof combines two standard results. First, from Gibbs' inequality, the KL-divergence between the softmax distribution $w = \{w_i\}$ and the uniform distribution $u = \{1/K\}$ is non-negative, which implies: $\sum_{i=1}^{K} w_i a_i \geq \log(Z/K)$ (14-1), where $Z = \sum_i e^{a_i}$ . Second, applying Jensen's inequality to the convex function $f(x) = e^x$ gives $\log(E[e^a]) \geq E[a]$ , which in our context is: + +$$ +\log (Z / K) = \log \left(\frac {1}{K} \sum_ {i = 1} ^ {K} e ^ {a _ {i}}\right) \geq \frac {1}{K} \sum_ {i = 1} ^ {K} a _ {i} \tag {14-2} +$$ + +Table 1: Quantitative comparison results on PIE benchmark. For each model group (diffusion-based and flow-based), the best and second-best values are indicated in bold and underlined, respectively. Ours† is the result with a fidelity-enhanced setting. + +
MethodModelStructureBackground PreservationCLIP Similarity
EditingDistance ×103↓PSNR ↑LPIPS ×103↓MSE ×104↓SSIM×102↑Whole ↑Edited ↑
DDIM [21]DiffusionP2P69.417.87208.80219.8871.1425.0122.44
DDIM [21]DiffusionPnP28.2222.28113.4683.6479.0525.4122.55
Null-text [16]DiffusionP2P13.4427.0360.6735.8684.1124.7521.86
PnP-Inv [10]DiffusionP2P11.6527.2254.5532.8684.7625.0222.10
PnP-Inv [10]DiffusionPnP24.2922.46106.0680.4579.6825.4122.62
RF-Inversion [19]Flux-40.620.82184.8129.171.9225.2022.11
RF-Solver [26]FluxRF-Solver31.122.90135.8180.1181.9026.0022.88
FireFlow [3]FluxRF-Solver28.323.28120.8270.3982.8225.9822.94
Flow Edit [11]Flux-27.721.91111.7094.083.3925.6122.70
FTEdit [29]SD3.5AdaLN18.1726.6280.5540.2491.5025.7422.27
iRFDS [31]SD3-62.7219.61186.39179.7674.5924.5421.67
FlowEdit [11]SD3-27.2422.13105.4687.3483.4826.8323.67
FlowEdit [11]SD3.5-11.8026.9753.6831.2389.7026.1822.88
SplitFlow(Ours)SD3-25.9622.45102.1481.9983.9126.9623.83
SplitFlow(Ours)†SD3-14.5525.2268.5344.9687.5426.2323.01
SplitFlow(Ours)SD3.5-11.6827.1252.9330.6189.7626.2922.89
+ +Chaining inequalities Eq. (14-1) and Eq. (14-2) directly yields the required result: + +$$ +\sum_ {i = 1} ^ {K} w _ {i} a _ {i} \geq \log (Z / K) \geq \frac {1}{K} \sum_ {i = 1} ^ {K} a _ {i} +$$ + +This proof formalises why VFA improves both fidelity and editability: It suppresses conflicts, steering the update toward high-consensus attributes. Empirically, this manifests as higher background preservation and better CLIP similarity in Table 1-2. + +When the decomposition phase is ended, the aggregated latent now follows the ODE formulation of the target prompt as described in Sec. 3.2. The decomposed flow stages facilitate fine-grained attribute manipulation without gradient entanglement, while the final unified flow phase ensures alignment with the holistic editing intent. This hybrid strategy improves editing stability and fidelity by integrating diversity-aware adjustments with prompt-level consistency. A detailed algorithmic description of SplitFlow is included in the supplementary material. + +# 5 Experiments + +# 5.1 Experimental Setup + +Implementation Details. To show the effectiveness of the proposed method and for a fair comparison with the prior works, we employed the commonly used Stable Diffusion (SD3, SD3.5) [4] rectified flow model. By following the protocol of the baseline [11], we use the same $T = 50$ steps with $\eta_{max} = 33$ , which skips the first one-third steps. The CFG values for the source and target are set to 3.5 and 13.5, respectively. The decomposition $\eta_{dec}$ is set to 28, which lasts for 5 steps $(\eta_{max} - \eta_{dec})$ . To decompose the target prompt, we used Mistral-7B [9]. + +Dataset. We evaluate our method on Prompt-based Image Editing (PIE) Benchmark [10], which contains 700 images of natural and artificial scenes. In PIE benchmark, ten categories span from random editing written by volunteers to change image style. Each image includes a source prompt, target prompt, editing directive, edit subjects, and editing mask. + +Baselines and Evaluation Metrics. As this work focuses on image editing based on rectified flow, we compare with the State-of-The-Art (SoTA) editing methods; RF-Inversion [19], RF-Solver [26], FireFlow [3], iRFDS [31], FTEdit [29], and FlowEdit [11]. Diffusion-based models such as DDIM [21, 16, 10] are also included for comparison. We evaluate our method on the PIE-Bench dataset using standard metrics. For assessing reconstruction quality and background preservation, we report image-level metrics: LPIPS [33], SSIM [27], MSE, PSNR, and Structure Distance [10]. To measure semantic alignment with the target prompt, we use CLIP similarity [28]. + +Table 2: Ablation study on PIE-benchmark. Here AVG denotes the case where mere average is applied for latent trajectory aggregation. + +
BaselineLTPVFAStructureBackground PreservationCLIP Similarity
Distance ×103↓PSNR ↑LPIPS ×103↓MSE ×104↓SSIM×102↑Whole ↑Edited ↑
✓27.2422.13105.4687.3483.4826.8323.67
AVG22.2823.3692.0068.2685.0026.8123.67
✓✓26.2222.37103.5883.6783.7626.9323.82
✓✓✓25.9622.45102.1481.9983.9126.9623.83
+ +![](images/a1be76d345006f490d2f9b7f70a46d9d776265f2b910046653d27489ade99b10.jpg) +Input + +![](images/eeafbd5d6d695cc5bde4e5d1dd2644700ce55acae85d1d2c3c6efc3336c42c3d.jpg) +Directinv+P2P + +![](images/856906427e4ba82f8cd778a822a942a2bbe86b988625f1682f3405b3509f515c.jpg) +RF solver + +![](images/04298a2cf60f5820b668fcc1ca3deb4528cf2f150098e3148ab38904bb7f49de.jpg) +FireFlow + +![](images/94454bc0333b76db974cbd2b641eb46aad652451dbdddab7bdc142f226b194b4.jpg) +FlowEdit-Flux + +![](images/b9ae14bbdc5d49da23bee985be4317aaaa611ac4b562e20f561a4e08fc4daafb.jpg) +FlowEdit-SD3 + +![](images/79d1fd2ccddd730e21831eeeb026f4e67ea9b21ff737a657cda82d278da4643f.jpg) +SplitFlow-SD3 + +![](images/364af5c3047a471f68a12ff648fde7f0dc914883f2fb694b130b96f21b0b02a7.jpg) + +![](images/52c6f77ba0e1d9d01a5b70c78446fa953c59b2008c76a1c1e364a9aedf3de9a9.jpg) + +![](images/94426a9e07732d28b26069da5a22c4010f47307c2b4ee0fcd0868b65e4fdc8de.jpg) +a) Change "plant" to "flower" + +![](images/05b07dc77199d6f3d44bdf4473f8c8ca60719db38e3cfcfe1b5dd48d0eefb40e.jpg) + +![](images/71b4d86c00d9e92e443abab4002879500b0d788568050fd3748ab36229a81e18.jpg) + +![](images/a9446cc8a2374af55bf6f62e16965fc3ec4d8c9bf65806cef8caec933cddf71e.jpg) + +![](images/94af3e447798d4a9d483b2a0842fc160869af54adbcd1088d56d9e4ae93b2f7c.jpg) + +![](images/8472e246141401d0de399cf065cebed520fd7132649e478c5e21f4910cc1acd5.jpg) + +![](images/ba76cb2b2f8d5a89cc99bfd84eb60718bc408efbdac93097bc8ba65a9ef9e3f0.jpg) + +![](images/2cb71baba134fe3c103a12cb27f31b38c4bc0d6bc0604f6418e129a80a037f31.jpg) +b) Change "dumplings" to "sushi" + +![](images/c1f784234b3f80d35b18fdb66a3da0a989473d34584551800ed8ae01e7d8eae2.jpg) +c) Change "red and white" to "blue and green" + +![](images/a878c9599b492b7659153ed95e0c35348a5321c70c14e0c90d858e3cc423f2d2.jpg) + +![](images/ec3f476f38f981d640c8ec606e6e296656843d8ddc4e229df19de0ecd5f0588c.jpg) + +![](images/32e973275442fff5521dfc2997ffe49b6db03a9a454116904932f5dce4313ec7.jpg) + +![](images/6e0b0a70a3a34f37173e8e95f6a40420f7acbe3e36eaf9a489460f2def27e729.jpg) + +![](images/187bfd1db71c7d8a0e9cd9bbb9fe2ee83cf45142bd632adef3808b03bf1d8ef2.jpg) + +![](images/8aa82f93d1f8c6bd2a83f4d7f2dc44e97c07f6f17ae800f49e833aace688d63d.jpg) +d) Add "with hat" + +![](images/03cf95d3e2ec95ff20fe12dcab864ee3cd06e1ee115e20bed2d745f0eb315ed8.jpg) + +![](images/c9b06641ef325e2a8c4c198f5b5aac2eb7e5e26541be6848cf50a3a7417a1971.jpg) + +![](images/3bb04c432261837277a98686d5555ff767df56c702729d757b6ef92304f11813.jpg) + +![](images/2f5a4ba2afafae95cd1920121d5fdf50170d8db95f651c0120a4305fcf713e8d.jpg) + +![](images/1d1e5d34a3b9c6e8be36a727fca801f89e879ee0a3615353fedc2d757573a666.jpg) +Figure 2: Qualitative comparison of prompt-based image editing methods. Each row corresponds to a specific editing instruction, where the source prompt is modified into a target prompt. From top to bottom, the tasks are: (a) change "plant" to "flower", (b) change "dumpling" to "sushi", (c) change "red and white" to "blue and green", (d) add "with hat", (e) delete "a single pink lotus flower is growing in the middle of". The columns show the input image and the results generated by different models, including Directinv+P2P, RF solver, FireFlow, FlowEdit-Flux, FlowEdit-SD3, and SplitFlow. + +![](images/605cb53ce1a3451f6ff8576c92c11f83fdd72366a66fef06b9438813de2ec488.jpg) +e) Delete "a single pink lotus flower is growing in the middle of" + +![](images/a5cf8a3eb3ed55e59725a3282a0dc2973dce2416d584b7872856e733ee2e2a04.jpg) + +![](images/51cb78ab59f58b2556e422bd0a293ceee277d7438d684540f1f27491eb875833.jpg) + +![](images/f10072d12f2e6f1630e8673d93b32939867e22b88b48e002001039ae3776ba9f.jpg) + +![](images/b37d0654dae4de555381350baa85786bcfcc1d63ecef19934db66f9f05ea5509.jpg) + +![](images/ad597cf3ebf4140da9fbd935d6196ffc97ecb8204b8d5c8e54e0bb89d4dde417.jpg) + +# 5.2 Main Results + +Comparison with State-of-the-art Methods. To demonstrate the effectiveness of the proposed SplitFlow, we conducted experiments as shown in Table 1. Compared to the FlowEdit [11] and iRFDS [31], within the same SD3 model, the proposed SplitFlow not only outperforms in preserving background but also in editing quality. To further demonstrate the effect of the trade-off between fidelity and editability, we omit the noise interpolation during the decomposition phase and directly use the source latent $x_0^{src}$ to better preserve structural details of the input image in the fidelity-enhanced setting†. Compared to the state-of-the-art method FTEdit [29], our SplitFlow† (a fidelity-enhanced variant) achieves superior performance in both Structure Distance and LPIPS, despite FTEdit employing a stronger backbone model. Moreover, it significantly outperforms FTEdit in both CLIP Similarity metrics, demonstrating better alignment with the target prompt. Also, compared + +Table 3: Ablation study of aggregation step ${\eta }_{dec}$ on PIE-benchmark. + +
ηdecStructureBackground PreservationCLIP Similarity
Distance ×103↓PSNR ↑LPIPS ×103↓MSE ×104↓SSIM×102↑Whole ↑Edited ↑
Baseline27.2422.13105.4687.3483.4826.8323.67
3025.9922.41102.3383.0683.8826.9223.79
2926.1322.41102.2682.6683.8526.9023.71
2825.9622.45102.1481.9983.9126.9623.83
2725.9622.44102.3982.1583.9126.9223.85
2625.9422.44102.7082.2783.9126.9523.84
+ +![](images/88b5de6855cd33aed7987be560a492b3fc5d62c7bd97d5cc8fc9e6a4281f6a55.jpg) +Input + +![](images/2f96c6abbab2931a9d279dd6f86f33a9801580f1b462f550d2baa41981e233d5.jpg) +Edit + +![](images/a3f904d2e270cbc469ee42769de067a730509dc407c53c72e24e5898a527b20d.jpg) +SplitFlow (Ours) + +Source: Three giraffes walk in a line through a lush, zoo-like forest path, while another animal rests near a pond. + +Target: Three elephants walk in a line through a lush, zoo-like forest path, while a tiger rests near a pond. + +![](images/d2f13823a23fe06578842dfd3cf670849d17c5e81e9969d2162264db2a85a4c2.jpg) +Figure 3: Qualitative comparison results with more complex prompts. + +![](images/c12ba3014e14e1df66ca68f07793d15c9fd57311a00f31e2781601c937241c76.jpg) + +![](images/e371b8a1ad351133e1d51c66c4c9f1a8e3d7dfe96b6cef5878b86718612ddeb1.jpg) +Editing prompts + +Source: A group of sheep stands on a green meadow with a lake and mountains in the background + +Target : Three dogs with sunglasses stands on a green meadow with a lake and snowy mountains in the background. The sky is covered with cloud. + +to the methods based on Flux such as RF-Inversion [19], RF-Solver [26], FireFlow [3], SplitFlow† outperforms the prior works. Within the SD3.5 model, SplitFlow demonstrates superior background preservation capability compared to prior works. + +Qualitative Comparison. The qualitative comparison results are presented in Fig. 2. Across various scenarios—including change, add, and delete object prompts—our proposed SplitFlow demonstrates superior editability while effectively preserving the background. For instance, as shown in Fig. 2-(d), which involves adding “with hat” to a portrait of Mozart, other methods fail to generate the hat or distort the original image, whereas our method successfully synthesizes the hat while maintaining the integrity of the source image. Although it is well known that enhancing editability often compromises fidelity, our approach achieves a favorable balance by disentangling gradients within the flow during the editing process with decomposition. In Fig. 3, to further demonstrate the effectiveness of our method, we provide qualitative comparison results with more complex scenario. In the first row, given the source prompt “Three giraffes walk in a line through a lush, zoo-like forest path, while another animal rests near a pond,” the editing prompt requires changing the three giraffes to elephants and the other animal to a tiger. While FlowEdit fails to convert “three giraffes” to “three elephants,” our method successfully performs the transformation and better preserves the semantic detail of “in a line.” Additional qualitative results under complex scenarios are provided in the supplementary material. + +# 5.3 Detailed Analysis + +Component Analysis. To validate the effect of the designed component, we conducted an ablation study on PIE-benchmark as shown in Table 2. The result of simply averaging the individual flows after decomposition is reported in the second column of Table 2, denoted as AVG. Interestingly, even naive averaging of sub-flows maintains CLIP similarity on par with the baseline [11], while significantly improving background preservation, as reflected in metrics such as PSNR, LPIPS, MSE, and SSIM. We attribute this to the semantic decoupling effect of sub-flows, which localize edits to specific attributes and reduce unintended changes in irrelevant regions. While semantic decomposition and flow separation are key contributions of our work, our overarching objective extends beyond fidelity enhancement. Our goal is to strike a balance between fidelity and editability, ensuring that complex, multi-attribute prompts are both faithfully represented and accurately reflected in the edited outputs. To this end, our proposed components—Latent Trajectory Projection (LTP) and Velocity Field Aggregation (VFA)—go beyond averaging by explicitly aligning sub-flows with the global editing direction and adaptively weighting their contributions. By applying LTP to align each + +sub-flow with the target flow, we observe meaningful improvements in CLIP similarity, particularly in the Edited metric, which is computed over the foreground mask. Although LTP results in lower background preservation compared to simple averaging, it still outperforms the baseline in all metrics. Furthermore, when VFA is applied, each sub-flow contributes more effectively to the final trajectory. This not only enhances background preservation—similar to the averaging strategy—but also improves CLIP similarity by promoting global semantic alignment while preserving diversity across sub-prompts. + +Ablation Study on Aggregation Timestep. In Table 3, we conduct an ablation study on the aggregation timestep $\eta_{dec}$ to evaluate the effectiveness and robustness of SplitFlow. Across all tested configurations, SplitFlow consistently outperforms the baseline. As $\eta_{dec}$ decreases—corresponding to a longer decomposition period—we observe improved editability at the expense of background preservation. Considering this trade-off, we set $\eta_{dec} = 28$ for our final configuration. Compared to the baseline, the total number of steps required by SplitFlow can be calculated as $N \times (\eta_{max} - \eta_{dec}) + \eta_{max}$ . Since $N \leq 3$ in most cases, the inference steps can be approximated as $3 \times (33 - 28) + 33 = 48$ . In practice, FlowEdit requires 57 minutes for inference, whereas SplitFlow takes 83 minutes to process 700 images on the PIE Benchmark. Additionally, prompt decomposition using an LLM takes approximately 20 minutes. Additional detailed ablation studies on LLM, cost analysis are provided in the supplementary material. + +Limitations A potential limitation of our method lies in its dependence on the decomposition of the target prompt. Since the editing flows are derived from sub-prompts, the quality and characteristics of the final output can vary depending on the choice of LLMs. Although SplitFlow incurs a higher inference time than the baseline, we emphasize that the LLM serves only as a proxy to facilitate our main contribution—demonstrating that decomposing the editing process into sub-target flows can significantly improve image-editing performance—and proposing a principled method to aggregate these flows effectively. By showing that flow decomposition substantially enhances both fidelity and editability in image editing, this work also opens up new directions for future research. These include developing more effective prompt decomposition techniques using LLMs or vision-language models, as well as exploring optimization-based approaches to mitigate gradient conflicts during flow composition. Additional discussions, including analyses of extreme cases, are provided in the supplementary materials. + +# 6 Conclusion + +In this paper, we proposed SplitFlow, a flow decomposition and composition framework designed to address gradient entanglement and semantic conflict that arise in image editing with complex and multi-attribute target prompts. SplitFlow computes independent editing flows for each sub-target prompt and forms a unified trajectory through projection and aggregation, thereby maintaining semantic alignment while mitigating interference between attributes. To this end, we introduced two aggregation strategies: Latent Trajectory Projection (LTP), which aligns the directional components of the latent trajectory to ensure coherence with the global target semantics, and Velocity Field Aggregation (VFA), which adaptively integrates sub-target flows while preserving their semantic diversity. These components enable SplitFlow to effectively balance fidelity and editability—two often conflicting objectives in image editing. Extensive experiments on the PIE-Bench benchmark demonstrate that our method consistently outperforms existing approaches in both visual quality and prompt alignment. Our results confirm that decomposing the editing process into semantically meaningful flows and carefully reassembling them provides a promising direction for accurate, and high-quality text-guided image editing. + +# References + +[1] Michael S. Albergo and Eric Vanden-Eijnden. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [Yes] + +Justification: Large Language Models (LLMs) were used as an integral part of the methodology, specifically to decompose editing prompts into semantically meaningful intermediate descriptions that guide the diffusion-based image editing process. 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